A class of Hamilton-Jacobi equations with constraint: uniqueness and constructive approach
aa r X i v : . [ m a t h . A P ] M a y A class of Hamilton-Jacobi equations with constraint: uniqueness andconstructive approach
Sepideh Mirrahimi ∗ Jean-Michel Roquejoffre †† December 1, 2018
Abstract
We discuss a class of time-dependent Hamilton-Jacobi equations, where an unknown function oftime is intended to keep the maximum of the solution to the constant value 0. Our main result isthat the full problem has a unique viscosity solution, which is in fact classical. The motivation is aselection-mutation model which, in the limit of small diffusion, exhibits concentration on the zerolevel set of the solution of the Hamilton-Jacobi equation.Uniqueness is obtained by noticing that, as a consequence of the dynamic programming principle,the solution of the Hamilton-Jacobi equation is classical. It is then possible to write an ODE forthe maximum of the solution, and treat the full problem as a nonstandard Cauchy problem.
Key-Words:
Hamilton-Jacobi equation with constraint, uniqueness, constructive existence result,selection-mutation models
AMS Class. No:
The purpose of this paper is to discuss existence and uniqueness for the following problem, withunknowns ( I ( t ) , u ( t, x )): ∂ t u = |∇ u | + R ( x, I ) ( t > , x ∈ R d ) , max x u ( t, x ) = 0 I (0) = I > ,u (0 , x ) = u ( x ) , (1)where I > u is a concave, quadratic function. For a given continuous function I ( t ), u ( t, x )solves a Hamilton-Jacobi equation. The unknown I may be thought of as a sort of regulator, or a sortof Lagrange multiplier, to maintain the maximum of u equal to 0. The constraint on the maximumof u ( t, . ) makes the problem nonstandard. ∗ Institut de Math´ematiques de Toulouse; UMR 5219, Universit´e de Toulouse; CNRS, UPS IMT, F-31062 ToulouseCedex 9, France; E-mail: [email protected] † Institut de Math´ematiques de Toulouse; UMR 5219, Universit´e de Toulouse; CNRS, UPS IMT, F-31062 ToulouseCedex 9, France; E-mail: jean-michel.roquejoff[email protected] u, I ) to (1) is not new, the first result is due to Perthame and Barles [21] (seealso Barles-Perthame-Mirrahimi [3] for a result with weaker assumptions). An important improvementis given by Lorz, Perthame and the first author in [16]; they indeed notice that a concavity assumptionson R - that we also make here - entail regularity. This allows them to derive the dynamics of themaximum point of a solution u ( t, x ). See also [17]. Both types of results rely on a special viscousapproximation of (1) - see equation (4) below. Uniqueness, however, has remained an open problem,apart from a very particular case [21].The main goal of the paper is to prove the missing uniqueness property; a result that we had alreadyannounced in [19]. We also provide a constructive existence proof which was not available in the pre-vious existence results [21, 3, 16]. Two important consequences, that we will present in a forthcomingpaper [20] (see also [19]), will be the convergence of the underlying selection-mutation model in astronger sense than what is known, and asymptotic expansion of the viscous solution. The asymptoticexpansion, which allows to approximate the phenotypical distribution of the population when the mu-tation steps are small but nonzero, is particularly interesting in view of biological applications. Oneof the main ingredients will be regularity under suitable concavity assumptions on R and u , which isfar from being available in general. Instead of relying on viscous approximations we will prove theseresults directly for the equation u t = |∇ u | + R ( t, x ) . (2)This will allow a much easier treatment than in the usual viscosity sense.The uniqueness result will also be helpful to develop the so-called Hamilton-Jacobi approach (see forinstance Diekmann et al. [9], [21, 16] and Subsection 1.2) to study more complex models describingselection and mutations. For instance, our result would allow to generalize a result due to Perthameand the first author in [18] on a selection model with spatial structure, where the proof relies on theuniqueness of the solution to a corresponding Hamilton-Jacobi equation with constraint. Model (1) arises in the limit ε → ∂ t n ε − ε ∆ n ε = n ε ε R (cid:0) x, I ε ( t ) (cid:1) ( t > , x ∈ R d ) , I ε ( t ) = Z R d ψ ( x ) n ε ( t, x ) dx, (3)where n ε ( t, x ) is the density of a population characterized by a d -dimensional biological trait x . Thepopulation competes for a single resource, this is represented by I ε ( t ), where ψ is a given positivesmooth function. The term R ( x, I ) is the reproduction rate; it is, as can be expected, very negativefor large x and decreases as the competition increases. The Laplace term corresponds to the muta-tions. The small parameter ε is introduced to consider the long time dynamics of the population whenthe mutation steps are small. Such models can be derived from individual based stochastic processesin the limit of large populations (see Champagnat-Ferri`ere-M´el´eard [5, 6]). There is a large litera-ture on the models of population dynamics under selection and mutations. We refer the interestedreader, for instance to Geritz et al. [15] and Diekmann [8] for an approach based on the study ofthe stability of differential systems (the so-called adaptive dynamics approach), to Champagnat [4]for the study of stochastic individual based models, to Raoul [22] and Mirrahimi [17] for the study ofintegro-differnetial models.The Hopf-Cole transformation n ε = exp ( u ε /ε ) yields the equation ∂ t u ε = ε ∆ u ε + |∇ u ε | + R ( x, I ε ) (4)2hich, in the limit ε →
0, yields the equation for u . Furthermore, I ε being uniformly positiveand bounded in ε , the Hopf-Cole transformation leads to the constraint on u . One expects that n ε concentrates at the points where u is close to 0 and the function I ε appears, in the limit, as a sort ofLagrange multiplier. One has indeed n ε ( x, t ) −− ⇀ ε → n ( x, t ) = ρ ( t ) δ ( x − ¯ x ( t )) , weakly in the sense of measures , with u ( t, x ( t )) = max x u ( t, x ) = 0 , ρ ( t ) = I ( t ) ψ ( x ( t )) . This method to study (3) has been introduced in [9] and then developed in different contexts. See forinstance Perthame-Barles [21] (convergence to Hamilton-Jacobi dynamics for (4), Barles-Mirrahimi-Perthame [3] (the same type of result, but with nonlinear, nonlocal diffusion), Champagnat-Jabin[7] (nonlinear integro-differential model with several resources), [16] (convergence improvement byintroduction of the concavity assumptions). This approach has a lot to do with the ’approximationof geometric optics’ for reaction-diffusion equations of the Fisher-KPP type. See Freidlin [11, 12] forthe probabilistic approach, and Evans-Souganidis [10], Barles-Evans-Souganidis [2] for the viscositysolutions approach.
The assumptions we are stating below are in the same spirit (but slightly weaker) as in [16], where theauthors noticed that this set of assumptions allowed them to work with smooth solutions, thus goingquite far in the study of (3). We believe that the results that we will prove would certainly be false ifsome of those assumptions were removed. • Assumptions on R ( x, I ) . We choose R to be smooth, and we suppose that there is I M > x appropriately)max x ∈ R d R ( x, I M ) = 0 = R (0 , I M ) , (5) − K | x | ≤ R ( x, I ) ≤ K − K | x | , for 0 ≤ I ≤ I M , (6) − K ≤ D R ( x, I ) ≤ − K < , (7) − K ≤ ∂R∂I ≤ − K , (8) | ∂ R∂I∂x i ( x, I ) | + | ∂ R∂I∂x i ∂x j ( x, I ) | ≤ K , for 0 ≤ I ≤ I M , and i, j = 1 , , · · · , d, (9) k D R ( · , I ) k L ∞ ( R d ) ≤ K , for 0 ≤ I ≤ I M . (10) • Assumptions on u ( . ) and I . We assume the existence of positive constants L , L , L , L such that − L − L | x | ≤ u ( x ) ≤ L − L | x | , (11) − L ≤ D u ≤ − L . (12)Note that this implies | Du ( x ) | ≤ L (1 + | x | ) , (13)3or a large constant L >
0. We also need that, for a positive constant L , k D u k L ∞ ( R d ) ≤ L . (14)Finally we assume that max x u ( x ) = u ( x ) = 0 , R ( x , I ) = 0 . (15)Note that the monotony assumption (8) means that the growth rate decreases as the competition in-creases, which is natural from the modeling point of view. The concavity assumption is a technical one.In Section 2 we will study an unconstrained Hamilton-Jacobi equation where we replace R ( x, I ) by R ( t, x ). To prove our results on this unconstrained problem we assume same type of regularity andconcavity assumptions on R that we state below: • Assumptions on R ( t, x ) . We choose R to be smooth, and we suppose that − K | x | ≤ R ( t, x ) ≤ K − K | x | , for t ∈ R + , (16) − K ≤ D R ( t, x ) ≤ − K < , (17) k D R ( t, · ) k L ∞ ( R d ) ≤ K , for t ∈ R + . (18) Our first result concerns the unconstrained Hamilton-Jacobi equation ( u t = |∇ u | + R ( t, x ) ( t > , x ∈ R d ) ,u (0 , x ) = u ( x ) . (19)The assumptions on u and R are those stated in the preceding subsection. Theorem 1.1
Equation (19) has a unique viscosity solution u that is bounded from above. Moreover,it is a classical solution: u ∈ L ∞ l oc (cid:0) R + ; W , ∞ l oc ( R d ) (cid:1) ∩ W , ∞ l oc (cid:0) R + ; L ∞ l oc ( R d ) (cid:1) , − max(2 L , p K ) ≤ D u ≤− min(2 L , p K ) and k D u k L ∞ ([0 ,T ] × R d ) ≤ L ( T ) where L ( T ) is a positive constant depending on L , K , K , L and T . Let us point out that the assumption of boundedness from above is, most certainly, irrelevant. Howeverthe constraint in (1) implies that all the solutions that we consider are bounded from above. Thisextra condition is thus legitimate, and will keep the length of the preliminary work to a minimum.
Theorem 1.2
The Hamilton-Jacobi equation with constraint (1) has a unique solution ( u, I ) . More-over we have ( u, I ) ∈ L ∞ l oc (cid:0) R + ; W , ∞ l oc ( R d ) (cid:1) ∩ W , ∞ l oc (cid:0) R + ; L ∞ l oc ( R d ) (cid:1) × W , ∞ ( R ) . The paper is organised as follows. In Section 2 we prove the Cauchy Problem for (19). In Section3 we reduce (1) to a (nonstandard) differential system. Theorem 1.2 is proved in Section 4. Section 5is devoted to the study of a particular example. 4
The Cauchy problem
In this section, we prove Theorem 1.1.We will first prove that the only solution to (2) that is bounded from above is the solution u ( t, x ) ofthe dynamic programming principle u ( t, x ) = sup ( γ ( s ) ,s ) ∈ R d × [0 ,t ] γ ( t )= x n F ( γ ) : γ ∈ C ([0 , t ]; R d ) o , (20)with F ( γ ) := u ( γ (0)) + Z t (cid:18) − | ˙ γ | s ) + R ( s, γ ( s )) (cid:19) ds. Such a solution is a viscosity solution to (1). We will prove, in addition, that it is classical and satisfiesthe properties claimed by Theorem 1.1.
Uniqueness.
This step essentially consists in showing that a viscosity solution of (2) does not growtoo wildly, which will reduce the problem to the application of classical arguments. We have alreadyassumed boundedness from above, so let us show that u goes to −∞ at most in a quadratic fashion.Due to the assumptions (16) on R , we have ∂ t v ≥ , v ( t, x ) = u ( t, x ) + tK | x | in the viscosity sense, which implies that v is time-increasing - thus the needed estimate. Let us -although this is elementary - explain why: choose T > < s < t ≤ T such that the inequality v ( t, x ) ≥ v ( s, x ) does not hold. In other words there is x such that v ( s, x ) >v ( t, x ). For ε > w ε ( t, x ) := v ( t, x ) − v ( s, x ) + εT − t + | x − x | ε . For ε > w ε , called ( t ε , x ε ), such that t ε is boundedaway from s and T , and x ε → x as ε →
0. At that minimum point the viscosity inequality implies − ∂ t T − t ≥
0, a contradiction.So there is a large
C > − C (1 + t )(1 + | x | ) ≤ u ( t, x ) ≤ C. And so, by an easy adaptation of Chap. 2 of Barles [1], where a uniqueness result for solutions thatgrow at most exponentially fast is provided (see also [13, 14]), there is at most one viscosity solutionto (2) that is bounded from above.
Existence.
We may thus turn to (20). Let us suppose that ( γ n ) ≤ n , with γ n ∈ C ([0 , t ]; R d ) and γ n ( t ) = x , is such that F ( γ n ) → u ( t, x ) as n → ∞ . Since R and u are bounded from above, we obtainthat, for some constant C Z t | ˙ γ n | ( s ) ds < C. γ n ( t ) = x we deduce, modifying the constant C if necessary, that k γ n k W , [0 ,t ] < C. It follows that, there exists γ ∈ W , ([0 , t ]; R d ), such that as n → ∞ , γ n → γ strongly in C ([0 , t ]; R d )and weakly in W , ([0 , t ]; R d ). We deduce that, as n → ∞ , u ( γ n (0)) → u ( γ ) , Z t R ( s, γ n ( s )) ds → Z t R ( s, γ ( s )) ds, Z t | ˙ γ | ( s ) ds ≤ lim inf n →∞ Z t | ˙ γ n | ( s ) ds. We conclude that u ( t, x ) = u ( γ (0)) + Z t (cid:18) − | ˙ γ | s ) + R ( s, γ ( s )) (cid:19) ds. (21)We claim that such a trajectory is unique, which entails uniqueness for the Cauchy problem (19). Wenote indeed that such trajectory γ satisfies the following Euler-Lagrange equation ¨ γ ( s ) = − ∇ R ( s, γ ( s )) , ˙ γ (0) = − ∇ u ( γ (0)) ,γ ( t ) = x. (22)However, from the concavity assumptions on R and u we obtain that the above elliptic problem iscoercive and hence the solution γ is unique. Regularity . Let us denote by γ x ( t ) the unique solution of (22). The function ( t, x ) ( γ x ( t ) , ˙ γ x ( t ))belongs to L ∞ loc ( R + , W , ∞ loc ( R d )) because ∇ u and ∇ R ( t, . ) are in W , ∞ loc ( R d ). Now, we have u ( t, x ) = u ( γ x (0)) + F ( γ x ) , yielding (this is a classical computation): ∂ i u ( t, x ) = ∇ u ( γ x (0)) .∂ i γ x (0) + Z t (cid:18) − ˙ γ x ( s ) .∂ i ˙ γ x ( s )2 + ∇ R ( s, γ x ( s )) ∂ i γ x ( s ) (cid:19) ds. Integrating by parts and using the Euler-Lagrange equation (22), we get: ∇ u ( t, x ) = − ˙ γ x ( t )2 . (23)Thus u ∈ L ∞ loc ( R + , W , ∞ loc ( R d )) . Strict concavity.
We will prove that u ( t, x ) is uniformly strictly concave, namely that D u ≤ − λI in the sense of symmetric matrices, for λ = min (cid:0) L , √ K (cid:1) . To this end, we show that, for all σ ∈ [0 , x, y ) ∈ R d × R d : σu ( t, x ) + (1 − σ ) u ( t, y ) + λσ (1 − σ ) | x − y | ≤ u ( t, σx + (1 − σ ) y ) . (24)Let γ x and γ y be optimal trajectories, solving (22), with γ x ( t ) = x and γ y ( t ) = y . Note from thechoice of γ x and γ y that we have u ( t, x ) = u ( γ x (0)) + Z t (cid:18) − | ˙ γ x ( s ) | R ( s, γ x ( s )) (cid:19) ds, ( t, y ) = u ( γ y (0)) + Z t (cid:18) − | ˙ γ y ( s ) | R ( s, γ y ( s )) (cid:19) ds, and u ( t, σx + (1 − σ ) y ) ≥ u ( σγ x (0) + (1 − σ ) γ y (0))+ Z t (cid:18) − | σ ˙ γ x + (1 − σ ) ˙ γ y ( s ) | R ( s, σγ x ( s ) + (1 − σ ) γ y ( s )) (cid:19) ds. Furthermore, from the concavity assumptions on R and u we have σu ( t, x ) + (1 − σ ) u ( t, y ) + L σ (1 − σ ) | γ x (0) − γ y (0) | ≤ u ( t, σx + (1 − σ ) y ) , and σ Z t R ( s, γ x ( s )) ds + (1 − σ ) Z t R ( s, γ y ( s )) ds + K σ (1 − σ ) Z t | γ x ( s ) − γ y ( s ) | ds ≤ Z t R ( s, σγ x ( s ) + (1 − σ ) γ y ( s )) ds. Moreover, from the strict concavity of µ
7→ −| µ | , we obtain that σ Z t − | ˙ γ x ( s ) | ds + (1 − σ ) Z t − | ˙ γ y ( s ) | ds + σ (1 − σ ) Z t | ˙ γ x ( s ) − ˙ γ y ( s ) | ds ≤ Z t − | σ ˙ γ x + (1 − σ ) ˙ γ y ( s ) | ds. We deduce that u ( t, σx + (1 − σ ) y )) ≥ σu ( t, x ) + (1 − σ ) u ( t, y )+ σ (1 − σ ) (cid:18)Z t ( 14 | ˙ γ x ( s ) − ˙ γ y ( s ) | + K | γ x ( s ) − γ y ( s ) | ) ds + L | γ x (0) − γ y (0) | (cid:19) (25)Next we have p K Z t dds | γ x ( s ) − γ y ( s ) | ds ≤ K Z t | γ x ( s ) − γ y ( s ) | ds + Z t | ˙ γ x − ˙ γ y | s ) ds. Writing | x − y | = | γ x ( t ) − γ y ( t ) | = | γ x (0) − γ y (0) | + Z t dds | γ x ( t ) − γ y ( t ) | ds we find | x − y | ≤ | γ x (0) − γ y (0) | + 2 q K Z t | γ x ( s ) − γ y ( s ) | ds + 12 p K Z t | ˙ γ x − ˙ γ y | ( s ) ds. Combing the above line with (25), we obtain (24) for λ = min (cid:0) L , p K (cid:1) . Bounds on u and ∇ u . The first thing to notice is a bound for ( γ x , ˙ γ x ). Indeed, the coercivity for(22), as well as the fact that x
7→ ∇ R ( t, x ) grows linearly with a constant only depending on t , impliesthe existence of a locally bounded constant K ( t ) such that | ( γ x ( s ) , ˙ γ x ( s )) | ≤ K ( t )(1 + | x | ) , for all s ∈ [0 , t ] . (26)7his implies, modifying the constant K ( t ) if necessary, thanks to (21): | u ( t, x ) | ≤ K ( t )(1 + | x | ) . Moreover, from (23), we have |∇ u ( t, x ) | ≤ K ( t )(1 + | x | ) . (27) Semi-convexity.
Because u is three times differentiable in x , then ∂ t u is locally W , ∞ in x andequation (19) may be differentiated twice with respect to x . So, let e be any unit vector, we have ∂ t ( ∂ ee u ) = 2 |∇ ∂ e u | + 2 ∇ u. ∇ ( ∂ ee u ) + ∂ ee R ( t, x ) ≥ | ∂ ee u | + 2 ∇ u. ∇ ( ∂ ee u ) + ∂ ee R ( t, x ) . Because of (27), the curve t γ x ( t ) becomes a characteristic curve for the equation ∂ t v = 2 v + 2 ∇ u ( t, x ) . ∇ v ( t, x ) + ∂ ee R ( t, x ) . Moreover, along this characteristics, we find ddt v ( t, γ x ( t )) = 2 v ( t, γ x ( t )) + ∂ ee R ( t, γ x ( t )) . We deduce, thanks to (12) and (17), that ∂ ee u ≥ − max(2 L , p K ) . Bounds on D u . As for the third derivative, we set v ( t, x ) = D u ( t, x ), the equation for v - that wemay obtain using differential quotients - is ∂ t v − ∇ v. ∇ u = S ( t, x, v ) := 6 v.D u + D R, where ∇ v denotes the column of tensors ( ∂ v, ..., ∂ d v ) and v.D u denotes the column of matrices( ∂ D u.D u, ..., ∂ d D u.D u ). This is a linear equation with (thanks to the bound on D u ) boundedcoefficients. Thus, local boundedness of k v ( t, . ) k L ∞ ( R d ) holds, and this concludes the proof of Theorem1.1. I The idea is to change the constrained problem (1) by the following slightly nonstandard differentialsystem: R ( x ( t ) , I ( t )) = 0 , for t ∈ R + , ˙ x ( t ) = (cid:0) − D u (cid:0) t, ¯ x ( t ) (cid:1)(cid:1) − ∇ R (cid:0) ¯ x ( t ) , I ( t ) (cid:1) , for t ∈ R + ,∂ t u = |∇ u | + R ( x, I ) , in R + × R d , (28)with initial conditions I (0) = I , u (0 , · ) = u ( · ) , x (0) = x , such that max x u ( x ) = u ( x ) = 0 and R ( x , I ) = 0. (29)Note that (28) is really a differential system because the assumptions on R imply that I ( t ) canimplicitely be expressed in terms of ¯ x ( t ). And it is slightly nonstandard because ¯ x solves an ODEwhose nonlinearity depends on u . The precise statement is the following8 heorem 3.1 Solving the constrained problem (1) is equivalent to solving the initial value ODE-PDEproblem (28) - (29) . Proof.
Let ( u, I ) be a solution of (1) with initial datum ( u , I ), the function I being continuous,and u a solution of the Hamilton-Jacobi equation in the sense of (20). Theorem 1.1 is applicable, andyields a solution u ( t, x ) which has at least three locally bounded spatial derivatives, locally uniformalyin time. Moreover, the D u is bounded uniformly in time and in x , and finally the function u ( t, . ) isstrictly concave. This allows a lot. • There is, at each time, a unique x ( t ) maximising u ( t, . ) over R d . Thus the trivial identity ∇ u ( t, x ( t )) = 0 (30)can (use differential quotients) be differentiated with respect to t , to yield that (i) x ( t ) is locally W , ∞ and (ii) the (a priori less trivial) identity ∂ t ( ∇ u )( t, x ( t )) + D u ( t, x ( t )) . dxdt ( t ) = 0 . (31) • The function u ( t, x ) has enough regularity so that we may take the gradient of (1) with respectto x , and evaluate the result at x = x ( t ). Because of (30) we have D u ( t, x ( t )) . ∇ u ( t, x ( t )) = 0and, because of (31), we have − D u ( t, x ( t )) . dxdt ( t ) = ∇ R ( x ( t ) , I ( t )) . (32) • The last item to take into account is the constraint u ( t, x ( t )) = 0 , which we may (still with the use of differential quotients) differentiate with respect to time, inorder to yield ∂ t u ( t, x ( t )) + ∇ u ( t, x ( t )) . dxdt ( t ) = 0 , thus entailing ∂ t u ( t, x ( t )) = 0 . This yields, from (1), R ( x ( t ) , I ( t )) = 0 . (33)Gathering (33), (1) and (32) shows that the constrained problem implies (28).We also prove that regularity plus (28) easily implies (1). Let ( u, I ) solve (28). The first line of (1)derives immediately. To prove the second line, note that, thanks to Theorem 1.1 and the third lineof (28) we deduce that u is strictly concave. It has hence, for all t ∈ R + , a unique strict maximumpoint, in the variable x , that we denote y ( t ). Following similar arguments as above, we obtain that ( ˙ y ( t ) = ( − D u ( t, y ( t )) − ∇ R ( y ( t ) , I ( t )) , for t ∈ R + , y (0) = x . x ( t ) = y ( t ), for all t ∈ R + . Finally,evaluating the third line of (28) at x ( t ) and using the first line of (28), we obtain that ∂ t u ( t, x ( t )) = 0.This equality together with ∇ u ( t, x ( t )) = 0 and (29) implies thatmax x u ( t, x ) = u ( t, x ( t )) = 0 . and the proof of Theorem 3.1 is complete. (cid:3) Remark.
What we have done here is nothing else than the derivation of the equation for x carried outin [9]. In particular, the equilibrium property R ( x, I ) = 0 would hold in a more general setting thanhere. The new point here is that we establish, in a mathematically rigorous fashion, the equivalencebetween the initial problem (1) and the coupled system (28) , and this equivalence holds because of allthe differentiations with respect to x and t that we are allowed to make. We fix
T >
0. To prove that (1) has a unique solution ( u, I ) in [0 , T ] × R d , it is enough to prove thatthere exists a unique solution to (28)–(29).We prove this using the Banach fixed point Theorem in a small interval and then iterate. Φ and its domain First, we define Ω = { x | R ( x, > } , Ω = { x | R ( x, I ) ≥ } , and A = n x ( · ) ∈ C (cid:16) [0 , δ ]; B (cid:0) x , r δ (cid:1)(cid:17) (cid:12)(cid:12)(cid:12) x (0) = x o , where B ( z, r ) is the ball of radius r centered at z , and δ is a positive constant such that δ < min( µ, c ( T )) , with µ = min(2 L , p K ) C M d (Ω , Ω) , (34)where d ( A, B ) is the distance between the sets A and B . The constant r δ is given by r δ = C M δ min(2 L , p K ) , the constant C M is chosen such that |∇ R ( x, I ) | ≤ C M , in Ω × [0 , I M ] , (35)and c ( T ), a constant depending only on T , will be chosen later. Note that, by the choice of δ andsince x ∈ Ω , we obtain that B ( x , r δ ) ⊂ Ω . Our theorem will be proved by the introduction of a mappingΦ :
A → A , Φ( x ) = y. We will prove the following theorem. 10 heorem 4.1
The mapping Φ is a strict contraction from A into itself. To define Φ, we first need to introduce some other mappings. Let x ( · ) ∈ A . We define I : A →
C ([0 , δ ]; [0 , I M ]) such that R ( x ( t ) , I [ x ]( t )) = 0 . From (5), (8) and since x ( t ) ∈ Ω for all t ∈ [0 , δ ], it follows that I can be defined in a unique way.Next, we define the following domain B = n v ∈ L ∞ (cid:0) [0 , δ ]; W , ∞ l oc ( R d ) (cid:1) ∩ W , ∞ (cid:0) [0 , δ ]; L ∞ l oc ( R d ) (cid:1) |− max(2 L , p K ) ≤ D v ≤ − min(2 L , p K ) , k D v k L ∞ ([0 ,δ ] × R d ) ≤ L ( T ) o , and the following mapping ( V : C ([0 , δ ]; [0 , I M ]) → B V ( I ) = v, where v solves ( ∂ t v = |∇ v | + R ( x, I ) , in [0 , δ ] × R d ,v (0 , x ) = u ( x ) , in R d . (36)It follows from Theorem 1.1 that the above mapping is well-defined.Finally, we introduce a last mapping: ( F : C ([0 , δ ]; [0 , I M ]) × A × B → A ,F ( I, x, v ) = y, where y ∈ A solves ( ˙ y ( t ) = (cid:0) − D v ( t, x ( t ) (cid:1) − ∇ R ( x ( t ) , I ( t )) , in [0 , δ ], y (0) = x . To prove that F is well-defined, we must verify that y ( t ) remains in B ( x , r δ ). We note that, since v ∈ B , we have 0 < L , p K ) ≤ (cid:0) − D v ( t, x ( t )) (cid:1) − ≤ L , p K ) . We deduce, thanks to (35), that y ( t ) ∈ B ( x , r δ ) . We are now ready to define mapping Φ: ( Φ :
A → A , Φ( x ) = F (cid:0) I ( x ) , x, V ( I ( x )) (cid:1) . It follows from the above arguments that Φ is well-defined.11 .2 Proof of Theorem 1.2
In this section we will explain why uniqueness to (28) holds. One main technical lemma will be stated,its proof will be postponed to a special section.Let us prove that Φ is a contraction for c ( T ) (and hence δ ) small enough. To this end, we first provethat I is Lipschitz: |I ( x ) − I ( x ) | ≤ C k x − x k L ∞ ([0 ,δ ]) , for all x , x ∈ C (cid:16) [0 , δ ]; B (cid:0) x , r δ (cid:1)(cid:17) . (37)We have indeed R ( x , I ( x )) = R ( x , I ( x )) = 0 . It follows that R ( x , I ( x )) − R ( x , I ( x )) = R ( x , I ( x )) − R ( x , I ( x )) , and thus ∇ R ( cx + (1 − c ) x , I ( x )) · ( x − x ) = ∂R∂I ( x , J )( I ( x ) − I ( x )) , with c ∈ (0 ,
1) and J ∈ ( I ( x ) , I ( x )). Finally, using (8), (35) and the fact that x ( t ) , x ( t ) ∈ Ω forall t ∈ [0 , δ ] we obtain (37) with C = C M K .Next, we have that V : C ([0 , δ ]; [0 , I M ]) → B is also Lipschitz. Lemma 4.2
Let I , I ∈ C ([0 , δ ]; [0 , I M ]) . Then k V ( I ) − V ( I ) k W , ∞ ([0 ,δ ] × R d ) ≤ C k I − I k L ∞ ([0 ,δ ]) δ. (38)This is a nontrivial lemma, whose proof will be given in the next section.From (7) and (9), and since x , x ∈ A and V , V ∈ B , we deduce that F : C ([0 , δ ]; [0 , I M ]) ×A×B →A is Lipschitz with respect to all the variables with Lipschitz constant Cδ : k F ( I , x , V ) − F ( I , x , V ) k L ∞ ([0 ,δ ]) ≤ Cδ h k x − x k L ∞ ([0 ,δ ]) + k I − I k L ∞ ([0 ,δ ]) + k V − V k W , ∞ ([0 ,δ ] × R d ) i . (39)Finally, we conclude from (37), (38) and (39) that Φ : A → A is a Lipschitz mapping with a Lipschitzconstant Cδ : k Φ( x ) − Φ( x ) k L ∞ ([0 ,δ ]) ≤ Cδ k x − x k L ∞ ([0 ,δ ]) . Choosing c ( T ) (and hence δ ) small enough, we deduce that Φ is a contraction.We deduce from the Banach fixed point Theorem that Φ has a unique fixed point and consequently(28)–(29) has a unique solution for t ∈ [0 , δ ].To prove that (28)–(29) has a unique solution in [0 , T ] we iterate the above procedure K = ⌈ Tδ ⌉ times. Let 2 ≤ i ≤ K and ( x, I, u ) be the unique solution of (28)-(29) for t ∈ [0 , ( i − δ ]. Then, atthe i -th step, we consider the same mapping Φ but as the initial data we choose e x = x (( i − δ ) , e u ( · ) = u (( i − δ, · ) , e I = I (( i − δ ) . (40)We claim that these initial conditions satisfy e x ∈ Ω , e u ∈ W , ∞ l oc ( R d ) , with − max(2 L , p K ) ≤ D e u ≤ − min(2 L , p K ) k D e u k L ∞ ( R d ) ≤ L (( i − δ ) , max x e u ( x ) = e x , R ( e x , e I ) = 0 . (41)12 emma 4.3 Let ( x, I, u ) be the unique solution of (28) - (29) in [0 , τ ] . Then, I ( t ) is increasing withrespect to t in [0 , τ ] and x ( t ) ∈ Ω , u ( t, · ) ∈ W , ∞ l oc ( R d ) , with − max(2 L , p K ) ≤ D u ( t, x ) ≤ − min(2 L , p K ) , k D u k L ∞ ([0 ,τ ] × R d ) ≤ L ( τ ) , max x u ( t, x ) = x ( t ) , R ( x ( t ) , I ( t )) = 0 , , for all t ∈ [0 , τ ] and x ∈ R d . Proof . The regularity estimates on u are immediate from Section 2. The two last claims also followimmediately from (28)-(29) and the arguments in Section 3. We only prove that I ( t ) is increasingwith respect to t and x ( t ) ∈ Ω .Differentiating the first line of (28) with respect to t we obtain ∇ R ( x ( t ) , I ( t )) · ˙ x ( t ) + ∂∂I R ( x ( t ) , I ( t )) ˙ I ( t ) = 0 . Moreover, multiplying the second line of (28) by ∇ R we obtain ∇ R ( x ( t ) , I ( t )) · ˙ x ( t ) = ∇ R ( x ( t ) , I ( t )) (cid:0) − D u ( t, x ( t ) (cid:1) − ∇ R ( x ( t ) , I ( t )) ≥ . Combining the above lines we obtain ∂∂I R ( x ( t ) , I ( t )) ˙ I ( t ) ≤ , for t ∈ [0 , τ ] , and hence, thanks to (8) we deduce ˙ I ( t ) ≥ , for t ∈ [0 , τ ] . Therefore, I ( t ) is increasing with respect to t and in particular I ( t ) ≥ I , for t ∈ [0 , τ ] . Consequently, from the first line of (28) and (8), we obtain that R ( x ( t ) , I ) > , for t ∈ [0 , τ ] . It follows that x ( t ) ∈ Ω , for t ∈ [0 , τ ] . which concludes the proof.It is then immediate that the initial data given by (40) verify (41). One can verify that the aboveconditions are the only properties that we have used to prove that Φ is well-defined and a contraction.Therefore, one can apply again the Banach fixed point Theorem and deduce that there exists a uniquesolution of (28)-(29) for t ∈ [( i − δ, iδ ]. (i) We first prove that k V ( I ) − V ( I ) k L ∞ ([0 ,δ ] × R d ) ≤ C k I − I k L ∞ ([0 ,δ ]) δ. (42)13et v = V ( I ) and v = V ( I ), and r = v − v . From (36) we obtain ( ∂ t r = ( ∇ v + ∇ v ) · ∇ r + R ( x, I ) − R ( x, I ) , in [0 , δ ] × R d r (0 , x ) = 0 , for all x ∈ R d . (43)Note that the above equation has a unique classical solution which can be computed by the methodof characteristics. The characteristics verify˙ γ ( t ) = −∇ v ( t, γ ) − ∇ v ( t, γ ) . (44)For any ( t , x ) ∈ [0 , δ ] × R d , there exists a unique characteristic curve γ which verifies γ ( t ) = x .Moreover, this characteristic curve is defined in [0 , t ].The local existence and uniqueness of the characteristic curve γ is derived from the Cauchy-LipschitzTheorem and the fact that ∇ v i is Lipschitz with respect to x , for i = 1 ,
2. The latter property derivesfrom the fact that 2 L ≤ D v i ≤ − L , for i = 1 , γ is defined in [0 , t ] we must prove that γ remains bounded inthis interval. However, this property follows using (44) and the fact that, since v , v ∈ B , |∇ v i ( γ ) | ≤ C | γ | + C , for i = 1 , . We are now ready to prove (38). To this end, we multiply (44) by ∇ r ( t, γ ) to obtain ∇ r ( t, γ ) · ˙ γ ( t ) = − ( ∇ v ( t, γ ) + ∇ v ( t, γ )) · ∇ r ( t, γ ) . Combining this with (43) we deduce that ∇ r ( t, γ ( t )) · ˙ γ ( t ) + ∂ t r ( t, γ ( t )) = R ( x, I ) − R ( x, I ) . We integrate this between 0 and t to find r ( t , γ ( t )) = Z t R ( x, I ( τ )) − R ( x, I ( τ )) dτ. It follows, using (8), that | r ( t , γ ( t )) | ≤ K | I − I | t , and hence (42).(ii) Next, we prove that k∇ r k L ∞ ([0 ,δ ] × R d ) ≤ C k I − I k L ∞ ([0 ,δ ]) δ. (45)We differentiate (43) in the direction e i , for i = 1 , · · · , d , to obtain ∂ t r ( i ) = (cid:0) ∇ v , ( i ) + ∇ v , ( i ) (cid:1) · ∇ r + ( ∇ v + ∇ v ) · ∇ r ( i ) + R ( i ) ( x, I ) − R ( i ) ( x, I ) , with the notation f ( i ) = ∇ f · e i . We multiply by r ( i ) , sum over i and divide by |∇ r | to obtain ∂ t |∇ r | ≤ ( ∇ v + ∇ v ) · ∇|∇ r | + n X i =1 (cid:12)(cid:12) ∇ v , ( i ) + ∇ v , ( i ) (cid:12)(cid:12) | r ( i ) | + |∇ R ( x, I ) − ∇ R ( x, I ) | . D v j , for j = 1 ,
2, is bounded, and using (9) we deduce that ∂ t |∇ r | ≤ ( ∇ v + ∇ v ) · ∇|∇ r | + C |∇ r | + K | I − I | . The characteristic curves corresponding to the above equation verify again (44). We multiply (44) by ∇|∇ r | ( t, γ ( t )) to obtain ∇|∇ r | ( t, γ ) · ˙ γ ( t ) = − ( ∇ v ( t, γ ) + ∇ v ( t, γ )) · ∇|∇ r | ( t, γ ) . Combining the above equations we obtain ddt |∇ r | ( t, γ ( t )) ≤ C |∇ r | ( t, γ ( t )) + K k I − I k L ∞ ([0 ,δ ]) . It follows that |∇ r ( t, x ) | ≤ (cid:0) e Ct − (cid:1) K C k I − I k L ∞ ([0 ,δ ]) . Hence (45), modifying the constant C if necessary.(iii) Finally we prove that k D r k L ∞ ([0 ,δ ] × R d ) ≤ C k I − I k L ∞ ([0 ,δ ]) δ. (46)Note that at every point ( t, x ) ∈ R + × R d , we can write mixed derivatives of the form r ξη in terms ofpure derivatives: r ξη = 12 ( ∂ ξ + η,ξ + η r − ∂ ξξ r − ∂ ηη r ) . (47)This implies the existence of ξ on the unit sphere of R d such that k D r k L ∞ ([0 ,δ ] × R d ) ≤ k r ξξ k L ∞ ([0 ,δ ] × R d ) . We differentiate (43) twice in the direction of ξ and obtain ∂ t r ξξ = ( ∇ v ,ξξ + ∇ v ,ξξ ) · ∇ r + 2 ( ∇ v ,ξ + ∇ v ,ξ ) · ∇ r ξ + ( ∇ v + ∇ v ) · ∇ r ξξ + R ξξ ( x, I ) − R ξξ ( x, I ) . Using the above arguments, the fact that D v i and D v i are bounded, (9) and (45) we deduce that ∂ t | r ξξ | ≤ C k I − I k L ∞ ([0 ,δ ]) δ + C k r ξξ k L ∞ ([0 ,δ ] × R d ) + ( ∇ v + ∇ v ) · ∇| r ξξ | + K | I − I | . Next we use the characteristic curves as previously. We multiply (44) by ∇| r ξξ ( t, γ ( t )) | and obtain ∇| r ξξ ( t, γ ( t )) | · γ ( t ) = − ( ∇ v ( t, γ ) + ∇ v ( t, γ )) · ∇| r ξξ ( t, γ ) | . We combine the above equations to obtain ddt | r ξξ ( t, γ ( t )) | ≤ C k I − I k L ∞ ([0 ,δ ]) + C k r ξξ k L ∞ ([0 ,δ ] × R d ) . We conclude that | r ξξ ( t, x ) | ≤ Ct ( k I − I k L ∞ ([0 ,δ ]) + k r ξξ k L ∞ ([0 ,δ ] × R d ) ) . Restricting c ( T ) (and hence δ ) if necessary we get k r ξξ k L ∞ ([0 ,δ ] × R d ) ≤ Cδ − Cδ k I − I k L ∞ ([0 ,δ ]) , hence (46). 15 An example with quadratic u and R Let us study the (instructive) example of a quadratic equation (1). In other words we choose (cid:26) u ( x ) = − A x · x,R ( x, I ) = − A x · x + b · x + I − I, where A and A are positive definite matrices and b ∈ R d . The Euler-Lagrange equation for thedynamic programming principle writes (cid:26) − ¨ γ + 2 A γ = 2 b, − ˙ γ (0) + 2 A γ (0) = 0 , γ ( t ) = x. (48)In order to translate the equivalent system (28), we need ∇ R ( I, x ) and D u ( t, x ). The first quantityis easily obtained: ∇ R ( I, x ) = − A x + b. We then solve (48), differentiate with respect to x and denote by Γ the differential of γ - the solutionof (48) with respect to x , to obtain (after an elementary but tedious computation)Γ( s ) = e s √ A B ( s ) B ( t ) − e − t √ A ,B ( s ) = I d + e − s √ A ( √ A + 2 A ) − ( √ A − A ) . (49)Notice that the result was expected because γ is linear function of x , and that B ( s ) is invertible for0 ≤ s ≤ t , and the norms of B and B − are bounded uniformly in s and t . And so we have D u ( t, x ) = − A Γ(0) − Z t (cid:18) ˙Γ( s ) A Γ( s ) . Γ( s ) (cid:19) ds := − C ( t ) , (50)and the matrix C ( t ) is bounded away from 0 and + ∞ , uniformly with respect to t . This, by the way,is not easily seen on the formula (50); the proof of Theorem 1.1 is still what one should use here. Theequation for x ( t ) and I ( t ) is thus straightforward: ( ˙ x ( t ) = C ( t ) − ( − A x ( t ) + b ) ,I ( t ) = I − A x ( t ) · x ( t ) + b · x ( t ) . Thus, existence and uniqueness of ( x, I ) is straightforward. For large t , examination of (50) and (49)yields D u ( t, x ) ∼ t → + ∞ − r A , and lim t → + ∞ x ( t ) = A − b, lim t → + ∞ I ( t ) = I + 12 A − b · b. This is consistent with the known behaviour of I ( t ) and x ( t ), as well as a closed form of the competitionincrease I ( t ) − I . 16 cknowledgements The authors thank G. Barles and B. Perthame for interesting comments on an earlier version of thispaper, which had a great influence on the present form of this work. They also thank M. Bardi forpointing out the possible tractability of the quadratic case. S. Mirrahimi was partially funded by theANR projects KIBORD ANR-13-BS01-0004 and MODEVOL ANR-13-JS01-0009. J.-M. Roquejoffrewas supported by the European Union’s Seventh Framework Programme (FP/2007-2013) / ERCGrant Agreement n. 321186 - ReaDi - “Reaction-Diffusion Equations, Propagation and Modelling”held by Henri Berestycki, as well as the ANR project NONLOCAL ANR-14-CE25-0013. Both authorsthank the Labex CIMI for a PDE-probability winter quarter in Toulouse, which provided a stimulatingscientific environment to this project.
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