Classical solutions to local first order Extended Mean Field Games
aa r X i v : . [ m a t h . A P ] F e b Classical solutions to local first order Extended MeanField Games
Sebastian Muñoz
Abstract
We study the existence of classical solutions to a broad class of local, first order, forward-backwardExtended Mean Field Games systems, that includes standard Mean Field Games, Mean Field Games withcongestion, and mean field type control problems. We work with a strictly monotone cost that may befully coupled with the Hamiltonian, which is assumed to have superlinear growth. Following previous workon the standard first order Mean Field Games system, we prove the existence of smooth solutions undera coercivity condition that ensures a positive density of players, assuming a strict form of the uniquenesscondition for the system. Our work relies on transforming the problem into a partial differential equationwith oblique boundary conditions, which is elliptic precisely under the uniqueness condition.
MSC: 35Q89, 35B65, 35J66, 35J70.Keywords: quasilinear elliptic equations; oblique derivative problems; Bernstein method; non-linear method of continu-ity; Hamilton-Jacobi equations.
Contents
References 15
In this paper, we prove existence of classical solutions to a broad class of first order Mean Field Games systems(MFG for short) with local coupling, which includes standard MFG, MFG with congestion, and mean fieldtype control problems. For this purpose, we study the MFG system: − u t + H ( x, D x u, m ) = 0 ( x, t ) ∈ Q T = T d × (0 , T ) ,m t − div ( B ( x, D x u, m )) = 0 ( x, t ) ∈ Q T ,m (0 , x ) = m ( x ) , u ( x, T ) = g ( x, m ( x, T )) x ∈ T d , (EMFG)where − H ( x, p, m ) : T d × R d × (0 , ∞ ) → R and g ( x, m ) : T d × (0 , ∞ ) → R are strictly increasing in m , and m : T d → R is a positive probability density.MFG were introduced by Lasry and Lions [16, 17], and at the same time, in a particular setting, by Caines,Huang, and Malhamé [4]. They are non-cooperative differential games with infinitely many players, in whichthe players find an optimal strategy by observing the distribution of the others.1he system (EMFG) was introduced by Lions and Souganidis in [18], who coined the term Extended MFG,to simultaneously study several MFG type problems for which, in contrast with the case of standard MFG,the vector field B does not equal mD p H . It was shown in [18] that (EMFG) has at most one classical solutionif − H m D p B > ( B m − D p H ) ⊗ ( B m − D p H ) . (1.1)In the case of standard MFG with a separated Hamiltonian, H ≡ H ( x, p ) − f ( x, m ) , (1.1) simply reduces tothe standard convexity assumption for H ( x, p ) in the second variable and the monotonicity of f .Before stating our results, we briefly describe some of the existing work on the well-posedness and regularityof (EMFG). For the case of standard MFG with a separated Hamiltonian, there exists a complete theory of weaksolutions (developed by Cardaliaguet, Graber, Porretta, and Tonon [5, 6, 7] in the degenerate case g m ≡ ,that is, when g is independent of m , and by the author [19] in the non-degenerate case considered here).Moreover, the solutions are known to be classical under the coercivity assumption lim m → + f ( x, m ) = −∞ (see [19]). For first order MFG systems with congestion, weak solutions were shown to exist by A. Porrettaand Y. Achdou [2], but classical solutions had not been obtained so far. Second order MFG systems withcongestion were also studied in [1, 8, 9, 10], where weak solutions and short-time existence result in the smoothsetting have been obtained. Finally, for second order mean field type control problems with congestion, weaksolutions were obtained in [3].In this paper, we follow the same methodology used in [19], and assume the aforementioned coercivitycondition, which is critical to our methods as it ensures the strict positivity of the density. Our contribution isstated below. It is a general result that yields classical solutions to (EMFG), assuming the strict form of theuniqueness condition (1.1), as well as growth assumptions on H and B which are modeled by Hamiltoniansof the type H ∼ ψ ( m ) | p | γ , γ > , ψ > , ψ ′ ≤ . (1.2)We remark that, in particular, we do not require the Hamiltonian to be quadratic or separated, and one of themain new applications of Theorem 1.1 is the existence of classical solutions to MFG systems with congestion.We refer to Section 2 for the exact assumptions (M), (H), (B) , (G), and (E). Theorem 1.1.
Let < s < , and assume that (M), (H), (B) , (G), and (E) hold. Then there exists a uniqueclassical solution ( u, m ) ∈ C ,s ( Q T ) × C ,s ( Q T ) to (EMFG). An important natural setting, covered by Theorem 1.1 in full generality, is when the derivatives H m and D x H satisfy growth conditions that are compatible with (1.2), namely mH m ∼ H ∼ ψ ( m ) | p | γ , and (1.3) | D x H | . ψ ( m ) | p | γ . (1.4)When lim m →∞ ψ ( m ) = 0 , these correspond to MFG systems with congestion, of which a typical example is − u t + | D x u | m + c ) α − V ( x ) = f ( m ) , u ( x, T ) = g ( x, m ( x, T )) m t − div ( m ( m + c ) α D x u ) = 0 , m (0 , x ) = m ( x ) (1.5)where the conditions < α < and c ≥ ensure that (1.1) and, hence, uniqueness, holds for (1.5) (see [1]).Now, Theorem 1.1 also allows for a more general behavior than (1.3), namely, for a constant ≤ γ ≤ γ , mH m ∼ ψ ( m ) | p | γ . One reason for working under such generality is that, despite (1.3) being the natural condition for fully generalHamiltonians H ( x, p, m ) satisfying (1.2), it is not satisfied by the important example of MFG systems with aseparated Hamiltonian. Such systems are nevertheless covered in Theorem 1.1, by setting γ = 0 .There are, however, two assumptions that must be strengthened when straying from (1.3). The first isthat, whereas in the case γ = γ , our result allows for “congestions” in which lim m →∞ ψ ( m ) = 0 , when γ < γ
2e must instead assume this limit to be positive. The second assumption is the control required for the x -dependence, and can be explained by comparing Theorem 1.1 with [19, Theorem 1.1]. In the latter work, weobtained classical solutions in the case of separated Hamiltonians, only requiring for D x H ( x, p ) the condition | D x H ( x, p ) | . | p | γ − ǫ , ǫ > , which allows the growth of D x H to be arbitrarily close to the natural one (1.4). This was achieved by exploitingin a crucial way the separated structure of the system. On the other hand, in (EMFG), no such structure isavailable, and therefore treating fully coupled Hamiltonians with ≤ γ < γ forces us to impose the strictercontrol | D x H | . ψ ( m ) | p | γ , where γ must satisfy γ < γ − γ + 2 . In other words, in the absence of additional structural assumptions,the more the growth γ of H m deviates from its natural value γ , the more we must restrict the growth γ ofthe space oscillation D x H .We will discuss now our methods of proof. The key insight that allows us to obtain classical solutions tothis first order system is the observation of Lions that, due to the strict monotonicity of H with respect to m ,one can eliminate the variable m and transform (EMFG) into a second order quasilinear equation in u withan oblique, non-linear boundary condition, ( Qu := − Tr ( A ( x, Du ) D u ) + b ( x, Du ) = 0 in Q T ,N u := N ( x, t, u, Du ) = 0 on ∂Q T , (Q)where Du = ( D x u, u t ) and, for ( x, z, p, s ) ∈ T d × R × R d × R , A ( x, p, s ) = (cid:18) B m + D p H , − (cid:19) ⊗ (cid:18) B m + D p H , − (cid:19) − (cid:18) B m − D p H ⊗ B m − D p H + H m D p B
00 0 (cid:19) , (Q1) b ( x, p, s ) = − D x H ( x, p, H − ) · B m ( x, p, H − ) + H m ( x, p, H − ) div x B ( x, p, H − ) , (Q2) N ( x, , z, p, s ) = − s + H ( x, p, m ( x )) , N ( x, T, z, p, s ) = − g ( x, H − ( x, p, s )) + z, (N)and the function H − ( x, p, s ) is the inverse of H with respect to m , defined by H − ( x, p, H ( x, p, m )) = m. An important observation that can be seen directly from the definition of A is that this equation is ellipticprecisely when (1.1) holds. For this reason, it is to be expected that the methods of quasilinear ellipticequations with oblique boundary conditions, which were successful in obtaining classical solutions to standardMFG systems in [17, 19], may also be applied in this more general setting. This is in fact the approach that wefollow here. Namely, we obtain a priori estimates for || u || C ( Q T ) and || Du || C ( Q T ) , and conclude the existenceof smooth solutions from the classical Hölder gradient estimates for oblique derivative problems (see [14]), theSchauder theory for linear oblique problems (see [13, 15]), and the non-linear method of continuity (see [13]). Notation
Let n, k ∈ N . Given x, y ∈ R n , x and y will always be understood to be row vectors, and their scalar product xy T will be denoted by x · y . For < s < , C k,s ( Q T ) refers to the space of k times differentiable real-valuedfunctions with s –Hölder continuous k th order derivatives. If u ∈ C ( Q T ) , the notation Du will always referto the full gradient in all variables, whereas D x u denotes the gradient in the space variable only. We write C = C ( K , K , . . . , K M ) for a positive constant C depending monotonically on the non-negative quantities K , . . . , K M . Assumptions
In what follows, C , γ, γ , γ are fixed constants satisfying C > , γ > , γ ≥ , γ ≤ γ ≤ γ, and γ < γ − γ + 2 . (2.1)The continuous functions C, ψ : (0 , ∞ ) → (0 , ∞ ) are also fixed, with ψ being non-increasing. If γ < γ , wefurther require that lim m →∞ ψ ( m ) > . (2.2)We note that the case γ = 0 , ψ ≡ , corresponds to a standard MFG system with a separated Hamiltonian,and the case γ = γ, ψ ( m ) ≡ m + c ) α corresponds to a MFG system with congestion.(M) (Assumptions on m ) The initial density m satisfies m ∈ C ( T d ) , m > , and Z T d m = 1 . (M1)(H) (Assumptions on H ) The function H : T d × R d × (0 , ∞ ) → R is four times continuously differentiableand satisfies H m < . Moreover, for ( x, m, p ) ∈ T d × (0 , ∞ ) × R d , C ψ ( m )(1 + | p | ) γ − I ≤ D pp H ≤ C ψ ( m )(1 + | p | ) γ − I, (H1) | D p H | ≤ C ψ ( m )(1 + | p | ) γ − , D p H · p ≥ (cid:18) C (cid:19) H − C ( m ) , (H2) C ψ ( m ) | p | γ ≤ − mH m ≤ C ψ ( m ) | p | γ + C ( m ) , (HM1) | mH mm | ≤ − C H m , | p || D p H m | ≤ − C H m , (HM2) | D x H | , | D xx H | ≤ C ψ ( m )(1 + | p | ) γ , | D xp H | ≤ C ψ ( m )(1 + | p | ) γ − , (HX1) m | D x H m | ≤ C ψ ( m )(1 + | p | ) γ , (HX2) | D x H ( x, , m ) | ≤ C . (HX3)(B) (Assumptions on B ) The function B : T d × R d × [0 , ∞ ) → R d is four times continuously differentiable, B ( · , · , ≡ and, mirroring the assumptions on H , B satisfies, for ( x, m, p ) ∈ T d × (0 , ∞ ) × R d , C mψ ( m ) | p | γ − I ≤ D p B ≤ C mψ ( m )(1 + | p | ) γ − I, (B1) | B m | ≤ C ψ ( m )(1 + | p | ) γ − , | D p B m | ≤ C ψ ( m )(1 + | p | ) γ − , | D pp B | ≤ C mψ ( m )(1 + | p | ) γ − , (B2) | B mm | ≤ − C (1 + | p | ) H m , (BM) | D x B | , | D xx B | ≤ mC ψ ( m )(1 + | p | ) γ − , | D x B m | ≤ C ψ ( m )(1 + | p | ) γ − , (BX1) | D xp B | ≤ C mψ ( m )(1 + | p | ) γ − , (BX2) | D x B ( x, , m ) | ≤ C m. (BX3)4G) (Assumptions on g ) The function g : T d × (0 , ∞ ) → R is four times continuously differentiable andsatisfies g m > . Furthermore, for each x ∈ T d , lim m →∞ g ( x, m ) = sup T d × [0 , ∞ ) g, and lim m → + g ( x, m ) = inf T d × [0 , ∞ ) g. (GX)(E) (Strict ellipticity of the system) The functions H and B satisfy the conditions lim m → + H ( x, p, m ) = + ∞ uniformly in ( x, p ) ∈ T d × R d , (E1) lim m →∞ H ( x, p, m ) − C ψ ( m ) | p | γ = −∞ uniformly in ( x, p ) ∈ T d × R d , (E2) − H m D p B > (cid:18) C (cid:19) ( B m − D p H ) ⊗ ( B m − D p H ) . (E3) Remark . In view of (H1), up to increasing the values of C and C , we may assume, with no loss ofgenerality, that, for ( x, p, m ) ∈ T d × R d × (0 , ∞ ) , ψ ( m ) C | p | γ − C ( m ) ≤ H ( x, p, m ) ≤ ψ ( m ) C | p | γ + C ( m ) . (2.3) max [min m , max m ] | H ( x, , · ) | ≤ C , max [min m , max m ] | B ( x, , · ) | ≤ C . (2.4)Moreover, in the case that (2.2) holds, we may also write ψ ( m ) ≥ C . (2.5) In the first result of this section, Lemma 3.1, we will estimate the L ∞ norms of u and the terminal density m ( · , T ) , where ( u, m ) is a solution to (EMFG). In order to provide an explicit form for the estimates of thissection, we consider the continuous, strictly increasing functions f , f , g , g : (0 , ∞ ) → R defined by f ( m ) = min x ∈ T d ( − H ( x, , m )) , f ( m ) = max x ∈ T d ( − H ( x, , m )) ,g ( m ) = min T d g ( · , m ) , g ( m ) = max T d g ( · , m ) , and the non-decreasing function h : (0 , ∞ ) → [0 , ∞ ) by h ( s ) = sup { m > ( x,p ) ∈ R d H ( x, p, m ) − C | p | γ ψ ( m ) ≥ − s } , (3.1)which is well-defined in view of (E2). Lemma 3.1.
There exists C = C ( C ) such that, for any solution ( u, m ) ∈ C ( Q T ) × C ( Q T ) of (EMFG),and every ( x, t ) ∈ Q T ,g f − ( − C ) − C ( e CT − e Ct ) ≤ u ( x, t ) ≤ g f − ( C ) + C ( e CT − e Ct ) , and (3.2) < g − g f − ( − C ) ≤ m ( x, T ) ≤ g − g f − ( C ) , (3.3)5 roof. The proof of this statement is analogous to [19, Lemma 3.1, Corollary 3.2]. We modify the function u in a way that ensures that its maximum value is achieved at t = T . This will allow us to conclude byexploiting the fact that the boundary condition (N) that holds at the terminal time is of “Robin type”. Forthis purpose, we set v ( x, t ) = u ( x, t ) + ζ ( t ) , where ζ ( t ) = M ( e Mt − e MT ) , for a large parameter M > .Conditions (HM2) and (BM) imply, respectively, the existence of a uniform Lipschitz bound for the maps w H − ( x, , w ) H m ( x, , H − ( x, , w )) and w B m ( x, , H − ( x, , w )) . Therefore, using (2.4), we obtain | mH m ( x, , m ) | ≤ C (1 + | H ( x, , m ) | ) and | B m ( x, , m ) | ≤ C (1 + | H ( x, , m ) | ) . In view of this, (Q), (Q2), (HX3), and (BX3) yield that, at any interior critical point ( x, t ) of v , − Tr ( A ( x, Du ) D v ) = − Tr ( A ( x, Du ) D u ) − ζ ′′ ( t ) = − b ( x, Du ) − ζ ′′ ( t ) = D x H ( x, , m ) · B m ( x, , m ) , − H m ( x, , m ) div x B ( x, , m ) − ζ ′′ ( t ) ≤ C (1 + | u t | ) − ζ ′′ ( t ) = C (1 + ζ ′ ( t )) − ζ ′′ ( t ) ≤ C (1 + M e Mt ) − M e Mt . Thus, if
M > C , one has − Tr ( A ( x, Du ) D v ) < at all interior critical points of v , and therefore v mustachieve its maximum value on the boundary ∂Q T . If the maximum is achieved at t = 0 , then D x v = 0 , v t ≤ ,and so M = ζ ′ (0) ≤ − u t = − H ( x, , m ( x )) ≤ C . Consequently, a sufficiently large value of M in fact forces the maximum to be achieved at { t = T } . At sucha point ( x, T ) , D x v = 0 , v t ≥ , and thus, since ζ ( T ) = 0 , M e MT = ζ ′ ( T ) ≥ − u t = − H ( x, , m ( x, T )) ≥ f ( m ( x, T )) = f ( g − ( x, u ( x, T )) = f ( g − ( x, v ( x, T )) , which yields v ( x, T ) ≤ g f − ( M e MT ) . Since ( x, T ) is a maximum point of v = u + ζ , this proves the upperbound in (3.2), with the lower bound being obtained through the same reasoning.The second inequality (3.3) then follows immediately by setting t = T in (3.2), using the fact that, by(GX), the functions g and g have the same range. To obtain the gradient estimate, we will make use of the Bernstein method, for which we will need to use thelinearization of (Q), namely L u ( v ) = − Tr ( A ( x, Du ) D v ) − D q Tr ( A ( x, Du ) D u ) · Dv + D q b ( x, Du ) · Dv, (3.4)where, for ( p, s ) ∈ R d × R , we denote q = ( p, s ) . The idea behind this classical method is the followinggeneral principle about elliptic equations: convex functions φ ( u ) and Φ( Du ) of the solution and its gradientare subsolutions of the linearized equation, up to an error that can often be controlled. More precisely, onehas L u ( φ ( u )) = − φ ′′ DuA · Du + E , L u (Φ( Du )) = − Tr ( D Φ D uAD u ) + E , (3.5)where E and E are regarded as error terms to be estimated. This observation can be exploited to bound || D x u || C ( Q T ) as follows. Since || u || C ( Q T ) is already known to be bounded a priori, the problem is equivalentto bounding v = φ ( u ) + Φ( D x u ) , as long as Φ : R d → R is coercive. At any interior maximum point ( x, t ) of v , one thus has ≤ L u ( v ) = − φ ′′ DuA · Du − Tr ( D Φ D uAD u ) + E + E , that is, φ ′′ DuA · Du ≤ − Tr ( D Φ D uAD u ) + E + E . (3.6)Thus, up to adequately estimating the error E + E in terms of the other two dominant signed terms, (3.6)leads naturally to a gradient bound.Now, we must note that the argument above applies only to interior maxima, so the possibility of themaximum being achieved on ∂Q T must be accounted for separately. In the usual case of Dirichlet boundaryconditions, the bound would follow automatically since u | ∂Q T would be an a priori given function, but since6N) defines an oblique boundary condition instead, an additional argument must be made. One can proceedby linearizing the boundary operator N and repeating the Bernstein process for this first order operator inplace of L u . Just like in the case of (3.5), the linearization is computed by differentiating both sides of theboundary equation. Whereas the ellipticity of (Q) is what allows E + E in (3.6) to be estimated, the errorat the boundary is instead controlled with the dominant signed term D p H · D x u by virtue of the superlineargrowth (H2) of H , the existing bounds on m | ∂Q T and the non-degeneracy of the boundary condition. Indeed,bounds on m ( · , and D x m ( · , are available because m is given a priori, and Lemma 3.1 provides boundsfor m ( · , T ) , albeit not for D x m ( · , T ) . The error terms that involve D x m ( · , T ) have, however, a favorable signthanks to the “Robin type” nature of (N) at time T that comes from the strict monotonicity of g . To carry out the strategy described above for the gradient estimate, we will require explicit computations ofthe error terms E and E described in (3.5), provided by the following Lemma. Lemma 3.2.
Let Φ( p, s ) ∈ C ( R d +1 ) , assume that ( u, m ) ∈ C ( Q T ) solves ( EM F G ) , and set v ( x, t ) =Φ( Du ( x, t )) . For each q = ( p, s ) ∈ R d +1 , and for each ( x, t ) ∈ Q T , define ζ ( p, s ) = − s + D p H ( x, D x u, m ) · p, Y + = B m + D p H, Y − = B m − D p H Then the following identities hold: D q Tr ( AD u ) · q = ( − D x u t + 12 Y + D xx u )( D p Y + p T − ( Y + m ) T H − m ζ ) − Y − D xx u ( D p Y − p T − ( Y − m ) T H − m ζ ) − ( D p H m · p − H mm H − m ζ ) Tr ( D p BD xx u ) − H m ( D p Tr ( D p BD xx u ) · p − Tr ( D p B m D xx u ) H − m ζ ) , (3.7) D q b ( x, Du ) · q = − B m ( D px Hp T − D x H Tm H − m ζ ) − D x H ( D p B m p T − B Tmm H − m ζ )+ H m ( D p div x B · p − div x B m H − m ζ ) + div x B ( D p H m · p − H mm H − m ζ ) , (3.8)Tr ( A x i D u ) = ( − D x u t + 12 Y + D xx u ) · ( Y + x i − Y + m H − m H x i ) − Y − D xx u · ( Y − x i − Y − m H − m H x i ) − ( H x i m − H mm H − m H x i ) Tr ( D p BD xx u ) − H m ( Tr ( D p B x i D xx u ) − Tr ( D p B m D xx u ) H − m H x i ) , (3.9) D x b ( x, Du ) · p = − B m ( D xx Hp T − D x H Tm H − m ( D x H · p )) − D x H ( D x B m p T − B mm H − m ( D x H · p ))+ H m ( D x div x ( B ) · p − div x B m H − m ( D x H · p )) + div x B ( D x H m · p − H mm H − m ( D x H · p )) , (3.10) L u v = − Tr ( D Φ D uAD u ) + d X i =1 Tr ( A x i D u )Φ p i − D p Φ · D x b. (3.11) Proof. (3.7) and (3.8) follow by differentiating, respectively, the expressions (Q1) and (Q2) with respect to q = ( p, s ) . Similarly, (3.9) and (3.10) result from differentiating (Q1) and (Q2) with respect to the spacevariables. Finally, (3.11) is obtained by applying D Φ · D to both sides of (Q) (see, for instance, [19, Lemma3.4]).We can now obtain the a priori gradient bound in terms of bounds for the solution u and the terminaldensity m ( · , T ) , which were already obtained in Subsection 3.1. Lemma 3.3.
Let ( u, m ) ∈ C ( Q T ) × C ( Q T ) be a solution to (EMFG), and set M = || m || C ( ∂Q T ) + || m − || C ( ∂Q T ) . (3.12)7 here exist constants C, C > , with C = C ( C , || ( ψ ◦ h ) − || C (0 ,C ] ) , C = C ( C , T, T − , || u || C ( Q T ) , M, || C || C [ M ,M ] , || ψ || C [ M ,M ] , || ψ − || C [ M ,M ] , || D x g || C ( T d × [ M ,M ]) , (2 γ − γ + 2 − γ ) − ) such that || Du || C ( Q T ) ≤ C. Proof.
We will consider first, for the sake of clarity, the natural case where γ = γ . First, we verify that it issufficient to bound the space gradient. Indeed, setting Φ( p, s ) = s in Lemma 3.2 yields L u ( u t ) = L u (Φ( Du )) = 0 , and thus, in view of the maximum principle and (2.3), − C ≤ u t ≤ C || D x u || γQ T + C. (3.13)We note that in (3.13), the constant C already depends on the upper and lower bounds for m on ∂Q T . Next,we will estimate || D x u || Q T through the Bernstein method. Let T u v = − v t + D p H ( x, D x u, m ) D x v, e u = u + || u || C ( Q T ) + 1 − || u || C ( Q T ) + 1) T ( T − t ) , and note that the function ˜ u has been constructed to satisfy | ˜ u | ≤ C, e u ( · , ≤ − , e u ( · , T ) ≥ . (3.14)Setting k = || D x u || / Q T , v ( x, t ) = k e u + 12 | D x u | , (3.15)we observe that the quantities || D x u || Q T and || v || Q T are comparable up to constants, so it is therefore sufficientto obtain a bound for the latter.Let ( x , t ) ∈ Q T be a point where v achieves its maximum value, and set p = D x u ( x , t ) . We assumewith no loss of generality that | p | ≥ and that || D x u || / Q T ≥ || u || Q T . The latter condition ensures that | p | ≥ || D x u || Q T − k || u || Q T ≥ || D x u || Q T . (3.16)Since the maximum may be achieved at the boundary of Q T , we must distinguish three cases. Case 1: t = T . Then D x v = 0 , v t ≥ . Therefore, in view of (2.3), (3.14), (HM1), (H2), and (3.16), ≥ T u v = T u (cid:18) | D x u | (cid:19) + k ˜ u ( − e u t + D p H · D x u )= − H m g m ( | p | − D x g · p ) − D x H · p + k ˜ u ( − u t + D p H · p − C ) ≥ ψ ( m ( T )) C m ( T ) g m | p | γ +2 − (cid:18) C ψ ( m ( T )) m ( T ) g m | p | γ + C ( m ( T )) (cid:19) | D x g || p | − C ( ψ ( m ( T )) | p | γ )) | p | + k ˜ u (cid:18) C ψ ( m ( T )) | p | γ − C (cid:19) ≥ C | p | γ +3 / + ψ ( m ( T )) C m ( T ) g m | p | γ +2 − C (1 + | p | γ +1 ) , Thus, since the second term is non-negative, we obtain C | p | γ +3 / ≤ C (1 + | p | γ +1 ) , | D x u | ≤ C. Case 2: t = 0 . Similarly to the first case, we obtain D x v = 0 , v t ≤ , and so ≤ T u v = T u (cid:18) | D x u | (cid:19) + k ˜ u ( − e u t + D p H · D x u )= − H m D x m ( x ) · p − D x H · p + k ˜ u ( − u t + D p H · p − C ) ≤ Cm ( | p | γ ψ ( m ) + C ( m )) | p | + Cψ ( m ) | p | γ +1 − ǫ ) + k ˜ u ( 1 C ψ ( m ) | p | γ − C ) ≤ − C ψ ( m ) | p | γ +3 / + C (1 + | p | γ +1 ) , and we conclude once more that | D x u | ≤ C. Case 3: < t < T . Then Dv = 0 , D v ≤ , which yields ≤ L u v. (3.17)By direct computation, we see from (Q) that L u (cid:18)
12 ˜ u (cid:19) = − D ˜ uAD ˜ u + ˜ uL u (˜ u ) = − D ˜ uAD ˜ u − ˜ uD q Tr ( AD u ) · D ˜ u + ˜ uD q b · D ˜ u − ˜ ub, whereas letting Φ( p, s ) = | p | in Lemma 3.2, L u (cid:18) | D x u | (cid:19) = − d X i =1 Du x i A · Du x i + d X i =1 Tr ( A x i D u ) u x i − D x b · p, and thus L u ( v ) = − kD ˜ uA · D ˜ u − d X i =1 Du x i A · Du x i + E, (3.18)where E is the error term, computed as follows. Setting Λ = D x + ˜ ukD p and using Lemma 3.2 once more, wehave E = E + E , with E = ( − D x u t + 12 Y + D xx u )Λ Y + · p − Y − D xx u Λ Y − · p − Tr ( D p BD xx u )Λ H m · p − H mm H − m ( D x H · p + ˜ ukζ )) Tr ( D p BD xx u ) − H m ( Tr (Λ D p BD xx u ) · p − Tr ( D p B m D xx u ) H − m ( D x H · p + ˜ ukζ )) , (3.19) E = − B m · (Λ D x Hp T − D x H m H − m ( D x H · p + ˜ ukζ )) − D x H · (Λ B m p T − B mm H − m ( D x H · p + ˜ ukζ ))+ H m (Λ div x Bp T − div x B m H − m ( D x H · p + ˜ ukζ )) + div x B (Λ H m · p − H mm H − m ( D x H · p + ˜ ukζ )) − k ˜ ub. (3.20)Before estimating the E i , we use (Q1) and (E3) to compute lower bounds for the dominant signed terms in(3.18), DuA · Du ≥ | − u t + 12 Y + · p | + 1 C | Y − · p | − C H m pD p B · p, (3.21)9 X i =1 Du x i A · Du x i = | − D x u t + 12 Y + D xx u | − | Y − D xx u | − H m Tr D xx uD p B · D xx u ≥ | − D x u t + 12 Y + D xx u | + 1 C | Y − D xx u | − C H m Tr ( D xx uD p B · D xx u ) . (3.22)Moreover, since ( Y + + Y − ) = D p H , | ζ | = | − u t + D p H · D x u | ≤ | − u t + 12 Y + · D x u | + | Y − · D x u | ) ≤ CDuA · Du. (3.23)On the other hand, (Q1) and (E3) yield D ˜ uA · D ˜ u = | − ˜ u t + 12 Y + · p | − | Y − · p | − H m pD p B · p ≥ DuA · Du − C. (3.24)Applying Young’s inequality and (3.22) in (3.19), we obtain | E | ≤ d X i =1 Du x i A · Du x i + C {| Λ Y + | | p | + | Λ Y − | | p | + | H m | − | D p B || Λ H m | | p | + | H mm | | H m | − ( | D x H | | p | + k ζ ) | D p B | + | H m || Λ D p B | | D p B | − | p | + | D p B m | | D p B | − | H m | − ( | D x H | | p | + k ζ ) } . (3.25)Now, the terms in (3.25) may all be estimated with the help of the growth assumptions (H) and (B). Indeed,in view of (H1), (HX1), (HX2), and (B2), we estimate | Λ Y + | , | Λ Y − | ≤ C ( | D xp H | + | D x B m | + k | D pp H | + k | D p B m | ) ≤ Cψ ( m ) ( | p | γ − + | p | γ − ) , (3.26) | Λ H m | ≤ C | D x H m | + Ck | D p H m | ≤ Cψ ( m ) m − ( | p | γ + | p | γ +1 ) , (3.27) | Λ D p B | ≤ C | D xp B | + Ck | D pp B | ≤ Cψ ( m ) m ( | p | γ − + | p | γ − ) . (3.28)Thus, using (HM1), (HM2), (B1), (B2), (HX1), (3.26), (3.27), and (3.28) in (3.25) yields | E | ≤ d X i =1 Du x i A · Du x i + Cψ ( m ) | p | γ +1 + C | p | − / kζ . (3.29)Similarly, for the second error term, we use Young’s Inequality and (3.23) in (3.20), obtaining | E | ≤ kDuA · Du + C {| B m || Λ D x H || p | + | B m || D x H m || H − m || D x H || p | + k | B m | | D x H m | | H m | − + | D x H | ( | Λ B m || p | + | B mm || H m | − | D x H || p | ) + k | D x H | | B mm | | H m | − + | H m || Λ div x B || p | + | div x B m || D x H || p | + | div x B m | k + | div x B | ( | Λ H m || p | + | H mm || H m | − | D x H || p | ) + k | div x B | | H mm | | H m | − + k | ˜ u | ( | H m || div x B | + | B m || D x H |} . (3.30)In view of (HX1), (B2), (BX2), and (BX1), we obtain | Λ D x H | ≤ C ( | D xx H | + k | D px H | ) ≤ Cψ ( m )( | p | γ + | p | γ +1 / ) , | Λ div x B | ≤ C ( | D xx B | + k | D px B | ) ≤ Cψ ( m )( | p | γ − + | p | γ − / ) , | Λ B m | ≤ C ( | D x B m | + k | D p B m | ) ≤ Cψ ( m )( | p | γ − + | p | γ − / ) , | Λ div x B | ≤ C ( | D xx B | + k | D px B | ) ≤ Cψ ( m )( | p | γ − + | p | γ − / ) . Consequently, (3.30), (HM1), (HM2), (B2), (HX1), (HX2), (BM), and (BX1) imply | E | ≤ kDuA · Du + Cψ ( m ) | p | γ +1 . (3.31)10aving estimated the error terms, (3.18), (3.24), (3.29), and (3.31) yield L u ( v ) = − kD ˜ uA · D ˜ u − d X i =1 Du x i A · Du x i + E ≤ − k DuA · Du − d X i =1 Du x i A · Du x i + Cψ ( m ) | p | γ +1 + C | p | − / kζ + Ck.
Therefore, in view of (3.16), (3.21), (3.23), (B1), and (HM1), L u ( v ) ≤ k H m pD p B · p − C k | ζ | + Cψ ( m ) | p | γ +1 + C | p | − / kζ + C | p | / ≤ − k C ψ ( m ) | p | γ +3 / − C k | ζ | + Cψ ( m ) | p | γ +1 + C | p | − / kζ + C | p | / ≤ − ψ ( m ) ( 18 C | p | γ +3 / − C | p | γ +1 ) − kζ ( 12 C − C | p | − / ) + C | p | / . So, given that ( x , t ) is a maximum point of v , we have L u ( v ) ≥ , and thus ψ ( m ) ( 18 C | p | γ +3 / − C | p | γ +1 ) + kζ ( 12 C − C | p | − / ) ≤ C | p | / . (3.32)This implies that either C − C | p | − / ≤ or ψ ( m ) ( C | p | γ +3 / − C | p | γ +1 ) ≤ C | p | / . If the former holds,there is nothing to prove, so we may assume the latter. We may further assume that | p | is large enough that C | p | γ +3 / − C | p | γ +1 ≥ C | p | γ +3 / . We thus obtain ψ ( m ) | p | γ ≤ C. (3.33)In view of (E2) and the fact that, by (3.13), H is bounded below, we conclude that m ≤ C . Hence ψ ( m ) isbounded below, which finally yields | p | ≤ C , concluding the proof when γ = γ .Now we describe the necessary changes in the proof to deal with the case γ < γ . Setting η = (2 γ − γ + 2) − γ , we see in view of (2.1) that η > . In (3.15), we replace k by k ′ = || D x u || κQ T , where κ = max (cid:18)
12 ( γ − γ ) + 1 , γ − γ + 32 (cid:19) . The proofs of Case 1 and Case 2 follow through with no change until the last step, leading in both cases tothe inequality C | p | γ + κ ≤ C (1 + | p | γ +1 + | p | κ + | p | γ +1 ) . (3.34)By definition, κ ≥ + γ − γ , so the left hand side of (3.34) has higher degree than the right hand side, andthus | p | ≤ C. The proof of Case 3 proceeds analogously as well. (3.25) and (3.30) are obtained with no change. To estimatethe errors, instead of (3.26), (3.27), and (3.28), we now have the bounds | Λ Y + | , | Λ Y − | ≤ C ( | D xp H | + | D x B m | + k | D pp H | + k | D p B m | ) ≤ Cψ ( m ) ( | p | γ − + | p | γ +2 κ − ) , | Λ H m | ≤ C | D x H m | + Ck | D p H m | ≤ Cψ ( m ) m − ( | p | γ + | p | γ +2 κ − ) , | Λ D p B | ≤ C | D xp B | + Ck | D pp B | ≤ Cψ ( m ) m ( | p | γ − + | p | γ +2 κ − ) . This allows us to estimate E as before, this time obtaining | E | ≤ d X i =1 Du x i ADu x i + Cψ ( m ) ( | p | γ +2 κ − + | p | γ + γ − γ + k | p | − (2+ γ − γ − κ ) ζ ) . (3.35)11ince the dominant power of | p | in (3.21) now has the exponent α = γ + γ + κ, we must verify that (3.35) does not have a higher degree. Indeed, α − (2 γ + 2 κ −
2) = 2 + γ − γ − κ ≥ min(2 + γ − γ − ( 12 ( γ − γ )+ 1) , γ − γ + 2 − ( γ − γ + 32 )) = 12 min( η, , (3.36) α − (2 γ + γ − γ ) = 2( γ − γ ) + κ ≥ γ − γ + κ ≥ γ − γ + 12 ( γ − γ ) + 1 = 12 η, (3.37)Hence, letting ǫ = min(1 , η ) , it follows from (3.35), (3.36) and (3.37), | E | ≤ d X i =1 Du x i ADu x i + Cψ ( m ) ( | p | α − ǫ + k | p | − ǫ ζ ) . (3.38)Moving now to E , we first obtain | Λ D x H | ≤ C ( | D xx H | + k | D px H | ) ≤ Cψ ( m )( | p | γ + | p | γ − κ ) , | Λ div x B | ≤ C ( | D xx B | + k | D px B | ) ≤ Cψ ( m )( | p | γ − + | p | γ − κ ) , | Λ B m | ≤ C ( | D x B m | + k | D p B m | ) ≤ Cψ ( m )( | p | γ − + | p | γ − κ ) , | Λ div x B | ≤ C ( | D xx B | + k | D px B | ) ≤ Cψ ( m )( | p | γ − + | p | γ − κ ) , and so, in place of (3.31), | E | ≤ kDuA · Du + Cψ ( m ) ( | p | γ + γ + | p | γ + γ + κ − + | p | γ + γ − γ + | p | κ +2 γ − γ − γ ) . (3.39)We again verify that the exponents do not exceed α , α − ( γ + γ ) = γ − γ + κ ≥ γ − γ + 12 ( γ − γ ) + 1 = 12 η, (3.40) α − ( γ + γ + κ −
1) = γ − γ + 1 ≥ , (3.41) α − ( κ + 2 γ − γ − γ ) = ( γ − γ ) + ((2 γ − γ + 2) − γ ) ≥ ((2 γ − γ + 2) − γ ) = η. (3.42)and thus, (3.39), (3.37), (3.40), (3.41), and (3.42) yield | E | ≤ kDuA · Du + Cψ ( m ) | p | α − ǫ . (3.43)Consequently, in view of (3.38) and (3.43), we obtain, instead of (3.32), ψ ( m ) ( 18 C | p | α − C | p | α − ǫ ) + kζ ( 12 C − C | p | − ǫ ) ≤ C | p | κ , and, thus, in place of (3.33), this time we conclude ψ ( m ) | p | γ + γ ≤ C. (3.44)Since γ < γ , (2.2) holds, and thus we have (2.5). This, together with (3.44), implies that | p | ≤ C , aswanted.The following Corollary provides global, positive two-sided bounds for the density.12 orollary 3.4. Let ( u, m ) ∈ C ( Q T ) × C ( Q T ) be a solution to (EMFG), and set, for K ∈ R ,δ K = inf ( x,p,s ) ∈ T d × R d × ( −∞ ,K ] H − ( x, p, s ) . There exist constants
C, C > , with C = C ( C , h ( C )) , C = C ( C , || Du || C ( Q T ) , δ − || Du || , || ψ || C [ δ || Du || , ∞ ) ) , such that || m || C ( Q T ) + || m − || C ( Q T ) ≤ C. Proof.
Due to (E1), δ K > is well-defined for each K ∈ R , and we may apply H − ( x, D x u, · ) to both sides ofthe inequality H ( x, D x u, m ) = u t ≤ || Du || C ( Q T ) , which yields, for ( x, t ) ∈ Q T , H − ( x, D x u, || Du || C ( Q T ) ) ≤ m ( x, t ) . Letting δ = δ || Du || , we thus obtain δ ≤ m ( x, t ) and, hence, || m − || C ( Q T ) ≤ δ − . On the other hand, H ( x, D x u, m ) − C ψ ( m ) | D x u | γ ≥ u t − C ψ ( m ) || D x u || γC ( Q T ) ≥ −|| Du || C ( Q T ) − C || ψ || [ δ, ∞ ) || D x u || γC ( Q T ) ≥ − C , which, in view of the definition of h (see 3.1), implies that m ≤ h ( C ) . We now summarize all of the a priori bounds obtained in this section.
Theorem 3.5.
Let ( u, m ) ∈ C ( Q T ) × C ( Q T ) be a solution to (EMFG), and let δ be defined as in Corollary3.4. Then there exist constants M, M , L, L , K, K > , with L = (cid:18) L , g ( f − ( L )) + , g ( f − ( − L )) − , g − g ( f − ( L )) , g − g ( f − ( − L )) (cid:19) , L = L ( C , T ) ,K = K ( C , || ( ψ ◦ h ) − || C (0 ,K ] ) ,K = K ( L, T − , || C || C [ L ,L ] , || ψ || C [ L ,L ] , || ψ − || C [ L ,L ] , || D x g || C ( T d × [ L ,L ]) , (2 γ − γ + 2 − γ ) − ) ,M = M ( M , h ( M )) , M = M ( K, δ − K , || ψ || C [ δ K , ∞ ) ) such that || u || C ( Q T ) ≤ L, || Du || C ( Q T ) ≤ K, and || m || C ( Q T ) + || m − || C ( Q T ) ≤ M. Proof.
This result follows from combining Lemma 3.1, Lemma 3.3, and Corollary 3.4.13 .4 Classical solutions
Having obtained the gradient bound, the existence result follows through the method of continuity.
Proof of Theorem 1.1.
We only sketch the proof, which follows the same steps as [19, Theorem 1.1]. We define,for θ ∈ [0 , and ( x, p, m ) ∈ T d × R d × (0 , ∞ ) ,H θ ( x, p, m ) = θH ( x, p, m ) + (1 − θ ) H (0 , p, m ) , B θ ( x, p, m ) = θB ( x, p, m ) + (1 − θ ) B (0 , p, m ) ,g θ ( x, m ) = θg ( x, m ) + (1 − θ ) m, m θ ( x ) = θm ( x ) + (1 − θ ) , and consider the family of Extended MFG systems − u t + H θ ( x, D x u, m ) = 0 ( x, t ) ∈ Q T ,m t − div ( B θ ( x, D x u, m )) = 0 ( x, t ) ∈ Q T ,m (0 , x ) = m θ ( x ) , u ( x, T ) = g θ ( x, m ( x, T )) x ∈ T d , (EMFG θ )together with the corresponding elliptic and boundary operators Q θ and N θ associated to them, accordingto (Q). We observe first that for θ = 0 , the solution is simply ( u, m ) ≡ (( t − T ) H (0 , ,
1) + 1 , . Now, bydefinition of g θ , g θ ◦ g − ◦ g = θg ◦ g − ◦ g + (1 − θ ) g − ◦ g ≥ θg + (1 − θ ) g − ◦ g = g θ , so we obtain ( g θ ) − g θ ≤ g − g , (3.45)and similarly g − g ≤ ( g θ ) − g θ . (3.46)Moreover, setting f θ ( m ) = min T d ( − H θ ( · , , m )) = θf ( m ) − (1 − θ ) H (0 , , m ) , and f θ ( m ) = max T d ( − H θ ( · , , m )) = θf ( m ) − (1 − θ ) H (0 , , m ) , it is readily seen that, by definition, ( f θ ) − ≤ f − , f − ≤ ( f θ ) − . (3.47)In view of (3.45), (3.46), and (3.47), Theorem 3.5 yields a constant C , independent of θ , such that || u θ || C ( Q T ) ≤ C. Moreover, the classical Hölder gradient estimates for oblique derivative problems (see, for instance, [14, Lemma2.3]) yield || u θ || C s ′ ( Q T ) ≤ C. (3.48)for some s ′ . Now we define the Banach spaces E = C ,s ( Q T ) , F = C ,s ( Q T ) × C ,s ( ∂Q T ) , and the continuously differentiable operator S : E × [0 , → F by S ( u, θ ) = ( Q θ u, N θ u ) , ( u, θ ) ∈ E × [0 , . Direct computation shows that, for fixed ( u, θ ) ∈ E × [0 , , the linearization S u of S with respect to u hasthe form ( L u,θ ) w, L u,θ ) w ) , where L u,θ ) is a linear, uniformly elliptic operator and L u,θ ) is a linear oblique14oundary operator. Moreover, the homogeneous problem ( L u,θ ) w, L u,θ ) w ) = (0 , has only the trivialsolution. The standard Fredholm alternative for linear oblique problems (see [13]) thus implies that S u isinvertible in C ,s ( Q T ) . Thus, by the Implicit Function Theorem, the set D = { θ ∈ [0 ,
1] : the equation S ( u, θ ) = (0 , has a unique solution u ∈ C ,s ( Q T ) } is open in [0 , . On the other hand, (3.48) together with the Schauder estimates for linear oblique problemsimply that D is also closed. Since ∈ D , we conclude that D = [0 , , and in particular ∈ D , which completesthe proof. References
Department of Mathematics, University of Chicago, Illinois, 60637, USA
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