Variational and viscosity operators for the evolutive Hamilton-Jacobi equation
VVARIATIONAL AND VISCOSITY OPERATORS FOR THEEVOLUTIVE HAMILTON-JACOBI EQUATION
VALENTINE ROOS
Abstract.
We study the Cauchy problem for the first order evolutive Hamilton-Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarilyconvex in the momentum variable and not a priori compactly supported. We build andstudy an operator giving a variational solution of this problem, and get local Lipschitzestimates on this operator. Iterating this variational operator we obtain the viscosityoperator and extend the estimates to the viscosity framework. We also check thatthe construction of the variational operator gives the Lax-Oleinik semigroup if theHamiltonian is convex or concave in the momentum variable.
Contents
1. Introduction 21.1. Classical solutions: the method of characteristics 31.2. Viscosity operator 41.3. Variational operator 41.4. A link between variational and viscosity operator 61.5. The convex case: Joukovskaia’s theorem and the Lax-Oleinik semigroup 72. Building a variational operator 92.1. Chaperon’s generating families 92.2. Critical value selector 132.3. Definition of R ts R ts . 203. Iterating the variational operator 243.1. Iterated operator and uniform Lipschitz estimates 243.2. Convergence towards the viscosity operator 274. The convex case 314.1. The Lax-Oleinik semigroup with broken geodesics 314.2. Proof of Joukovskaia’s Theorem 32Appendices 33A. Generating families of the Hamiltonian flow 33A.1. Generating family in the general case 36A.2. Generating family in the convex case 39B. Minmax: a critical value selector 44B.1. Definition of the minmax for smooth functions 45 The research leading to these results has received funding from the European Research Council underthe European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement 307062and from the French National Research Agency via ANR-12-BLAN-WKBHJ. a r X i v : . [ m a t h . S G ] J a n VALENTINE ROOS
B.2. Minmax properties for smooth functions 47B.3. Extension to non-smooth functions 51C. Deformation lemmas 55C.1. Global deformation of sublevel sets 55C.2. Sending sublevel sets to sublevel sets 57D. Uniqueness of viscosity solution: doubling variables 58E. Graph selector 63E.1. Graph selector 63E.2. Application to the evolutive Hamilton-Jacobi equation 65References 651.
Introduction
We study the Cauchy problem associated with the evolutive Hamilton-Jacobi equation(HJ) ∂ t u ( t, q ) + H ( t, q, ∂ q u ( t, q )) = 0 , where H : R × T (cid:63) R d → R is a C Hamiltonian, u : R × R d → R is the unknown function,and the initial condition is given by u (0 , · ) = u Lipschitz.The Cauchy problem does not admit classical solutions in large time even for smooth u and H . Two different types of generalized solutions, namely viscosity and variationalsolutions, were then defined, respectively by P.-L. Lions and M.G. Crandall (see [CL83])and by J.-C. Sikorav and M. Chaperon (see [Sik90], [Cha91]). T. Joukovskaia showedthat the two solutions match for compactly supported fiberwise convex Hamiltonians(see [Jou91]), which is not necessarily true in the non convex case. Examples where thesolutions differ were proposed in [Che75], [Vit96], [BC11] and [Wei14].In order to compare these solutions in the framework of Lipschitz initial conditions,we will define two notions of operators, denoted respectively by V ts and R ts , defined onthe space of Lipschitz functions on R d and giving respectively viscosity and variationalsolutions of the Cauchy problem. In [Wei14], Q. Wei obtains for compactly supportedHamiltonians the viscosity operator via a limiting process on the iterated variationaloperator, which has a simple expression when the Hamiltonian does not depend on t : (cid:16) R tn (cid:17) n → n →∞ V t . This result is extended to the contact framework in [CC17].The assumption of compactness on the Hamiltonian’s support is due to the symplec-tic origin of the variational solution. This paper is aimed at removing this constraint,by replacing it with the following set of assumptions on the Hamiltonian that is morestandard in the framework of viscosity solution theory and thus more natural when oneis interested in making the link between variational and viscosity solutions.
Hypothesis . There is a
C > such that for each ( t, q, p ) in R × R d × R d , (cid:107) ∂ q,p ) H ( t, q, p ) (cid:107) < C, (cid:107) ∂ ( q,p ) H ( t, q, p ) (cid:107) < C (1 + (cid:107) p (cid:107) ) , | H ( t, q, p ) | < C (1 + (cid:107) p (cid:107) ) , where ∂ ( q,p ) H and ∂ q,p ) H denote the first and second order spatial derivatives of H . ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 3
This hypothesis implies a finite propagation speed principle in both viscosity andvariational contexts, which allows to work with non compactly supported Hamiltonians.We refer for example to [Bar94] for the viscosity side, where in particular the uniquenessof the viscosity operator (see also Proposition D.5) is studied, and to Appendix B of[CV08] for the existence of variational solutions for Hamiltonians satisfying this finitepropagation speed principle.In this paper we propose under Hypothesis 1.1 a complete and elementary constructionof both variational and viscosity operators, and extend Joukovskaia’s and Wei’s results tothis framework. The proof of the convergence of the iterated variational operator relieson the computation of local Lipschitz estimates for the variational operator R ts that obeya semigroup-type property with respect to s and t . The same Lipschitz estimates thenalso apply to the viscosity operator via the limiting process.A special care was provided in an attempt to produce a self-contained text, accessibleto a reader with no specific background on symplectic geometry.1.1. Classical solutions: the method of characteristics.
Under Hypothesis 1.1, the
Hamiltonian system (HS) ® ˙ q ( t ) = ∂ p H ( t, q ( t ) , p ( t )) , ˙ p ( t ) = − ∂ q H ( t, q ( t ) , p ( t )) admits a complete Hamiltonian flow φ ts , meaning that t (cid:55)→ φ ts ( q, p ) is the unique solutionof (HS) with initial conditions ( q ( s ) , p ( s )) = ( q, p ) . We denote by (cid:0) Q ts , P ts (cid:1) the coordi-nates of φ ts . We call a function t (cid:55)→ ( q ( t ) , p ( t )) solving the Hamiltonian system (HS) a Hamiltonian trajectory . The
Hamiltonian action of a C path γ ( t ) = ( q ( t ) , p ( t )) ∈ T (cid:63) R d is denoted by A ts ( γ ) = (cid:90) ts p ( τ ) · ˙ q ( τ ) − H ( τ, q ( τ ) , p ( τ )) dτ. The next lemma states the existence of characteristics for C solutions of the Hamilton-Jacobi equation (HJ). Lemma 1.2. If u is a C solution of (HJ) on [ T − , T + ] × R d and γ : τ (cid:55)→ ( q ( τ ) , p ( τ )) is a Hamiltonian trajectory satisfying p ( s ) = ∂ q u ( s, q ( s )) for some s ∈ [ T − , T + ] , then p ( t ) = ∂ q u ( t, q ( t )) for each t ∈ [ T − , T + ] and u ( t, q ( t )) = u ( s, q ( s )) + A ts ( γ ) ∀ t ∈ [ T − , T + ] . Proof. If f ( t ) denotes the quantity ∂ q u ( t, q ( t )) , one can show that both f and p solve theODE ˙ y ( t ) = − ∂ q H ( t, q ( t ) , y ( t )) and p ( s ) = f ( s ) implies that p ( t ) = f ( t ) for each time t ∈ [ T − , T + ] . Then, differentiating the function t (cid:55)→ u ( t, q ( t )) gives the result. (cid:3) Here is another formulation of the first statement of Lemma 1.2: if Γ s denotes the graphof ∂ q u s , then the set φ ts Γ s is included in the graph of ∂ q u t for each T − ≤ s ≤ t ≤ T + .In particular, if φ Ts Γ s is not a graph for some time T > s , then the existence of classicalsolution on [ s, T ] × R d is not possible, hence the introduction of generalized solutions. VALENTINE ROOS
Viscosity operator.
A family of operators (cid:0) V ts (cid:1) s ≤ t mapping C , ( R d ) (the spaceof Lipschitz functions) into itself is called a viscosity operator if it satisfies the followingconditions: Hypotheses . (1) Monotonicity: if u ≤ v are Lipschitz on R d , then V ts u ≤ V ts v on R d for each s ≤ t ,(2) Additivity: if u is Lipschitz on R d and c ∈ R , then V ts ( c + u ) = c + V ts u ,(3) Regularity: if u is Lipschitz, then for each τ ≤ T , (cid:8) V tτ u, t ∈ [ τ, T ] (cid:9) is Lipschitz in q uniformly w.r.t. t and ( t, q ) (cid:55)→ V tτ u ( q ) is locally Lipschitz on ( τ, ∞ ) × R d ,(4) Compatibility with Hamilton-Jacobi equation: if u is a Lipschitz C solution of theHamilton-Jacobi equation, then V ts u s = u t for each s ≤ t ,(5) Markov property: V ts = V tτ ◦ V τs for all s ≤ τ ≤ t .In Appendix D, we state that such an operator gives the viscosity solutions as in-troduced by P.-L. Lions and M. G. Crandall in 1981 (see [CL83]), and we prove theuniqueness of such an operator when H satisfies Hypothesis 1.1 (Consequence D.5).The existence of the viscosity operator for our framework was already granted by thework of Crandall, Lions and Ishii (see [CIL92]) and it is proved again in this paper, wherewe obtain a viscosity operator by a limiting process.1.3. Variational operator.
A family of operators (cid:0) R ts (cid:1) s ≤ t mapping C , ( R d ) into itselfis called a variational operator if it satisfies the monotonicity, additivity and regularityproperties (1), (2), (3) of Hypotheses 1.3 and the following one, requiring the existenceof characteristics (see Lemma 1.2):(4’) Variational property: for each Lipschitz C function u , Q in R d and s ≤ t , thereexists ( q, p ) such that p = d q u , Q ts ( q, p ) = Q and if γ denotes the Hamiltoniantrajectory issued from ( q ( s ) , p ( s )) = ( q, p ) , R ts u ( Q ) = u ( q ) + A ts ( γ ) . We call variational solution to the Cauchy problem associated with u a function givenby a variational operator as follows: u ( t, q ) = R t u ( q ) . Remark . Variational property (4’) implies Compatibility property (4). This impliesin particular that if a variational operator satisfies the Markov property (5) of Hypotheses1.3, it is a viscosity operator.
Proof.
We fix s and Q and a Lipschitz C solution u of (HJ). Variational property (4’)implies that there exists q and p = ∂ q u s with Q ts ( q, p ) = Q such that, if γ ( τ ) = φ τs ( q, p ) denotes the Hamiltonian trajectory issued from ( q, p ) at time s , R ts u s ( Q ) = u s ( q )+ A ts ( γ ) .Since p = ∂ q u s and Q = Q ts ( q, p ) , Lemma 1.2 can be applied and states that u ( t, Q ) = u s ( q ) + A ts ( γ ) , hence the conclusion: R ts u s ( Q ) = u t ( Q ) . (cid:3) The uniqueness of such an operator is not granted a priori. See Appendix E for apresentation of the associated notion of graph selector and the different ways to defineone. In the same Appendix is also proved the following result.
Proposition 1.5. If u is C and R ts is a variational operator, ( t, q ) (cid:55)→ R t u ( q ) solves (HJ) in the classical sense for almost every ( t, q ) in (0 , ∞ ) × R d . ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 5
This is weaker than what happens in the viscosity case, since the viscosity solution u solves the equation on its domain of differentiability (which is of full measure since thesolution is Lipschitz) even for u only Lipschitz. We do not know either if ( t, q ) (cid:55)→ R t u ( q ) solves the equation almost everywhere when u is only Lipschitz.In this paper, we present a complete construction of the variational operator underHypothesis 1.1, which comes down to build a graph selector directly for the suspendedgeometric solution and its wavefront introduced in Appendix E. We follow the idea ofJ.-C. Sikorav (see [Sik86] or [Vit96]) consisting in selecting suitable critical values of a generating family describing this geometric solution. In order to get Lipschitz estimatesfor this operator, we work with the explicit generating family constructed by M. Chap-eron via the broken geodesics method (see [Cha90] and [Cha91]), whose critical pointsand values are related to the Hamiltonian objects of the problem. We use a general critical value selector σ defined from an axiomatic point of view (see Proposition 2.7),for functions which differ by a Lipschitz function from a nondegenerate quadratic form.An obstacle is that the generating family of Chaperon is of this form only for Hamil-tonians that are quadratic for large (cid:107) p (cid:107) , so we need to modify the Hamiltonian for large (cid:107) p (cid:107) into a quadratic form Z to be able to use the critical value selector, and check thatthe choice of Z does not matter in the definition of the operator.We denote by R ts the obtained operator, keeping in mind that it depends a priori onthe choice of a critical value selector σ . The explicit derivatives of the generating familyallow to prove the estimates of the following statement. Theorem 1.6. If H satisfies Hypothesis 1.1 with constant C , there exists a variationaloperator, denoted by ( R ts ) s ≤ t , such that for ≤ s ≤ s (cid:48) ≤ t (cid:48) ≤ t and u and v two L -Lipschitz functions,(1) R ts u is Lipschitz with Lip( R ts u ) ≤ e C ( t − s ) (1 + L ) − ,(2) (cid:107) R t (cid:48) s u − R ts u (cid:107) ∞ ≤ Ce C ( t − s ) (1 + L ) | t (cid:48) − t | ,(3) (cid:107) R ts (cid:48) u − R ts u (cid:107) ∞ ≤ C (1 + L ) | s (cid:48) − s | ,(4) ∀ Q ∈ R d , (cid:12)(cid:12) R ts u ( Q ) − R ts v ( Q ) (cid:12)(cid:12) ≤ (cid:107) u − v (cid:107) ¯ B ( Q, ( e C ( t − s ) − L ) ) ,where ¯ B ( Q, r ) denotes the closed ball of radius r centered in Q and (cid:107) u (cid:107) K := sup K | u | . The interest of these estimates is that they behave well with the iteration of theoperator R ts , and Theorem 1.6 allows then to prove Theorem 1.9 with no compactnessassumption on H .With the same method we are also able to quantify the dependence of the constructedoperator R ts with respect to the Hamiltonian: Proposition 1.7.
Let H and H be two C Hamiltonians satisfying Hypothesis 1.1 withconstant C , u be a L -Lipschitz function, Q be in R d and s ≤ t . Then | R ts,H u ( Q ) − R ts,H u ( Q ) | ≤ ( t − s ) (cid:107) H − H (cid:107) ¯ V , where ¯ V = [ s, t ] × ¯ B Ä Q, ( e C ( t − s ) − L ) ä × ¯ B Ä , e C ( t − s ) (1 + L ) − ä . An other formulation of the two last estimates is a localized version of the monotonicityof this variational operator with respect to the initial condition or to the Hamiltonian:
Proposition 1.8. If H and H are two C Hamiltonians satisfying Hypothesis 1.1 withconstant C , then for s ≤ t , Q in R d and u and v two L -Lipschitz functions, VALENTINE ROOS • R ts u ( Q ) ≤ R ts v ( Q ) if u ≤ v on the set ¯ B Ä Q, ( e C ( t − s ) − L ) ä , • R ts,H u ( Q ) ≤ R ts,H u ( Q ) if H ≥ H on the set [ s, t ] × ¯ B Ä Q, ( e C ( t − s ) − L ) ä × ¯ B Ä , e C ( t − s ) (1 + L ) − ä . A link between variational and viscosity operator.
If the variational and vis-cosity operators do not coincide in general, Q. Wei showed in [Wei14] that, for compactlysupported Hamiltonians, it is possible to obtain the viscosity operator by iterating thevariational operator along a subdivision of the time space and letting then the maximalstep of this subsequence tend to . This result fits in the approximation scheme proposedby Souganidis in [Sou85] for a slightly different set of assumptions, where the variationaloperator acts like a generator . We also refer to [BS91] for a presentation of this approxi-mation scheme method in a wider framework that includes second order Hamilton-Jacobiequations.Let us fix a sequence of subdivisions of [0 , ∞ ) Ä ( τ Ni ) i ∈ N ä N ∈ N such that for all N , τ N , τ Ni → i →∞ ∞ and i (cid:55)→ τ Ni is increasing. Let us also assume that for all N , i (cid:55)→ τ Ni +1 − τ Ni is bounded by a constant δ N such that δ N → when N → ∞ . For t in R + , we denote by i N ( t ) the unique integer such that t belongs to [ τ Ni N ( t ) , τ Ni N ( t )+1 ) . If u isLipschitz on R d , and ≤ s ≤ t , let us define the iterated operator at rank N by R ts,N u = R tτ NiN ( t ) R τ NiN ( t ) τ NiN ( t ) − · · · R τ NiN ( s )+1 s u, where R ts is any variational operator satisfying the Lipschitz estimate of Theorem 1.6. Theorem 1.9 (Wei’s theorem) . For each Hamiltonian H satisfying Hypothesis 1.1, thesequence of iterated operators ( R ts,N ) converges simply when N → ∞ to the viscosity oper-ator V ts . Furthermore, for each Lipschitz function u , ¶ ( s, t, Q ) (cid:55)→ R ts,N u ( Q ) © N convergesuniformly towards ( s, t, Q ) (cid:55)→ V ts u ( Q ) on every compact subset of { ≤ s ≤ t } × R d . Theorem 1.9 implies amongst other things the existence of the viscosity operator,and the local uniform convergence allows to transfer Lipschitz estimates to the viscosityframework:
Proposition 1.10. If H satisfies Hypothesis 1.1 with constant C , the viscosity operator ( V ts ) s ≤ t satisfies the following estimates: for ≤ s ≤ s (cid:48) ≤ t (cid:48) ≤ t and u and v twoLipschitz functions with Lipschitz constant L ,(1) V ts u is Lipschitz with Lip( V ts u ) ≤ e C ( t − s ) (1 + L ) − ,(2) (cid:107) V t (cid:48) s u − V ts u (cid:107) ∞ ≤ Ce C ( t − s ) (1 + L ) | t (cid:48) − t | ,(3) (cid:107) V ts (cid:48) u − V ts u (cid:107) ∞ ≤ C (1 + L ) | s (cid:48) − s | ,(4) ∀ Q ∈ R d , (cid:12)(cid:12) V ts u ( Q ) − V ts v ( Q ) (cid:12)(cid:12) ≤ (cid:107) u − v (cid:107) ¯ B ( Q, ( e C ( t − s ) − L ) ) .Moreover, if H and H are two Hamiltonians satisfying Hypothesis 1.1 with constant C , u is a L -Lipschitz function, Q is in R d and s ≤ t , the associated operators satisfy | V ts,H u ( Q ) − V ts,H u ( Q ) | ≤ ( t − s ) (cid:107) H − H (cid:107) ¯ V , where ¯ V = [ s, t ] × ¯ B Ä Q, ( e C ( t − s ) − L ) ä × ¯ B Ä , e C ( t − s ) (1 + L ) − ä .Furthermore, ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 7 • V ts u ( Q ) ≤ V ts v ( Q ) if u ≤ v on the set ¯ B Ä Q, ( e C ( t − s ) − L ) ä , • V ts,H u ( Q ) ≤ V ts,H u ( Q ) if H ≥ H on the set [ s, t ] × ¯ B Ä Q, ( e C ( t − s ) − L ) ä × ¯ B Ä , e C ( t − s ) (1 + L ) − ä . These estimates are not a priori very surprising since they are satisfied for classicalsolutions, but due to their dynamical origin they are likely to be sharper than the onesobtained using viscosity arguments. For example, the Lipschitz estimate with respectto u gives a better speed of propagation than the one obtained in Proposition D.4 with e CT (1 + L ) − as uniform Lipschitz constant.1.5. The convex case: Joukovskaia’s theorem and the Lax-Oleinik semigroup. If H is strictly convex w.r.t. p , the Lagrangian function, defined on the tangent bundle,is the Legendre transform of H : L ( t, q, v ) = sup p ∈ ( R d ) (cid:63) p · v − H ( t, q, p ) . The Legendre inequality writes L ( t, q, v ) + H ( t, q, p ) ≥ p · v for all t , q , p and v , and is an equality if and only if p = ∂ v L ( t, q, v ) or equivalently v = ∂ p H ( t, q, p ) . In particular, if ( q ( τ ) , p ( τ )) is a Hamiltonian trajectory, ˙ q ( τ ) = ∂ p H ( τ, q ( τ ) , p ( τ ) and (cid:90) ts L ( τ, q ( τ ) , ˙ q ( τ )) dτ = (cid:90) ts p ( τ ) · ˙ q ( τ ) − H ( τ, q ( τ ) , p ( τ ) dτ. In other words, the Hamiltonian action of a Hamiltonian trajectory coincides with theso-called Lagrangian action of its projection on the position space.The Lax-Oleinik semigroup (cid:0) T ts (cid:1) s ≤ t is usually expressed with this Lagrangian action:if u is a Lipschitz function on R d , then T ts u is defined by(1) T ts u ( q ) = inf c u ( c ( s )) + (cid:90) ts L ( τ, c ( τ ) , ˙ c ( τ )) dτ, where the infimum is taken over all the Lipschitz curves c : [ s, t ] → R d such that c ( t ) = q .Under this form, it is straightforward that the Markov property (5) is satisfied by theoperator. The Lax-Oleinik semigroup is known to be the viscosity operator when theHamiltonian is Tonelli, i.e. strictly convex and superlinear w.r.t. p , and also to satisfy theVariational property (4’), see for example [Fat12], [Ber12]. It is hence both a variationaland a viscosity operator for Tonelli Hamiltonians: T ts = V ts . The following theorem states that the variational operator construction of this papergives effectively the Lax-Oleinik semigroup for uniformly strictly convex Hamiltonian,and the viscosity operator in the convex case. We assume for this result that the criticalvalue selector σ satisfies two additional assumptions, presented in Proposition 4.4. VALENTINE ROOS
Theorem 1.11 (Joukovskaia’s theorem) . If p (cid:55)→ H ( t, q, p ) is convex for each ( t, q ) orconcave for each ( t, q ) , the variational operator R ts associated with the critical value selec-tor σ is the viscosity operator. In particular, it coincides with the Lax-Oleinik semigroupif H is uniformly strictly convex w.r.t. p . The last part of this statement was proved by T. Joukovskaia in the case of a compactmanifold, see [Jou91].This theorem was generalized to convex-concave type Hamiltonians, see [Wei13a] and[BC11], but only when both the Hamiltonian and the initial condition are in the form ofsplitting variables: H ( t, q, p ) = H ( t, q , p ) + H ( t, q , p ) and u ( q ) = u ( q ) + u ( q ) where d = d + d , ( q i , p i ) denotes the variables in T (cid:63) R d i , H (resp. H ) is a Hamiltonianon R × R d (resp. on R × R d ) convex in p (resp. concave in p ), and u and u areLipschitz functions on R d and R d .The paper is organized as follows: in Section 2 we build the variational operatorand prove Theorem 1.6. We first describe the construction of Chaperon’s generatingfamily and its properties (§2.1), and introduce the notion of critical value selector and itsproperties (§2.2). Then, we address carefully the difficulty related to the bahaviour of theHamiltonian for large p in order to define the variational operator without compactnessassumption (§2.3). We finally collect some properties of the variational operator and itsLipschitz estimates, proving Theorem 1.6 and Propositions 1.7 and 1.8 (§2.4).In Section 3 we prove Theorem 1.9. We study the uniform Lipschitz estimates of theiterated operator R ts,N (§3.1), and then show that the limit of any subsequence is theviscosity operator (§3.2). Section 3 can be read independently from Section 2, once theLipschitz constants of Theorem 1.6 are granted.In Section 4 we give a direct proof of Joukovskaia’s theorem, while describing theLax-Oleinik semigroup with the broken geodesics method (§4.1).Appendix A details the construction and properties of Chaperon’s generating familiesfor the Hamiltonian flow, both in the general (§A.1) and in the convex case (§A.2). Ap-pendix B proposes a functorial construction of a critical value selector as needed in theconstruction of the variational operator. It requires two deformation lemmas proved inAppendix C. Appendix D is about viscosity solutions, and gives an elementary proof ofthe uniqueness for Lipschitz initial data and under Hypothesis 1.1, via a standard dou-bling variables argument. Appendix E presents the notion of graph selector and containsa proof of Proposition 1.5. Acknowledgement.
I am grateful to my supervisor Patrick Bernard for his advices andcareful reading. I also thank Qiaoling Wei, Marc Chaperon and Alain Chenciner forfruitful discussions in the cheerful Observatoire de Paris. This paper was improved bymany suggestions of the referees of my PhD thesis, Jean-Claude Sikorav and Guy Barles,and by the anonymous referee of the paper.
ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 9 Building a variational operator
In this section we present the complete construction of the variational operator, follow-ing the idea proposed by J.-C. Sikorav in [Sik90] and M. Chaperon in [Cha91]. We workwith an explicit generating family of the geometric solution defined by Chaperon via the broken geodesics method (see [Cha84]). We gather its properties in the next paragraph,referring to Appendix A for some of the proofs. Then we apply on this generating familya critical value selector , which we handle only via a few axioms, see Proposition 2.7. Theexistence of a selector satisfying these axioms is proved in Appendix B. This selector canonly be directly applied to generating families associated with Hamiltonians coincidingwith a quadratic form at infinity, so we need to handle this difficulty by modifying theHamiltonian for large p , see Proposition 2.17 and Definition 2.18. The rest of the chap-ter consists in verifying that the obtained operator is a variational operator, and that itsatisfies the Lipschitz estimates of Theorem 1.6.2.1. Chaperon’s generating families.
We first build a generating family of the Hamil-tonian flow, following Chaperon’s broken geodesics method introduced in [Cha84] anddetailed in [Cha90], and then adapt it to the Cauchy problem. The results of this sectionare detailed and proved in Appendix A.Under Hypothesis 1.1, it is possible to find a δ > depending only on C (for example δ = ln(3 / C ) such that φ ts − id is -Lipschitz (see Proposition A.2), and as a consequence ( q, p ) (cid:55)→ ( Q ts ( q, p ) , p ) is a C -diffeomorphism for each | t − s | ≤ δ , where ( Q ts , P ts ) denotesthe components of the Hamiltonian flow φ ts .For a Hamiltonian H satisfying Hypothesis 1.1 and ≤ t − s ≤ δ , let F ts : R d → R be the C function defined by(2) F ts ( Q, p ) = (cid:90) ts ( P τs ( q, p ) − p ) · ∂ τ Q τs ( q, p ) − H ( τ, φ τs ( q, p )) dτ, where q is the only point satisfying Q ts ( q, p ) = Q . The function F ts is called a generatingfunction for the flow φ ts , meaning that ( Q, P ) = φ ts ( q, p ) ⇐⇒ ® ∂ p F ts ( Q, p ) = q − Q,∂ Q F ts ( Q, p ) = P − p, which is proved in Proposition A.5.Note that if H ( t, q, p ) = H ( p ) is integrable, Hamiltonian trajectories have constantimpulsion p and F ts ( Q, p ) = − ( t − s ) H ( p ) does not depend on Q .When t − s is large, we choose a subdivision of the time interval with steps smaller than δ and add intermediate coordinates along this trajectory. For each s ≤ t and ( t i ) suchthat t = s ≤ t ≤ · · · ≤ t N +1 = t and t i +1 − t i ≤ δ for each i , let G ts : R d (1+ N ) → R bethe function defined by(3) G ts ( p , Q , p , Q , · · · , Q N − , p N , Q N ) = N (cid:88) i =0 F t i +1 t i ( Q i , p i ) + p i +1 · ( Q i +1 − Q i ) where indices are taken modulo N + 1 . In Proposition A.7 we prove that G ts is a generating function for the flow φ ts , meaningthat if ( Q, p ) = ( Q N , p ) and ν = ( Q , p , · · · , Q N − , p N ) , ( Q, P ) = φ ts ( q, p ) ⇐⇒ ∃ ν ∈ R dN , ∂ p G ts ( p, ν, Q ) = q − Q,∂ Q G ts ( p, ν, Q ) = P − p,∂ ν G ts ( p, ν, Q ) = 0 , and in this case ( Q i , p i +1 ) = φ t i +1 s ( q, p ) for all ≤ i ≤ N − . Furthermore, if Q = Q ts ( q, p ) and γ denotes the Hamiltonian trajectory issued from ( q, p ) ,(4) G ts ( p, ν, Q ) = A ts ( γ ) − p · ( Q − q ) for critical points ν of ν (cid:55)→ G ts ( p, ν, Q ) .This is called the broken geodesics method : G ts is actually the sum of the actions of theunique Hamiltonian trajectories γ i such that γ i ( t i ) = ( (cid:63), p i ) and γ i ( t i +1 ) = ( Q i , (cid:63) ) andof boundary terms (of the form p i +1 · ( q i +1 − Q i ) ) smartly arranged in order that takingcritical values for G ts is equivalent to sew the pieces of trajectories γ i at the intermediatepoints into a nonbroken geodesic on the whole time interval.Note that if H ( t, q, p ) = H ( p ) , this function is quite simple:(5) G ts ( p , Q , p , Q , · · · , Q N − , p N , Q N ) = N (cid:88) i =0 − ( t i +1 − t i ) H ( p i ) + p i +1 · ( Q i +1 − Q i ) . Now let us use the generating family G ts of the flow to build what is called a generatingfamily for the Cauchy problem associated with the Hamilton-Jacobi equation (HJ) andan initial condition u , using a composition formula proposed by Chekanov. If u : R d → R is Lipschitz and s ≤ t , let us define S ts u by(6) S ts u : R d × R d × R d × R dN → R ( Q, q, p, ν (cid:124) (cid:123)(cid:122) (cid:125) ξ ) (cid:55)→ u ( q ) + G ts ( p, ν, Q ) + p · ( Q − q ) . Proposition 2.1.
Let u : R d → R be a Lipschitz C initial condition and ≤ t − s ≤ T .If Q is fixed in R d , ( q, p, Q , p , · · · , p N ) is a critical point of S ts u ( Q, · ) if and only if p = du ( q ) ,Q ts ( q, p ) = Q, ( Q i − , p i ) = φ t i s ( q, p ) ∀ ≤ i ≤ N, and in that case, ∂ Q S ts u ( Q, q, p, Q , · · · , p N ) = P ts ( q, p ) .Furthermore, the critical value of S ts u ( Q, · ) associated with a critical point ( q, p, ν ) isequal to u ( q ) + A ts ( γ ) , where γ denotes the Hamiltonian trajectory τ (cid:55)→ φ τs ( q, p ) .Proof. The point ( q, p, ν ) is a critical point of S ts u ( Q, · ) , if and only if ∂ q S ts u ( Q, q, p, ν ) = du ( q ) − p, ∂ p S ts u ( Q, q, p, ν ) = ∂ p G ts ( p, ν, Q ) + Q − q, ∂ ν S ts u ( Q, q, p, ν ) = ∂ ν G ts ( p, ν, Q ) . Since G is a generating family of the flow, the two last lines implies that Q ts ( q, p ) = Q and φ t i s ( q, p ) = ( Q i − , p i ) , hence P ts ( q, p ) = ∂ Q G ts u ( p, ν, Q ) + p = ∂ Q S ts u ( Q, ξ ) . The formof the critical values directly follows from the form of the critical values of G , see (4). (cid:3) ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 11
In other words, if Γ denotes the graph of du { ( q, du ( q )) , q ∈ R d } , the generating familythat we built describes the so-called geometric solution φ ts (Γ) as follows: φ ts (Γ) = ¶ ( Q, ∂ q S ts u ( Q, ξ )) | Q ∈ R d , ∂ ξ S ts u ( Q, ξ ) = 0 © , meaning that above each point Q , a point ( Q, P ) is in φ ts (Γ) if and only if there is acritical point ξ of ξ (cid:55)→ S ts u ( Q, ξ ) such that P = ∂ Q S ts u ( Q, ξ ) .Let us state the values of the other derivatives of S ts u at the points of interest: Proposition 2.2.
Let u a C L -Lipschitz function and Q in R d be fixed.(1) If ξ = ( q, p, ν ) is a critical point of ξ (cid:55)→ S ts u ( Q, ξ ) , then ® ∂ t S ts u ( Q, ξ ) = − H ( t, Q, P ts ( q, p )) ,∂ s S ts u ( Q, ξ ) = H ( s, q, p ) . (2) If H µ is a C family of Hamiltonians satisfying Hypothesis 1.1 with constant C ,the same subdivision can be chosen to build the associated generating families S ts,µ u , and then µ (cid:55)→ S ts,µ u ( Q, ξ ) is C and if ξ = ( q, p, ν ) is a critical point of ξ (cid:55)→ S ts,µ u ( Q, ξ ) , ∂ µ S ts,µ u ( Q, ξ ) = − (cid:90) ts ∂ µ H µ ( τ, φ τs ( q, p )) dτ. Proof.
We obtain these derivatives using Proposition A.5 and A.6, and the fact that acritical point ξ = ( q, p, ν ) of the generating family ξ (cid:55)→ S ts u ( Q, ξ ) describes steps of anonbroken Hamiltonian trajectory from ( q, p ) to ( Q, P ts ( q, p )) (Proposition 2.1). (cid:3) Propositions 2.1 and 2.2 imply that if ξ is a critical point of S ts u ( Q, · ) , the Hamilton-Jacobi equation is satisfied at this one point: ∂ t S ts u ( Q, ξ ) + H ( t, Q, ∂ Q S ts u ( Q, ξ )) = 0 . Inparticular if ( t, Q ) (cid:55)→ ξ ( t, Q ) is a differentiable function giving for each ( t, Q ) a criticalpoint of S ts u ( Q, · ) , then ( t, Q ) (cid:55)→ S ts u ( Q, ξ ( t, Q )) is a differentiable solution of the Cauchyproblem. An idea to build a generalized solution is then to select adequatly critical valuesof S ts u ( Q, · ) , which we are going to do in the next paragraphs.Until now, we only used the part of Hypothesis 1.1 stating that (cid:107) ∂ q,p ) H (cid:107) is uniformlybounded. The two next propositions requires the fact that (cid:107) ∂ ( q,p ) H ( t, q, p ) (cid:107) ≤ C (1+ (cid:107) p (cid:107) ) .The first one states that if H is nearly quadratic at infinity, so is ξ (cid:55)→ S ts u ( Q, ξ ) , and thesecond one allows to localize the critical points of S ts u . Proposition 2.3.
Let Z be a (possibly degenerate) quadratic form on R d . If both H and ( t, q, p ) (cid:55)→ Z ( p ) satisfy Hypothesis 1.1 with the same constant C , and H ( t, q, p ) = Z ( p ) for all (cid:107) p (cid:107) ≥ R , then S ts u ( Q, ξ ) = Z ( ξ ) + (cid:96) ( Q, ξ ) , where ξ (cid:55)→ (cid:96) ( Q, ξ ) is a Lipschitzfunction with constant (cid:107) Q (cid:107) + Lip( u ) + 4(1 + R ) and Z is the nondegenerate quadratic form with associated matrix τ Z · · · − Id Id 0 · · ·
00 2 τ Z · · · − Id Id . . . ... τ Z . . . . . . . . . ... ... . . . . . . ... ... ... . . . − Id Id0 0 · · · τ N Z · · · − Id − Id 0 0 · · · · · · − Id 0 · · ·
00 Id . . . . . . ... ... ...... . . . . . . − Id 00 · · · − Id 0 · · · when written in the basis ( p, p , · · · , p N , q, Q , · · · , Q N − ) , where τ i = t i +1 − t i .Proof. Let us denote ˜ H ( t, q, p ) = Z ( p ) , and apply Proposition A.8, noticing that since H = ˜ H for (cid:107) p (cid:107) ≥ R , (cid:107) d q,p ( H − ˜ H )( t, q, p ) (cid:107) ≤ C (1 + (cid:107) p (cid:107) ) ≤ C (1 + R ) . It gives that asubdivision can be chosen for both H and ˜ H and that ˜ G ts − G ts is then R ) -Lipschitz.For ˜ H , it directly follows from (5) that ˜ S ts u ( Q, q, p, ν ) = u ( q ) + Z ( ξ ) + p N · Q . Thequadratic form Z is nondegenerate as the associated matrix is invertible.Since ξ (cid:55)→ ˜ S ts u ( Q, ξ ) − S ts u ( Q, ξ ) = ˜ G ts ( Q, p, ν ) − G ts u ( Q, p, ν ) , it is R ) -Lipschitz,which proves the point. (cid:3) Proposition 2.4.
Let H be a Hamiltonian satisfying Hypothesis 1.1 with constant C , u be a C L -Lipschitz function, s < t and Q be in R d . If ξ = ( q, p, ν ) is a critical point of ξ (cid:55)→ S ts u ( Q, ξ ) , then for all τ in [ s, t ] , φ τs ( q, p ) ∈ B Ä Q, ( e C ( t − s ) − L ) ä × B Ä , e C ( t − s ) (1 + L ) − ä , where B ( x, r ) denotes the open ball of radius r centered on x .As a consequence, if H and ˜ H are two Hamiltonians satisfying Hypothesis 1.1 with con-stant C and coinciding on [ s, t ] × B Ä Q, ( e C ( t − s ) − L ) ä × B Ä , e C ( t − s ) (1 + L ) − ä ,the functions ξ (cid:55)→ S ts,H u ( Q, ξ ) and ξ (cid:55)→ S ts, ˜ H u ( Q, ξ ) have the same critical points and thesame associated critical values.Proof. We need to quantify the maximal distance covered by Hamiltonian trajectories.Hypothesis 1.1 gives an estimate which is uniform with respect to the initial position q : Lemma 2.5. If H satisfies Hypothesis 1.1 with constant C , then for each ( q, p ) , s ≤ t , (cid:107) P ts ( q, p ) − p (cid:107) < (1 + (cid:107) p (cid:107) )( e C ( t − s ) − , (cid:107) Q ts ( q, p ) − q (cid:107) < (1 + (cid:107) p (cid:107) )( e C ( t − s ) − . In other words, φ ts ( q, p ) belongs to B ( q, (1+ (cid:107) p (cid:107) )( e C ( t − s ) − × B ( p, (1+ (cid:107) p (cid:107) )( e C ( t − s ) − .Proof. The Hamiltonian system gives that (cid:107) P ts ( q, p ) − p (cid:107) ≤ (cid:82) ts (cid:107) ∂ q H ( τ, φ τs ( q, p )) (cid:107) dτ andusing the hypothesis, we get(7) (cid:107) P ts ( q, p ) − p (cid:107) < C (cid:90) ts (1 + (cid:107) P τs ( q, p ) (cid:107) ) dτ ≤ C (cid:90) ts ( (cid:107) P τs ( q, p ) − p (cid:107) + 1 + (cid:107) p (cid:107) ) dτ. ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 13
Lemma A.3 applied to f ( t ) = (cid:107) P ts ( q, p ) − p (cid:107) with K = C (1 + (cid:107) p (cid:107) ) gives the first estimate.Since (cid:107) Q ts ( q, p ) − q (cid:107) is bounded by the same inequality (7), it is easy to check the secondone. (cid:3) Now, if ξ = ( q, p, ν ) is a critical point, Proposition 2.1 states that p = du ( q ) , whence (cid:107) p (cid:107) ≤ L . Lemma 2.5 hence implies that for all s ≤ τ ≤ t , (cid:107) P τs ( q, p ) (cid:107) ≤ (cid:107) p (cid:107) + (1 + (cid:107) p (cid:107) )( e C ( τ − s ) − ≤ e C ( τ − s ) (1 + L ) − . Now using Lemma 2.5 between τ and t gives, since Q = Q tτ ( Q τs ( q, p ) , P τs ( q, p )) : (cid:107) Q − Q τs ( q, p ) (cid:107) ≤ (1 + (cid:107) P τs ( q, p ) (cid:107) )( e C ( t − τ ) − , and since (cid:107) P τs ( q, p ) (cid:107) ≤ e C ( τ − s ) (1 + L ) , we get (cid:107) Q − Q τs ( q, p ) (cid:107) ≤ (1 + L )( e C ( t − s ) − e C ( τ − s ) ) ≤ (1 + L )( e C ( t − s ) − . To prove the second statement, let us recall that if ˜ φ ts = ( ˜ Q ts , ˜ P ts ) denotes the Hamil-tonian flow for ˜ H , Proposition 2.1 states that ξ = ( q, p, Q , p , · · · , p N ) is a critical pointof ξ (cid:55)→ S ts,H u ( Q, ξ ) (resp. of ξ (cid:55)→ S ts, ˜ H u ( Q, ξ ) ) if and only if p = du ( q ) ,Q ts ( q, p ) = Q, ( resp. ˜ Q ts ( q, p ) = Q, )( Q i − , p i ) = φ t i s ( q, p ) ( resp. ( Q i − , p i ) = ˜ φ t i s ( q, p )) ∀ ≤ i ≤ N. But if ξ is a critical point of ξ (cid:55)→ S ts,H u ( Q, ξ ) , the previous work shows that the trajectory γ ( τ ) = φ τs ( q, p ) stays in B Ä Q, ( e C ( t − s ) − L ) ä × B Ä , e C ( t − s ) (1 + L ) − ä . It is hencea Hamiltonian trajectory both for H and ˜ H and ˜ φ τs ( q, p ) = φ τs ( q, p ) for all s ≤ τ ≤ t ,which hence shows that ξ is a critical point of ξ (cid:55)→ S ts, ˜ H u ( Q, ξ ) . The associated criticalvalue u ( q ) + A ts ( γ ) is also the same for H and ˜ H since γ stays in the set where H and ˜ H coincide. (cid:3) Remark . If H ( p ) is an integrable Hamiltonian satisfying Hypothesis 1.1 with constant C , then for each ( q, p ) , s ≤ t , P ts ( q, p ) = p and Lemma 2.5 may be improved: (cid:107) Q ts ( q, p ) − q (cid:107) < C ( t − s )(1 + (cid:107) p (cid:107) ) . As a consequence, if u is a C L -Lipschitz function, s < t and Q is in R d , and ξ = ( q, p, ν ) is a critical point of ξ (cid:55)→ S ts u ( Q, ξ ) , then for all τ in [ s, t ] , φ τs ( q, p ) ∈ B ( Q, C ( t − s )(1 + L )) × B (0 , L ) . Critical value selector.
Let us denote by Q m the set of functions on R m that canbe written as the sum of a nondegenerate quadratic form and of a Lipschitz function. Proposition 2.7.
There exists a function σ : (cid:83) m ∈ N Q m → R that satisfies:(1) if f is C , then σ ( f ) is a critical value of f ,(2) if c is a real constant, then σ ( c + f ) = c + σ ( f ) ,(3) if φ is a Lipschitz C ∞ -diffeomorphism of R m such that f ◦ φ is in Q m , then σ ( f ◦ φ ) = σ ( f ) , (4) if f − f is Lipschitz and f ≤ f on R d , then σ ( f ) ≤ σ ( f ) , (5) if ( f µ ) µ ∈ [ s,t ] is a C family of Q m with ( Z − f µ ) µ equi-Lipschitz for some nonde-generate quadratic form Z , then for all µ (cid:54) = ˜ µ ∈ [ s, t ] , min µ ∈ [ s,t ] min x ∈ Crit ( f µ ) ∂ µ f µ ( x ) ≤ σ ( f ˜ µ ) − σ ( f µ )˜ µ − µ ≤ max µ ∈ [ s,t ] max x ∈ Crit ( f µ ) ∂ µ f µ ( x ) . (6) if g ( x, η ) = f ( x ) + Z ( η ) where f is in Q m and Z is a nondegenerate quadraticform, then σ ( g ) = σ ( f ) .We call such an object a critical value selector . Such a critical value selector, named minmax , was introduced by Chaperon in 1991,see [Cha91]. Its construction and properties are detailed in Appendix B, which provesProposition 2.7. The uniqueness of such a selector is not guaranteed, see [Wei14].
Remark . Additional assumptions, which are satisfied by the minmax, will be made onthe critical value selector (see Proposition 4.4) in order to prove Joukovskaia’s theorem.They are not needed to prove Theorems 1.6 and 1.9, so we choose not to require themuntil then.
Remark . Properties 2.7-(2), 2.7-(3) and 2.7-(6) coupled with Viterbo’s uniquenesstheorem on generating functions (see [Vit92] and [Thé99]) imply that the variationaloperator we are going to obtain does not depend on the choice of generating family. SeeRemark B.2 for more details. Property 2.7-(3) implies in particular that σ ( f ◦ τ ) = σ ( f ) for each affine transformation τ of R d , which would be sufficient to prove Theorems 1.6and 1.9.Let us fix a critical value selector σ for the rest of the discussion. We gather here threeconsequences of the properties of the critical value selector. Consequence . If f and g are two functions of Q m with difference bounded andLipschitz on R m , then | σ ( f ) − σ ( g ) | ≤ (cid:107) f − g (cid:107) ∞ . This is obtained by combining 2.7-(4) and 2.7-(2).
Consequence . If f is a coercive function of Q m , then σ ( f ) = min( f ) . Proof.
Since f is in Q m , there exist a nondegenerate quadratic form Z and an L -Lipschitzfunction (cid:96) on R m such that f = Z + (cid:96) . Since f is coercive, it attains a global minimumat some point x , and necessarily Z is coercive, hence convex. Without loss of generality,we assume that x = 0 .We are going to use the following regularization of the norm: for each ε > , thefunction x (cid:55)→ (cid:107) x (cid:107) + εe −(cid:107) x (cid:107) /ε is C , strictly convex, -Lipschitz and attains its globalminimum ε at which is its only critical point.We have necessarily σ ( f ) ≥ min( f ) = f (0) (if f is C , this is true because σ ( f ) is acritical value of f - see Proposition 2.7-(1) - and we get the result for a general f bycontinuity - see Consequence 2.10). Let us prove the other inequality. For each x , f ( x ) = Z ( x ) + (cid:96) ( x ) ≤ Z ( x ) + (cid:96) (0) + L (cid:107) x (cid:107) ≤ Z ( x ) + (cid:96) (0) + L Ä (cid:107) x (cid:107) + εe −(cid:107) x (cid:107) /ε ä . The function x (cid:55)→ Z ( x )+ (cid:96) (0)+ L Ä (cid:107) x (cid:107) + εe −(cid:107) x (cid:107) /ε ä is convex as a sum of convex functionsand admits as a critical point, hence its only critical value is (cid:96) (0)+ ε . Since the difference ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 15 with f is L -Lipschitz, we may apply the Monotonicity property (Proposition 2.7-(4))which gives σ ( f ) ≤ (cid:96) (0)+ ε = f (0)+ ε . Letting ε tend to gives the wanted inequality. (cid:3) Consequence . If f µ = Z µ + (cid:96) µ is a C family of Q m with (cid:96) µ equi-Lipschitz, such thatthe set of critical points f µ does not depend on µ and such that µ (cid:55)→ f µ is constant onthis set, then µ (cid:55)→ σ ( f µ ) is constant. Proof.
Let us take µ in some bounded set [ s, t ] . Since µ (cid:55)→ Z µ is C and Z µ is nondegenerate for all µ , the index of Z µ does not depend on µ and for all µ there existsa linear isomorphism φ µ : R m → R m such that Z µ ◦ φ µ = Z s , and µ (cid:55)→ φ µ is C . Letus define ˜ f µ = f µ ◦ φ µ = Z s + (cid:96) µ ◦ φ µ and observe that ˜ f µ satisfies the hypotheses ofProposition 2.7-(5): to do so, we only need to check that (cid:96) µ ◦ φ µ is equi-Lipschitz, whichfollows from the fact that φ µ is equi-Lipschitz for µ in the compact set [ s, t ] .Now, let us check that ∂ µ ˜ f µ ( x ) = 0 for each critical point x of ˜ f µ , so that both boundsof Proposition 2.7-(5) are zero. Since φ µ is a C -diffeomorphism, x is a critical pointof ˜ f µ if and only if φ µ ( x ) is a critical point of f µ , i.e. df µ ( φ µ ( x )) = 0 . Then since µ (cid:55)→ f µ is constant on its critical points, ∂ µ f µ ( φ µ ( x )) = 0 . As a consequence, ∂ µ ˜ f µ ( x ) = ∂ µ f µ ( φ µ ( x )) + ∂ µ φ µ ( x ) df µ ( φ µ ( x )) = 0 and µ (cid:55)→ σ ( ˜ f µ ) is constant by Proposition 2.7-(5).Proposition 2.7-(3) ends the proof, stating that for all µ , σ ( ˜ f µ ) = σ ( f µ ◦ φ µ ) = σ ( f µ ) . (cid:3) Definition of R ts . In this section, we will say that a Hamiltonian is fiberwise com-pactly supported if there exists a
R > such that H ( t, q, p ) = 0 for (cid:107) p (cid:107) ≥ R . If Z ( p ) isa quadratic form, we denote by H CZ the set of C Hamiltonians H satisfying Hypothesis1.1 with constant C and such that H ( t, q, p ) − Z ( p ) is fiberwise compactly supported.If Z is a (possibly degenerate) quadratic form, Proposition 2.3 proves that the gener-ating family associated with a Hamiltonian in H CZ differs by a Lipschitz function from anondegenerate quadratic form. For Hamiltonians in H CZ , we are then able to define theoperator R ts directly by applying the critical value selector σ on the generating family.The localization of the critical points of the generating family (Proposition 2.4) allowsthen to show that the value of the operator does only depend on the behaviour of H ona large enough strip R × R d × B (0 , R ) .For general Hamiltonians satisfying Hypothesis 1.1, the generating family is a priorinot in any Q m , so we cannot select a critical value with the selector σ . To get aound thisdifficulty, we modify the Hamiltonian outside a large enough strip into some Z ( p ) . It isremarkable that the choice of Z has no incidence on the value of the operator: we henceobtain exactly the same operator by making the Hamiltonian compactly supported withrespect to p or by setting it on (cid:107) p (cid:107) , for example. To prove Theorems 1.6 and 1.9, we willsimply use Z = 0 , but when dealing with fiberwise convex Hamiltonians, for example toprove Theorem 1.11, the choice of a convex nondegenerate quadratic form will be moreadequate. Definition 2.13. If H is in H CZ and s ≤ t , let the operator ( R ts ) be defined for Lipschitzfunctions u on R d by R ts u ( Q ) = σ ( S ts u ( Q, · )) ∀ Q ∈ R d , where S ts u ( Q, · ) is the function ξ (cid:55)→ S ts u ( Q, ξ ) and S is the generating family defined at(6). In particular, if u is C , R ts u ( Q ) is a critical value of ξ (cid:55)→ S ts u ( Q, ξ ) . Proof.
Proposition 2.3 states that ξ (cid:55)→ S ts u ( Q, ξ ) is in some Q m . (cid:3) Proposition 2.14.
The operator R ts does not depend on the choice of subdivision of [ s, t ] in the definition of G , see (3) .Proof. It is enough to consider two cases: either the subdivisions are identical with onlyone intermediate step t i changing, or one subdivision is obtained from the other by addingartificially an intermediate step of length zero.In the first case, we observe that if the subdivision is fixed except for one interme-diate step t i , the function t i (cid:55)→ S ts u ( Q, ξ ) is C , hence uniformly continuous, and byConsequence 2.10 this implies that t i (cid:55)→ R ts u ( Q ) is continuous. But the set of criticalvalues of ξ (cid:55)→ S ts u ( Q, ξ ) does not depend on t i (see Proposition 2.1) and is discrete, hence t i (cid:55)→ R ts u ( Q ) must be a constant function.In the second case, let us artificially add an intermediate step t ι equal to t i : thesubdivision is now s = t ≤ t ≤ · · · ≤ t i − ≤ t ι = t i ≤ · · · ≤ t N +1 = t andthe variables ( Q, p, Q , p , Q , · · · , Q i − , p ι , Q ι , p i , · · · p N ) . We denote by G (resp. ˜ G )the family associated with the subdivision without (resp. with) t ι , that takes variables ( Q, p, Q , · · · , Q i − , p i , · · · , p N ) (resp. ( Q, p, Q , · · · , Q i − , p ι , Q ι , p i , · · · , p N ) ).Since F t i t ι = 0 and F t i t i − = F t ι t i − , we may observe that: ˜ G ( Q, · · · , Q i , p ι , Q ι , p i +1 , · · · , p N ) = G ( Q, · · · , Q i , p i +1 , · · · , p N ) − ( p i − p ι ) · ( Q ι − Q i − ) , and the same holds for the associated families S and ˜ S : ˜ S ( Q, q, ·· , Q i , p ι , Q ι , p i +1 , ·· , p N ) = S ( Q, q, ·· , Q i , p i +1 , ·· , p N ) − ( p i − p ι ) · ( Q ι − Q i − ) . The affine transformation mapping p ι to ˜ p ι = p i − p ι , Q ι to ˜ Q ι = Q ι − Q i − and keepingthe other variables fixed preserves the value of the selector by property 2.7-(3) of σ . Inthese new coordinates, the family writes: ˜ S ( Q, q, · · · , Q i , ˜ p ι , ˜ Q ι , p i +1 , · · · , p N ) = S ( Q, q, · · · , Q i , p i +1 , · · · , p N ) − ˜ p ι · ˜ Q ι and since (˜ p ι , ˜ Q ι ) (cid:55)→ − ˜ p ι · ˜ Q ι is a nondegenerate quadratic function of (˜ p ι , ˜ Q ι ) , theinvariance by stabilization 2.7-(6) for σ of the critical value selector concludes the proof. (cid:3) The following basic continuity result for R ts , which is improved in Theorem 1.6, is onlythere to allow to work with u os class C and extend the results by density: Proposition 2.15 (Weak contraction) . If H is in H CZ and u and v are two Lipschitzfunctions such that u − v is bounded, then R ts u − R ts v is bounded by (cid:107) u − v (cid:107) ∞ .Proof. Let us fix s , t and Q , and note that the quantity S ts u ( Q, ξ ) − S ts v ( Q, ξ ) = u ( q ) − v ( q ) is a Lipschitz and bounded function of ξ . The continuity of σ established in Consequence2.10 gives that (cid:107) R ts u ( Q ) − R ts v ( Q ) (cid:107) ≤ (cid:107) S ts u ( Q, · ) − S ts v ( Q, · ) (cid:107) ∞ ≤ (cid:107) u − v (cid:107) ∞ . (cid:3) The following proposition implies that the value of the operator depends only on thevalue of H on a large enough compact set: ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 17
Proposition 2.16.
Let Z and ˜ Z be two quadratic forms, and H (resp. ˜ H ) be a Hamil-tonian in H CZ (resp. H C ˜ Z ). For each L -Lipschitz function u and s ≤ t , if H = ˜ H on R × R d × B Ä , e C ( t − s ) (1 + L ) − ä , then R ts,H u = R ts, ˜ H u .Proof. Let us first assume that u is a C L -Lipschitz function and s ≤ t . Let us define H µ = µH +(1 − µ ) ˜ H . Observe that H µ is in H C ˜ Z µ where Z µ = µZ +(1 − µ ) ˜ Z is a quadraticform, and that there exists R > such that for all µ in [0 , , H µ ( t, q, p ) = Z µ ( p ) if (cid:107) p (cid:107) ≥ R .Proposition 2.3 hence guarantees that for all µ , S ts,H µ u ( Q, ξ ) = Z µ ( ξ ) + (cid:96) µ ( Q, ξ ) where Z µ is a nondegenerate quadratic form and ξ (cid:55)→ (cid:96) µ ( Q, ξ ) is Lipschitz with constant Lip( u ) + (cid:107) Q (cid:107) + 4(1 + R ) . Note that if Q is fixed, the family ξ (cid:55)→ (cid:96) µ ( Q, ξ ) is henceequi-Lipschitz when µ is in [0 , .As H µ is constant on R × R d × B Ä , e C ( t − s ) (1 + L ) − ä , the second part of Proposition2.4 states that the set of critical points of ξ (cid:55)→ S ts,H µ u ( Q, ξ ) does not depend on µ , andneither do the associated critical values.So if Q is fixed, the family of functions f µ = S ts,H µ u ( Q, · ) satisfies the conditions ofConsequence 2.12, and hence R ts,H µ u ( Q ) = σ ( f µ ) does not depend on µ . As a conse-quence, R ts,H u = R ts, ˜ H u .The result extends to every L -Lipschitz u thanks to Proposition 2.15 and the fact that u can be L ∞ -approximated by a C L -Lipschitz function. (cid:3) We now want to extend the definition to a Hamiltonian that is not quadratic at infinity,by modifying it outside some large enough strip R × R d × B (0 , R ) into some Z ( p ) . Wecannot make sure that the modified Hamiltonian still satisfies Hypothesis 1.1 with thesame constant C than H , so we have to be cautious since the width of the strip dependson C . Lemma 2.19 shows that the constant of the modified Hamiltonian can be arbitrarilyclose to C , and this independently from the width of the strip, which avoids any trouble. Proposition 2.17.
Let H be a C Hamiltonian satisfying Hypothesis 1.1 with constant C , u be a L -Lipschitz function and s ≤ t . For all δ > , and for each quadratic form Z such that (cid:107) d Z (cid:107) ≤ C , there exists a Hamiltonian H δ,Z in H C (1+ δ ) Z that coincides with H on R × R d × B Ä , e C (1+ δ )( t − s ) (1 + L ) − ä . Then, R ts,H δ,Z u does neither depend on thechoice of H δ,Z , nor on the choice of Z , nor on δ > . This proposition allows to define the variational operator for general Hamiltonians:
Definition 2.18.
Let H be a C Hamiltonian satisfying Hypothesis 1.1 with constant C . For each L -Lipschitz function u and s ≤ t , we define R ts,H u = R ts,H δ,Z u , where δ > and H δ,Z is a Hamiltonian of H C (1+ δ ) Z for some quadratic form Z such that (cid:107) d Z (cid:107) ≤ C ,which coincides with H on R × R d × B Ä , e C (1+ δ )( t − s ) (1 + L ) − ä . Proof of Proposition 2.17.
Let us show that for all δ > , there exists H δ in H C (1+ δ ) Z coinciding with H on R × R d × B (0 , R δ ) , where R δ = e C (1+ δ )( t − s ) (1 + L ) − . To do so,we use the following lemma: Lemma 2.19. If R > and ε > , there exists a compactly supported C function ϕ : R + → [0 , , equal to on [0 , R ] , such that for all r ≥ , | ϕ (cid:48) ( r ) | ≤ ε r ) , | ϕ (cid:48)(cid:48) ( r ) | ≤ ε r ) and | ϕ (cid:48) ( r ) | r ≤ ε r ) . For such a function ϕ , if H and ˜ H are two Hamiltonians satisfying Hypothesis 1.1 withconstant C , the Hamiltonian H ϕ : ( t, q, p ) (cid:55)→ ϕ ( (cid:107) p (cid:107) ) H ( t, q, p ) + (1 − ϕ ( (cid:107) p (cid:107) )) ˜ H ( t, q, p ) satisfies Hypothesis 1.1 with constant C (1 + ε ) , is equal to H on R × R d × B (0 , R ) and H ϕ − ˜ H is fiberwise compactly supported.Proof. Take some R (cid:48) > max(1 , R ) and let us define ϕ ( r ) = max Å , − ε
12 max (cid:0) , ln(1 + r ) − ln(1 + R (cid:48) ) (cid:1) ã . If r ≤ R (cid:48) , ϕ ( r ) = 1 . If r ≥ (1 + R (cid:48) ) e /ε − , ϕ ( r ) = 0 . For all r ≥ , ≤ ϕ ( r ) ≤ .The function ϕ is C ∞ except at r = R (cid:48) or r = (1 + R (cid:48) ) e /ε − . Let us evaluate itsderivatives on ( R (cid:48) , (1 + R (cid:48) ) e /ε − , where f ( r ) = 1 − ε (ln(1 + r ) − ln(1 + R (cid:48) )) : ϕ (cid:48) ( r ) = − ε r ) , ϕ (cid:48)(cid:48) ( r ) = ε r ) . Furthermore, as long as r ≥ R (cid:48) > , this implies that | ϕ (cid:48) ( r ) | = ε r ) ≤ εr r ) . Hence the three wanted estimates are satisfied on ( R (cid:48) , (1 + R (cid:48) ) e /ε − . Since ϕ (cid:48) and ϕ (cid:48)(cid:48) are zero if r < R (cid:48) or r > (1 + R (cid:48) ) e /ε − , it is possible to smooth ϕ by below at R (cid:48) and by above at (1 + R (cid:48) ) e /ε − without increasing the derivative bounds, keeping ϕ = 1 for r ≤ R and ϕ compactly supported.Now if H and ˜ H are two Hamiltonians satisfying Hypothesis 1.1 with constant C , letus define H ϕ by H ϕ ( t, q, p ) = ϕ ( (cid:107) p (cid:107) ) H ( t, q, p ) + (1 − ϕ ( (cid:107) p (cid:107) )) ˜ H ( t, q, p ) . It is C , coincideswith H on R × R d × B (0 , R δ ) , and H ϕ ( t, q, p ) − ˜ H ( t, q, p ) = ϕ ( (cid:107) p (cid:107) )( H ( t, q, p ) − ˜ H ( t, q, p )) is fiberwise compactly supported since ϕ ( r ) = 0 for r large enough.In order to verify that H ϕ satisfies Hypothesis 1.1 with constant C (1 + ε ) , let us boundthe derivatives of φ ( p ) = ϕ ( (cid:107) p (cid:107) ) : (cid:107) dφ ( p ) (cid:107) = | ϕ (cid:48) ( (cid:107) p (cid:107) ) | ≤ ε (cid:107) p (cid:107) ) , (cid:107) d φ ( p ) (cid:107) ≤ max Ç | ϕ (cid:48)(cid:48) ( (cid:107) p (cid:107) ) | , | ϕ (cid:48) ( (cid:107) p (cid:107) ) |(cid:107) p (cid:107) å ≤ ε (cid:107) p (cid:107) ) . Now, since both H and ˜ H satisfy | H ( t, q, p ) | ≤ C (1 + (cid:107) p (cid:107) ) and φ ( p ) ∈ [0 , for all p , | H ϕ ( t, q, p ) | ≤ φ ( p ) | H ( t, q, p ) | + (1 − φ ( p )) | ˜ H ( t, q, p ) | ≤ C (1 + (cid:107) p (cid:107) ) , ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 19
Since H and ˜ H satisfies Hypothesis 1.1 with constant C , H − ˜ H satisfies Hypothesis 1.1with constant C , and the following holds: (cid:107) dH ϕ (cid:107) ≤ φ ( p ) (cid:107) dH (cid:107) (cid:124) (cid:123)(cid:122) (cid:125) ≤ C (1+ (cid:107) p (cid:107) ) +(1 − φ ( p )) (cid:107) d ˜ H (cid:107) (cid:124) (cid:123)(cid:122) (cid:125) ≤ C (1+ (cid:107) p (cid:107) ) + | dφ ( p ) | (cid:124) (cid:123)(cid:122) (cid:125) ≤ ε (cid:107) p (cid:107) ) | H − ˜ H | (cid:124) (cid:123)(cid:122) (cid:125) ≤ C (1+ (cid:107) p (cid:107) ) ≤ C (1 + (cid:107) p (cid:107) ) + ε C (1 + (cid:107) p (cid:107) ) ≤ C (1 + ε )(1 + (cid:107) p (cid:107) ) , (cid:107) d H ϕ (cid:107) ≤ φ (cid:107) d H (cid:107) + (1 − φ ) (cid:107) d ˜ H (cid:107) + 2 (cid:107) dφ (cid:107)(cid:107) dH − d ˜ H (cid:107) + (cid:107) d φ (cid:107)| H − ˜ H |≤ φC + (1 − φ ) C + 2 ε (cid:107) p (cid:107) ) · C (1 + (cid:107) p (cid:107) ) + ε (cid:107) p (cid:107) ) · C (1 + (cid:107) p (cid:107) ) ≤ C + 2 ε C + ε C ≤ C (1 + ε ) . (cid:3) To build H δ,Z in H C (1+ δ ) Z coinciding with H on R × R d × B (0 , R δ ) , it is enough toapply Lemma 2.19 with ˜ H ( t, q, p ) = Z ( p ) , ε = δ and R = R δ = e C (1+ δ )( t − s ) (1 + L ) − .Let us now check that R ts,H δ,Z u is independent from the choice of H δ,Z and Z : if H δ,Z in H C (1+ δ ) Z and ˜ H δ, ˜ Z in H C (1+ δ )˜ Z coincide on R × R d × B Ä , e C (1+ δ )( t − s ) (1 + L ) − ä ,Proposition 2.16 applies and R ts,H δ,Z u = R ts, ˜ H δ, ˜ Z u .From now on, we may take Z = 0 , hence the set H C is exactly the set of C fiberwisecompactly supported Hamiltonians satisfying Hypothesis 1.1 with constant C . Let usprove the independence with respect to δ .Let s ≤ t and u a L -Lipschitz function be fixed, and still denote by R δ the radiusgiven by e C (1+ δ )( t − s ) (1 + L ) − , which is increasing with respect to δ . Take δ > ˜ δ > ,and H δ (resp. H ˜ δ ) a Hamiltonian in H C (1+ δ )0 (resp. H C (1+˜ δ )0 ) coinciding with H on R × R d × B (0 , R δ ) (resp. × B (cid:0) , R ˜ δ (cid:1) ), so that R t,δs,H u ( Q ) = R ts,H δ u ( Q ) and R t, ˜ δs,H u ( Q ) = R ts,H ˜ δ u ( Q ) .Lemma 2.19 applied with R = R δ , ε = ˜ δ and ˜ H = 0 gives a Hamiltonian H ϕ in H C (1+˜ δ )0 coinciding with H (hence H δ ) on R × R d × B (0 , R δ ) , and therefore since B (cid:0) , R ˜ δ (cid:1) ⊂ B (0 , R δ ) , with H ˜ δ on R × R d × B (cid:0) , R ˜ δ (cid:1) . Proposition 2.16 gives on the one hand that R ts,H δ u = R ts,H ϕ u , and on the other hand that R ts,H ϕ u = R ts,H ˜ δ u , hence the result. (cid:3) Addendum 2.20. If H is uniformly strictly convex with respect to p (i.e. there exists m > such that ∂ p H ( t, q, p ) ≥ m id for all ( t, q, p ) ) and Z is a strictly positive quadraticform such that m id ≤ Z ≤ C id , then the function H δ,Z of Proposition 2.17 can be chosenuniformly strictly convex w.r.t. p .Proof. In the proof of Lemma 2.19, we assume that H and ˜ H are uniformly strictlyconvex with respect to p with a constant m > . Then following the construction of H ϕ ,we may estimate its second derivative with respect to p : ∂ p H ϕ ≥ φ∂ p H + (1 − φ ) ∂ p ˜ H − Ä (cid:107) dφ (cid:107)(cid:107) ∂ p H − ∂ p ˜ H (cid:107) + (cid:107) d φ (cid:107)| H − ˜ H | ä id ≥ ( m − Cε )id using the estimates on the derivatives of ϕ , H and ˜ H . So, if ε < m/C , the obtainedfunction is uniformly strictly convex. (cid:3) Properties and Lipschitz estimates of R ts . Let us prove that ( R ts ) s ≤ t is a vari-ational operator. Monotonicity and additivity properties are straightforward: Proposition 2.21 (Monotonicity) . If u ≤ v are Lipschitz functions on R d , then for each s ≤ t , R ts u ≤ R ts v on R d .Proof. Let L be a Lipschitz constant for both u and v , and fix s ≤ t , δ > . Let H δ be aHamiltonian in H C (1+ δ )0 coinciding with H on R × R d × B Ä , e C (1+ δ )( t − s ) (1 + L ) − ä asin Definition 2.18, so that R ts,H u ( Q ) = R ts,H δ u ( Q ) and R ts,H v ( Q ) = R ts,H δ v ( Q ) .Since S ts,H δ v ( Q, ξ ) − S ts,H δ u ( Q, ξ ) = v ( q ) − u ( q ) is a non negative and Lipschitz functionof ξ , the monotonicity 2.7-(4) of σ applies and R ts,H δ u ( Q ) ≤ R ts,H δ v ( Q ) , thus R ts,H u ( Q ) ≤ R ts,H v ( Q ) . (cid:3) Proposition 2.22 (Additivity) . If c is a real constant, then R ts ( c + u ) = c + R ts u foreach Lipschitz function u .Proof. The additivity property 2.7-(2) of σ and the form of S ts u conclude, as in theprevious proof. (cid:3) Proposition 2.23 (Variational property) . For each C Lipschitz function u , Q in R d and s ≤ t , there exists ( q, p ) such that p = d q u , Q ts ( q, p ) = Q and if γ denotes theHamiltonian trajectory issued from ( q ( s ) , p ( s )) = ( q, p ) , R ts u ( Q ) = u ( q ) + A ts ( γ ) , Proof.
Let us fix u , s ≤ t and δ > and take as in Definition 2.18 a Hamiltonian H δ in H C (1+ δ )0 equal to H on R × R d × B Ä , e C (1+ δ )( t − s ) (1 + L ) − ä , such that R ts,H u ( Q ) = R ts,H δ u ( Q ) .Since u is C , R ts,H δ u ( Q ) is a critical value of χ (cid:55)→ S ts,H δ u ( Q, χ ) . Proposition 2.1,which describes the critical points and values of S , gives the existence of ( q, p ) suchthat Q ts,H δ ( q, p ) = Q and p = du ( q ) , and states that if γ δ ( τ ) = φ τs,H δ ( q, p ) denotes theHamiltonian trajectory issued from ( q, p ) for the Hamiltonian H δ , R ts,H δ u ( Q ) = u ( q ) + A ts,H δ ( γ δ ) . Proposition 2.4, which localizes the critical points of S under Hypothesis 1.1, gives that γ δ ( τ ) belongs to the set R × R d × B Ä , e C (1+ δ )( t − s ) (1 + L ) − ä for all τ in [ s, t ] .Since H and H δ coincide on that set for each time in [ s, t ] , γ δ is also a Hamiltoniantrajectory for H on [ s, t ] , the Hamiltonian action of γ δ has the same expression for H and H δ , and the conclusion holds: Q = Q ts,H δ ( q, p ) = Q ts,H ( q, p ) and R ts,H u ( Q ) = R ts,H δ u ( Q ) = u ( q ) + A ts,H ( γ δ ) . (cid:3) We now prove the Lipschitz estimates of Theorem 1.6, which imply that R ts satisfiesthe regularity property (3) of Hypotheses 1.3. ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 21
Proof of Theorem 1.6.
Suppose to begin with that u is C and that H is fiberwise com-pactly supported, meaning that there exists R > such that H ( t, q, p ) = 0 for (cid:107) p (cid:107) ≥ R .Under that assumption, in Proposition 2.3, the nondegenerate quadratic form Z doesnot depend on s or t .For each item of this proof, we are going to use Property 2.7-(5) on a suitable homotopy f µ , the form of the derivatives of S ts u given in Propositions 2.1 and 2.2 and the localizationof the critical points of S ts u described in Proposition 2.4.(1) Let us show that R ts u is Lipschitz with Lip( R ts u ) ≤ e C ( t − s ) (1 + L ) − . Let us fix Q and h in R d and define f µ ( ξ ) = S ts u ( Q + µh, ξ ) for µ in [0 , . The aim is toestimate | R ts u ( Q + h ) − R ts u ( Q ) | = | σ ( f ) − σ ( f ) | .Proposition 2.3 states that the family f µ is of the form required in Property2.7-(5), i.e. f µ ( ξ ) = Z ( ξ ) + (cid:96) µ ( ξ ) , where the family (cid:96) µ is equi-Lipschitz withconstant Lip( u ) + (cid:107) Q (cid:107) + (cid:107) h (cid:107) + 4(1 + R ) .Let us then estimate ∂ µ f µ : ∂ µ f µ ( q, p, ν ) = h · ∂ Q S ts ( Q + µh, ξ ) . If ξ µ = ( q µ , p µ , ν µ ) is a critical point of f µ , Proposition 2.1 gives on one handthat ∂ Q S ts ( Q + µh, ξ µ ) = P ts ( q µ , p µ ) and Proposition 2.4, on the other hand, that (cid:107) P ts ( q µ , p µ ) (cid:107) ≤ e C ( t − s ) (1 + L ) − .To sum it up, we have just proved that (cid:107) ∂ µ f µ (cid:107) ≤ (cid:107) h (cid:107) ( e C ( t − s ) (1 + L ) − foreach critical point of f µ . This implies by Property 2.7-(5) of the selector that | σ ( f ) − σ ( f ) | ≤ (cid:107) h (cid:107) ( e C ( t − s ) (1 + L ) − , hence the result.(2) Let us show that (cid:107) R t (cid:48) s u − R ts u (cid:107) ∞ ≤ Ce C ( t − s ) (1 + L ) | t (cid:48) − t | . It is enough toprove the result for | t − t (cid:48) | < δ / . We may therefore assume that ( t , · · · , t N ) is a subdivision suitable both between s and t and between s and t (cid:48) , since thechoice of the subdivision does not change the value of the variational operator R (see Proposition 2.14).Let us fix Q , t (cid:48) < t and s and define f µ ( ξ ) = S µs u ( Q, ξ ) for µ in [ t (cid:48) , t ] . The aimis to estimate | R ts u ( Q ) − R t (cid:48) s u ( Q ) | = | σ ( f t ) − σ ( f t (cid:48) ) | .By Proposition 2.3, the family f µ is as required in Property 2.7-(5), thanks tothe fact that the nondegenerate quadratic form Z does not depend on t ( = µ ).If ξ µ = ( q µ , p µ , ν µ ) is a critical point of f µ , Proposition 2.2-(1) gives on onehand that ∂ µ S µs ( Q, ξ µ ) = − H ( µ, Q, P µs ( q µ , p µ )) and Proposition 2.4 gives on theother hand that (cid:107) P µs ( q µ , p µ ) (cid:107) ≤ e C ( µ − s ) (1 + L ) − .By Hypothesis 1.1, we hence get that | ∂ µ S µs ( Q, ξ µ ) | ≤ C (1 + (cid:107) P µs ( q µ , p µ ) (cid:107) ) ≤ Ce C ( µ − s ) (1 + L ) . To sum it up, we have just proved that (cid:107) ∂ µ f µ (cid:107) ≤ Ce C ( t − s ) (1+ L ) for each µ in [ t (cid:48) , t ] and each critical point of f µ . Property 2.7-(5) hence states that µ (cid:55)→ σ ( f µ ) is Lipschitz with constant Ce C ( t − s ) (1 + L ) on [ t (cid:48) , t ] , hence the result.(3) Let us show that (cid:107) R ts (cid:48) u − R ts u (cid:107) ∞ ≤ C (1 + L ) | s (cid:48) − s | . Again we may assume that | s − s (cid:48) | is small enough to choose a subdivision suitable both between s and t andbetween s (cid:48) and t .Let us fix Q , t and s ≤ s (cid:48) and define f µ ( ξ ) = S tµ u ( Q, ξ ) for µ in [ s, s (cid:48) ] . Theaim is to estimate | R ts (cid:48) u ( Q ) − R ts u ( Q ) | = | σ ( f s (cid:48) ) − σ ( f s ) | . By Proposition 2.3, the family f µ is, again, as required in Property 2.7-(5).If ξ µ = ( q µ , p µ , ν µ ) is a critical point of f µ , Proposition 2.2-(1) gives on onehand that ∂ µ S tµ ( Q, ξ µ ) = H ( µ, q µ , p µ ) and Proposition 2.1 on the other hand that (cid:107) p µ (cid:107) = (cid:107) du ( q µ ) (cid:107) ≤ L .By Hypothesis 1.1, we hence get that | ∂ µ S tµ ( Q, ξ ) | ≤ C (1 + L ) . To sum it up, we have just proved that (cid:107) ∂ µ f µ (cid:107) ≤ C (1 + L ) for each µ in [ s, s (cid:48) ] and each critical point of f µ , hence µ (cid:55)→ σ ( f µ ) is Lipschitz with constant C (1 + L ) on [ s, s (cid:48) ] and the result holds.(4) Let us show that ∀ Q ∈ R d , (cid:12)(cid:12) R ts u ( Q ) − R ts v ( Q ) (cid:12)(cid:12) ≤ (cid:107) u − v (cid:107) ¯ B ( Q, ( e C ( t − s ) − L ) ) .For Q fixed, let us again define f µ = S ts ((1 − µ ) u + µv ) ( Q, · ) for µ in [0 , .The aim is to estimate | R ts v ( Q ) − R ts u ( Q ) | = | σ ( f ) − σ ( f ) | .By Proposition 2.3, since (1 − µ ) u + µv is L -Lipschitz, the family f µ is, again,as required in Property 2.7-(5). Let us then estimate ∂ µ f µ : ∂ µ f µ ( q, p, ν ) = v ( q ) − u ( q ) . If ξ µ = ( q µ , p µ , ν µ ) is a critical point of f µ , Proposition 2.4 gives that q µ belongsto B Ä Q, ( e C ( t − s ) − L ) ä , so that (cid:107) ∂ µ f µ (cid:107) ≤ (cid:107) u − v (cid:107) ¯ B ( Q, ( e C ( t − s ) − L ) ) foreach critical point of f µ , hence the result. Remark . The proof of the alternative Proposition 1.8 is contained here: if u ≤ v on B Ä Q, ( e C ( t − s ) − L ) ä , then ∂ µ f µ ( q, p, ν ) = v ( q ) − u ( q ) ≥ foreach critical point of f µ , hence R ts v ( Q ) − R ts u ( Q ) = σ ( f ) − σ ( f ) ≥ .If u is only Lipschitz with constant L , for all ε > we may find a C and L -Lipschitzfunction u ε such that (cid:107) u − u ε (cid:107) ∞ ≤ ε , and then by weak contraction (Proposition2.15) R ts u − R ts u ε is also bounded by ε for each s ≤ t . Writing the previous resultsfor u ε and then letting ε tend to zero gives us the wanted estimates.If H is not fiberwise compactly supported, let us fix L , T , and δ > and take aHamiltonian H δ in H C (1+ δ )0 that coincides with H on R × R d × B Ä , e C (1+ δ ) T (1 + L ) − ä as in Definition 2.18, so that if u is L -Lipschitz and ≤ s ≤ t ≤ T , R ts u = R ts,H δ u .The previous Lipschitz estimates, applied to R ts,H δ , give that:(1) R ts u is Lipschitz with constant Lip( R ts u ) ≤ e C (1+ δ )( t − s ) (1 + L ) − ,(2) (cid:107) R t (cid:48) s u − R ts u (cid:107) ∞ ≤ C (1 + δ ) e C (1+ δ )( t − s ) (1 + L ) | t (cid:48) − t | ,(3) (cid:13)(cid:13) R ts (cid:48) u ( Q ) − R ts u ( Q ) (cid:13)(cid:13) ∞ ≤ C (1 + δ )(1 + L ) | s (cid:48) − s | ,(4) (cid:12)(cid:12) R ts u ( Q ) − R ts v ( Q ) (cid:12)(cid:12) ≤ (cid:107) u − v (cid:107) ¯ B ( Q, ( e C (1+ δ )( t − s ) − L ) ) ,and we conclude the proof by letting δ tend to . (cid:3) Let us end this section with the analogous proof of Proposition 1.7, which describesthe dependence of the constructed operator with respect to the Hamiltonian.
Proof of Proposition 1.7.
Let H and H be two C Hamiltonians satisfying Hypothesis1.1 with constant C , u be a L -Lipschitz function, Q be in R d and s ≤ t . We are going toshow that | R ts,H u ( Q ) − R ts,H u ( Q ) | ≤ ( t − s ) (cid:107) H − H (cid:107) ¯ V , ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 23 where ¯ V = [ s, t ] × ¯ B Ä Q, ( e C ( t − s ) − L ) ä × ¯ B Ä , e C ( t − s ) (1 + L ) − ä .Let us first assume that u is a C function, and that H and H are fiberwise compactlysupported. Let us define H µ = (1 − µ ) H + µH for µ in [0 , and observe that H µ is in H C , and that there exists a R > such that H µ ( t, q, p ) = 0 for all (cid:107) p (cid:107) ≥ R and all µ in [0 , . Let us denote by φ ts,µ = ( Q ts,µ , P ts,µ ) the Hamiltonian flow for H µ .Let us fix Q and h in R d and define f µ ( ξ ) = S ts,H µ u ( Q, ξ ) for µ in [0 , . The aim is toestimate | R ts,H u ( Q ) − R ts,H u ( Q ) | = | σ ( f ) − σ ( f ) | .Proposition 2.3 states that the homotopy f µ is of the form required in the condition2.7-(5): f µ ( ξ ) = Z ( ξ ) + (cid:96) µ ( ξ ) , where the family ( (cid:96) µ ) is equi-Lipschitz with constant Lip( u ) + (cid:107) Q (cid:107) + 4(1 + R ) .Let ξ = ( q, p, ν ) be a critical point of f µ . On the one hand, Proposition 2.4 gives that φ τs,µ ( q, p ) is in B Ä Q, ( e C ( t − s ) − L ) ä × B Ä , e C ( t − s ) (1 + L ) − ä for all s ≤ τ ≤ t ,since H µ satisfies Hypothesis 1.1 with constant C . On the other hand, Proposition 2.2-(2)gives that ∂ µ f µ ( ξ ) = ∂ µ S ts,H µ u ( Q, q, p, ν ) = − (cid:90) ts ∂ µ H µ ( τ, φ τs,µ ( q, p )) dτ. Since ∂ µ H µ = H − H , we have just proved that (cid:107) ∂ µ f µ (cid:107) ≤ ( t − s ) (cid:107) H − H (cid:107) V for eachcritical point of f µ . This implies that | σ ( f ) − σ ( f ) | ≤ ( t − s ) (cid:107) H − H (cid:107) V by Property2.7-(5) of the selector, hence the result. Remark . The proof of the alternative Proposition 1.8 is contained here: if H ≤ H on V , then ∂ µ f µ ( ξ ) = − (cid:82) ts ( H − H )( τ, φ τs,µ ( q, p )) ≤ for each critical point of f µ , hence R ts,H u ( Q ) − R ts,H u ( Q ) = σ ( f ) − σ ( f ) ≤ .If u is only Lipschitz with constant L , for all ε > we may find a C and L -Lipschitzfunction u ε such that (cid:107) u − u ε (cid:107) ∞ ≤ ε , and then by continuity (Proposition 2.15) R ts u − R ts u ε is also bounded by ε for each s ≤ t . Writing the previous results for u ε and then letting ε tend to zero gives us the wanted estimates.If H and H are not fiberwise compactly supported, take δ > and H ,δ (resp. H ,δ )in H C (1+ δ )0 coinciding with H (resp. with H ) on R × R d × B Ä , e C (1+ δ )( t − s ) (1 + L ) − ä as in Definition 2.18, so that R ts,H u = R ts,H ,δ u and R ts,H u = R ts,H ,δ u . The previouswork applied to H ,δ and H ,δ gives that (cid:12)(cid:12)(cid:12) R ts,H u ( Q ) − R ts,H u ( Q ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) R ts,H ,δ u ( Q ) − R ts,H ,δ u ( Q ) (cid:12)(cid:12)(cid:12) ≤ ( t − s ) (cid:107) H ,δ − H ,δ (cid:107) V δ (cid:124) (cid:123)(cid:122) (cid:125) = (cid:107) H − H (cid:107) Vδ , where V δ = [ s, t ] × B Ä Q, ( e C (1+ δ )( t − s ) − L ) ä × B Ä , e C (1+ δ )( t − s ) (1 + L ) − ä . Theresult is then obtained by letting δ tend to . (cid:3) Let us add here the considerably simpler Lipschitz estimates obtained for integrableHamiltonians, using Remark 2.6 instead of Proposition 2.4 in the previous proofs.
Addendum 2.26. If H ( p ) (resp. ˜ H ( p ) ) satisfies Hypothesis 1.1 with constant C , thenfor ≤ s ≤ s (cid:48) ≤ t (cid:48) ≤ t and u and v two L -Lipschitz functions,(1) R ts u is L -Lipschitz,(2) (cid:107) R t (cid:48) s u − R ts u (cid:107) ∞ ≤ C (1 + L ) | t (cid:48) − t | , (3) (cid:107) R ts (cid:48) u − R ts u (cid:107) ∞ ≤ C (1 + L ) | s (cid:48) − s | ,(4) ∀ Q ∈ R d , (cid:12)(cid:12) R ts u ( Q ) − R ts v ( Q ) (cid:12)(cid:12) ≤ (cid:107) u − v (cid:107) ¯ B ( Q,C ( t − s )(1+ L )) ,(5) (cid:107) R ts, ˜ H u − R ts,H u (cid:107) ∞ ≤ ( t − s ) (cid:107) ˜ H − H (cid:107) ¯ B (0 ,L ) . where ¯ B ( Q, r ) denotes the closed ball of radius r centered in Q and (cid:107) u (cid:107) K := sup K | u | . Iterating the variational operator
A variational operator does a priori not satisfy the Markov property (5) of Hypotheses1.3, and in that case it cannot coincide with the viscosity operator. Yet we may obtain theviscosity operator from the variational operator we have just constructed by iterating italong a subdivision of the time space and letting then the maximal step of the subdivisiontend to zero. Doing so preserves the monotonicity, additivity, regularity and compatibilityproperties of the operator and the limit operator satisfies the Markov property, hence isthe viscosity operator.3.1.
Iterated operator and uniform Lipschitz estimates.
Let us recall the defini-tion of the iterated operator. We fix a sequence of subdivisions of [0 , ∞ ) Ä ( τ Ni ) i ∈ N ä N ∈ N such that for all N , τ N , τ Ni → i →∞ ∞ and i (cid:55)→ τ Ni is increasing. Assume also that forall N , i (cid:55)→ τ Ni +1 − τ Ni is bounded a constant δ N such that δ N tends to zero when N tendsto the infinite. Definition 3.1.
Let N be fixed and omitted in the notations. For t in R + , denote by i ( t ) the unique integer such that t belongs to [ τ i ( t ) , τ i ( t )+1 ) . Now, if u is a Lipschitz functionon R d , and ≤ s ≤ t , let us define the iterated operator at rank N by R ts,N u = R tτ i ( t ) R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s u, where R ts is any variational operator satisfying the Lipschitz estimate of Theorem 1.6.Let us now sum up the Lipschitz estimates of the iterated operator: note that thanksto the semigroup form of Lipschitz constants for the non iterated operator in Theorem1.6, the new estimates do not depend on N . Proposition 3.2.
Let ≤ s ≤ s (cid:48) ≤ t (cid:48) ≤ t ≤ T and u and v two L -Lipschitz functions.The Lipschitz constants for the iterated operator are:(1) Lip( R ts,N u ) ≤ e CT (1 + L ) − ,(2) (cid:107) R t (cid:48) s,N u − R ts,N u (cid:107) ∞ ≤ Ce CT (1 + L ) | t (cid:48) − t | ,(3) (cid:107) R ts (cid:48) ,N u − R ts,N u (cid:107) ∞ ≤ C (1 + L ) | s (cid:48) − s | ,(4) ∀ Q ∈ R d , (cid:12)(cid:12)(cid:12) R ts,N u ( Q ) − R ts,N v ( Q ) (cid:12)(cid:12)(cid:12) ≤ (cid:107) u − v (cid:107) ¯ B ( Q, ( e CT − L )) .Proof. This whole proof consists in exploiting the results of Theorem 1.6 while keepingthe Lipschitz estimates independent of N . ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 25 (1) Since
Lip( R ts u ) ≤ e C ( t − s ) (1+Lip( u )) − and R ts,N u = R tτ i ( t ) ( R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s u ) : Lip( R ts,N u ) ≤ e C ( t − τ i ( t ) ) (1 + Lip( R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s u )) − ≤ e C ( t − τ i ( t ) ) e C ( τ i ( t ) − τ i ( t ) − ) (1 + Lip( R τ i ( t ) − τ i ( t ) − · · · R τ i ( s )+1 s u )) − ≤ e C ( t − τ i ( t ) + τ i ( t ) −···− s ) (1 + Lip( u )) − ≤ e CT (1 + L ) − . (2) Assume that ≤ s ≤ t (cid:48) ≤ t ≤ T . It is enough to prove the result for | t − t (cid:48) | ≤ δ N ,and in that case either i ( t ) = i ( t (cid:48) ) , or i ( t ) = i ( t (cid:48) ) + 1 . If i ( t ) = i ( t (cid:48) ) , then (cid:107) R ts,N u − R t (cid:48) s,N u (cid:107) ∞ = (cid:107) R tτ i ( t ) Ä R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s u ä − R t (cid:48) τ i ( t ) Ä R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s u ä (cid:107) ∞ ≤ Ce C ( t − τ i ( t ) ) Ä Ä R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s u ää | t (cid:48) − t | . Now since Ä R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s u ä ≤ e C ( τ i ( t ) − s ) (1 + L ) , (cid:107) R ts,N u − R t (cid:48) s,N u (cid:107) ∞ ≤ Ce C ( t − s ) (1 + L ) | t − t (cid:48) | ≤ Ce CT (1 + L ) | t − t (cid:48) | . Else, assume that i ( t ) = i ( t (cid:48) ) + 1 . Then (cid:107) R ts,N u − R t (cid:48) s,N u (cid:107) ∞ = (cid:107) R ts,N u − R τ i ( t ) s,N u + R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s u − R t (cid:48) τ i ( t ) − · · · R τ i ( s )+1 s u (cid:107) ∞ and we may use the previous case to estimate both quantities: (cid:107) R ts,N u − R t (cid:48) s,N u (cid:107) ∞ ≤ Ce C ( t − s ) (1 + L ) | t − τ i ( t ) | + Ce C ( t − s ) (1 + L ) | τ i ( t ) − t (cid:48) |≤ Ce C ( t − s ) (1 + L ) | t − t (cid:48) | ≤ Ce CT (1 + L ) | t − t (cid:48) | since in that case t (cid:48) ≤ τ i ( t ) ≤ t .(3) Again, it is enough to prove the result for | s − s (cid:48) | ≤ δ N . We freely use a conse-quence of the estimate proved in the next point: (cid:107) R ts,N u − R ts,N v (cid:107) ∞ ≤ (cid:107) u − v (cid:107) ∞ If i ( s (cid:48) ) = i ( s ) , (cid:107) R ts,N u − R ts (cid:48) ,N u (cid:107) ∞ = (cid:107) R tτ i ( s )+1 ,N R τ i ( s )+1 s u − R tτ i ( s )+1 ,N R τ i ( s )+1 s (cid:48) u (cid:107) ∞ ≤ (cid:107) R τ i ( s )+1 s (cid:48) u − R τ i ( s )+1 s u (cid:107) ∞ ≤ C (1 + L ) | s − s (cid:48) | . If i ( s (cid:48) ) = i ( s ) + 1 , (cid:107) R ts (cid:48) ,N u − R ts,N u (cid:107) ∞ ≤ (cid:107) R ts (cid:48) ,N u − R tτ i ( s (cid:48) ) ,N u (cid:107) ∞ + (cid:107) R tτ i ( s (cid:48) ) ,N u − R ts,N u (cid:107) ∞ ≤ C (1 + L ) (cid:0) ( s (cid:48) − i ( s (cid:48) )) + ( i ( s (cid:48) ) − s ) (cid:1) ≤ C (1 + L ) | s − s (cid:48) | . (4) Let Q be fixed. Note that R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s u and R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s v are bothLipschitz with constant ( e C ( τ i ( t ) − s ) (1 + L ) − . Then | R ts,N u ( Q ) − R ts,N v ( Q ) | = | R tτ i ( t ) Ä R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s u ä ( Q ) − R tτ i ( t ) Ä R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s v ä ( Q ) |≤ (cid:107) R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s u − R τ i ( t ) τ i ( t ) − · · · R τ i ( s )+1 s v (cid:107) ¯ B Ä Q, ( e C ( t − τi ( t )) − e C ( τi ( t ) − s ) (1+ L )) ä . Estimating the Lipschitz constant of R τ i ( t ) − τ i ( t ) − · · · R τ i ( s )+1 s u and R τ i ( t ) − τ i ( t ) − · · · R τ i ( s )+1 s v gives the next step: | R ts,N u ( Q ) − R ts,N v ( Q ) |≤ (cid:107) R τ i ( t ) − τ i ( t ) − · · · R · s u − R τ i ( t ) − τ i ( t ) − · · · R · s v (cid:107) ¯ B Ä Q, ( e C ( t − s ) − e C ( τi ( t ) − − s ) )(1+ L )) ä ≤ · · · ≤ (cid:107) u − v (cid:107) ¯ B ( Q, ( e C ( t − s ) − L )) ) . (cid:3) Let us gather the Lipschitz dependence in s and t to obtain an estimation of hownon-Markov the iterated operator is: Proposition 3.3.
Take ≤ s ≤ r ≤ t ≤ T and u L -Lipschitz. Then for all integer N , (cid:107) R ts,N u − R tr,N R rs,N u (cid:107) ∞ ≤ Ce CT (1 + L ) δ N where δ N is the upper bound of i (cid:55)→ τ Ni +1 − τ Ni .Proof. Let us first show that if s ≤ r ≤ t , then (cid:107) R ts u − R tr R rs u (cid:107) ∞ ≤ Ce C ( t − s ) (1 + Lip( u )) | r − s | for each Lipschitz function u . Since R ss u = u , we might write (cid:107) R ts u − R tr R rs u (cid:107) ∞ ≤ (cid:107) R ts u − R tr u (cid:107) ∞ + (cid:107) R tr R ss u − R tr R rs u (cid:107) ∞ ≤ C (1 + Lip( u )) | r − s | + (cid:107) R ss u − R rs u (cid:107) ∞ ≤ C (1 + Lip( u )) | r − s | + Ce C ( r − s ) (1 + Lip( u )) | r − s |≤ C (1 + e C ( t − s ) ) (1 + Lip( u )) | r − s |≤ Ce C ( t − s ) (1 + Lip( u )) | r − s | . The second line is obtained by applying the Lipschitz estimates w.r.t. s and u of Theorem1.6, the third line by applying the Lipschitz estimate w.r.t. t (same Theorem).Now, let us fix N and estimate (cid:107) R ts,N u − R tr,N R rs,N u (cid:107) ∞ . The fourth point of Proposition3.2 implies that (cid:107) R ts,N u − R tr,N R rs,N u (cid:107) ∞ ≤ (cid:107) R τ i ( r )+1 s,N u − R τ i ( r )+1 r R rs,N u (cid:107) ∞ ≤ (cid:107) R τ i ( r )+1 τ i ( r ) R τ i ( r ) s,N u − R τ i ( r )+1 r R rτ i ( r ) R τ i ( r ) s,N u (cid:107) ∞ . Using the previous result gives that (cid:107) R ts,N u − R tr,N R rs,N u (cid:107) ∞ ≤ Ce C ( τ i ( r )+1 − τ i ( r ) ) Ä R τ i ( r ) s,N u ) ä | r − τ i ( r ) | and since Ä R τ i ( r ) s,N u ) ä ≤ e C ( τ i ( r ) − s ) (1 + Lip( u )) , we get (cid:107) R ts,N u − R tr,N R rs,N u (cid:107) ∞ ≤ Ce C ( τ i ( r )+1 − s ) (1 + Lip( u )) | r − τ i ( r ) | . Then the result comes by using the definition of δ N . (cid:3) Let us add a word on the dependence with respect to H , extending Proposition 1.7: ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 27
Proposition 3.4.
Let H and H be two C Hamiltonians satisfying Hypothesis 1.1 withconstant C , u be a L -Lipschitz function, Q be in R d and s ≤ t . Then | R ts,H ,N u ( Q ) − R ts,H ,N u ( Q ) | ≤ ( t − s ) (cid:107) H − H (cid:107) ¯ V , where ¯ V = [ s, t ] × ¯ B Ä Q, ( e C ( t − s ) − L ) ä × ¯ B Ä , e C ( t − s ) (1 + L ) − ä .Proof. To lighten the notation, let us prove that for the non iterated operator, | R tτ,H R τs,H u ( Q ) − R tτ,H R τs,H u ( Q ) |≤ ( t − s ) (cid:107) H − H (cid:107) [ s,t ] × ¯ B ( Q, ( e C ( t − s ) − L ) ) × ¯ B ( ,e C ( t − s ) (1+ L ) − ) . The result is then obtained for the iterated operator by induction on the number of stepsbetween s and t .For both H and H , R τs u ) ≤ e C ( τ − s ) (1 + L ) by Theorem 1.6. Hence, on theone hand, Proposition 1.7 gives that | R tτ,H R τs,H u ( Q ) − R tτ,H R τs,H u ( Q ) |≤ ( t − τ ) (cid:107) H − H (cid:107) [ τ,t ] × ¯ B ( Q, ( e C ( t − τ ) − e C ( τ − s ) (1+ L ) ) × ¯ B ( ,e C ( t − τ ) e C ( τ − s ) (1+ L ) − ) ≤ ( t − τ ) (cid:107) H − H (cid:107) ¯ V . On the other hand, using the Lipschitz estimate with respect to u of Theorem 1.6, | R tτ,H R τs,H u ( Q ) − R tτ,H R τs,H u ( Q ) | ≤ (cid:107) R τs,H u − R τs,H u (cid:107) ¯ B ( Q, ( e C ( t − s ) − e C ( τ − s ) (1+ L ) ) Proposition 1.7 gives that for each q of ¯ B Ä Q, ( e C ( t − s ) − e C ( τ − s ) (1 + L ) ä , | R τs,H u ( q ) − R τs,H u ( q ) | ≤ ( τ − s ) (cid:107) H − H (cid:107) [ s,τ ] × ¯ B ( q, ( e C ( τ − s ) − L ) ) × ¯ B ( ,e C ( τ − s ) (1+ L ) − ) , and then summing up the radius of the balls gives | R tτ,H R τs,H u ( Q ) − R tτ,H R τs,H u ( Q ) |≤ ( τ − s ) (cid:107) H − H (cid:107) [ s,τ ] × ¯ B ( Q, ( e C ( t − s ) − L ) ) × ¯ B ( ,e C ( τ − s ) (1+ L ) − ) ≤ ( τ − s ) (cid:107) H − H (cid:107) ¯ V . Summing up the two estimates concludes the proof. (cid:3)
Convergence towards the viscosity operator.
In this section we prove thatthe iterated operator sequence ( R ts,N ) N converges to a limit operator when the maximalstep of the subdivision tends to . To do so, we first use a compactness argumentto get a converging subsequence (Theorem 3.9), then show that the limit of such asubsequence is the viscosity operator (Proposition 3.10) and finally prove Theorem 1.9with the uniqueness of this operator. Definition 3.5.
Let (cid:107) · (cid:107)
Lip be the norm on the sets of real-valued Lipschitz functionson R d given by (cid:107) u (cid:107) Lip = | u (0) | + Lip( u ) . Definition 3.6.
We denote by L L ( K ) the set of Lipschitz functions on R d supported bythe compact set K and with Lipschitz norm (cid:107) · (cid:107) Lip bounded by the constant L : L L ( K ) = ® u ∈ C , ( R d , R ) (cid:12)(cid:12)(cid:12)(cid:12) supp ( u ) ⊂ K (cid:107) u (cid:107) Lip ≤ L ´ Proposition 3.7.
The set L L ( K ) is a compact set for the uniform norm.Proof. The Arzelà-Ascoli theorem immediately gives that the closure of L L ( K ) is com-pact. Then, it is easy to check that L L ( K ) is closed. Hence, it is compact. (cid:3) Proposition 3.8.
For each
T > , R > , L > , the family ¶ ( s, t, Q, u ) (cid:55)→ R ts,N u ( Q ) © N is equi-Lipschitz on the set { ≤ s ≤ t ≤ T } × ¯ B (0 , R ) × L L ( ¯ B (0 , R )) .Proof. It is enough to observe that the Lipschitz constants obtained in Proposition 3.2depend only on T , R , L , and that if u and v are compactly supported Lipschitz functions, (cid:107) R ts,N u − R ts,N v (cid:107) ≤ (cid:107) u − v (cid:107) ∞ . (cid:3) Theorem 3.9.
There exists a subsequence N k such that for all ≤ s ≤ t , Q ∈ R d , u Lipschitz function on R d , R ts,N k u ( Q ) has a limit when k tends to ∞ , denoted ¯ R ts u ( Q ) .Furthermore, the sequence of functions ¶ ( s, t, Q ) (cid:55)→ R ts,N k u ( Q ) © k converges uniformlytowards ( s, t, Q ) (cid:55)→ ¯ R ts u ( Q ) on every compact subset of { ≤ s ≤ t } × R d .Proof. The first step consists in applying Arzelà-Ascoli theorem with ( s, t, Q, u ) livingin the compact set { ≤ s ≤ t ≤ T } × ¯ B (0 , R ) × L L (cid:0) ¯ B (0 , R ) (cid:1) , where T , R and L arefixed. The second step is to get a subsequence working for all T , R and L . The third stepconsists in extending the result to Lipschitz functions which are not compactly supported. First step.
Since Proposition 3.8 gives that ¶ ( s, t, Q, u ) (cid:55)→ R ts,N v ( Q ) © N is equi-Lipschitzon { ≤ s ≤ t ≤ T } × ¯ B (0 , R ) × L L (cid:0) ¯ B (0 , R + CT ) (cid:1) , it is enough to prove that it isuniformly bounded at one point - for example ( s, s, Q, - to gather all the conditionsrequired to apply Arzelà-Ascoli theorem. | R ss,N Q ) | = | Q ) | = 0 , hence, there exists a subsequence N k (a priori depending on T , R and L ) such thatthe sequence ¶ ( s, t, Q, u ) (cid:55)→ R ts,N k u ( Q ) © k converges uniformly to a limit ( s, t, Q, u ) (cid:55)→ ¯ R ts u ( Q ) on the compact set { ≤ s ≤ t ≤ T } × ¯ B (0 , R ) × L L (cid:0) ¯ B (0 , R ) (cid:1) . Second step.
In this paragraph we will describe a subsequence by the diagonal process.Note that the first step also applies on every subsequence of ( R ts,N ) N .Let T i = R i = L i = i for each integer i .For i = 1 , let ψ be the subsequence given by the Arzelà-Ascoli theorem for thesequence ( R ts,N ) N ∈ N and the constants T = L = R = 1 .For i > , let ψ i be the subsequence given by the Arzelà-Ascoli theorem for thesequence ( R ts,ψ i − ( N ) ) N ∈ N and the constants T i = L i = R i = i .Now define the diagonal subsequence N k = ψ k ( k ) : for all k , ( N i ) i ≥ k is extracted from ψ k .For each T , R , L , there exists i such that T ≤ i , R ≤ i and L ≤ i . Since R ts,ψ i ( k ) converges on { ≤ s ≤ t ≤ i } × ¯ B (0 , i ) × L i (cid:0) ¯ B (0 , i ) (cid:1) , it converges on { ≤ s ≤ t ≤ T } × ¯ B (0 , R ) × L L (cid:0) ¯ B (0 , R ) (cid:1) , and so does R ts,N k since N k is a subsequence of ψ i ( k ) . ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 29
Hence we have constructed a subsequence that works for all L , R , T positive constants.If L c denotes the set of compactly supported Lipschitz functions, (cid:91) T,L,R { ≤ s ≤ t ≤ T } × ¯ B (0 , R ) × L L (cid:0) ¯ B (0 , R ) (cid:1) = { ≤ s ≤ t } × R d × L c , and the subsequence we have constructed converges for all s ≤ t , Q ∈ R d and u compactlysupported Lipschitz function. Third step.
Now take T and R two constants and u a Lipschitz function on R d , withLipschitz constant L . For all ¯ L > L , we build a compactly supported ¯ L -Lipschitz function ¯ u such that ¯ u = u on ¯ B Ä , R + ( e CT − L ) ä : to do so, let us take a compactlysupported C function φ : R + → [0 , such that ® φ = 1 on [0 , R + ( e CT − L )] , | φ (cid:48) ( x ) | ≤ L (cid:48) − L | u (0) | + Lx ∀ x ≥ , and ¯ u ( q ) = φ ( (cid:107) q (cid:107) ) · u ( q ) .If u is C , so is ¯ u , and since (cid:107) d q ( φ ( (cid:107) q (cid:107) )) (cid:107) = | φ (cid:48) ( (cid:107) q (cid:107) ) | ≤ L (cid:48) − L | u (0) | + L (cid:107) q (cid:107) , the differential of ¯ u is bounded by ¯ L : (cid:107) d ¯ u ( q ) (cid:107) ≤ (cid:107) d q ( φ ( (cid:107) q (cid:107) )) (cid:107) · | u ( q ) | (cid:124) (cid:123)(cid:122) (cid:125) ≤ ¯ L − L + | φ ( q ) | (cid:124) (cid:123)(cid:122) (cid:125) ≤ · (cid:107) du ( q ) (cid:107) (cid:124) (cid:123)(cid:122) (cid:125) ≤ L ≤ ¯ L. If u is not C , one can show that ¯ u is ¯ L -Lipschitz by applying the mean value theoremto φ .For all Q in the ball ¯ B (0 , R ) , since u and ¯ u are ¯ L -Lipschitz and coincide on the ballcentered in Q of radius ( e CT − L ) , the Lipschitz property 3.2-(4) gives R ts,N ¯ u ( Q ) = R ts,N u ( Q ) ∀ N ∈ N , ∀ ≤ s ≤ t ≤ T. Since ¯ u is a compactly supported function, ¶ ( s, t, Q ) (cid:55)→ R ts,N k ¯ u ( Q ) © k uniformly convergeson { ≤ s ≤ t ≤ T }× ¯ B (0 , R ) , and thus the same holds for ¶ ( s, t, Q ) (cid:55)→ R ts,N k u ( Q ) © k . (cid:3) Proposition 3.10.
The limit operator ¯ R ts is the viscosity operator: ¯ R ts = V ts .Proof. (1) Monotonicity property follows from the monotonicity of R ts , for s ≤ t .(2) Same thing for the additivity property.(3) Regularity: since the convergence of ¶ ( s, t, Q ) (cid:55)→ R ts,N k v ( Q ) © k is uniform on everycompact subset of { ≤ s ≤ t } × R d , and the family is equi-Lipschitz in time andspace, the limit satisfies that (cid:8) ¯ R tτ u, t ∈ [ τ, T ] (cid:9) is uniformly Lipschitz for each τ ≤ T and ( t, q ) (cid:55)→ ¯ R tτ u ( q ) is locally Lipschitz on ( τ, ∞ ) × R d .(4) Compatibility with Hamilton-Jacobi equation: Remark 1.4 and Proposition 2.23give the compatibility property for the operator R ts . Hence if u is a Lipschitz C solution of the Hamilton-Jacobi equation, for all N : R ts,N u s = R tτ i ( t ) · · · R τ i ( s )+1 s u s (cid:124) (cid:123)(cid:122) (cid:125) = u τi ( s )+1 = R tτ i ( t ) u τ i ( t ) = u t , and the limit satisfies ¯ R ts u s = u t . (5) Markov property: take u Lipschitz, and ≤ s ≤ τ ≤ t ≤ T . Let us show theequality ¯ R tτ ◦ ¯ R τs u = ¯ R ts u . Let Q be fixed in R d .Since Q (cid:55)→ ¯ R τs u ( Q ) is Lipschitz, Ä R tτ,N k ¯ R τs u ( Q ) ä k converges to ¯ R tτ ¯ R τs u ( Q ) .Let us first show that R tτ,N k R τs,N k u ( Q ) tends to ¯ R tτ ¯ R τs u ( Q ) . (cid:12)(cid:12)(cid:12) R tτ,N k R τs,N k u ( Q ) − ¯ R tτ ¯ R τs u ( Q ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) R tτ,N k R τs,N k u ( Q ) − R tτ,N k ¯ R τs u ( Q ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) R tτ,N k ¯ R τs u ( Q ) − ¯ R tτ ¯ R τs u ( Q ) (cid:12)(cid:12)(cid:12)(cid:124) (cid:123)(cid:122) (cid:125) → . Now, the uniform Lipschitz estimates of property 3.2-(4) give (cid:12)(cid:12)(cid:12) R tτ,N k R τs,N k u ( Q ) − R tτ,N k ¯ R τs u ( Q ) (cid:12)(cid:12)(cid:12) ≤ (cid:107) R τs,N k u − ¯ R τs u (cid:107) ¯ B ( Q,r ) for some radius r depending only on C , T , L ; as the convergence is uniform onevery compact subset of R d , the right hand side tends to when k tends to ∞ .Now, since δ N k → k →∞ , Proposition 3.3 implies that R tτ,N k R τs,N k u ( Q ) and R ts,N k u ( Q ) have the same limit, hence the conclusion: ¯ R ts u ( Q ) = ¯ R tτ ¯ R τs u ( Q ) . (cid:3) Consequence . We have proved, for every Hamiltonian satisfying Hypothesis 1.1, thatthe viscosity operator exists. In particular, for such a Hamiltonian and for a Lipschitzinitial condition, there exists a viscosity solution of (HJ) on (0 , ∞ ) × R d that coincideswith the initial condition at time , see Proposition D.2. Proof of Theorem 1.9.
Since every subsequence of R ts,N u admits a subsequence uniformlyconverging to the viscosity solution V ts u on every compact set, the whole family ( R ts,N u ) N converge to V ts u by uniqueness of the viscosity solution. (cid:3) The local Lipschitz estimates on the viscosity operator V and the local monotonicityproperties stated in Proposition 1.10 are directly deduced from this uniform convergenceand the estimates on the variational operator R . In the integrable case, the iteratedoperator R ts,N satisfies the same Lipschitz estimate than the variational operator R ts (seeAddendum 2.26), whence the following result. Addendum 3.12. If H ( p ) (resp. ˜ H ( p ) ) satisfies Hypothesis 1.1 with constant C , thenfor ≤ s ≤ s (cid:48) ≤ t (cid:48) ≤ t and u and v two L -Lipschitz functions,(1) V ts u is L -Lipschitz,(2) (cid:107) V t (cid:48) s u − V ts u (cid:107) ∞ ≤ C (1 + L ) | t (cid:48) − t | ,(3) (cid:107) V ts (cid:48) u − V ts u (cid:107) ∞ ≤ C (1 + L ) | s (cid:48) − s | ,(4) ∀ Q ∈ R d , (cid:12)(cid:12) V ts u ( Q ) − V ts v ( Q ) (cid:12)(cid:12) ≤ (cid:107) u − v (cid:107) ¯ B ( Q,C ( t − s )(1+ L )) ,(5) (cid:107) V ts, ˜ H u − V ts,H u (cid:107) ∞ ≤ ( t − s ) (cid:107) ˜ H − H (cid:107) ¯ B (0 ,L ) . where ¯ B ( Q, r ) denotes the closed ball of radius r centered in Q and (cid:107) u (cid:107) K := sup K | u | . ARIATIONAL AND VISCOSITY OPERATORS FOR THE EVOLUTIVE HJ EQUATION 31 The convex case
The purpose of this chapter is to prove Theorem 1.11, that states in particular that forstrictly convex Hamiltonians, the variational operator constructed in this paper coincideswith the Lax-Oleinik semigroup. To do so, we give a description of the Lax-Oleinik semi-group in terms of broken geodesics, and discuss the link between the so-called
Lagrangiangenerating family involved in this description and the generating family used for generalHamiltonians.4.1.
The Lax-Oleinik semigroup with broken geodesics.
The Lax-Oleinik semi-group defined by the equation (1) in the introduction may also be written as a finitedimensional optimization problem. If H is strictly uniformly convex w.r.t. p and satis-fies Hypothesis 1.1, we fix δ > such that ( q, p ) (cid:55)→ ( q, Q ts ( q, p )) is a C -diffeomorphismfor each | t − s | ≤ δ (see Proposition A.9). Proposition 4.1. If s = t ≤ t ≤ · · · ≤ t N = t is a subdivision such that t i +1 − t i < δ for all i , then T ts u ( Q ) = min q,Q , ··· ,Q N − A ts u ( Q, q, Q , · · · , Q N − ) , with the Lagrangian generating family A defined by A ts u ( Q, q, Q , · · · , Q N − ) = u ( q ) + N (cid:88) i =0 (cid:90) t i +1 t i L Ä τ, Q τt i ( Q i − , p i ) , ∂ τ Q τt i ( Q i − , p i ) ä dτ where p i is uniquely defined by Q t i +1 t i ( Q i − , p i ) = Q i and while denoting q = Q − and Q = Q N . A proof of this statement can be found in [Ber12], Lemma 48 and Proposition 49.The two next propositions gather properties of the Lagrangian generating family A . Proposition 4.2. If H is uniformly strictly convex w.r.t. p , for δ small enough, A ts u ( Q, q, Q , · · · , Q N − ) = max p,p , ··· ,p N S ts u ( Q, q, p, Q , · · · , p N ) . Proof.
This is a direct consequence of Proposition A.12, since by definition A ts u ( Q, q, Q , · · · , Q N − ) = u ( q ) + A ts ( q, Q , · · · , Q ) and S ts u ( Q, q, p, Q , · · · , p N ) = u ( q ) + G ts ( p, Q , · · · , p N , Q ) + p · ( Q − q ) , with the notations of Appendix A. (cid:3) Proposition 4.3. If H satisfies Hypothesis 1.1 with constant C , is uniformly strictlyconvex w.r.t. p and H ( t, q, p ) = (cid:107) p (cid:107) outside of a band R × R d × B (0 , R ) , then thefunction ( q, Q , · · · , Q N − ) (cid:55)→ A ts u ( Q, q, Q , · · · , Q N − ) is coercive and in some Q m . Proof.
We are first going to prove the result for H ( t, q, p ) = (cid:107) p (cid:107) . In that case, L ( t, q, v ) = (cid:107) v (cid:107) and Q t i +1 t i ( Q i − , p i ) = Q i if and only if Q i = Q i − + ( t i +1 − t i ) p i . Thus A ts u ( Q, q, Q , · · · , Q N − ) = u ( q ) + N (cid:88) i =0 (cid:90) t i +1 t i L Ä τ, Q τt i ( Q i − , p i ) , ∂ τ Q τt i ( Q i − , p i ) ä dτ = u ( q ) + 12 N (cid:88) i =0 (cid:90) t i +1 t i (cid:107) Q i − Q i − (cid:107) ( t i +1 − t i ) dτ = u ( q ) + 12 N (cid:88) i =0 (cid:107) Q i − Q i − (cid:107) t i +1 − t i always denoting q = Q − and Q = Q N . To see that the considered function is coer-cive and in some Q m , we may then use the affine diffeomorphism ( q, Q , · · · , Q N − ) (cid:55)→ ( Q − q √ t − s , Q − Q √ t − t , · · · , Q − Q N − √ t − t N ) .Now, if H ( t, q, p ) = (cid:107) p (cid:107) outside of a band R × R d × B (0 , R ) , and if ˜ H denotesthe quadratic form ˜ H ( p ) = (cid:107) p (cid:107) , H and ˜ H satisfy the hypotheses of Proposition A.14with constants C and K = C (1 + R ) , and thus ˜ A ts u − A ts u = ˜ A ts − A ts is a Lip-schitz function of ( q, Q , · · · , Q N − ) . The previous part hence proves that the function ( q, Q , · · · , Q N − ) (cid:55)→ A ts u ( Q, q, Q , · · · , Q N − ) is coercive and in some Q m . (cid:3) Proof of Joukovskaia’s Theorem.
To prove that the variational operator R ts constructed in Chapter 2 is the viscosity operator, it is enough to prove that it satisfiesthe Markov property (5), see Remark 1.4. In that purpose, we need the critical valueselector to satisfy the two additional following properties - properties that are actuallysatisfied by the minmax constructed in Appendix B. Proposition 4.4.
There exists a critical value selector σ : (cid:83) m ∈ N Q m → R , as defined inProposition 2.7, that satisfies:(1) σ ( − f ) = − σ ( f ) ,(2) if f ( x, y ) is a C function of Q m such that ∂ y f ≥ c id for some c > , and if g defined by g ( x ) = min y f ( x, y ) is in some Q ˜ m , then σ ( g ) = σ ( f ) . We assume σ to be such a critical value selector. Proof of Theorem 1.11. First step.
We assume that the Hamiltonian H is uniformlystrictly convex w.r.t. p ( ∂ p H ≥ m id), satisfies Hypothesis 1.1 with some constant C andcoincides with the quadratic form Z ( p ) = (cid:107) p (cid:107) outside of a band R × R d × B (0 , R ) . Thenthe variational operator constructed in Chapter 2 is the Lax-Oleinik operator: R ts = T ts .To see this, we apply the last item to the function f ( x, y ) = S ts u ( Q, q, p, Q , · · · , p N ) where x = ( q, Q , Q , · · · , Q N − ) and y = ( p, · · · , p N ) . Proposition A.11 gives, since S ts u ( Q, q, p, Q , · · · , p N ) = u ( q ) + G ts ( p, Q , · · · , p N , Q ) + p · ( Q − q ) , that y (cid:55)→ f ( x, y ) isuniformly strictly concave, and Proposition 4.2 gives that g ( x ) = max y f ( x, y ) = u ( q ) + N (cid:88) i =0 (cid:90) t i +1 t i L Ä τ, Q τt i ( Q i − , p i ) , ∂ τ Q τt i ( Q i − , p i ) ä dτ. Proposition 4.3 states that g is a coercive function of some Q ˜ m . Since g is coercive,Consequence 2.11 states that σ ( g ) = min g , so we have that T ts u ( Q ) = min g = σ ( g ) = σ ( f ) = R ts u ( Q ) . Second step.
We only assume that the Hamiltonian H is uniformly strictly convex w.r.t. p ( ∂ p H ≥ m id) and satisfies Hypothesis 1.1 with some constant C . It does not a prioricoincide with a quadratic form at infinity.Let us prove the Markov property: we fix u , s ≤ τ ≤ t and Q and we are going toshow that R tτ R τs u ( Q ) = R ts u ( Q ) . If Z denotes the quadratic form Z ( p ) = (cid:107) p (cid:107) , we maychoose δ > and build as in Definition 2.18 a Hamiltonian H δ in H C (1+ δ ) Z such that both R ts u ( Q ) = R ts,H δ u ( Q ) and R tτ,H δ R τs,H δ u ( Q ) = R tτ R τs u ( Q ) . Addendum 2.20 states that H δ can moreover be constructed uniformly strictly convex w.r.t. p .The previous work applies to H δ , and hence R tτ R τs u ( Q ) = R tτ,H δ R τs,H δ u ( Q ) = T tτ,H δ T τs,H δ u ( Q ) = T ts,H δ u ( Q ) = R ts,H δ u ( Q ) = R ts u ( Q ) since T ts,H δ is a semigroup. We hence showed that R ts satisfies the Markov property (5).The uniqueness of the viscosity operator concludes: R ts = V ts = T ts . Third step. If H is convex with respect to p and satisfies Hypothesis 1.1 with constant C , H ε ( t, q, p ) = H ( t, q, p ) + ε (cid:107) p (cid:107) is uniformly strictly convex w.r.t. p ( ∂ p H ε ≥ ε id )and satisfies Hypothesis 1.1 with constant C + ε .Now for all ε ≤ , the estimates of Propositions 1.7 and 1.10 give, for all s ≤ t andLipschitz function u : (cid:107) R ts,H ε u − R ts,H u (cid:107) ∞ ≤ ( t − s ) (cid:107) H ε − H (cid:107) ¯ V , (cid:107) V ts,H ε u − V ts,H u (cid:107) ∞ ≤ ( t − s ) (cid:107) H ε − H (cid:107) ¯ V , where V = R × R d × ¯ B Ä , e ( C +1)( t − s ) (1 + Lip ( u )) − ä . In other words, (cid:107) R ts,H ε u − R ts,H u (cid:107) ∞ ≤ ε ( t − s ) Ä e ( C +1)( t − s ) (1 + Lip ( u )) − ä , (cid:107) V ts,H ε u − V ts,H u (cid:107) ∞ ≤ ε ( t − s ) Ä e ( C +1)( t − s ) (1 + Lip ( u )) − ä . The second step applied to H ε states that R ts,H ε u = V ts,H ε u , and hence letting ε tend tozero gives the conclusion: R ts,H u = V ts,H u .The result is obtained analogously in the concave case, where the Lax-Oleinik semi-group is defined as a maximum, see Remark A.13. (cid:3) Appendices A. Generating families of the Hamiltonian flow
All the results and proofs of this appendix are inspired from [Cha90]. We write themdown here only to explicit the time derivatives of the generating families (see PropositionA.5) and the Lipschitz constant in Proposition A.8.We first state a useful basic technical lemma. Lemma A.1. If u, v : R n → R n are C and such that Lip( u ) < and Lip( v ) < , then f = id − u and g = id − v are C -diffeomorphisms of R n . If f − g is bounded, then f − − g − is bounded by (cid:107) f − g (cid:107) ∞ − Lip( u ) .Proof. Let us first proof that f is a C -diffeomorphism of R n . It is clearly C , and is alocal diffeomorphism since (cid:107) df (cid:107) = (cid:107) id − du (cid:107) ≥ − Lip( u ) > . To see that it is invertible,we observe that f ( q ) = θ can be rewritten as a fixed point problem q = u ( q ) + θ , wherethe map q (cid:55)→ u ( q ) + θ is contracting.Now, if f − g is bounded, so is u − v , with (cid:107) u − v (cid:107) ∞ = (cid:107) f − g (cid:107) ∞ . Let us denote x = f − ( z ) and y = g − ( z ) for some z in R n . Then x = u ( x ) + z and y = v ( y ) + z and | x − y | ≤ | u ( x ) − v ( y ) | ≤ | u ( x ) − u ( y ) | + | u ( y ) − v ( y ) | ≤ Lip( u ) | x − y | + (cid:107) u − v (cid:107) ∞ , whence | x − y | ≤ (cid:107) f − g (cid:107) ∞ − Lip( u ) . (cid:3) Let us now state two Grönwall-type estimates on Hamiltonian flows:
Proposition A.2.
Let H and ˜ H be two C Hamiltonians on R × T (cid:63) R d such that (cid:107) ∂ q,p H (cid:107) and (cid:107) ∂ q,p ˜ H (cid:107) are uniformly bounded by a constant C and (cid:107) ∂ q,p H − ∂ q,p ˜ H (cid:107) is uniformlybounded by a constant K . Then, if φ and ˜ φ denote the Hamiltonian flows respectivelyassociated with H and ˜ H , we have for all s ≤ t :(1) (cid:107) φ ts − ˜ φ ts (cid:107) ≤ KC ( e C ( t − s ) − ,(2) (cid:107) dφ ts − id (cid:107) ≤ e C ( t − s ) − .In particular if t − s ≤ δ = ln(3 / C , φ ts − id is -Lipschitz. Lemma A.3 (Grönwall’s lemma, elementary version) . If t (cid:55)→ f ( t ) is a continuousnon negative function such that f ( s ) = 0 and f ( t ) ≤ (cid:82) ts ( Cf ( u ) + K ) du , then f ( t ) ≤ KC ( e C ( t − s ) − .Proof. Observe that the assumed inequality can be written ∂ t Ç e − C ( t − s ) (cid:90) ts f ( u ) du å ≤ e − C ( t − s ) K ( t − s ) , and integrating this between s and t we get (cid:90) ts f ( u ) du ≤ Ke C ( t − s ) (cid:90) ts e − C ( u − s ) ( u − s ) du = KC ( e C ( t − s ) − C ( t − s ) − . Reinjecting this into f ( t ) ≤ (cid:82) ts ( Cf ( u ) + K ) du gives the result. (cid:3) Proof of Proposition A.2.
Let us define Γ t ( q, p ) = ( ∂ p H ( t, q, p ) , − ∂ q H ( t, q, p )) , so thatthe Hamiltonian system (HS) can be rewritten ∂ t φ ts = Γ t ( φ ts ) , and ˜Γ associated similarlywith ˜ H . (1) Since (cid:107) ∂ q,p H − ∂ q,p ˜ H (cid:107) ≤ K , (cid:107) Γ u − ˜Γ u (cid:107) ≤ K for all u and since Γ u is C -Lipschitz, (cid:107) φ ts − ˜ φ ts (cid:107) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:90) ts Γ u ( φ us ) − ˜Γ u ( ˜ φ us ) du (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:90) ts (cid:107) Γ u ( φ us ) − Γ u ( ˜ φ us ) (cid:107) + (cid:107) Γ u ( ˜ φ us ) − ˜Γ u ( ˜ φ us ) (cid:107) du ≤ (cid:90) ts C (cid:107) φ us − ˜ φ us (cid:107) + K du.
So, f ( t ) = (cid:107) φ ts − ˜ φ ts (cid:107) satisfies the conditions of Lemma A.3 and hence (cid:107) φ ts − ˜ φ ts (cid:107) ≤ KC ( e C ( t − s ) − . (2) Since (cid:107) ∂ q,p H (cid:107) ≤ C , d Γ t is bounded by C and hence (cid:107) ∂ t dφ ts (cid:107) = (cid:107) d (Γ t ( φ ts )) (cid:107) = (cid:107) d Γ t ( φ ts ) ◦ dφ ts (cid:107) ≤ C (cid:107) dφ ts (cid:107) . which implies that (cid:107) dφ ts − id (cid:107) ≤ (cid:82) ts C (cid:0) (cid:107) dφ ts − id (cid:107) + 1 (cid:1) du .Since dφ ss = id , f ( t ) = (cid:107) dφ ts − id (cid:107) satisfies the conditions of Lemma A.3 with K = C , and hence (cid:107) dφ ts − id (cid:107) ≤ e C ( t − s ) − . (cid:3) If γ = ( q, p ) is a path on T (cid:63) R d , its Hamiltonian action is given by A ts ( γ ) = (cid:90) ts p ( τ ) · ∂ τ q ( τ ) − H ( τ, γ ( τ )) dτ. We give here a simple element of calculus of variations, giving for a parametrized familyof Hamiltonian trajectories the link between the dependence of the Hamiltonian actionwith respect to the parameter and the behaviour of the family at the endpoints. It isuseful to understand the construction of the generating family of the flow in the nextparagraph.
Lemma A.4. If γ u = ( q u , p u ) : R → T (cid:63) R d is a C family of Hamiltonian trajectories, ∂ u A ts ( γ u ) = p u ( t ) · ∂ u q u ( t ) − p u ( s ) · ∂ u q u ( s ) . Proof.
We recall the Hamiltonian system satisfied by the Hamiltonian trajectory γ u : ® ∂ τ q u ( τ ) = ∂ p H ( t, q u ( τ ) , p u ( τ )) ,∂ τ p u ( τ ) = − ∂ q H ( t, q u ( τ ) , p u ( τ )) . As a consequence, ∂ u A ts ( γ u ) = ∂ u (cid:90) ts p u ( τ ) · ∂ τ q u ( τ ) − H ( τ, q u ( τ ) , p u ( τ )) dτ = (cid:90) ts ∂ u p u ( τ ) · ∂ τ q u ( τ ) + p u ( τ ) · ∂ u ∂ τ q u ( τ ) − ∂ q H ( τ, q u ( τ ) ,p u ( τ )) · ∂ u q u ( τ ) − ∂ p H ( τ, q u ( τ ) , p u ( τ )) · ∂ u p u ( τ ) dτ = (cid:90) ts p u ( τ ) · ∂ u ∂ τ q u ( τ ) + ∂ τ p u ( τ ) · ∂ u q u ( τ ) dτ = [ p u · ∂ u q u ] ts . (cid:3) A.1.
Generating family in the general case.
As a consequence of Lemma A.1 andProposition A.2, if we choose a δ ≤ δ = ln (3 / C , the map g ts : ( q, p ) (cid:55)→ ( Q ts ( q, p ) , p ) is a C -diffeomorphism for all ≤ t − s ≤ δ , since we have Lip( g ts − id) ≤ Lip( φ ts − id) ≤ / .If ≤ t − s ≤ δ , let F ts : R d → R be the function defined by(8) F ts ( Q, p ) = (cid:90) ts ( P τs ( q, p ) − p ) · ∂ τ Q τs ( q, p ) − H ( τ, φ τs ( q, p )) dτ, where q is the only point satisfying Q ts ( q, p ) = Q , i.e. the first coordinate of ( g ts ) − ( Q, p ) .In other terms, if γ ( τ ) = ( q ( τ ) , p ( τ )) is the unique Hamiltonian trajectory such that ( q ( t ) , p ( s )) = ( Q, p ) ,(9) F ts ( Q, p ) = p · ( q ( s ) − Q ) + A ts ( γ ) = p · ( q ( s ) − Q ) + (cid:90) ts p ( τ ) · ∂ τ q ( τ ) − H ( τ, γ ( τ )) dτ. Proposition A.5.
The family of functions ( F ts ) s ≤ t ≤ s + δ is C with respect to s , t , Q , p and its derivatives are given by ® ∂ p F ts ( Q, p ) = q − Q, ∂ t F ts ( Q, p ) = − H ( t, Q, P ) ,∂ Q F ts ( Q, p ) = P − p, ∂ s F ts ( Q, p ) = H ( s, q, p ) , where P and q are uniquely defined by ( Q, P ) = φ ts ( q, p ) . In particular, ( Q, P ) = φ ts ( q, p ) ⇐⇒ ® ∂ p F ts ( Q, p ) = q − Q,∂ Q F ts ( Q, p ) = P − p. Furthermore, if Q = Q ts ( q, p ) and γ denotes the Hamiltonian trajectory issued from ( q, p ) , F ts ( Q, p ) = A ts ( γ ) − p · ( Q − q ) . The generating family is constructed by adding boundary terms to the Hamiltonianaction of a Hamiltonian trajectory depending on parameters.
Proof of Proposition A.5.
Let us differentiate F with respect to s , t , Q and p . The restof the proposition is a straightforward consequence of the form of the derivatives of F .In terms of Lemma A.4, let us denote by u = ( s, t, Q, p ) and by γ u = ( q u , p u ) the uniqueHamiltonian trajectory such that p u ( s ) = p and q u ( t ) = Q . Let us gather the derivativesof q u at the endpoints in view of applying Lemma A.4: we differentiate q u ( t ) = Q withrespect to s , t , Q and p , while denoting by τ the time variable of the trajectory γ u :(10) ∂ s q u ( t ) = 0 , ∂ t q u ( t ) + ∂ τ q u ( t ) = 0 , ∂ Q q u ( t ) = id , ∂ p q u ( t ) = 0 . The equation (9) defining F may now be written as: F ts ( Q, p ) = p · ( q u ( s ) − Q ) + A ts ( γ u ) . Lemma A.4 gives the dependence of A ts ( γ u ) with respect to u . We differentiate thisexpression with respect to s , t , Q and p , cautiously denoting by τ the time variable of the trajectory γ u = ( q u , p u ) , and taking into account the term p · ( q u ( s ) − Q ) and theboundaries of the integral defining the action: ∂ s F ts ( Q, p ) = p · ( ∂ s q u ( s ) + ∂ τ q u ( s )) − ( p u ( s ) · ∂ τ q u ( s ) − H ( s, q u ( s ) , p u ( s ))) + [ p u · ∂ s q u ] ts = H ( s, q u ( s ) , p u ( s )) + ( p − p u ( s )) · ( ∂ s q u ( s ) + ∂ τ q u ( s )) + p u ( t ) · ∂ s q u ( t )= H ( s, q, p ) ,∂ t F ts ( Q, p ) = p · ∂ t q u ( s ) + ( p u ( t ) · ∂ τ q u ( t ) − H ( t, q u ( t ) , p u ( t ))) + [ p u · ∂ t q u ] ts = ( p − p u ( s )) · ∂ t q u ( s ) + p u ( t ) · ( ∂ τ q u ( t ) + ∂ t q u ( t )) − H ( t, q u ( t ) , p u ( t ))= − H ( t, Q, P ) ,∂ Q F ts ( Q, p ) = p · ∂ Q q u ( s ) − p + [ p u · ∂ Q q u ] ts = ( p − p u ( s )) · ∂ Q q u ( s ) − p + p u ( t ) · ∂ Q q u ( t ) = − p + P,∂ p F ts ( Q, p ) = p · ∂ p q u ( s ) + q u ( s ) − Q + [ p u · ∂ p q u ] ts = ( p − p u ( s )) · ∂ p q u ( s ) + q u ( s ) − Q + p u ( t ) · ∂ p q u ( t ) = q − Q if we denote by ( P, q ) = ( p u ( t ) , q u ( s )) , using (10) and ( p u ( s ) , q u ( t )) = ( p, Q ) . (cid:3) Proposition A.6. If H µ is a C family of Hamiltonians such that (cid:107) ∂ q,p H µ (cid:107) is boundedby C , let us denote by F ts,µ associated with H µ as previously for t − s ≤ δ . Then ∂ µ F ts,µ ( Q, p ) = − (cid:90) ts ∂ µ H µ ( τ, γ µ ( τ )) dτ where γ µ = ( q µ , p µ ) is the unique Hamiltonian trajectory for H µ with q µ ( t ) = Q and p µ ( s ) = p .Proof. Let us fix Q , p , s and t , and take γ µ as in the statement. By definition (9), F ts,µ ( Q, p ) = p · ( q µ ( s ) − Q )) + A ts,H µ ( γ µ ) and thus differentiating w.r.t. µ gives the following, using Lemma A.4: ∂ µ F ts,µ ( Q, p ) = p · ∂ µ q µ ( s ) + [ p µ · ∂ µ q µ ] ts − (cid:90) ts ∂ µ H µ ( τ, γ u ( τ )) dτ. Now, since q µ ( t ) = Q for all µ , ∂ µ q µ ( t ) = 0 , and since p = p µ ( s ) , the two first terms ofthe right hand side cancel, hence the conclusion. (cid:3) When t − s is large, we choose a subdivision of the time interval with steps smallerthan δ and add intermediate coordinates along this trajectory. For each s ≤ t and ( t i ) such that t = s ≤ t ≤ · · · ≤ t N +1 = t and t i +1 − t i ≤ δ for each i , let G ts : R d (1+ N ) → R be the function defined by G ts ( p , Q , p , Q , · · · , Q N − , p N , Q N ) = N (cid:88) i =0 F t i +1 t i ( Q i , p i ) + p i +1 · ( Q i +1 − Q i ) where indices are taken modulo N + 1 . Proposition A.7.
The family of functions ( G ts ) s ≤ t is C with respect to s , t , t i , Q i and p i , and its derivatives are given by ® ∂ p i G ts ( p , · · · , Q N ) = ∂ p F t i +1 t i ( Q i , p i ) + Q i − Q i − = q i − Q i − ,∂ Q i G ts ( p , · · · , Q N ) = ∂ Q F t i +1 t i ( Q i , p i ) + p i − p i +1 = P i − p i +1 , where P i and q i are uniquely defined by ( Q i , P i ) = φ t i +1 t i ( q i , p i ) and indices are takenmodulo N + 1 .It is hence a generating family for the flow φ , meaning that if we denote ( Q, p ) =( Q N , p ) and ν = ( Q , p , · · · , Q N − , p N ) , ( Q, P ) = φ ts ( q, p ) ⇐⇒ ∃ ν ∈ R dN , ∂ p G ts ( p, ν, Q ) = q − Q,∂ Q G ts ( p, ν, Q ) = P − p,∂ ν G ts ( p, ν, Q ) = 0 , and in this case ( Q i , p i +1 ) = φ t i +1 s ( q, p ) for all ≤ i ≤ N − .Furthermore, if Q = Q ts ( q, p ) and γ denotes the Hamiltonian trajectory issued from ( q, p ) , G ts ( p, ν, Q ) = A ts ( γ ) − p · ( Q − q ) if ∂ ν G ts ( p, ν, Q ) = 0 .Proof. The derivatives of G , which are directly obtained from the ones of F , give that,if p and Q are fixed, ∂ p G ts ( p, ν, Q ) = q − Q,∂ Q G ts ( p, ν, Q ) = P − p,∂ ν G ts ( p, ν, Q ) = 0 , ⇐⇒ q = q ,P N = P, ( Q i , p i +1 ) = φ t i +1 t i ( Q i − , p i ) ∀ ≤ i ≤ N − . If this is satisfied, ν describes a non broken Hamiltonian geodesic, ( Q i , p i +1 ) = φ t i +1 s ( q, p ) and ( Q, P ) = φ ts ( q, p ) . If ( Q, P ) = φ ts ( q, p ) , then ν is given by φ t i s ( q, p ) and the righthand system holds.The critical value of ν (cid:55)→ G ts ( p, ν, Q ) is obtained by summing up the result obtainedfor F in Proposition A.5. (cid:3) The last statement compares the generating families of flows related to Hamiltonianswith Lipschitz difference.
Proposition A.8.
Let H and ˜ H be two C Hamiltonians on R × T (cid:63) R d such that (cid:107) ∂ q,p H (cid:107) and (cid:107) ∂ q,p ˜ H (cid:107) are uniformly bounded by a constant C and (cid:107) ∂ q,p H − ∂ q,p ˜ H (cid:107) is uniformlybounded by a constant K . We can find a δ > suiting both ˜ H and H and build ˜ G ts and G ts with the same subdivision ( t i ) , and then ˜ G ts − G ts is Lipschitz with constant KC ( e C ( t − s ) − and also with constant KC .Proof. Let δ ≤ δ = ln(3 / C so that both φ ts − id and ˜ φ ts − id are -Lipschitz if ≤ t − s ≤ δ , see Proposition A.2, and in that case g ts : ( q, p ) (cid:55)→ ( Q ts ( q, p ) , p ) satisfies also Lip( g ts − id) ≤ / .Proposition A.2 states that (cid:107) ˜ φ t i +1 t i − φ t i +1 t i (cid:107) ∞ ≤ KC ( e C ( t i +1 − t i ) − under the assumptionsmade on H and ˜ H . We are hence going to check that for all i , (cid:107) ∂ Q i ˜ G ts − ∂ Q i G ts (cid:107) and (cid:107) ∂ p i ˜ G ts − ∂ p i G ts (cid:107) are both bounded by (cid:107) ˜ φ t i +1 t i − φ t i +1 t i (cid:107) ∞ in order to get the wanted Lipschitz constants. Proposition A.7 states that (cid:107) ∂ Q i ˜ G ts − ∂ Q i G ts (cid:107) = (cid:107) ˜ P i − P i (cid:107) and (cid:107) ∂ p i ˜ G ts − ∂ p i G ts (cid:107) = (cid:107) ˜ q i − q i (cid:107) , where P i and q i (resp. ˜ P i and ˜ q i ) are uniquely defined by ( Q i , P i ) = φ t i +1 t i ( q i , p i ) (resp. ( Q i , ˜ P i ) = ˜ φ t i +1 t i (˜ q i , p i ) ). Since ( q i , p i ) = ( g t i +1 t i ) − ( Q i , p i ) and (˜ q i , p i ) = (˜ g t i +1 t i ) − ( Q i , p i ) , Lemma A.1 gives (cid:107) ˜ q i − q i (cid:107) ≤ (cid:107) (˜ g t i +1 t i ) − − ( g t i +1 t i ) − (cid:107) ∞ ≤ (cid:107) ˜ g t i +1 t i − g t i +1 t i (cid:107) ∞ − Lip( g ts − id) ≤ (cid:107) ˜ φ t i +1 t i − φ t i +1 t i (cid:107) ∞ since Lip( g t i +1 t i − id) ≤ / . Now, (cid:107) ˜ P i − P i (cid:107) ≤ (cid:107) ˜ φ t i +1 t i (˜ q i , p i ) − φ t i +1 t i ( q i , p i ) (cid:107)≤ (cid:107) ˜ φ t i +1 t i (˜ q i , p i ) − φ t i +1 t i (˜ q i , p i ) (cid:107) + Lip( φ t i +1 t i ) (cid:107) ˜ q i − q i (cid:107)≤ (cid:107) ˜ φ t i +1 t i − φ t i +1 t i (cid:107) ∞ + Lip( φ t i +1 t i )2 (cid:107) ˜ φ t i +1 t i − φ t i +1 t i (cid:107) ∞ ≤ (cid:107) ˜ φ t i +1 t i − φ t i +1 t i (cid:107) ∞ since φ t i +1 t i is -Lipschitz.Since t i +1 − t i is smaller than t − s and than δ for all i , we have proved that (cid:107) d ˜ G ts − dG ts (cid:107) is bounded by KC ( e Cδ − ≤ KC and by KC ( e C ( t − s ) − . (cid:3) A.2.
Generating family in the convex case.
In this section we assume that theHamiltonian H satisfies Hypothesis 1.1 with constant C , and that there exists m > such that for each ( t, q, p ) , ∂ p H ( t, q, p ) ≥ m id in the sense of quadratic forms. Proposition A.9.
The following holds in the sense of quadratic forms: ∂ p Q ts ≥ m ( t − s )id − Ä e C ( t − s ) − − C ( t − s ) ä id . In particular there exists δ > depending only on C and m such that if | t − s | ≤ δ , ∂ p Q ts ≥ m t − s )id which implies that the function p (cid:55)→ Q ts ( q, p ) is m ( t − s )2 -monotone, meaning that ( Q ts ( q, ˜ p ) − Q ts ( q, p )) · (˜ p − p ) ≥ m t − s ) (cid:107) ˜ p − p (cid:107) . In particular, if | t − s | ≤ δ , ( q, p ) (cid:55)→ ( q, Q ts ( q, p )) is a C -diffeomorphism.Remark A.10 . For A a not necessarily symmetric matrix, we say that A ≥ c id in thesense of quadratic forms if Ax · x ≥ c (cid:107) x (cid:107) for all x . If (cid:107) A (cid:107) ≤ a , then in particular − a id ≤ A ≤ a id . Proof.
Let us recall the variational equation ∂ p ˙ Q ts = ∂ p H∂ p P ts + ∂ q,p H∂ p Q ts that we write under the form ∂ p ˙ Q ts − ∂ p H = ∂ p H ( ∂ p P ts − id) + ∂ q,p H∂ p Q ts . Lemma A.2 gives that (cid:107) ∂ q Q ts − id (cid:107) , (cid:107) ∂ p Q ts (cid:107) and (cid:107) ∂ p P ts − id (cid:107) are smaller than e C ( t − s ) − .Adding the estimate on ∂ H , we get (cid:107) ∂ p ˙ Q ts − ∂ p H (cid:107) ≤ C ( e C ( t − s ) − , which implies that ∂ p ˙ Q ts ≥ ∂ p H − C ( e C ( t − s ) − ≥ Ä m − C ( e C ( t − s ) − ä id in the sense of quadratic forms, see Remark A.10. Integrating the result between s and t we obtain ∂ p Q ts ≥ m ( t − s )id − Ä e C ( t − s ) − − C ( t − s ) ä id . Since the second term of the right hand side is second order, there exists a constant δ > depending only on C and m such that if | t − s | ≤ δ , ∂ p Q ts ≥ m t − s )id , which means that for all z , ∂ p Q ts ( q, p ) z · z ≥ m t − s ) (cid:107) z (cid:107) . Applying this to z = ˜ p − p we get ( Q ts ( q, ˜ p ) − Q ts ( q, p )) · (˜ p − p ) = (cid:90) ∂ p Q ts ( q, p + τ (˜ p − p ))(˜ p − p ) dτ · (˜ p − p )= (cid:90) ∂ p Q ts ( q, p + τ (˜ p − p ))(˜ p − p ) · (˜ p − p ) dτ ≥ (cid:90) m t − s ) (cid:107) ˜ p − p (cid:107) dτ ≥ m t − s ) (cid:107) ˜ p − p (cid:107) . We have proved that the function p (cid:55)→ Q ts ( q, p ) is m ( t − s )2 -monotone. It is then a classicalresult that p (cid:55)→ Q ts ( q, p ) is a global C -diffeomorphism (see for example Proposition of [Ber12]), and therefore ( q, p ) (cid:55)→ ( q, Q ts ( q, p )) is also a global C -diffeomorphism. (cid:3) Proposition A.11.
There exists δ > depending only on C and m such that if G ts is constructed with a maximal step smaller than δ , the function ( p , p , · · · , p N ) (cid:55)→ G ts ( p , Q , p , Q , · · · , Q N − , p N , Q N ) is uniformly strictly concave.Proof. Let us denote the function ( p , p , · · · , p N ) (cid:55)→ G ts ( p , Q , p , · · · , Q N − , p N , Q N ) by g . Proposition A.7 gives that ∂ p i G ts ( p , · · · , Q N ) = q i − Q i − , where q i is the onlypoint such that Q i = Q t i +1 t i ( q i , p i ) . On one hand, we get that if i (cid:54) = j , ∂ p i p j G ts is zero.On the other hand, ∂ p i G ts = ∂ p i q i .Differentiating Q t i +1 t i ( q i , p i ) = Q i w.r.t. p i gives ∂ q Q t i +1 t i ( q i , p i ) ∂ p i q i + ∂ p Q t i +1 t i ( q i , p i ) = 0 , so we have ∂ p i G ts = − ( ∂ q Q t i +1 t i ) − ∂ p Q t i +1 t i . Lemma A.2 gives that (cid:107) ∂ p Q t i +1 t i (cid:107) ≤ e C ( t i +1 − t i ) − and (cid:107) ∂ q Q t i +1 t i − id (cid:107) ≤ e C ( t i +1 − t i ) − ,and hence ∂ q Q t i +1 t i is invertible as long as e C ( t i +1 − t i ) < and satisfies(11) (cid:13)(cid:13)(cid:13) ( ∂ q Q t i +1 t i ) − − id (cid:13)(cid:13)(cid:13) ≤ e C ( t i +1 − t i ) − − e C ( t i +1 − t i ) . Using (11) and the estimate of Proposition A.9 we get ∂ p i G ts = − (( ∂ q Q t i +1 t i ) − − id) ∂ p Q t i +1 t i − ∂ p Q t i +1 t i ≤ e C ( t i +1 − t i ) − − e C ( t i +1 − t i ) ( e C ( t i +1 − t i ) − − m ( t i +1 − t i )id + 2 Ä e C ( t i +1 − t i ) − − C ( t i +1 − t i ) ä id . Since the only first order term is − m ( t i +1 − t i )id , there exists a δ > depending onlyon C and m such that if t i +1 − t i ≤ δ , ∂ p i G t i +1 t i ≤ − m t i +1 − t i )id . If δ ≤ δ , then d g , which is a blockwise diagonal matrix, is smaller than − mδ id and g is hence uniformly strictly concave. (cid:3) When the Hamiltonian H is strictly convex w.r.t. p , the Lagrangian function on thetangent bundle is associated as follows: L ( t, q, v ) = sup p ∈ ( R d ) (cid:63) p · v − H ( t, q, p ) . Assume that δ < min( δ , δ , δ ) , and let h i be the inverse of ( q, p ) (cid:55)→ Ä q, Q t i +1 t i ( q, p ) ä (see Proposition A.9). We define A ts ( q, Q , · · · , Q N − , Q ) = N (cid:88) i =0 (cid:90) t i +1 t i L Ä τ, Q τt i ( h i ( Q i − , Q i )) , ∂ τ Q τt i ( h i ( Q i − , Q i )) ä dτ with the notations q = Q − and Q = Q N . Proposition A.12.
The so-called
Lagrangian generating family A is C and satisifies :(1) A ts ( q, Q , · · · , Q N − , Q ) = max ( p , ··· ,p N ) G ts ( p , Q , · · · , Q N − , p N , Q ) + p · ( Q − q ) . (2) ∂ Q i A ts ( q, Q , · · · , Q N − , Q ) = P i − p i +1 ∀ i = 0 · · · N − ,∂ q A ts ( q, Q , · · · , Q N − , Q ) = − p ,∂ Q A ts ( q, Q , · · · , Q N − , Q ) = P N , where P i and p i are uniquely defined by ( Q i , P i ) = φ t i +1 t i ( Q i − , p i ) . This function is indeed a generating family for the flow, in the sense that if v =( Q , · · · , Q N − ) , the graph of the flow φ ts is the set (cid:110) Ä ( q, − ∂ q A ts ( q, v, Q )) , ( Q, ∂ Q A ts ( p, v, Q )) ä (cid:12)(cid:12)(cid:12) ∂ v A ts ( p, v, Q ) = 0 (cid:111) . Proof. (1) The function ( p , p , · · · , p N ) (cid:55)→ G ts ( p , Q , p , Q , · · · , Q N − , p N , Q )+ p · ( Q − q ) is uniformly strictly concave by Proposition A.11, and its maximum ishence attained by a unique point.For i from to N , this is a consequence of the derivative of G ts given inProposition A.7: ∂ p i G ts ( p , Q , p , Q , · · · , Q N − , p N , Q ) = q i − Q i − = 0 if andonly if Q t i +1 t i ( Q i − , p i ) = Q i . For i = 0 , the derivative with respect to p is q − q where q is the only point such that Q t s ( q , p ) = Q , and consequently ∂ p (cid:0) G ts + p · ( Q − q ) (cid:1) = 0 if and only if Q t s ( q, p ) = Q .The maximum is hence uniquely attained by the C function p : ( q, Q · · · , Q ) (cid:55)→ Ä h ( q, Q ) , h ( Q , Q ) , · · · , h N ( Q N − , Q ) ä , where h i denotes the second coordinate of h i . In other terms, its coordinatessatisfy Q t i +1 t i ( Q i − , p i ) = Q i for all i from to N , with the notations q = Q − and Q = Q N .By definition of the Lagrangian, if ( q ( t ) , p ( t )) is a Hamiltonian trajectory as-sociated with H , then L ( t, q ( t ) , ˙ q ( t )) = p ( t ) · ˙ q ( t ) − H ( t, q ( t ) , p ( t )) . In particular the function F defined in (9) can be written in Lagrangian terms: F ts ( Q, p ) = p · ( q − Q ) + (cid:90) ts L ( τ, Q τs ( q, p ) , ∂ τ Q τs ( q, p )) dτ. where q is the only point such that Q ts ( q, p ) = Q , and the function G is hencethe following: G ts ( p , Q , p , Q , · · · , Q N − , p N , Q N ) = N (cid:88) i =0 F t i +1 t i ( Q i , p i ) + p i +1 · ( Q i +1 − Q i )= N (cid:88) i =0 (cid:90) t i +1 t i L ( τ, Q τt i ( q i , p i ) , ∂ τ Q τt i ( q i , p i )) dτ + p i · ( q i − Q i ) + p i +1 · ( Q i +1 − Q i )= N (cid:88) i =0 (cid:90) t i +1 t i L ( τ, Q τt i ( q i , p i ) , ∂ τ Q τt i ( q i , p i )) dτ + p i · ( q i − Q i − ) , where q i is the only point such that Q t i +1 t i ( q i , p i ) = Q i .Now, if ( p , · · · , p N ) is the critical point, we have on one hand that q i = Q i − and on the other hand that Q t i +1 t i ( q i , p i ) = Q i if and only if ( q i , p i ) = h i ( q i , Q i ) ,hence the result.(2) Since A ts ( q, · · · , Q ) = G ts ( Q , · · · , Q, p ( q, · · · , Q )) + p ( q, · · · , Q ) · ( Q − q ) whilereorganising the variables, we have for all i from − to N∂ Q i A ts ( q, ·· , Q ) = ∂ Q i Ä G ts ( Q , ·· , Q, p ( q, ·· , Q )) + p ( q, ·· , Q ) · ( Q − q ) ä + ∂ p Ä G ts ( Q , · , Q, p ( q, ·· , Q )) + p ( q, ·· , Q ) · ( Q − q ) ä (cid:124) (cid:123)(cid:122) (cid:125) =0 ∂ Q i p since p ( q, ·· , Q ) is the critical point. The result is then a straightforward conse-quence of Proposition A.7 and of the second point. (cid:3) Let us state what happens in the case of a uniformly strictly concave Hamiltonian.
Remark
A.13 . If H is uniformly strictly concave (which means that − H is uniformlystrictly convex), Proposition A.9 analogous statement is that − Q ts is m ( t − s ) / monotone,which implies the twist property: ( q, p ) (cid:55)→ ( q, Q ts ( q, p )) is a C -diffeomorphism for | t − s | ≤ δ . Proposition A.11 analogous statement is that − G ts is strictly concave with respect toits p variable for | t − s | ≤ δ . The Lagrangian is now defined by L ( t, q, v ) = inf p ∈ ( R d ) (cid:63) p · v − H ( t, q, p ) , and the analogous statement of Proposition A.12 is that A ts ( q, Q , · · · , Q N − , Q ) = min ( p ,p , ··· ,p N ) G ts ( p , Q , · · · , Q N − , p N , Q ) + p · ( Q − q ) , where A is defined as in the convex case. Finally, the next Proposition holds in bothconvex and concave cases. Proposition A.14.
Let H and ˜ H be two C Hamiltonians on R × T (cid:63) R d such that • ∂ q,p H and ∂ q,p ˜ H are uniformly bounded by a constant C , • ∂ p H ≥ m id , ∂ p ˜ H ≥ m id (or ≤ − m id in the concave case), • ∂ q,p H − ∂ q,p ˜ H is uniformly bounded by a constant K .We fix a subdivision s ≤ t ≤ · · · ≤ t N +1 = t such that < t i +1 − t i < δ , with δ smallerthan δ , δ and δ , and build the Lagrangian generating families A ts and ˜ A ts as previously,respectively for H and ˜ H . Then the difference ˜ A ts − A ts is Lipschitz.Proof. We denote by ˜ · the objects defined for ˜ H instead of H . Given the form of thederivatives of A ts obtained in Proposition A.12, it is enough to prove that ˜ p i − p i and ˜ P i − P i are bounded uniformly with respect to ( q, · · · , Q ) for all i , where P i and p i (resp. ˜ P i and ˜ p i ) are uniquely defined by ( Q i , P i ) = φ t i +1 t i ( Q i − , p i ) (resp. ( Q i , ˜ P i ) = ˜ φ t i +1 t i ( Q i − , ˜ p i ) ).Proposition A.9 states that p (cid:55)→ Q t i +1 t i ( q, p ) is m ( t i +1 − t i )2 -monotone, meaning that forall p and ˜ p ( Q t i +1 t i ( q, ˜ p ) − Q t i +1 t i ( q, p )) · (˜ p − p ) ≥ m t i +1 − t i ) (cid:107) ˜ p − p (cid:107) . Applying the Cauchy-Schwarz inequality and dividing by (cid:107) ˜ p − p (cid:107) we get (cid:107) ˜ p − p (cid:107) ≤ m ( t i +1 − t i ) (cid:13)(cid:13)(cid:13) Q t i +1 t i ( q, ˜ p ) − Q t i +1 t i ( q, p ) (cid:13)(cid:13)(cid:13) . Take q = Q i − , ˜ p = ˜ p i and p = p i . Since Q t i +1 t i ( Q i − , p i ) = ˜ Q t i +1 t i ( Q i − , ˜ p i ) , we have (cid:107) ˜ p i − p i (cid:107) ≤ m ( t i +1 − t i ) (cid:13)(cid:13)(cid:13) Q t i +1 t i ( Q i − , ˜ p i ) − ˜ Q t i +1 t i ( Q i − , ˜ p i ) (cid:13)(cid:13)(cid:13) ≤ mµ (cid:13)(cid:13)(cid:13) φ t i +1 t i − ˜ φ t i +1 t i (cid:13)(cid:13)(cid:13) ∞ where µ denotes the minimum of t i +1 − t i . The first estimate of Proposition A.2 gives: (cid:107) ˜ p i − p i (cid:107) ≤ mµ KC ( e Cδ − . Finally, since P i = P t i +1 t i ( Q i − , p i ) and ˜ P i = ˜ P t i +1 t i ( Q i − , ˜ p i ) , (cid:107) ˜ P i − P i (cid:107) ≤ (cid:13)(cid:13)(cid:13) φ t i +1 t i − ˜ φ t i +1 t i (cid:13)(cid:13)(cid:13) ∞ + (cid:13)(cid:13)(cid:13) P t i +1 t i ( Q i − , p i ) − P t i +1 t i ( Q i − , ˜ p i ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) φ t i +1 t i − ˜ φ t i +1 t i (cid:13)(cid:13)(cid:13) ∞ + Lip ( φ t i +1 t i ) (cid:107) ˜ p i − p i (cid:107) is uniformly bounded since φ t i +1 t i is -Lipschitz (see Proposition A.2). (cid:3) B. Minmax: a critical value selector
We denote by Q m the set of functions on R m that can be written as the sum of anondegenerate quadratic form and of a Lipschitz function. The aim of this appendix isto build a function σ : (cid:83) m ∈ N Q m → R , named minmax , satisfying:(1) if f is C , then σ ( f ) is a critical value of f ,(2) if c is a real constant, then σ ( c + f ) = c + σ ( f ) ,(3) if φ is a Lipschitz C ∞ -diffeomorphism on R m such that f ◦ φ is in Q m , then σ ( f ◦ φ ) = σ ( f ) , (4) if f − f is Lipschitz and f ≤ f on R d , then σ ( f ) ≤ σ ( f ) ,(5) if ( f µ ) µ ∈ [ s,t ] is a C family of Q m with ( Z − f µ ) µ equi-Lipschitz for some nonde-generate quadratic form Z , then for all µ (cid:54) = ˜ µ ∈ [ s, t ] , min µ ∈ [ s,t ] min x ∈ Crit ( f µ ) ∂ µ f µ ( x ) ≤ σ ( f ˜ µ ) − σ ( f µ )˜ µ − µ ≤ max µ ∈ [ s,t ] max x ∈ Crit ( f µ ) ∂ µ f µ ( x ) . (6) σ ( − f ) = − σ ( f ) ,(7) if f ( x, y ) is a C function of Q m such that ∂ y f ≥ c id for a c > , and if g ( x ) =min y f ( x, y ) is in some Q ˜ m , then σ ( g ) = σ ( f ) .For smooth functions, (1), (3) and (2) are proved in Proposition B.8, (4) is implied byProposition B.11, and (6) and (7) are proved respectively in Propositions B.13 and B.15.They are extended to non smooth functions in Propositions B.17 and B.18, and (5) isproved in Proposition B.19. Consequences
B.1 . These properties imply the following consequences:(1) If f and g are two functions of Q m with difference bounded and Lipschitz on R m ,then | σ ( f ) − σ ( g ) | ≤ (cid:107) f − g (cid:107) ∞ . This is a consequence of properties (2) and (4).(2) If g ( x, η ) = f ( x ) + Z ( η ) where Z is a nondegenerate quadratic form and f isin Q m , then σ ( g ) = σ ( f ) . This is a consequence of properties (6) and (7) forsmooth functions, which may be extended by continuity thanks to the previouspoint.(3) If f µ = Z µ + (cid:96) µ is a C family of Q m with (cid:96) µ equi-Lipschitz, such that the set ofcritical points f µ does not depend on µ and such that µ (cid:55)→ f µ is constant on thisset, then µ (cid:55)→ σ ( f µ ) is constant. This is a consequence of properties (3) and (5).(4) If f is bounded below, then σ ( f ) = min( f ) . This is a consequence of properties(1) and (4).Consequences B.1-(3) and B.1-(4) are proved in the main corpus, see respectivelyConsequences 2.12 and 2.11.The construction of such a critical value selector proves Propositions 2.7 and 4.4. We will use two deformation lemmas proved in Appendix C, and we refer to [Wei13b]for a survey of minmax related subtleties, including an example due to F. Laudenbachwhere the minmax is not uniquely defined.
Remark
B.2 . In this paper we describe the geometric solution associated with the con-sidered Cauchy problem with a particular generating family proposed by Chaperon. In amore general setting, Viterbo’s uniqueness theorem on generating functions state that if S and ˜ S are two generating functions quadratic at infinity describing a same Lagrangiansubmanifold which is Hamiltonianly isotopic to the zero section, they may be obtainedone from another via a combination of the three following transformations: • Addition of a constant: ˜ S = S + c for some c ∈ R , • Diffeomorphism operation: ˜ S = S ◦ φ for some fiber C ∞ -diffeomorphism φ , • Stabilization: ˜ S ( x, ξ, ν ) = S ( x, ξ ) + Z ( ν ) for a nondegenerate quadratic form Z .The proof of D. Theret in [Thé99] puts forward the fact that the diffeomorphism φ maybe chosen affine outside a compact set - in particular such a diffeomorphism is Lipschitzand if f is in Q m , so does f ◦ φ . Hence, the invariance of the minmax by additivity(property (2)), by diffeomorphism action (property (3)) and by stabilization (propertyB.1-(2)) gives that the minmax behave well when applied to generating functions. Up toadding a constant, it is the same for generating functions describing the same Lagrangiansubmanifold.B.1. Definition of the minmax for smooth functions.
Let us denote by Q ∞ m the set of C ∞ functions of Q m . The critical points and values of C functions of Q m are bounded: Proposition B.3. If Z is a nondegenerate quadratic form and (cid:96) is a C Lipschitz functionwith constant L , then the set of critical points of the function f = Z + (cid:96) is closed andcontained in the ball ¯ B (0 , L/m ) where m = inf (cid:107) x (cid:107) =1 (cid:107) d Z ( x ) (cid:107) . The set of critical valuesof f is hence closed and bounded. Notation B.4.
For f a function and c a real number, let f c = { x ∈ R m | f ( x ) ≤ c } bethe sublevel set of f associated with the value c . Note that f c ⊂ f c (cid:48) if c ≤ c (cid:48) . Definition B.5.
Let f be a function of Q ∞ m and a be a real constant. Since the criticalvalues of f are bounded, we can find c ≥ | a | greater than any critical values of f inmodulus. For a ≤ c , let i ca be the canonical injection ( f a , f − c ) (cid:44) → ( f c , f − c ) . It induces a morphism i c(cid:63)a in relative cohomology: H • ( f c , f − c ) i c(cid:63)a → H • ( f a , f − c ) . We assume that the cohomology is calculated with coefficients in a field, which allows tochoose a simplified definition.Let the minmax of f be defined by σ ( f ) = inf { a ∈ R | i c(cid:63)a (cid:54) = 0 } = sup { a ∈ R | i c(cid:63)a = 0 } . This definition does not depend on the choice of c when c is large enough. Proof.
The fact that σ ( f ) does not depend on the choice of c when it is large enough isa consequence of the following lemma: Lemma B.6. If c ≥ | a | and c ≥ a are two real constants greater than any criticalvalues of f in modulus, i c (cid:63)a and i c (cid:63)a are conjugate in cohomology. Therefore they aresimultaneously zero or non-zero.Proof. Suppose c > c . If a = − c , let us check that i c (cid:63)a = i c (cid:63)a = 0 : H • ( f c , f − c ) i c (cid:63)a → H • ( f − c , f − c ) (cid:124) (cid:123)(cid:122) (cid:125) = { } and therefore i c (cid:63) − c = 0 . We can prove that i c (cid:63) − c = 0 in the same way: H • ( f c , f − c ) i c (cid:63)a → H • ( f − c , f − c ) (cid:124) (cid:123)(cid:122) (cid:125) = { } where the nullity of H • ( f − c , f − c ) is guaranteed by the retraction constructed in LemmaC.1.Now, if a > − c , there is an ε > such that − c + ε ≤ a , and f has no critical valuein [ − c − ε, − c + ε ] or in [ c − ε, c + ε ] . Since − c + ε ≤ a ≤ c , Deformation lemmaC.1 gives two homotopy equivalences Φ + and Φ − such that: ® Φ + ( f c ) = f c Φ + ( f a ) = f a and ® Φ − ( f − c ) = f − c Φ − ( f a ) = f a . The homotopy equivalences give isomorphisms in cohomology, and the following diagramcommutes: H • ( f c , f − c ) i c (cid:63)a → H • ( f a , f − c ) (cid:111) ↓ (Φ (cid:63) + ) − (cid:111) ↓ (Φ (cid:63) + ) − H • ( f c , f − c ) (cid:8) H • ( f a , f − c ) (cid:111) ↓ Φ (cid:63) − (cid:111) ↓ Φ (cid:63) − H • ( f c , f − c ) → i c (cid:63)a H • ( f a , f − c ) which proves that i c (cid:63)a and i c (cid:63)a are conjugate in cohomology. (cid:3) Let us now fix c large enough and prove that inf { a ∈ R | i c(cid:63)a (cid:54) = 0 } = sup { a ∈ R | i c(cid:63)a = 0 } .To do so, we are going to prove that any element of the set { a ∈ R | i c(cid:63)a (cid:54) = 0 } is bigger thanany element of its complement set { a ∈ R | i c(cid:63)a = 0 } . Let a be such that i c(cid:63)a (cid:54) = 0 and b besuch that i c(cid:63)b = 0 . Assume that b > a . The following diagram commutes: ( f a , f − c ) i (cid:44) → ( f b , f − c ) (cid:38) i ca (cid:8) ↓ i cb ( f c , f − c ) where i denotes the canonical injection from ( f a , f − c ) to ( f b , f − c ) . It induces a commu-tative diagram in cohomology: H • ( f a , f − c ) i (cid:63) ← H • ( f b , f − c ) (cid:45) i c(cid:63)a (cid:8) ↑ i c(cid:63)b H • ( f c , f − c ) Since i c(cid:63)b is zero, i c(cid:63)a is necessarily zero which is excluded. We have proved that a ≥ b (and then a > b since i c(cid:63)a (cid:54) = i c(cid:63)b ), and consequently: inf { a ∈ R | i c(cid:63)a (cid:54) = 0 } = sup { a ∈ R | i c(cid:63)a = 0 } . (cid:3) Theorem B.7.
The minmax σ ( f ) is a critical value of f .Proof. Suppose that σ ( f ) is not a critical value of f . Then, since the set of critical valuesof f is closed (see Proposition B.3), there is a ε > such that f has no critical valuein [ σ ( f ) − ε, σ ( f ) + ε ] . Since σ ( f ) is finite, by definition, there exist a and b such that σ ( f ) − ε < a ≤ σ ( f ) ≤ b < σ ( f ) + ε , i (cid:63)a = 0 and i (cid:63)b (cid:54) = 0 . Taking c strictly bigger than | a | , | b | and any critical value of f , Proposition B.6 states that i c(cid:63)a = 0 and i c(cid:63)b (cid:54) = 0 .One can find an ε (cid:48) > such that [ a − ε (cid:48) , b + ε (cid:48) ] ⊂ [ σ ( f ) − ε, σ ( f )+ ε ] and b + ε (cid:48) ≤ c , so that [ a − ε (cid:48) , b + ε (cid:48) ] does not contain any critical point of f , and Deformation lemma C.1 builds acontinuous function Φ such that Φ( f b , f − c ) = ( f a , f − c ) and also Φ( f c , f − c ) = ( f c , f − c ) since b + ε (cid:48) ≤ c . Since Φ is a homotopy equivalence, it defines an isomorphism incohomology. The following diagram should then commute: H • ( f c , f − c ) i c(cid:63)a =0 → H • ( f a , f − c ) (cid:111) ↓ Φ (cid:63) (cid:8) (cid:111) ↓ Φ (cid:63) H • ( f c , f − c ) → i c(cid:63)b (cid:54) =0 H • ( f b , f − c ) which is impossible. Hence, σ ( f ) is necessarily a critical value of f . (cid:3) B.2.
Minmax properties for smooth functions.Proposition B.8.
Let f be in Q ∞ m . Then the minmax satisfies:(1) σ ( f ) is a critical value of f ,(2) if c is a real number, σ ( c + f ) = c + σ ( f ) ,(3) if φ is a Lipschitz C ∞ -diffeomorphism on R m such that f ◦ φ is in Q m , then σ ( f ◦ φ ) = σ ( f ) . Proof. (1) has already been proved (see Theorem B.7).(2) If b > is a real number, g = b + f is in Q ∞ m . For all c ∈ R , f c = g c + b . Choose c big enough so that c − b is strictly greater than | a | and than the critical valuesof f . Take a in R and let us show that i c,f(cid:63)a (cid:54) = 0 ⇐⇒ i c − b,g(cid:63)a + b (cid:54) = 0 . There is an ε > such that f has no critical value of f in [ c + ε, c − b − ε ] . Now take thehomotopy equivalence constructed in Lemma C.1 and satisfying: ® Φ( f c ) = f c − b Φ( f u ) = f u ∀ u ≤ c − b. This gives the following commutative diagram, since a and − c are smaller than c − b : H • ( f c , f − c ) i c,f(cid:63)a → H • ( f a , f − c ) (cid:111) ↑ Φ (cid:63) (cid:111) ↑ Φ (cid:63) H • ( f c − b , f − c ) (cid:8) H • ( f a , f − c ) (cid:107) (cid:107) H • ( g c − b , g − c + b ) → i ( c − b ) ,g(cid:63)a + b H • ( g a + b , g − c + b ) which proves that i c,f(cid:63)a = 0 ⇐⇒ i ( c − b ) ,g(cid:63)a + b = 0 . But since the critical values of g arethe critical values of f added to the constant b , c − b is greater than any criticalvalue of g in modulus since c − b is greater in modulus than the critical valuesof f . Lemma B.6 states that the nullity of i c,f(cid:63)a (resp. i c,g(cid:63)a ) does not depend on c large enough, hence: σ ( f ) = inf ¶ a ∈ R | i c,f(cid:63)a (cid:54) = 0 © = inf (cid:110) a ∈ R | i ( c − b ) ,g(cid:63)a + b (cid:54) = 0 (cid:111) = σ ( g ) − b. (3) Let φ be a Lipschitz C ∞ -diffeomorphism of R m such that g = f ◦ φ is in Q ∞ m .Note that f and g have the same critical values. Take a in R and c ≥ | a | greaterthan any critical value of f (hence g ).For all u ∈ R , f u = φ ( g u ) . Since φ is a C ∞ -diffeomorphism mapping thepair ( g u (cid:48) , g u ) to ( f u (cid:48) , f u ) for all real numbers u < u (cid:48) , φ gives an isomorphism incohomology. The following diagram commutes: H • ( f c , f − c ) i c,f(cid:63)a → H • ( f a , f − c ) (cid:111) ↓ φ (cid:63) (cid:8) (cid:111) ↓ φ (cid:63) H • ( g c , g − c ) → i c,g(cid:63)a H • ( g a , g − c ) which shows that i c,f(cid:63)a (cid:54) = 0 ⇐⇒ i c,g(cid:63)a (cid:54) = 0 , hence σ ( f ) = σ ( g ) . (cid:3) Now let us focus on the monotonicity of the minmax.
Definition B.9. If f and f are two functions of Q ∞ m with Lipschitz difference, let usconsider the homotopy f t = (1 − t ) f + tf between f and f and denote by C f ,f theset of critical points C f ,f = { x ∈ R m |∃ t ∈ [0 , , df t ( x ) = 0 } . Proposition B.10.
Under these assumptions, the set C f ,f is compact.Proof. Let us denote by f = Z + (cid:96) and f = Z + (cid:96) . If L is a Lipschitz constant suitingboth (cid:96) and (cid:96) , note that (cid:96) + t ( (cid:96) − (cid:96) ) is also L -Lipschitz. The critical points of f t arehence in the ball ¯ B (0 , L/m ) by Proposition B.3, and C f ,f is a bounded set.Let ( x n ) be a converging sequence of C f ,f and denote by x its limit. By definitionof C f ,f , there is a sequence ( t n ) ∈ [0 , such that df t n ( x n ) = 0 for all n . Since ( t n ) isbounded, it is possible to find a subsequence of t n converging to some t ∈ [0 , . Since ( t, x ) (cid:55)→ f t ( x ) is C , df t ( x ) is zero, and C f ,f is closed. (cid:3) Proposition B.11.
Let f and f be two functions of Q ∞ m with Lipschitz difference. If U is a set containing C f ,f and f ≥ f on U , then σ ( f ) ≥ σ ( f ) . In particular if f ≥ f on C f ,f (or if f ≥ f on R m ), then σ ( f ) ≥ σ ( f ) . Consequence
B.12 . If f and f are two functions of Q ∞ m with Lipschitz difference: inf U ( f − f ) ≤ inf C f ,f ( f − f ) ≤ σ ( f ) − σ ( f ) ≤ sup C f ,f ( f − f ) ≤ sup U ( f − f ) . for each set U containing the set C f ,f . In particular if f − f is Lipschitz and boundedon R m , then | σ ( f ) − σ ( f ) | ≤ (cid:107) f − f (cid:107) ∞ . Proof.
Since f + inf C f ,f ( f − f ) ≤ f ≤ f + sup C f ,f ( f − f ) on C f ,f and the threefunctions are in Q ∞ m with Lipschitz difference, Proposition (B.11) gives σ ( f + inf C f ,f ( f − f )) ≤ σ ( f ) ≤ σ ( f + sup C f ,f ( f − f )) . The additivity (B.8-2) then concludes: inf C f ,f ( f − f ) ≤ σ ( f ) − σ ( f ) ≤ sup C f ,f ( f − f ) . (cid:3) Proof.
Let us first prove Proposition B.11 in the case of an open and bounded set U . Take a in R and C = max t ∈ [0 , sup U | f t | , and choose a c bigger than C and | a | . Note that c is biggerin modulus than the critical values of f and f (which are contained in U ). LemmaC.2 gives a C -diffeomorphism Ψ : ( f c , f − c ) → ( f c , f − c ) , sending the pair ( f a , f − c ) intothe pair ( f a , f − c ) (since Ψ( f a ) ⊂ f a and Ψ( f − c ) = f − c ). This results in the followingcommutative diagram: H • ( f c , f − c ) i c,f (cid:63)a → H • ( f a , f − c ) (cid:111) ↓ Ψ (cid:63) (cid:8) ↓ Ψ (cid:63) H • ( f c , f − c ) → i c,f (cid:63)a H • ( f a , f − c ) Hence, if i c,f (cid:63)a is zero, since the left arrow is one-to-one, i c,f (cid:63)a is necessarily zero. Thisproves that { a ∈ R | i c,f (cid:63)a (cid:54) = 0 } ⊂ { a ∈ R | i c,f (cid:63)a (cid:54) = 0 } and then σ ( f ) ≤ σ ( f ) .Now, if U is not open anymore, but bounded, it is contained for all δ > in the openand bounded set U δ = { x ∈ R d | d ( x, U ) < δ } . Furthermore since f ≥ f on U andsince U δ is bounded, we have by continuity of f and f that f ≥ f + w ( δ ) on U δ with w ( δ ) → when δ → . The previous work states that σ ( f ) ≥ σ ( f + w ( δ )) = σ ( f )+ w ( δ ) by additivity of the minmax, and letting δ tend to finishes the proof.Finally, we get rid of the boundness assumption by observing that since C f ,f iscompact (Proposition B.10), we may always replace U by the intersection of U with aball large enough to contain C f ,f , which ends the proof. (cid:3) Proposition B.13.
If the cohomology is calculated with coefficients in a field, σ ( − f ) = − σ ( f ) for each function f of Q ∞ m .Proof. If f is in Q ∞ m with an associate nondegenerate form Z of index λ , take c bigger inmodulus than the critical values of f . The homology calculation for the quadratic formgives that H k ( f c , f − c ) = 0 if k (cid:54) = λ and H λ ( f c , f − c ) is one dimensional. In particular,if the homology is calculated with coefficients in a field, the homology morphism i ca(cid:63) : H • ( f a , f − c ) → H • ( f c , f − c ) induced by i ca is non zero if and only if it is one-to-one. Since i ca(cid:63) is the transposition of i c(cid:63)a , they are simultaneously non zero.Alexander duality gives the following commutative diagram, with exact columns: H • ( f a , f − c ) (cid:39) H • ( R m \ f − c , R m \ f a ) i ca(cid:63) ↓ (cid:8) ↓ H • ( f c , f − c ) (cid:39) H • ( R m \ f − c , R m \ f c ) ↓ (cid:8) ↓ H • ( f c , f a ) (cid:39) H • ( R m \ f a , R m \ f c ) If a is not a critical value of f , for ε > small enough R m \ f a = {− f < − a } retractson − f − a − ε via the homotopy equivalence constructed in Lemma C.1, just as − f − a . Thesame can be done for c and − c , and composing the cohomology induced isomorphismswe get an isomorphism Φ (cid:63) , completing the previous diagram as follows: H • ( f a , f − c ) (cid:39) H • ( R m \ f − c , R m \ f a ) i ca(cid:63) ↓ (cid:8) ↓ H • ( f c , f − c ) (cid:39) H • ( R m \ f − c , R m \ f c ) Φ (cid:63) (cid:39) H • (( − f ) c , ( − f ) − c ) ↓ (cid:8) ↓ (cid:8) ↓ ( i − ac, − f ) (cid:63) H • ( f c , f a ) (cid:39) H • ( R m \ f a , R m \ f c ) (cid:39) Φ (cid:63) H • (( − f ) − a , ( − f ) − c ) If a is larger than σ ( f ) , i c(cid:63)a is non zero, hence i ca(cid:63) is non zero and it is then one-to-one.Since the first column is exact, this implies that ( i − ac, − f ) (cid:63) is zero, hence − a ≤ σ ( − f ) . Thisbeing true for each a larger than σ ( g ) , it comes that − σ ( f ) ≤ σ ( − f ) .If a is smaller than σ ( f ) , i c(cid:63)a , hence i ca(cid:63) , are zero and it follows that ( i − ac, − f ) (cid:63) is nonzero, hence − a ≥ σ ( − f ) . As before this implies that − σ ( f ) ≥ σ ( − f ) , and the resultholds. (cid:3) Remark
B.14 . The proof of Proposition B.13 is the only place where we need to workwith coefficients in a field.
Proposition B.15. If f : ( x, y ) ∈ R d × R k → R is a function of Q ∞ d + k such that ∂ y f ≥ c id for some c > , and if g ( x ) = min y f ( x, y ) is in Q d , then σ ( g ) = σ ( f ) .Proof. If ∂ y f ≥ c id , y (cid:55)→ f ( x, y ) attains for each x a strict minimum at a point y ( x ) and x (cid:55)→ y ( x ) is C by implicit differentiation of ∂ y f ( x, y ( x )) = 0 . Note that g ( x ) = f ( x, y ( x )) and f have the same critical values and choose c larger in modulus than these criticalvalues.We denote by ˜ g a the set { ( x, y ( x )) | g ( x ) ≤ a } . It is the restriction of the graph of x (cid:55)→ y ( x ) on g a . Hence Ψ : x (cid:55)→ ( x, y ( x )) , which is a C -diffeomorphism from R d to thegraph of x (cid:55)→ y ( x ) , maps for all a g a on ˜ g a , and it induces an isomorphism in relativecohomology.For all a in R , the sublevel set f a retracts to ˜ g a via Φ t ( x, y ) = ( x, (1 − t ) y + ty ( x )) whichis a deformation retraction. One can indeed check, using the convexity of y (cid:55)→ f ( x, y ) and the fact that y ( x ) is the minimum of this function, that: Φ = id , Φ ( f a ) ⊂ ˜ g a , Φ t ( f a ) ⊂ f a ∀ t ∈ [0 , , Φ t = id on ˜ g a . Since this retraction does not depend on a , the following diagram commutes: H • ( f c , f − c ) i c,f(cid:63)a → H • ( f a , f − c ) (cid:111) ↑ Φ (cid:63) (cid:111) ↑ Φ (cid:63) H • (˜ g c , ˜ g − c ) (cid:8) H • (˜ g a , ˜ g − c ) (cid:111) ↑ Ψ − (cid:63) (cid:111) ↑ Ψ − (cid:63) H • ( g c , g − c ) → i c,g(cid:63)a H • ( g a , g − c ) Hence i c,g(cid:63)a and i c,f(cid:63)a are simultaneously nonzero and therefore σ ( g ) = σ ( f ) . (cid:3) B.3.
Extension to non-smooth functions.
From now on the aim is to extend bycontinuity the definition and properties of the minmax to non-smooth functions.
Definition B.16. If f is in Q m , there exists by definition a nondegenerate quadraticform Z and a Lipschitz function (cid:96) such that f = Z + (cid:96) . Since (cid:96) is Lipschitz, there existsan equi-Lipschitz sequence ( (cid:96) n ) of C ∞ functions such that (cid:96) n converge uniformly towards (cid:96) . Then the minmax of f = Z + (cid:96) is defined by σ ( f ) = lim n →∞ σ ( Z + (cid:96) n ) . This does not depend on the choice of ( (cid:96) n ) . Proof.
Let us show that the limit exists, and that it does not depend on the choice ofthe sequence ( (cid:96) n ) . • Let ε > be fixed. Since (cid:96) n converges uniformly, it is a Cauchy sequence andthere is a N > such that: (cid:107) (cid:96) n − (cid:96) m (cid:107) ∞ ≤ ε ∀ n, m ≥ N. Then, since Z + (cid:96) n and Z + (cid:96) m are in Q ∞ m with Lipschitz and bounded difference,Consequence B.12 gives: | σ ( Z + (cid:96) n ) − σ ( Z + (cid:96) m ) | ≤ (cid:107) (cid:96) n − (cid:96) m (cid:107) ∞ ≤ ε ∀ n, m ≥ N and ( σ ( Z + (cid:96) n )) is a Cauchy sequence in R , hence has a limit denoted σ ( f ) . • Let ( (cid:96) n ) and (˜ (cid:96) n ) be two equi-Lipschitz sequences of C ∞ functions, and assumethat (cid:96) n and ˜ (cid:96) n admit the same uniform limit (cid:96) . Let us show that σ ( Z + (cid:96) n ) and σ ( Z + ˜ (cid:96) n ) tend to the same limit.Let ε > . Since (cid:96) n and ˜ (cid:96) n have the same limit, there is a N > such that: (cid:107) (cid:96) n − ˜ (cid:96) n (cid:107) ∞ ≤ ε ∀ n ≥ N. Then, since Z + (cid:96) n and Z + ˜ (cid:96) n are in Q ∞ m with Lipschitz and bounded difference,Consequence B.12 gives: | σ ( Q + (cid:96) n ) − σ ( Q + ˜ (cid:96) n ) | ≤ ε ∀ n ≥ N. Letting n tend to ∞ shows that the limit does not depend on the choice of thesequence ( (cid:96) n ) . (cid:3) Let us gather the properties satisfied for continuous functions of Q m : Proposition B.17. If f is in Q m , the properties of the smooth minmax still hold: (1) if c is a real constant, then σ ( c + f ) = c + σ ( f ) ,(2) if f ≤ f on R m and if f − f is Lipschitz, then σ ( f ) ≤ σ ( f ) ,(3) if φ is a Lipschitz C ∞ -diffeomorphism on R m such that f ◦ φ is in Q m , then σ ( f ◦ φ ) = σ ( f ) , (4) σ ( − f ) = − σ ( f ) .Proof. (1) It is enough to notice that if Z + (cid:96) n converges to f as in the definition,then Z + (cid:96) n + c converges to f + c . Then, σ ( Z + c + (cid:96) n ) = c + σ ( Z + (cid:96) n ) by theadditivity property (B.8-2), and the statement holds when n tends to ∞ .(2) If f ≤ f are in Q m and if their difference is Lipschitz, then there exist twosequences of equi-Lipschitz C ∞ functions ( (cid:96) n ) and ( (cid:96) n ) such that Z + (cid:96) n (resp. Z + (cid:96) n ) converges uniformly to f (resp. f ) with (cid:96) n ≤ (cid:96) n for n big enough.Then, Proposition B.11 states that σ ( Z + (cid:96) n ) ≤ σ ( Z + (cid:96) n ) for n big enough, andthe statement holds when n tends to ∞ .(3) Since (cid:107) ( Z + (cid:96) n ) ◦ φ − f ◦ φ (cid:107) ∞ ≤ (cid:107)Z + (cid:96) n − f (cid:107) ∞ , if ( Z + (cid:96) n ) converges uniformlyto f = Z + (cid:96) , then ( Z + (cid:96) n ) ◦ φ converges uniformly to f ◦ φ . Moreover, since φ is Lipschitz, (cid:96) n ◦ φ and (cid:96) ◦ φ are (equi-)Lipschitz. Now since f ◦ φ = Z ◦ φ + (cid:96) ◦ φ is in Q m and (cid:96) ◦ φ is Lipschitz, Z ◦ φ is in Q ∞ m (as Z is C ∞ ) and the sequence (( Z + (cid:96) n ) ◦ φ ) is still in Q ∞ m .Thus, (( Z + (cid:96) n ) ◦ φ ) is a sequence converging uniformly to f ◦ φ , as requiredin the definition. Since Property (B.8-3) states that σ (( Z + (cid:96) n ) ◦ φ ) = σ ( Z + (cid:96) n ) for all n , the statement holds when n tends to ∞ .(4) This is a direct consequence of Proposition B.13. (cid:3) Proposition B.18.
The properties involving critical elements hold for C func-tions of Q m :(1) If f ∈ Q m is C , then σ ( f ) is a critical value of f .(2) If f , f ∈ Q m are C with Lipschitz difference, and C f ,f is the set of criticalpoints of the homotopy f t = (1 − t ) f + tf , then inf C f ,f ( f − f ) ≤ σ ( f ) − σ ( f ) ≤ sup C f ,f ( f − f ) . Proof. (1) If f = Z + (cid:96) is C , then (cid:96) is C and there exists an equi-Lipschitz sequence ( (cid:96) n ) of C ∞ functions such that (cid:96) n uniformly converges towards (cid:96) and d(cid:96) n convergeuniformly towards d(cid:96) , hence Z + (cid:96) n (resp. d Z + d(cid:96) n ) ) uniformly converges towards f (resp. df ).For all n , σ ( Z + (cid:96) n ) is a critical value of Z + (cid:96) n , hence there exists x n in R m such that d Z ( x n ) + (cid:96) n ( x n ) = 0 and σ ( Z + (cid:96) n ) = ( Z + (cid:96) n )( x n ) .Since the sequence ( (cid:96) n ) is equi-Lipschitz, the sequence ( x n ) is contained inthe closed ball ¯ B ( L/m ) where L denotes a Lipschitz constant suiting all (cid:96) n and m = inf (cid:107) x (cid:107) =1 (cid:107) d Z ( x ) (cid:107) , see Proposition B.3.Hence x n admits a subsequence converging to some x in R m . On the one hand,since d ( Z + (cid:96) n ) converges uniformly towards df and d ( Z + (cid:96) n )( x n ) = 0 , x is acritical point of f . On the other hand, since Z + (cid:96) n converges uniformly towards f and ( Z + (cid:96) n )( x n ) = σ ( Z + (cid:96) n ) , f ( x ) = σ ( f ) . Thus σ ( f ) is a critical value of f . (2) Take f and f in Q m , C and with Lipschitz difference. There exists an equi-Lipschitz sequence ( (cid:96) n ) of C ∞ functions such that Z + (cid:96) n (resp. d Z + d(cid:96) n )converges uniformly to f (resp. df ). Note that if t is in [0 , , the sequence ( (cid:96) tn ) = ( (cid:96) n + t ( f − f )) is equi-Lipschitz uniformly with respect to t , and f tn = Z + (cid:96) tn converges uniformly to f t = (1 − t ) f + tf , and the derivative sequence ( df tn ) converges uniformly to df t .For all n , Consequence B.12 states that: inf C f n,f n ( f n − f n ) ≤ σ ( f n ) − σ ( f n ) ≤ sup C f n,f n ( f n − f n ) . Let us focus on the second inequality. Since C f n ,f n is compact (Proposition B.10),the supremum is attained at some x n in C f n ,f n . By definition of C f n ,f n , thereexists a sequence ( t n ) of [0 , such that x n is a critical point of f t n n .Now, since the sequence ( (cid:96) tn ) n is equi-Lipschitz uniformly with respect to t ,there exists a ball B (0 , R ) , where R depends only on the Lipschitz constants andon Z , containing C f n ,f n for all n . The sequence ( t n , x n ) is hence bounded and wemay assume it converges to some ( t, x ) . Since df t n n converges uniformly towards df t , the fact that x n is a critical point of f t n n implies that x is a critical point of f t , hence x is in C f ,f .But then letting n tend to ∞ in σ ( f n ) − σ ( f n ) ≤ sup C f n,f n ( f n − f n ) = f n ( x n ) − f n ( x n ) gives that σ ( f n ) − σ ( f n ) ≤ f ( x ) − f ( x ) ≤ sup C f ,f ( f − f ) , using first the uniform convergence of f n − f n towards f − f and then the factthat x is in C f ,f . (cid:3) The next proposition is the improved version of Proposition B.18-(2) that we requirein the definition of a critical value selector, see Definition 2.7.
Proposition B.19.
Let ( f t ) t ∈ [0 , be a C homotopy of Q m such that there exists anondegenerate quadratic function Z and an equi-Lipschitz family of C functions ( (cid:96) t ) t ∈ [0 , with f t = Z + (cid:96) t . Then for all s (cid:54) = t in [0 , t ∈ [0 , min x ∈ Crit ( f t ) ∂ t f t ( x ) ≤ σ ( f t ) − σ ( f s ) t − s ≤ max t ∈ [0 , max x ∈ Crit ( f t ) ∂ t f t ( x ) . Let ( f t ) t ∈ [0 , be as in the proposition. Note that if m = inf (cid:107) x (cid:107) =1 (cid:107) d Z ( x ) (cid:107) , the crit-ical points of f t are contained for each t in the compact set C = ¯ B (0 , L/m ) . The set { ( t, x ) , t ∈ [0 , , ∂ x f t ( x ) = 0 } is also compact: it is contained in the bounded set [0 , × C and is closed by continuity of ∂ x f w.r.t. t and x . Both min t ∈ [0 , min x ∈ Crit ( f t ) ∂ t f t ( x ) and max t ∈ [0 , max x ∈ Crit ( f t ) ∂ t f t ( x ) are hence attained, and we denote them respectivelyby a and b . Lemma B.20.
For all ε > , there exists α > such that for all t in [0 , , (cid:107) ∂ x f t ( x ) (cid:107) ≤ α implies a − ε ≤ ∂ t f t ( x ) ≤ b + ε .Proof. Assume that there exists an ε > and a sequence ( t n , x n ) such that (cid:107) ∂ x f t n ( x n ) (cid:107) ≤ /n and ∂ t f t n ( x n ) / ∈ ( a + ε, b + ε ) . Since f t n = Z + (cid:96) t n , (cid:107) ∂ x f t n ( x n ) (cid:107) ≥ m (cid:107) x n (cid:107) − L andthe sequence x n is necessarily bounded. Since t n is in [0 , , there exists a subsequenceof ( t n , x n ) converging to some ( t, x ) . The continuity of df gives then a contradiction atthe point ( t, x ) . (cid:3) Proof of Proposition B.19.
Let us define w ( δ ) = sup x ∈ C, | t − s |≤ δ { ∂ t f s ( x ) − ∂ t f t ( x ) , ∂ x f s ( x ) − ∂ x f t ( x ) } . The continuity of df and the compacity of C grants that w ( δ ) → when δ → .Let us fix ε > and prove that ( a − ε )( t − s ) ≤ σ ( f t ) − σ ( f s ) ≤ ( b + 2 ε )( t − s ) forall s ≤ t in [0 , . Take α as in Lemma B.20 and δ > such that both w ( δ ) < ε and w ( δ ) < α . We first show the result for t − s ≤ δ , and it is immediately extended to large t − s by iteration.For all x in R d , we have ( t − s ) inf τ ∈ [ s,t ] ∂ t f τ ( x ) ≤ f t ( x ) − f s ( x ) ≤ ( t − s ) sup τ ∈ [ s,t ] ∂ t f τ ( x ) . Now if C f s ,f t denotes the set of critical points of the functions g u = (1 − u ) f s + uf t for u in [0 , , on the one hand, one has that C f s ,f t ⊂ C = ¯ B (0 , L/m ) , while on the other handProposition B.18-(2) states that: inf C fs,ft ( f t − f s ) ≤ σ ( f t ) − σ ( f s ) ≤ sup C fs,ft ( f t − f s ) , which implies ( t − s ) inf τ ∈ [ s, t ] x ∈ C f s ,f t ∂ t f τ ( x ) ≤ σ ( f t ) − σ ( f s ) ≤ ( t − s ) sup τ ∈ [ s, t ] x ∈ C f s ,f t ∂ t f τ ( x ) . Since C f s ,f t and [ s, t ] are compact, the right hand side supremum is attained for some τ and x , where x is the critical point of a function g u = (1 − u ) f s + uf t , and consequentlysatisfies ∂ x f s ( x ) = u ( ∂ x f s ( x ) − ∂ x f t ( x )) . Since x is in C and u is in [0 , , we get (cid:107) ∂ x f s ( x ) (cid:107) ≤ w ( | t − s | ) ≤ α by definition of w and δ . Lemma B.20 then implies that ∂ t f s ( x ) ≤ b + ε .Now let us estimate ∂ t f τ ( x ) : since x is in C and w ( δ ) ≤ ε , ∂ t f τ ( x ) ≤ ∂ t f s ( x ) + w ( | τ − s | ) ≤ b + 2 ε. Putting it altogether we get that for all ε > , σ ( f t ) − σ ( f s ) ≤ f t ( x ) − f s ( x ) ≤ ( t − s ) ∂ t f τ ( x ) ≤ ( t − s )( b + 2 ε ) for t − s ≤ δ , and hence for all t and s . The same work for the left hand side infimumgives that for all ε > , ( t − s )( a − ε ) ≤ σ ( f t ) − σ ( f s ) ≤ ( t − s )( b + 2 ε ) , and letting ε tend to gives the wanted estimate. (cid:3) C. Deformation lemmas
C.1.
Global deformation of sublevel sets.
We still work with functions of Q ∞ m , i.e. with functions that can be written as the sum of a nondegenerate quadratic function andof a C ∞ Lipschitz function.
Lemma C.1 (Strong deformation retraction) . Let f be a function of Q ∞ m . Take ε > and a < b in R . If [ a − ε, b + ε ] does not contain any critical value of f , then there is a strongdeformation retraction mapping f b to f a , i.e. a continuous function Φ : [0 , × R m → R m such that Φ = id R m , Φ ( f b ) ⊂ f a , Φ t (cid:12)(cid:12)(cid:12) f a = id f a ∀ t ∈ [0 , t ( f c ) ⊂ f c ∀ t ∈ [0 , , ∀ c ∈ R satisfying the additional requirement Φ t ( f c ) = f c for all t ∈ [0 , and c > b + ε .Proof. First step. We build a continuous function
Ψ : [0 , × R m → R m such that(12) Ψ = id R m , Ψ ( f b ) ⊂ f a , Ψ t ( f c ) ⊂ f c ∀ t ∈ [0 , , ∀ c ∈ R Ψ t ( f c ) = f c , ∀ t ∈ [0 , , ∀ c > b + ε, without requiring that Ψ t is the identity on f a for all t .Let X be the locally Lipschitz vector field defined for x in R m by X ( x ) = (cid:40) ∇ f ( x ) si (cid:107)∇ f ( x ) (cid:107) ≤ ∇ f ( x ) (cid:107)∇ f ( x ) (cid:107) si (cid:107)∇ f ( x ) (cid:107) > Let us take a C ∞ function φ : R → [0 , satisfying φ = 1 on ( −∞ , b ] and φ = 0 on [ b + ε, ∞ ) , and consider the following vector field: Y ( x ) = φ ( f ( x )) X ( x ) , defined such that Y = X on f b and Y ( x ) = 0 if f ( x ) ≥ b + ε .Let us denote by Ψ t ( x ) the flow associated with − Y as follows: ® ∂ t Ψ t ( x ) = − Y (Ψ t ( x ))Ψ ( x ) = x. As (cid:107) Y (cid:107) is locally Lipschitz and bounded by the constant , Ψ is defined on R + × R m and Ψ t is a homeomorphism of R m for all t . Let us check that t (cid:55)→ f (Ψ t ( x )) is non-increasing: ∂ t ( f (Ψ t ( x ))) = − φ ( f (Ψ t ( x ))) (cid:124) (cid:123)(cid:122) (cid:125) ≥ X (Ψ t ( x )) · ∇ f (Ψ t ( x )) (cid:124) (cid:123)(cid:122) (cid:125) ≥ min {(cid:107)∇ f (Ψ t ( x )) (cid:107) , (cid:107)∇ f (Ψ t ( x )) (cid:107)}≥ ≤ . In particular, Ψ t ( f c ) ⊂ f c for all t ≥ , and c ∈ R .Let us prove that Ψ t ( f c ) = f c for all c > b + ε . It is enough to prove that f c ⊂ Ψ t ( f c ) since the other inclusion is true for all c . Since Y = 0 on R m \ f b + ε , Ψ t (cid:12)(cid:12)(cid:12) R m \ f c = id R m \ f c for all c > b + ε . Then, if x ∈ f c , there is a y ∈ R m such that Ψ t ( y ) = x (since Ψ t isonto), and y cannot be in R m \ f c since x ∈ f c . Hence, x belongs to Ψ t ( f c ) .The aim is now to find a T > such that Ψ T ( f b ) ⊂ f a . Let us prove that there is a real constant M > such that: (cid:107) df ( x ) (cid:107) ≥ M ∀ x ∈ f b \ f a − ε . Suppose that ( x n ) is a sequence of f b \ f a − ε such that df ( x n ) → . Since f = Z + (cid:96) with Z nondegenerate quadratic and (cid:96) Lipschitz, ( x n ) is hence bounded an admits aconverging subsequence ; let x be the limit. Since f and df are continuous, df ( x ) = 0 and f ( x ) belongs to [ a − ε, b ] . As this is excluded, the existence of M is proved.Let x be in f b . If t ≥ , Ψ t ( x ) is in f b too. Hence, we have (cid:107)∇ f (Ψ t ( x ) (cid:107) ≥ M as longas f (Ψ t ( x )) > a − ε , and the estimation of ddt f (Ψ t ( x )) can be improved: ∂ t ( f (Ψ t ( x ))) ≤ − φ ( f (Ψ t ( x ))) (cid:124) (cid:123)(cid:122) (cid:125) =1 since Ψ t ( x ) ∈ f b min ¶ (cid:107)∇ f (Ψ t ( x )) (cid:107) , (cid:107)∇ f (Ψ t ( x )) (cid:107) © ≤ − min ¶ M , M © < . Let K = min( M , M ) > . As long as f (Ψ t ( x )) > a − ε , the previous calculation gives: f (Ψ t ( x )) ≤ f (Ψ ( x )) (cid:124) (cid:123)(cid:122) (cid:125) = f ( x ) ≤ b − Kt ≤ b − Kt.
Let T = b − aK . Assume that for all t ∈ [0 , T ] , f (Ψ t ( x )) > a . The previous calculationshows that f (Ψ T ( x )) ≤ b − KT = a , which is absurd. Hence there exists t ∈ [0 , T ] suchthat f (Ψ t ( x )) ≤ a and then since t (cid:55)→ f (Ψ t ( x )) is non increasing, Ψ T ( x ) ⊂ f a .Up to a time rescaling sending T to ( ˜Ψ t ( x ) = Ψ t/T ( x ) ), we have just constructed adeformation retraction satisfying (12). Second step.
Let us now build the strong deformation retraction. For all x in R m , let τ ( x ) be defined by τ ( x ) = inf { t ∈ [0 , | Ψ t ( x ) ∈ f a } . It is a continuous function on R m . If Ψ t ( x ) stays out of f a for all t in [0 , (this is thecase for all x in R m \ f b + ε ), then τ ( x ) is by convention equal to . Since t (cid:55)→ f (Ψ t ( x )) is non-increasing, if Ψ t ( x ) is not in f a , t ≤ τ ( x ) .Let us define the mapping Φ : Φ : [0 , × R m → R m ( t, x ) (cid:55)→ Φ t ( x ) = Ψ min( t,τ ( x )) ( x ) so that in particular Φ = Ψ and Φ ( x ) = Ψ τ ( x ) ( x ) . The continuity of τ and Ψ impliesthe continuity of Φ . Let us check that Φ is as required in the Lemma: • Φ = Ψ = id R m , • for all x in f b , Ψ ( x ) is in f a , and as a consequence Φ ( x ) = Ψ τ ( x ) ( x ) is in f a , • since τ = 0 on f a , Φ t = Ψ = id on f a , • for all t in [0 , , Φ t ( f c ) ⊂ ∪ u ∈ [0 , Ψ u ( f c ) ⊂ f c . • let us fix t in [0 , and show that if c > b + ε , f c ⊂ Φ t ( f c ) . Since f c ⊂ Ψ t ( f c ) for such a c , for all x in f c there exists y in f c such that Ψ t ( y ) = x . If x is notin f a , τ ( y ) ≥ t , and hence x = Ψ t ( y ) = Φ t ( y ) is in Φ t ( f c ) . If x is in f a , since Φ t is the identity on f a , x = Φ t ( x ) is in Φ t ( f a ) ⊂ Φ t ( f c ) . (cid:3) C.2.
Sending sublevel sets to sublevel sets.Lemma C.2 (Deformation of big sublevel sets of Q ∞ m functions with Lipschitz difference) . Let (cid:96) and (cid:96) be two C ∞ Lipschitz functions, Z be a nondegenerate quadratic form on R m , and define f t = Z + (cid:96) + t ( (cid:96) − (cid:96) ) the homotopy between f = Z + (cid:96) and f = Z + (cid:96) .Let U be an open and bounded set of R m containing C = { x ∈ R m |∃ t ∈ [0 , , df t ( x ) = 0 } .There exists a C ∞ -diffeomorphism Ψ of R m such that: Ψ( f c ) = f c ∀ c > max t ∈ [0 , sup U f t , ∀ c < min t ∈ [0 , inf U f t . Moreover, if f ≥ f on U , Ψ can be constructed so that Ψ( f a ) ⊂ f a for all a ∈ R .Proof. Since C is compact (see Proposition B.10), there exists an open set Ω containing C such that Ω ∩ U c is empty ( Ω is an open set which is "strictly included" in the openset U ). Let X t be the vector field defined on R m \ Ω by X t ( x ) = − ∂ t ( f t ( x )) ∇ f t ( x ) (cid:107)∇ f t ( x ) (cid:107) for t ∈ [0 , . Lemma C.3. If γ ( t ) is a trajectory for the vector field X t , that is if γ ( t ) stays in R m \ Ω and ˙ γ ( t ) = X t ( γ ( t )) , then f t ( γ ( t )) does not depend on t .Proof. This is proved by the following calculation: ∂ t ( f t ( γ ( t ))) = ˙ γ ( t ) · ∇ f t ( γ ( t )) (cid:124) (cid:123)(cid:122) (cid:125) = − ∂ t f t ( γ ( t )) + ∂ t f t ( γ ( t )) = 0 . (cid:3) Since ¯Ω and R m \ U are closed and disjoint, it is possible to find g : R m → [0 , smoothsuch that ® g = 0 on ¯Ω g = 1 on R m \ U . Let us define Y t ( x ) = g ( x ) X t ( x ) . The vector field Y iswell-defined, C ∞ on R m . It satisfies: ® Y t = X t on R m \ UY t = 0 on Ω . Lemma C.4.
The vector field Y is bounded.Proof. If m = inf (cid:107) x (cid:107) =1 (cid:107) d Z ( x ) (cid:107) , we get that (cid:107)∇ f t ( x ) (cid:107) ≥ m (cid:107) x (cid:107) − L for all x in R d . As aconsequence, if (cid:107) x (cid:107) ≥ L/m , (cid:107) Y t ( x ) (cid:107) ≤ | ∂ t f t ( x ) |(cid:107)∇ f t ( x ) (cid:107) ≤ Lm (2 L/m ) − L ≤ . Now, define M = sup t ∈ [0 , (cid:107) x (cid:107) ≤ L/m (cid:107) Y t ( x ) (cid:107) . Then Y is bounded by max(1 , M ) on R m . (cid:3) The flow ψ of Y is hence defined on R × R m ; it is the C ∞ solution of the Cauchyproblem: ® ∂ t ψ ( t, x ) = Y t ( ψ ( t, x )) ψ (0 , x ) = x. Let Ψ be the C ∞ -diffeomorphism mapping x to ψ (1 , x ) .Let us denote by C + (resp. C − ) the quantity max t ∈ [0 , sup U f t (resp. min t ∈ [0 , inf U f t ), and provethat for c ∈ R \ [ C − , C + ] , Ψ ( { f = c } ) = { f = c } . Take x and y such that Ψ( x ) = y , anddenote by γ ( t ) the trajectory t (cid:55)→ ψ ( t, x ) . Since Y and X coincide on R m \ U ⊂ R m \ Ω ,Lemma C.3 states that as long as γ ( t ) is in R m \ U , f t ( γ ( t )) is constant. By definitionof C + and C − , { f t = c } is included in R m \ U for all t in [0 , , and as a consequence ∃ t ∈ [0 , , f t ( γ ( t )) = c = ⇒ ∀ t ∈ [0 , , f t ( γ ( t )) = c. This means that f ( x ) = c if and only if f ( y ) = c , and since Ψ is one-to-one we henceproved that Ψ ( { f = c } ) = { f = c } .As a consequence, we obtain applying the previous work to a suitable union of levelsetsthat Ψ ( { f ≤ c } ) = { f ≤ c } for c < C − , and that Ψ ( { f > c } ) = { f > c } for c > C + .Since Ψ is one-to-one, this implies Ψ ( { f ≤ c } ) = { f ≤ c } for c > C + .Finally, assume that f ≥ f on U . Let us again estimate the evolution of f t ( γ ( t )) fora trajectory ˙ γ ( t ) = Y t ( γ ( t )) : ∂ t f t ( γ ( t )) = ˙ γ ( t ) · ∇ f t ( γ ( t )) + ∂ t f t ( γ ( t ))= g ( γ ( t ))( f − f )( γ ( t )) + ( f − f )( γ ( t ))= (1 − g ( γ ( t ))( f − f )( γ ( t )) = ® = 0 if γ ( t ) ∈ R m \ U ≤ if γ ( t ) ∈ U. since g = 1 on R m \ U , − g ≥ and f ≤ f on U . Now, for a ∈ R , let x ∈ f a .Since t (cid:55)→ f t ( ψ ( t, x )) is non-increasing, f (Ψ( x )) ≤ f ( x ) ≤ a and we have proved that Ψ( f a ) ⊂ f a . (cid:3) D. Uniqueness of viscosity solution: doubling variables
Let us first recall a possible definition of viscosity solutions in a continuous setting:
Definition D.1.
A continuous function u is a subsolution of (HJ) on the set (0 , T ) × R d if for each C ∞ function φ : (0 , T ) × R d → R such that u − φ admits a (strict) localmaximum at a point ( t, q ) ∈ (0 , T ) × R d , ∂ t φ ( t, q ) + H ( t, q, ∂ q φ ( t, q )) ≤ . A continuous function u is a supersolution of (HJ) on the set (0 , T ) × R d → R if for each C ∞ function φ : (0 , T ) × R d such that u − φ admits a (strict) local minimum at a point ( t, q ) ∈ (0 , T ) × R d , ∂ t φ ( t, q ) + H ( t, q, ∂ q φ ( t, q )) ≥ . A viscosity solution is both a sub- and supersolution of (HJ). The following proposition justifies the name of viscosity operator for an operator sat-isfying Hypotheses 1.3.
Proposition D.2.
Let H be a C Hamiltonian with uniformly bounded second spatialderivative and V ts : C , ( R d , R ) → C , ( R d , R ) be a viscosity operator defined for each ≤ s ≤ t . Then for each Lipschitz function u : R d → R , u ( t, q ) = V t u ( q ) solves the Hamilton-Jacobi equation in the viscosity sense on (0 , ∞ ) × R d . This characterization and its proof may be found in [Ber12] (Proposition 20). A sim-ilar axiomatic description of the viscosity solutions was initially proposed in [AGLM93](Theorem 2) for multiscale analysis, see also [FS06] (Theorem 5.1).The uniqueness of the viscosity operator for H satisfying Hypothesis 1.1 is a con-sequence of a stronger uniqueness result for unbounded solutions stated by H. Ishii in[Ish84] (Theorem 2.1 with Remark 2.2), see also [CL87]. It is also a consequence of thefollowing finite speed of propagation argument proposed by G. Barles in [Bar94] (Theo-rem 5.3). We write the proof here for the sake of completeness, adopting his argumentsand notations, and using only the second estimate of Hypothesis 1.1. Proposition D.3 (Finite speed of propagation) . If H satisfies (cid:107) ∂ q,p H (cid:107) ≤ C (1 + (cid:107) p (cid:107) ) forsome C > , and u and v are respectively sub- and supersolutions of (HJ) on [0 , T ] × R d which are L -Lipschitz uniformly in time with respect to the space variable, then: u (0 , · ) ≤ v (0 , · ) on B (0 , R ) = ⇒ u ≤ v on [0 , T ] × B (0 , R − C (1 + 2 L ) T ) as long as R is strictly larger than C (1 + 2 L ) T .Consequence D.4 . If u and v are two viscosity solutions of (HJ) which are L -Lipschitzwith respect to q on [0 , T ] × R d , then for each t in [0 , T ] : | u ( t, q ) − v ( t, q ) | ≤ (cid:107) u (0 , · ) − v (0 , · ) (cid:107) ¯ B ( q,C (1+2 L ) t ) Proof.
We apply Proposition D.3 with R = C (1 + 2 L ) t + δ to the subsolution u and thesupersolution v + (cid:107) u (0 , · ) − v (0 , · ) (cid:107) ¯ B ( q,R ) , use the symmetry and let δ tend to . (cid:3) Consequence
D.5 . If u and v are both viscosity solutions on [0 , T ] × R d that satisfy u (0 , · ) = v (0 , · ) on R d and are Lipschitz uniformly in time, they coincide on [0 , T ] × R d .In particular, there exists at most one viscosity operator. Lemma D.6. If u is a continuous function of (0 , T ] × R d and also a subsolution of (HJ) on (0 , T ) × R d , then it is a subsolution on (0 , T ] × R d , meaning that if u − φ attains astrict maximum on (0 , T ] × R d at some ( T, q ) , the derivatives of φ satisfy the requiredinequality.Proof. Take φ C ∞ on (0 , T ] × R d such that u − φ attains its strict maximum at some ( T, q ) . Let us consider the functions ( t, q ) (cid:55)→ u ( t, q ) − φ ( t, q ) − ηT − t for small η > . Since u − φ attains a strict maximum at ( T, q ) , there exists a sequence ( t η , q η ) in (0 , T ) × R d of local maximal points of u − φ − ηT − t such that ( t η , q η ) tends to ( T, q ) when η → .Since u is a subsolution on (0 , T ) × R d , this implies that: ∂ t Å φ ( t, q ) + ηT − t ã + H Å t η , q η , ∂ q Å φ ( t, q ) + ηT − t ãã ≤ hence ∂ t φ ( t η , q η ) + η ( T − t η ) + H ( t η , q η , ∂ q φ ( t η , q η )) ≤ . The positive term η ( T − t η ) may be dropped, and then the continuity of φ gives that: ∂ t φ ( T, q ) + H ( T, q , ∂ q φ ( T, q )) ≤ . (cid:3) Lemma D.7.
If the assumptions of Proposition D.3 are satisfied, the function w = u − v is a subsolution on (0 , T ] × R d of ∂ t w − C (1 + 2 L ) (cid:107) ∂ q w (cid:107) = 0 . Proof.
Let us assume that φ is a C ∞ function such that w − φ attains a strict localmaximum at a point ( t , q ) in (0 , T ) × R d . The aim is to show that ∂ t φ ( t , q ) ≤ C (1 + 2 L ) (cid:107) ∂ q φ ( t , q ) (cid:107) . Here is where the variables are doubled: let us define the function Ψ ε,α : ( t, q, s, p ) (cid:55)→ u ( t, q ) − v ( s, p ) − (cid:107) q − p (cid:107) ε − | t − s | α − φ ( t, q ) . In particular Ψ ε,α ( t , q , t , q ) = w ( t , q ) − φ ( t , q ) is the local maximum of w − φ forall ε > and α > .Take r > such that the maximum of w − φ on ¯ B (( t , q ) , r ) is attained only at ( t , q ) . Then Ψ ε,α attains a maximum on the compact set ¯ B (( t , q ) , r ) × ¯ B (( t , q ) , r ) ,and we denote by (¯ t, ¯ q, ¯ s, ¯ p ) a point reaching this maximum, without forgetting that thesequantities depend on ε and α . Lemma D.8.
The point (¯ t, ¯ q, ¯ s, ¯ p ) satisfies:(1) (¯ t, ¯ q ) , (¯ s, ¯ p ) → ( t , q ) when ε, α → ,(2) (cid:107) ¯ q − ¯ p (cid:107) ε ≤ L .Proof. (1) Since (¯ t, ¯ q, ¯ s, ¯ p ) belongs to the compact set ¯ B (( t , q ) , r ) × ¯ B (( t , q ) , r ) ,accumulation points ( t, q, s, p ) exist when ε and α tend to zero. These accumu-lation points must satisfy ( t, q ) = ( s, p ) : else, the value of Ψ ε,α (¯ t, ¯ q, ¯ s, ¯ p ) explodestowards −∞ while it is supposed to remain larger than Ψ ε,α ( t , q , t , q ) whichis the maximum of w − φ and does not therefore depend on ε and α .Now, let us denote by ( t, q ) ∈ ¯ B (( t , q ) , r ) an accumulation point of both (¯ t, ¯ q ) and (¯ s, ¯ p ) . Since Ψ ε,α (¯ t, ¯ q, ¯ s, ¯ p ) ≥ Ψ ε,α ( t , q , t , q ) = w ( t , q ) − φ ( t , q ) , we alsohave using the sign of − (cid:107) ¯ q − ¯ p (cid:107) ε − | ¯ t − ¯ s | α that u (¯ t, ¯ q ) − v (¯ s, ¯ p ) − φ (¯ t, ¯ q ) ≥ w ( t , q ) − φ ( t , q ) . Hence if ε and α tend to zero, w ( t, q ) − φ ( t, q ) ≥ w ( t , q ) − φ ( t , q ) , and the fact that ( t , q ) is the only point of ¯ B (( t , q ) , r ) where the maximum isattained concludes the proof. (2) Since (¯ t, ¯ q, ¯ s, ¯ q ) is in the set ¯ B (( t , q ) , r ) × ¯ B (( t , q ) , r ) , Ψ ε,α (¯ t, ¯ q, ¯ s, ¯ q ) ≤ Ψ ε,α (¯ t, ¯ q, ¯ s, ¯ p ) hence u (¯ t, ¯ q ) − v (¯ s, ¯ q ) − | ¯ t − ¯ s | α − φ (¯ t, ¯ q ) ≤ u (¯ t, ¯ q ) − v (¯ s, ¯ p ) − (cid:107) ¯ q − ¯ p (cid:107) ε − | ¯ t − ¯ s | α − φ (¯ t, ¯ q ) and since v is L -Lipschitz, (cid:107) ¯ q − ¯ p (cid:107) ε ≤ v (¯ s, ¯ q ) − v (¯ s, ¯ p ) ≤ L (cid:107) ¯ q − ¯ p (cid:107) . (cid:3) Now, since (¯ t, ¯ q, ¯ s, ¯ p ) converge to ( t , q , t , q ) , it is in B (( t , q ) , r ) × B (( t , q ) , r ) for ε and α small enough, and the fact that it maximizes Ψ ε,α tells us that: • (¯ t, ¯ q ) is a maximum point of ( t, q ) (cid:55)→ u ( t, q ) − Ç φ ( t, q ) + v (¯ s, ¯ p ) + (cid:107) q − ¯ p (cid:107) ε + | t − ¯ s | α å (cid:124) (cid:123)(cid:122) (cid:125) = φ ( t,q ) , and since u is a subsolution, the derivatives of φ satisfy ∂ t φ (¯ t, ¯ q ) + H (¯ t, ¯ q, ∂ q φ (¯ t, ¯ q )) ≤ , hence ∂ t φ (¯ t, ¯ q ) + 2 · ¯ t − ¯ sε + H Å ¯ t, ¯ q, ∂ q φ (¯ t, ¯ q ) + 2 · ¯ q − ¯ pε ã ≤ . Note also that since u is L -Lipschitz with respect to q , the q -derivative of φ ata point of maximum of u − φ is necessarily bounded by L , hence:(13) (cid:107) ∂ q φ (¯ t, ¯ q ) + 2 · ¯ q − ¯ pε (cid:107) ≤ L. • (¯ s, ¯ p ) is a minimum point of ( s, p ) (cid:55)→ v ( s, p ) − Ç u (¯ t, ¯ q ) − φ (¯ t, ¯ q ) − (cid:107) ¯ q − p (cid:107) ε − | ¯ t − s | α å (cid:124) (cid:123)(cid:122) (cid:125) = φ ( s,p ) , and since v is a supersolution, the derivatives of φ satisfy ∂ s φ (¯ s, ¯ p ) + H (¯ s, ¯ p, ∂ p φ (¯ s, ¯ p )) ≤ , hence · ¯ t − ¯ sε + H Å ¯ s, ¯ p, · ¯ q − ¯ pε ã ≥ . Combining the two previous points gives that ∂ t φ (¯ t, ¯ q ) ≤ H Å ¯ s, ¯ p, · ¯ q − ¯ pε ã − H Å ¯ t, ¯ q, ∂ q φ (¯ t, ¯ q ) + 2 · ¯ q − ¯ pε ã ≤ H Å ¯ s, ¯ p, · ¯ q − ¯ pε ã − H Å ¯ t, ¯ q, · ¯ q − ¯ pε ã + H Å ¯ t, ¯ q, · ¯ q − ¯ pε ã − H Å ¯ t, ¯ q, ∂ q φ (¯ t, ¯ q ) + 2 · ¯ q − ¯ pε ã (cid:124) (cid:123)(cid:122) (cid:125) . ≤ C (1 + 2 L ) (cid:107) ∂ q φ (¯ t, ¯ q ) (cid:107) Let us explain the last point: the estimate (13) and the second result of Lemma D.8state that both ∂ q φ (¯ t, ¯ q ) + 2 · ¯ q − ¯ pε and · ¯ q − ¯ pε are bounded by L . The assumption madeon (cid:107) ∂ p,q H (cid:107) implies that ∂ p H is bounded by C (1 + 2 L ) on the set [0 , T ] × R d × ¯ B (0 , L ) ,and hence (cid:12)(cid:12)(cid:12)(cid:12) H Å ¯ t, ¯ q, · ¯ q − ¯ pε ã − H Å ¯ t, ¯ q, ∂ q φ (¯ t, ¯ q ) + 2 · ¯ q − ¯ pε ã (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (1 + 2 L ) (cid:107) ∂ q φ (¯ t, ¯ q ) (cid:107) . Lemma D.8 implies that the quantity H Ä ¯ s, ¯ p, · ¯ q − ¯ pε ä − H Ä ¯ t, ¯ q, · ¯ q − ¯ pε ä tends to when ε and α tend to . To finish, since (¯ t, ¯ q ) tends to ( t , q ) : ∂ t φ ( t , q ) ≤ C (1 + 2 L ) (cid:107) ∂ q φ ( t , q ) (cid:107) . We then extend the subsolution property to { T } × R d with Lemma D.6. (cid:3) Proof of Proposition D.3.
Take
R > C (1 + 2 L ) T and let us denote by M the maximumof w on the set [0 , T ] × ¯ B (0 , R ) . We are going to prove that for all δ > such that R > δ + C (1 + 2 L ) T , w ( t, q ) ≤ δt on the set [0 , T ] × B (0 , R − C (1 + 2 L ) T − δ ) , using acomparison with an ad hoc smooth solution of ∂ t w − C (1 + 2 L ) (cid:107) ∂ q w (cid:107) = 0 .For such a δ > , it is possible to find a smooth and increasing function χ δ : R → R such that χ δ ( r ) = 0 if r ≤ R − δ and χ δ ( r ) = M if r ≥ R . Then φ δ : ( t, q ) (cid:55)→ χ δ ( (cid:107) q (cid:107) + C (1 + 2 L ) t ) is a smooth solution of ∂ t w − C (1 + 2 L ) (cid:107) ∂ q w (cid:107) = 0 on [0 , T ] × ¯ B (0 , R ) . Let us then showthat the function ( t, q ) (cid:55)→ w ( t, q ) − φ δ ( t, q ) − δt on [0 , T ] × ¯ B (0 , R ) is non positive.The maximum of this function cannot be attained at a point ( t, q ) of (0 , T ] × B (0 , R ) ,or else the fact that w is a subsolution on (0 , T ] × B (0 , R ) (Lemma D.7) gives that: ∂ t φ δ ( t, q ) + δ − C (1 + 2 L ) (cid:107) ∂ q φ δ ( t, q ) (cid:107) ≤ . Since φ δ solves the equation in the classical way and δ is positive, this is impossible.So, either the maximum is attained at a point (0 , q ) , or at a point ( t, q ) with (cid:107) q (cid:107) = R .In the first case, the maximum is of the form w (0 , q ) − φ δ ( (cid:107) q (cid:107) ) and is hence non positivesince u ≤ v on { } × R d and φ δ is non negative.In the second case, φ δ ( t, q ) = M and the maximum is of the form w ( t, q ) − M − δt .Since w is smaller than M on [0 , T ] × ¯ B (0 , R ) , the maximum is non positive.Hence, for each ( t, q ) in [0 , T ] × ¯ B (0 , R ) , w ( t, q ) ≤ φ δ ( t, q ) + δt. Since φ δ ( t, q ) is zero on [0 , T ] × B (0 , R − C (1 + 2 L ) T − δ ) , on this set we have: w ( t, q ) ≤ δt. Letting δ tend to zero gives that w = u − v ≤ on [0 , T ] × B (0 , R − C (1 + 2 L ) T ) . (cid:3) E. Graph selector
In this appendix we present the graph selector notion in the usual symplectic frame-work and its application to the variational resolution of the evolutive Hamilton-Jacobiequation. The graph selector can also be used to address other dynamical questions, see[PPS03], [Arn10] and [BdS12].E.1.
Graph selector.
Let us settle in a usual symplectic framework: we assume that M is a closed Riemannian d -manifold and look at its cotangent bundle π : T (cid:63) M → M . If q = ( q , · · · , q d ) are the coordinates of a chart on M , the dual coordinates p =( p , · · · , p d ) ∈ T (cid:63)q M are defined by p i ( e j ) = δ ij , where e j is the j th vector of the canonicalbasis and δ i,j is the Kronecker symbol. The manifold T (cid:63) M is endowed with the Liouville -form λ , which writes λ = pdq in this dual chart. The symplectic structure on T (cid:63) M isgiven by the symplectic form ω = dλ = dp ∧ dq in the dual chart.A submanifold L of T (cid:63) M is called Lagrangian if it is d -dimensional and if i (cid:63) L w = 0 ,where i L : L → T (cid:63) M is the inclusion. It is exact if i (cid:63) L λ is exact, i.e. if there exists asmooth function S : L → R such that dS = i (cid:63) L λ . Such a function is called a primitive of L ,and is uniquely determined up to the addition of a constant. If L is an exact Lagrangiansubmanifold, we call wavefront for L a set of the form W = { ( π ( x ) , S ( x )) , x ∈ L} for S a primitive of L , see Figure 1.If L is an exact Lagrangian submanifold and W is a wavefront for L , we call graphselector a Lipschitz function u whose graph is included in W . Since a possible primitive S of the Lagrangian submanifold is given by an underlying action, the existence of agraph selector can be deduced under reasonable hypotheses from the existence of actionselectors . These action selectors are obtained by using either generating family techniques(see [Cha91]), via Floer homology (see [Flo88] and [Oh97]) or lately by microlocal sheaftechniques (see [Gui12]). In [MO97], the link between the invariants constructed withgenerating families and via the Floer homology is studied, which leads to the conclusionthat they give the same graph selector under a suitable normalization (see also [MVZ12]).A graph selector provides simultaneously a continuous section of the wavefront and adiscontinuous section of the Lagrangian submanifold: Proposition E.1 (Graph selector) . Let L be an exact Lagrangian submanifold of T (cid:63) M such that π |L is proper, W be a wavefront for L , and u be a graph selector. Then ( q, du ( q )) ∈ L for almost every q . The author was unable to locate the proof of this statement in the literature, yet itis close to Proposition . in [PPS03] and to Proposition II in [OV94], which both dealwith the graph selector in terms of generating family. We present a proof improved byJ.-C. Sikorav. a continuous function with graph is included in W is automatically Lipschitz if L is uniformlybounded in the fiber variable. S W p L q Figure 1.
A Lagrangian submanifold and an associate wavefront. Thetwo greyed domains delimited by the position of the intersection in thewavefront have the same area.
Proof.
Let S : L → R be a primitive of L and u be a graph selector of the associatedwavefront. If x is in L , we will denote by p x ∈ T (cid:63)π ( x ) L the second coordinate of x =( π ( x ) , p x ) .We are going to prove that if q ∈ M is a regular value of π |L and a point of differentia-bility of u , ( q, du ( q )) is in L . Then combining Rademacher’s theorem (on u ) and Sard’stheorem (on π |L ) imply that the statement holds for almost every q .Let us fix such a point q . We denote by L q the fiber π − |L ( { q } ) , which is finite set since q is a regular value of the proper map π |L . We are going to prove that for all v in S d − ,there exists x = ( q, p ) ∈ L q such that du ( q ) .v = p.v .Let v ∈ S d − . We work in a local chart in the neighbourhood of q ∈ M : take asequence q n such that lim n →∞ q n − q (cid:107) q n − q (cid:107) = v . For all n , there exists x n in L q n such that u ( q n ) = S ( x n ) . Since π |L is proper, we may assume without loss of generality that x n admits a limit x in L . We again work in the local chart to write x n = x + x n − x , where x n − x is a sequence of T x L converging to zero. We have on one hand u ( q n ) − u ( q ) = du ( q )( q n − q ) + o ( (cid:107) q n − q (cid:107) ) = (cid:107) q n − q (cid:107) du ( q ) v + o ( (cid:107) q n − q (cid:107) ) and on the other hand u ( q n ) − u ( q ) = S ( x n ) − S ( x ) = dS ( x )( x n − x )+ o ( (cid:107) x n − x (cid:107) ) = p x dπ ( x )( x n − x )+ o ( (cid:107) x n − x (cid:107) ) . Now, since π ( x n ) = q n for each n , we have since dπ |L ( x ) is invertible dπ ( x )( x n − x ) = q n − q + o ( (cid:107) q n − q (cid:107) ) = (cid:107) q n − q (cid:107) v + o ( (cid:107) q n − q (cid:107) ) . Putting these three equations together we get (cid:107) q n − q (cid:107) du ( q ) v = (cid:107) q n − q (cid:107) p x v + o ( (cid:107) q n − q (cid:107) ) , and dividing by (cid:107) q n − q (cid:107) and letting n tend to + ∞ gives that du ( q ) .v = p x .v .Now we define E x = { v ∈ S d − | du ( q ) v = p x v } . The previous result implies that { E x } x ∈L q is a finite cover of S d − , hence { Vect ( E x ) } x ∈L q is a finite cover of R d madeof vector subspaces: one of them is hence the whole space R d , and the corresponding x ∈ L q hence satisfies du ( q ) = p x . (cid:3) E.2.
Application to the evolutive Hamilton-Jacobi equation.
We follow [Vit96]to explicit the link between the variational operator and the graph selector introducedin the previous paragraph for a C initial condition u . We define the autonomoussuspension of H by K ( t, s, q, p ) = s + H ( t, q, p ) on the cotangent T (cid:63) ( R × R d ) , identifiedwith T (cid:63) R × T (cid:63) R d , and denote by Φ its Hamiltonian flow. The Hamiltonian system for K writes ® ˙ t = 1 , ˙ q = ∂ p H ( t, q, p ) , ˙ s = − ∂ t H ( t, q, p ) , ˙ p = − ∂ q H ( t, q, p ) , hence t can be taken as the time variable.The submanifold Γ = { (0 , − H (0 , q , du ( q )) , q , du ( q )) , q ∈ R d } is contained in thelevel set K − ( { } ) , and since K is autonomous, it is constant along its trajectories, andas a consequence Φ t (Γ ) = ¶ ( t, − H ( t, φ t ( q , du ( q ))) , φ t ( q , du ( q ))) , q ∈ R d © . Wecall suspended geometric solution of the Cauchy problem the Lagrangian submanifold L = ∪ t ∈ R Φ t (Γ ) , and the following set is a wavefront for L : W = ®Ä t, q, u ( q ) + A t ( φ τ ( q , du ( q ))) ä (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t ∈ R , q ∈ R d , q ∈ R d ,Q t ( q , du ( q )) = q. ´ Proof of Proposition 1.5.
The axioms required to be a variational operator implies thatthe function u : ( t, q ) (cid:55)→ R t u ( q ) is a graph selector for L : it is Lipschitz, and thevariational property asks that its graph is contained in W . Also, Proposition E.1 statesthat for almost every ( t, q ) , ( t, ∂ t u ( t, q ) , q, ∂ q u ( t, q )) belongs to L ⊂ K − ( { } ) , whichproves Proposition 1.5. (cid:3) References [AGLM93] L. Alvarez, F. Guichard, P.-L. Lions, and J.-M. Morel. Axioms and fundamental equationsof image processing.
Arch. Rational Mech. Anal. , 123(3):199–257, 1993.[Arn10] M.-C. Arnaud. On a theorem due to Birkhoff.
Geom. Funct. Anal. , 20(6):1307–1316, 2010.[Bar94] G. Barles.
Solutions de viscosité des équations de Hamilton-Jacobi , volume 17 of
Mathéma-tiques & Applications (Berlin) [Mathematics & Applications] . Springer-Verlag, Paris, 1994.[BC11] O. Bernardi and F. Cardin. On C -variational solutions for Hamilton-Jacobi equations. Dis-crete Contin. Dyn. Syst. , 31(2):385–406, 2011.[BdS12] P. Bernard and J. O. dos Santos. A geometric definition of the Mañé-Mather set and atheorem of Marie-Claude Arnaud.
Math. Proc. Cambridge Philos. Soc. , 152(1):167–178, 2012. [Ber12] P. Bernard. The Lax-Oleinik semi-group: a Hamiltonian point of view. Proc. Roy. Soc.Edinburgh Sect. A , 142(6):1131–1177, 2012.[BS91] G. Barles and P. E. Souganidis. Convergence of approximation schemes for fully nonlinearsecond order equations.
Asymptotic Anal. , 4(3):271–283, 1991.[CC17] J. D. Castillo Colmenares. Geometric and viscosity solutions for the cauchy problem of firstorder. 2017. arXiv:1611.10293.[Cha84] M. Chaperon. Une idée du type “géodésiques brisées” pour les systèmes hamiltoniens.
C. R.Acad. Sci. Paris Sér. I Math. , 298(13):293–296, 1984.[Cha90] M. Chaperon.
Familles génératrices . 1990. Cours donné à l’école d’été Erasmus de Samos.[Cha91] M. Chaperon. Lois de conservation et géométrie symplectique.
C. R. Acad. Sci. Paris Sér. IMath. , 312(4):345–348, 1991.[Che75] A. Chenciner. Aspects géométriques de l’études des chocs dans les lois de conservation.
Problèmes d’évolution non linéaires, Séminaire de Nice , (15):1–37, 1975.[CIL92] M. G. Crandall, H. Ishii, and P.-L. Lions. User’s guide to viscosity solutions of second orderpartial differential equations.
Bull. Amer. Math. Soc. (N.S.) , 27(1):1–67, 1992.[CL83] M. G. Crandall and P.-L. Lions. Viscosity solutions of Hamilton-Jacobi equations.
Trans.Amer. Math. Soc. , 277(1):1–42, 1983.[CL87] M. G. Crandall and P.-L. Lions. Remarks on the existence and uniqueness of unboundedviscosity solutions of Hamilton-Jacobi equations.
Illinois J. Math. , 31(4):665–688, 1987.[CV08] F. Cardin and C. Viterbo. Commuting Hamiltonians and Hamilton-Jacobi multi-time equa-tions.
Duke Math. J. , 144(2):235–284, 2008.[Fat12] A. Fathi. Weak KAM from a PDE point of view: viscosity solutions of the Hamilton-Jacobiequation and Aubry set.
Proc. Roy. Soc. Edinburgh Sect. A , 142(6):1193–1236, 2012.[Flo88] A. Floer. Morse theory for Lagrangian intersections.
J. Differential Geom. , 28(3):513–547,1988.[FS06] W. H. Fleming and H. M. Soner.
Controlled Markov processes and viscosity solutions , vol-ume 25 of
Stochastic Modelling and Applied Probability . Springer, New York, second edition,2006.[Gui12] S. Guillermou. Quantization of conic Lagrangian submanifolds of cotangent bundles. 2012.arXiv:1212.5818.[Ish84] H. Ishii. Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations.
IndianaUniv. Math. J. , 33(5):721–748, 1984.[Jou91] T. Joukovskaia.
Singularités de Minimax et Solutions Faibles d’Équations aux Dérivées Par-tielles . 1991. Thèse de Doctorat, Université de Paris VII, Denis Diderot.[MO97] D. Milinković and Y.-G. Oh. Floer homology as the stable Morse homology.
J. Korean Math.Soc. , 34(4):1065–1087, 1997.[MVZ12] A. Monzner, N. Vichery, and F. Zapolsky. Partial quasimorphisms and quasistates on cotan-gent bundles, and symplectic homogenization.
J. Mod. Dyn. , 6(2):205–249, 2012.[Oh97] Y.-G. Oh. Symplectic topology as the geometry of action functional. I. Relative Floer theoryon the cotangent bundle.
J. Differential Geom. , 46(3):499–577, 1997.[OV94] A. Ottolenghi and C. Viterbo. Solutions généralisées pour l’équation de Hamilton-Jacobidans le cas d’évolution. ,1994.[PPS03] G. P. Paternain, L. Polterovich, and K. F. Siburg. Boundary rigidity for Lagrangian subman-ifolds, non-removable intersections, and Aubry-Mather theory.
Mosc. Math. J. , 3(2):593–619,745, 2003. Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday.[Sik86] J.-C. Sikorav. Sur les immersions lagrangiennes dans un fibré cotangent admettant une phasegénératrice globale.
C. R. Acad. Sci. Paris Sér. I Math. , 302(3):119–122, 1986.[Sik90] J.-C. Sikorav. Exposé au Séminaire de Géométrie et Analyse.
Université Paris-VII , 1990.[Sou85] P. E. Souganidis. Approximation schemes for viscosity solutions of Hamilton-Jacobi equa-tions.
J. Differential Equations , 59(1):1–43, 1985.[Thé99] D. Théret. A complete proof of Viterbo’s uniqueness theorem on generating functions. vol-ume 96, pages 249–266. 1999. [Vit92] C. Viterbo. Symplectic topology as the geometry of generating functions. volume 292, pages685–710. 1992.[Vit96] C. Viterbo. Solutions of Hamilton-Jacobi equations and symplectic geometry. Addendum to: Séminaire sur les Équations aux Dérivées Partielles. 1994–1995 [école Polytech., Palaiseau,1995; MR1362548 (96g:35001)]. In
Séminaire sur les Équations aux Dérivées Partielles, 1995–1996 , Sémin. Équ. Dériv. Partielles, page 8. École Polytech., Palaiseau, 1996.[Wei13a] Q. Wei.
Solutions de viscosité des équations de Hamilton-Jacobi et minmax itérés . 2013. Thèsede Doctorat, Université de Paris VII, Denis Diderot.[Wei13b] Q. Wei. Subtleties of the minimax selector.
Enseign. Math. , 59(3-4):209–224, 2013.[Wei14] Q. Wei. Viscosity solution of the Hamilton-Jacobi equation by a limiting minimax method.