Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions

Transcription

1 Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions Item Type Preprint Authors Ferreira, Rita; Gomes, Diogo A.; Tada, Teruo Eprint version Pre-print Publisher arxiv Rights Archived with thanks to arxiv Download date 6/1/018 05:18:33 Link to Item

2 EXISTENCE OF WEAK SOLUTIONS TO FIRST-ORDER STATIONARY MEAN-FIELD GAMES WITH DIRICHLET CONDITIONS RITA FERREIRA, DIOGO GOMES, AND TERUO TADA arxiv: v1 math.ap] 19 Apr 018 Abstract. In this paper, we study first-order stationary monotone mean-field games (MFGs) with Dirichlet boundary conditions. While for Hamilton Jacobi equations Dirichlet conditions may not be satisfied, here, we establish the existence of solutions of MFGs that satisfy those conditions. To construct these solutions, we introduce a monotone regularized problem. Applying Schaefer s fixed-point theorem and using the monotonicity of the MFG, we verify that there exists a unique weak solution to the regularized problem. Finally, we take the limit of the solutions of the regularized problem and using Minty s method, we show the existence of weak solutions to the original MFG. 1. Introduction Mean-field games (MFGs) were introduced in the mathematical community in 7], 8], and 9] and, independently, around the same time in the engineering community in 5] and 6]. These games model the behavior of large populations of rational agents who seek to optimize an individual utility. Here, we consider the following first-order stationary MFG with a Dirichlet boundary condition. Problem 1. Suppose that R d is an open and bounded set with smooth boundary,, and γ > 1. Let V,φ,h C (), g C (R + 0 ), and H C ( R d ) be such that g is increasing, φ 0 in, and φdx = 1. Find (m,u) L1 () W 1,γ () satisfying m 0 and u H(x,Du)+g(m) V(x) = 0 in, m div ( md p H(x,Du) ) φ = 0 in, u = h on. In Problem 1, H is the Hamiltonian and its Legendre transform, given by L(v) = sup{ p v H(x,p)}, p gives the agent s cost of movement at speed v; the potential, V, determines spatial preferences of each agent, and the coupling, g, encodes the interactions between agents and the mean field. When agents leave the domain through a point x, they incur a charge h(x). Finally, the source, φ, represents an incoming flow of agents replacing the ones leaving through or due to the unit discount encoded in the term u in the Hamilton Jacobi equation and in the term m in the transport equation. We note that m is the density of the distribution of the agents and u(x) the value function of an agent in the state x. In general, Hamilton Jacobi equations with Dirichlet boundary conditions do not admit continuous solutions up to the boundary. This fact can be illustrated by the equation u (x) = 0 Date: April 0, 018. Key words and phrases. Mean-Field Games; Stationary Problems; Weak solutions; Dirichlet boundary conditions. R. Ferreira, D. Gomes, and T. Tada were partially supported baseline and start-up funds from King Abdullah University of Science and Technology (KAUST).. 1

3 RITA FERREIRA, DIOGO GOMES, AND TERUO TADA for 0 < x < 1 with u(0) = 0 and u(1) = 1. However, keeping the boundary conditions and coupling the previous equation with a transport equation { u (x) = m(x) m (x) = 0, we obtain a model that has a unique solution continuousup to the boundary, (u,m) = (x,1). This model motivates our main result, the existence of solutions for Problem 1 satisfying the Dirichlet boundary condition in the trace sense. MFGs have been studied intensively, see, for example, the monograph ], the surveys 1], 4], the note 6], and the lectures 30]. Because there are few known explicit solutions for stationary MFGs 0], 19], and 14], a substantial effort has been undertaken to develop numerical methods 3] and 5], find special transformations8], and to establish the existence of solutions. The existence of solution for second-order, stationary MFGs without congestion was investigated in 3], 1], 3], 4], and 10]; problems with congestion were examined in 13], 18]. The theory for first-order MFGs is less developed. The existence of solutions for first or second-order stationary MFGs was examined in 16] (also see ]) using monotone operators and, using a variational approach, certain first-order MFGs with congestion were examined in 1]. For first-order MFGs and often for second-order MFGs, existing publications consider only periodic boundary conditions. However, in applications, MFGs with boundary conditions are quite natural: for example, Dirichlet boundary conditions arise in models where agents can leave the domain and are charged an exit fee. Here, our goal is to prove the existence of weak solutions for the first-order stationary monotone MFG with Dirichlet boundary conditions. Thus, we introduce a notion of weak solutions to Problem 1 similar to the one considered in 16]. Here, however, we account for the Dirichlet boundary conditions. We recall that in 16], the authors consider stationary monotone MFGs with periodic boundary conditions, and their notion of weak solutions is induced by monotonicity. Monotonicity plays an essential role in the uniqueness of solutions 30], and in its absence, the theory becomes substantially harder, see the examples in 0], and the non-monotone second-order MFGs in 11], 9], and 31]. Throughout this paper, k N is a fixed natural number such that k > d +, and A and H k h () are the sets given, respectively, by A := { } m H k () m 0 (1.1) and H k h () := { w H k () w h H k 0 () }. (1.) Definition 1.1. A weak solution toproblem 1 is a pair (m,u) L 1 () W 1,γ () satisfying (D1) u = h on, m 0 in, η η m (D) F, 0 for all (η,v) A H u] h k (), where, for (η,v) H k () H k () fixed, Fη, : L 1 () L 1 () R is the functional given by ] η w1 ( ) F, := v H(x,Dv)+g(η) V w1 dx w ( + η div ( ηd p H(x,Dv) ) ) (1.3) φ w dx. Next, we state our main theorem that, under the assumptions detailed in Section, establishes the existence of weak solutions to Problem 1. Theorem 1.. Consider Problem 1 and suppose that Assumptions 1 7 hold. Then, there exists a weak solution (m,u) L 1 () W 1,γ () to Problem 1 in the sense of Definition 1.1.

4 MEAN-FIELD GAMES WITH DIRICHLET CONDITIONS 3 In 16], the method of continuity was used to prove the existence of a weak solution to stationary monotone MFGs with periodic boundary conditions. Here, we use a different approach: we apply Schaeffer s fixed-point theorem and extend the results in 16] to Dirichlet boundary conditions. To prove Theorem 1., we introduce a regularized problem, Problem, that we believe to be of interest on its own. This regularized problem preserves the monotonicity of the original MFG in the sense of Assumption 7 (see Section and Lemma 7.). Note that the choice of boundary conditions is critical to preserve monotonicity. Moreover, because of the regularizing terms (see the ǫ-terms in (1.4) below), it is simpler to prove the existence and uniqueness of weak solutions to this problem. Then, by letting ǫ 0, we can construct a weak solution to Problem 1 (see Section 7). Problem. Let be an open and bounded set with smooth boundary,, and outward pointing unit normal vector n. Let V,φ,ξ,h C (), g C (R + 0 ), and H C ( R d ) be such that g is increasing, φ 0 in, and φdx = 1. Fix ǫ (0,1). Find (m,u) H k () H k () satisfying m 0 and u H(x,Du)+g(m) V(x)+ǫ ( m+ k m ) = 0 in, m div ( md p H(x,Du) ) φ+ǫ ( u+ξ + k (u+ξ) ) = 0 in, n α i m = 0 on for all α N d 0 and i N 0 such that α +i = k 1, β u = β h on for all β N d 0 and such that β k 1. (1.4) In the preceding problem, ξ is a technical term used to cancel the boundary conditions in u so that we can work with vanishing Dirichlet boundary conditions, see Section 6. Since the regularizing terms are of order greater than two, we cannot apply the maximum principle. Thus, Problem may not have classical solutions with m 0. Hence, in the following definition, we introduce a notion of weak solution to Problem that requires positivity and relaxes the equality in the Hamilton Jacobi equation. This definition is related to the ones in 7] and in 16], where u is only required to be a subsolution of the Hamilton Jacobi equation. Definition 1.3. A weak solution to Problem is a pair (m,u) H k () H k () satisfying, for all w A and v H k 0 (), (E1) u Hh k (), m 0 in, ( ) (E) u H(x,Du)+g(m) V (w m)dx + ǫm(w m)+ǫ ] α m( α w α m) dx 0, ( ( (E3) m div mdp H(x,Du) ) φ ) vdx + ǫ (uv + ) ] α u α v +ǫ(ξ + k ξ)v dx = 0. Remark 1.4. Assume that (m,u) is a weak solution to Problem. Let := {x m(x) > 0}, and fix w 1 C c ( ). For all τ R with τ sufficiently small, we have w = m+τw 1 A. Then, from (E), we obtain τ ( ) u H(x,Du)+g(m) V w1 dx+τ ( ǫmw1 +ǫ Because the sign of τ is arbitrary, we conclude that m satisfies α m α w 1 ) dx 0. u H(x,Du)+g(m) V(x)+ǫ(m+ k m) = 0 pointwise in.

5 4 RITA FERREIRA, DIOGO GOMES, AND TERUO TADA Moreover, let w Cc () be such that w 0; taking w = m + w A in (E) and integrating by parts, we obtain ( ) ( τ u H(x,Du)+g(m) V(x) w dx+τ ǫmw +ǫ k ) mw dx 0. Thus, u H(x,Du)+g(m) V(x)+ǫ(m+ k m) 0 in the sense of distributions in. Also, by (E3), we have m div ( md p H(x,Du) ) φ+ǫ ( u+ξ+ k (u+ξ) ) = 0 in the sense of distributions in. Under the same assumptions of Theorem 1., we prove the existence and uniqueness of the weak solutions to Problem. Theorem 1.5. ConsiderProblemandsupposethatAssumptions1 5and7holdforsome γ > 1. Then, there exists a unique weak solution (m,u) H k () H k () to Problem in the sense of Definition 1.3. We begin by addressing Theorem 1.5. First, in Section 3, we prove a priori estimates for classical and weak solutions to Problem. Second, in Sections 4 and 5, we introduce two auxiliary problems: a variational problem whose Euler Lagrange equation is the first equation in Problem and a problem associated with a bilinear form corresponding to the second equation in that problem. In each of these two sections, we show that there exists a unique solution. Finally, in Section 6, we prove Theorem 1.5 using Schaefer s fixedpoint theorem together with the results established in Sections 3 5. Finally, the proof of Theorem 1. is given in Section 7 using Minty s method.. Assumptions To prove our main results, we need the following assumptions on the functions that arise in Problems 1 and. The first three assumptions prescribe standard growth conditions on the Hamiltonian, H. For instance, H(x,p) = a(x)(1+ p ) γ +b(x) p, where a C(), a(x) > 0 for all x, and b : R d is a C function satisfies our assumptions. Assumption 1. There exists a constant, C > 0, such that, for all (x,p) R d, H(x,p)+D p H(x,p) p 1 C p γ C. Assumption. There exists a constant, C > 0, such that, for all (x,p) R d, H(x,p) 1 C p γ C. Assumption 3. There exists a constant, C > 0, such that, for all (x,p) R d, D p H(x,p) C p γ 1 +C. The next three assumptions impose growth conditions on g. For example, these are satisfied for g(m) = m α,α > 0, or by g(m) = lnm (apart from small changes). Assumption 4. The function g is increasing in R + 0. Moreover, for all δ > 0, there exists a constant, C δ > 0, such that, for all m L 1 (), { } max g(m) dx, mdx δ mg(m)dx+c δ.

6 MEAN-FIELD GAMES WITH DIRICHLET CONDITIONS 5 Assumption 5. There exists a constant, C > 0, such that, for all m L 1 (), mg(m)dx C. Assumption 6. If {m j } j=1 L1 () is a sequence satisfying sup m j g(m j )dx < +, j N then there exists a subsequence of {m j } j=1 that converges weakly in L1 (). Remark.1. If g is an increasing function with g 0 and lim t g(t) =, then g satisfies Assumption 6. This fact is a consequence of the De la Vallée Poussin lemma together with the Dunford Pettis theorem; see, for instance, Theorems.9 and.54 in 17]. Our final assumption concerns the monotonicity of the functional F introduced in Definition 1.1. Monotonicity is the key ingredient in the proof of Theorem 1. from Theorem 1.5 through Minty s method. Assumption 7. The functional F introduced in Definition 1.1 is monotone with respect to the L L -inner product; that is, for all (η 1,v 1 ), (η,v ) A Hh (), F satisfies ] ] ] ] η1 η η1 η F F, 0. v 1 v v 1 v 3. Properties of weak solutions In this section, we investigate properties of weak solutions, (m, u), to Problem. First, we prove an a priori estimate for classical solutions. Then, we verify that this a priori estimate also holds for weak solutions. Finally, we show that u is bounded in W 1,γ (), and that ( ǫm, ǫu) is bounded in H k () H k (). In Section 6, we combine these estimates with Schaefer s fixed-point theorem to prove the existence of weak solutions to Problem. To simplify the notation, throughout this section, we use the same letter C to denote any positive constant that depends only on the problem data; that is, may depend on, d, γ, H,V,φ,ξ, and h, on the constants in the Assumptions 1 5, or on universal constants such as the constant in Morrey s theorem or Gagliardo Nirenberg interpolation inequality. In particular, these constants do not depend on the particular choice of solutions to Problem nor on ǫ. Proposition 3.1. Consider Problem and suppose that Assumptions 1 4 hold for some γ > 1. Then, there exists a positive constant, C, that depends only on the problem data such that for any solution (m,u) to Problem in the classical sense, we have ( mg(m)+ 1 C m Du γ + 1 C φ Du γ) dx+ǫ m +u + ( ( α m) +( α u) )] dx C. (3.1) Proof. Multiplying the first equation in(1.4) by(m φ) and the second one by(u h), adding and integrating over, and then using integration by parts and the boundary conditions,

7 6 RITA FERREIRA, DIOGO GOMES, AND TERUO TADA we have mg(m)+m ( H(x,Du)+D p H(x,Du) Du ) +φh(x,du) = +ǫ (m +u + ( ( α m) +( α u) ))] dx φg(m)+(ǫφ+v +h)m+(ǫξh Vφ φh)+md p H(x,Du) Dh +ǫ (u(h ξ)+ ( α φ α m+ α u( α h α ξ)+ α ξ α h ))] dx. By Assumptions 1 3, Young s inequality, and the positivity of m and φ, we have m ( H(x.Du)+D p H(x,Du) Du ) ( m Du γ ) dx Cm dx, C ( ) φ Du γ φh(x,du)dx Cφ dx, C ( ) m Du md p H(x,Du) Dhdx Cm( Du γ 1 γ +1) +Cm dx. C Therefore, from Assumption 4, Young s inequality, and (3.), we obtain mg(m)+ m Du γ + φ Du γ + ǫ (m +u + ( ( α m) +( α u) ))] dx C C ) (φg(m)+cm+ m Du γ dx+c 1 ) (mg(m)+ m Du γ dx+c, C C from which Proposition 3.1 follows. (3.) Corollary 3.. Consider Problem and suppose that Assumptions 1 4 hold for some γ > 1, and let (m,u) be a weak solution to Problem in the sense of Definition 1.3. Then, (m, u) satisfies the estimate (3.1). Proof. Let (m,u) be a weak solution to Problem in the sense of Definition 1.3. Taking v = u h H0 k () and w = φ A in (E) and (E3) in Definition 1.3, and then summing the resulting inequalities, we obtain (3.) with = replaced by. Thus, arguing as in Proposition 3.1, we conclude that (m, u) satisfies (3.1). Corollary 3.3. Consider Problem and suppose that Assumptions 1 5 hold for some γ > 1. Then, there exists a positive constant, C, that depends only on the problem data such that for any weak solution (m,u) to Problem, we have u W 1,γ () C. Proof. By Corollary 3., the positivity of m and φ, and Assumption 5, we have mg(m) dx C. Ontheotherhand, using(e)in Definition1.3with w := m+1 A, ǫ 1, Assumption4, the previous estimate, and a weighted Young s inequality, we obtain ( ) H(x,Du)dx ǫm u+g(m) V dx σ u γ dx+c σ, where σ > 0 is arbitrary. Moreover, because u h = 0 on, Poincaré s inequality yields u γ dx C Du γ dx+c. (3.3) Then, invoking Assumption and using the estimates above with σ chosen appropriately, we obtain Du γ dx C.

8 MEAN-FIELD GAMES WITH DIRICHLET CONDITIONS 7 Using (3.3) once more, we conclude that u W 1,γ () C, where C is a positive constant that depends only on the problem data. Corollary 3.4. Consider Problem and suppose that Assumptions 1 5 hold for some γ > 1. Then, there existsapositiveconstant, C, that depends onlyonthe problemdatasuch that for any weak solution (m,u) to Problem, we have ǫm H k () + ǫu H k () C. Proof. Using Corollary 3., Assumption 5, and the positivity of m and φ, we obtain ǫ m +u + ( ( α m) +( α u) )] dx C, where C is a positive constant that depends only on the problem data. The conclusion follows from the Gagliardo Nirenberg interpolation inequality. 4. An auxiliary variational problem Here, we introduce a variational problem whose Euler Lagrange equation is the first equation in (1.4). We prove the existence and uniqueness of a minimizer, m, to this variational problem. Then, we investigate properties of m that enable us to prove the existence and uniqueness of a weak solution to Problem in Section 6. Given (m,u) H k () H k 1 () with m 0, let I (m,u) : H k () R be the functional defined, for w H k (), by ǫ I (m,u) w] := (w + ( α w) ) + ( u H(x,Du)+g(m) V ) ] w dx. (4.1) Note that by Morrey s embedding theorem, H k () is compactly embedded in C 0,l () for some l (0,1). In particular, there exists a positive constant, C = C(,k,d,l), such that for all ϑ H k (), we have ϑ C 0,l () C ϑ H k (). (4.) Next, we fix (m 0,u 0 ) H k () H k 1 () with m 0 0, and set I 0 = I (m0,u 0). We address the variational problem finding m A such that where A is defined in (1.1). I 0 m] = inf w A I 0w], (4.3) Proposition 4.1. Let H,g, and V be as in Problem, and fix (m 0,u 0 ) H k () H k 1 () with m 0 0. Then, there exists a unique m A satisfying (4.3). Proof. Let {m n } n=1 A be a minimizing sequence for (4.3), and fix δ (0,1). Then, there exists N N such that for all n N, I 0 m n ] < inf w A I 0w]+δ I 0 0]+1 = 1. (4.4) By (4.), we have m 0 C 0,l () and u 0 C 1,l () for some l (0,1), and C 0 := u 0 H(x,Du 0 )+g(m 0 ) V L () R. Then, using Young s inequality and (4.4), for all n N, we have ǫ m n + ( α m n ) ] dx 0 m n dx+1 C ǫ 4 By the Gagliardo Nirenberg interpolation inequality, we have m ndx+ C 0 ǫ +1. (4.5) α m n L () C( m n L () + Dk m n L ()), (4.6) where α is any multi-index such that α k. Hence, from (4.5) and (4.6), we conclude that {m n } n=1 is bounded in H k (). Consequently, m n m weakly in H k () for some m H k (), extracting a subsequence if necessary. Because m n 0, also m 0; so, m A.

9 8 RITA FERREIRA, DIOGO GOMES, AND TERUO TADA Moreover, m n m in L () and D k m L () liminf n D k m n L () ; hence, I 0 m] liminf n I 0 m n ] = infi 0 w] I 0 m], which shows that m is a minimizer of I 0 over A. We now prove uniqueness. Assume that m, m A are minimizers of I 0 over A with m+ m m m. Then, A and, recalling that m m C 0 (), (m m) dx > 0. Moreover, I 0 m+ m ] = ǫ (( m+ m ) + ( α m+ α m ) ) + ( u 0 H(x,Du 0 )+g(m 0 ) V )( m+ m ) ] dx = 1 ǫ (m + ( α m) ) + ( u 0 H(x,Du 0 )+g(m 0 ) V ) ] m dx + 1 ǫ ( m + ( α m) ) + ( u 0 H(x,Du 0 )+g(m 0 ) V ) m ] dx ǫ (m m) + ( α m α m) ] dx 8 < 1 I 0m]+ 1 I 0 m] = min I 0w], w A (4.7) which contradicts the fact that m+ m A. Thus, m = m. Corollary 4.. Let H,g, and V be as in Problem, fix (m 0,u 0 ) H k () H k 1 () with m 0 0, and let m A be the unique solution to (4.3). Set C 0 := u 0 H(x,Du 0 )+ g(m 0 ) V L (). Then, there exists a positive constant, C, depending only on the problem data and on C 0, such that m H k () C. Proof. Because I 0 m] I 0 0], (4.5) and (4.6) hold with m n replaced by m, from which Corollary 4. follows. Proposition 4.3. LetH,g,andV beasinproblem, fix(m 0,u 0 ) H k () H k 1 () with m 0 0, and let m A be the unique solution to (4.3). Then, for all w A, m satisfies ( u0 H(x,Du 0 )+g(m 0 ) V ) (w m)dx + ǫm(w m)+ǫ ] α m( α w α m) dx 0. (4.8) Proof. Let w A. If τ 0,1], then m + τ(w m) = (1 τ)m + τw A; hence, the mapping i : 0,1] R given by iτ] := I 0 m+τ(w m) ] is a well-defined C 1 -function. Becausei(0) i(τ) forall 0 τ 1, wehavei (0) 0. On the other hand, for 0 < τ 1, we have 1( ) ( i(τ) i(0) = u0 H(x,Du 0 )+g(m 0 ) V ) (w m)dx τ m(w m)+ +ǫ +τ ǫ (w m) + ] α m( α w α m) dx ( α w α m) ] dx. Consequently, letting τ 0 + in this equality and using i (0) 0, we obtain (4.8).

10 MEAN-FIELD GAMES WITH DIRICHLET CONDITIONS 9 Proposition 4.4. LetH,g,andV beasinproblem, fix(m 0,u 0 ) H k () H k 1 () with m 0 0, and let m be the unique solution of (4.3). Set 1 = {x m(x) > 0}; then, m satisfies and u 0 H(x,Du 0 )+g(m 0 ) V +ǫ(m+ k m) = 0 pointwise in 1 u 0 H(x,Du 0 )+g(m 0 ) V +ǫ(m+ k m) 0 in the sense of distributions in. Proof. To prove Proposition 4.4, it suffices to argue as in Remark 1.4, invoking (4.8) in place of (E) and recalling the embedding H k () C 0,l () for some l (0,1). 5. A problem given by a bilinear form Here, we consider an auxiliary problem determined by a bilinear form related to the second equation in (1.4). Using the Lax Milgram Theorem, we prove the existence and uniqueness of a solution, u, to this auxiliary problem, and we establish a uniform bound on u. These results are used in Section 6 to study the existence and uniqueness of a weak solution to Problem. With H,φ, and ξ as in Problem, given (m,u) H k () H k 1 () with m 0, we define a bilinear form, B : H0 k() Hk 0 () R, and a linear functional, f (m,u) : L () R, by setting, for v 1,v H0 k () and v L (), Bv 1,v ] := ǫ (v 1 v + α v 1 α v )dx, f(m,u),v := m+div ( mdp H(x,Du) ) +φ ǫ(ξ + k ξ) ] vdx. (5.1) Fix (m 0,u 0 ) H k () H k 1 () with m 0 0, and set f 0 := f (m0,u 0). We address next the problem of finding u H0 k () satisfying Bu, = f 0,v for all v H0 k (). (5.) Proposition 5.1. Let H,φ, and ξ be as in Problem, and fix (m 0,u 0 ) H k () H k 1 () with m 0 0. Then, there exists a unique solution, u H k 0 (), to (5.). Proof. Because (m 0,u 0 ) ( H k () H k 1 () ) ( C 0,l () C 1,l () ) for some l (0,1) (see (4.)), we have ( m 0 +div ( m 0 D p H(x,Du 0 ) ) +φ ǫ(ξ + k ξ)) L (). Hence, by Hölder s inequality, f 0 is bounded in L (). Using Hölder s and Poincaré s inequalities, we have Bv 1,v ] c v 1 H k 0 () v H k 0 () for all v 1,v H0 k (), where c > 0 is a constant independent of v 1 and v. Moreover, we clearly have Bv 1,v 1 ] ǫ v 1 H0 k() for all v 1 H0 k(). Therefore, by the Lax Milgram Theorem, there exists a unique u H0 k () satisfying (5.). Lemma 5.. Let H,φ, andξ be asin Problem, fix (m 0,u 0 ) H k () H k 1 () with m 0 0, and let u be the unique solution to (5.) in H0 k (). Then, there exists a positive constant, C, depending only on the problem data and on m 0 H k () and u 0 H k 1 (), such that u H k () C. Proof. ArguingasintheproofoftheProposition5.1, wehavec 0 := m 0 +div ( m 0 D p H(x,Du 0 ) ) + φ ǫ(ξ + k ξ) L () R+ 0. Then, using Young s inequality, we obtain ( ) ǫ u L () + u H0 k() = Bu,u] = f 0,u ǫ u L () + c 0 4ǫ. Hence, u H k 0 () c 0/(4ǫ ), and the conclusion follows by Poincaré s inequality.

11 10 RITA FERREIRA, DIOGO GOMES, AND TERUO TADA 6. Proof of Theorem 1.5 This section is devoted to the proof of Theorem 1.5. First, using Schaefer s fixed-point theorem, we prove existence and uniqueness of a weak solution to (1.4) with h 0. Next, we apply this result to address the case of an arbitrary h C (). Let à be the subset of A (see (1.1)) given by à := {w H k () w 0}, and consider the mapping A : à H k 1 () à Hk 1 () defined, for (m 0,u 0 ) à H k 1 (), by ] ] m0 m A := 0, (6.1) u 0 where m 0 A is the unique solution to (4.3) and u 0 Hk 0 () is the unique solution to (5.). In the following proposition, we show that A is continuous and compact. Proposition 6.1. Let H,g,V,φ, and ξ be as in Problem. Then, the mapping A : à H k 1 () à Hk 1 () defined by (6.1) is continuous and compact. Proof. We start by proving that A is continuous. Let (m 0,u 0 ), (m n,u n ) à Hk 1 () be such that m n m 0 in H k () and u n u 0 in H k 1 (). We want to show that m n m 0 in Hk () and u n u 0 in Hk 1 (), where ] m 0 = A u 0 ] m0 u 0 and u 0 ] m n = A u n ] mn. u n Recalling (4.1) and (5.1), we set I n := I (mn,u n) and f n := f (mn,u n). By the definition of A, we have that (m 0,u 0) and (m n,u n) belong to A H k 0 () and satisfy, for all v L (), I 0 m 0] = min w A I 0w], I n m n] = min w A I nw], Bu 0, = f 0,v, Bu n, = f n,v. Also, using (4.), there exists a positive constant, c > 0, independent of n N, such that sup( m 0 L () + m n L () + u 0 W 1, () + u n W 1, ()) < c. (6.) n N Then, because H, D p H, and g are locally Lipschitz functions, we have c := max { Lip(H; B(0,c)),Lip(D p H; B(0,c)),Lip(g;B(0,c)) } R + 0. (6.3) Using the fact that m n and m 0 are minimizers, we have m I 0 m 0 ]+I nm n ] I 0 +m n 0 ] +I n m 0 +m n Then, exploiting the second equality in (4.7) in the preceding estimate first, and then using Young s inequality and (6.3), we obtain ǫ (m 4 0 m ( ] n) + α m 0 α m n) dx 1 m 0 m n ( u 0 u n + H(x,Du0 ) H(x,Du n ) + g(m0 ) g(m n ) ) dx (6.4) ǫ 8 (m 0 m n ) + 6 ( (u0 u n ) + c Du 0 Du n + c (m 0 m n ) )] dx. ǫ Because m n m 0 in H k () and u n u 0 in H k 1 (), (6.4) yields lim n m 0 m n L () = 0, lim α m n 0 α m n L () = 0. Then, by the Gagliardo Nirenberginterpolation inequality, we have m n m 0 in Hk (). ].

12 MEAN-FIELD GAMES WITH DIRICHLET CONDITIONS 11 Next, we show that u n converges to u 0 in H k (). By (5.1), (6.), and (6.3), we have ǫ ( u 0 u n L () + ) α u 0 α u n L () = Bu 0 u n,u 0 u n ] = f 0 f n,u 0 u n mn m 0 u 0 u n + m 0 D p H(x,Du 0 ) m n D p H(x,Du n ) Du 0 Du n ] dx mn m 0 u 0 u n + c m 0 m n Du 0 Du n +c Du 0 Du n Du 0 Du n ] dx. (6.5) Using Gagliardo Nirenberg interpolation inequality together with Young s inequality, we obtain from (6.5) that u 0 u n L () + α u 0 α u n L () C ( m 0 m n L () + Du 0 Du n ) L () forsomeconstantc > 0independent ofn N. Arguingasbefore, weconcludethatu n u 0 in H k (). Finally, we address the compactness of A. We want to show that if {(m n,u n )} n=1 is a bounded sequence in à Hk 1 (), then {A(m n,u n )} n=1 is pre-compact in à Hk 1 (). This is an immediate consequence of (4.), Corollary 4., Lemma 5., and the compact embedding H k () H k () H k () H k 1 () due to the Rellich Kondrachov theorem. As we mentioned before, the existence of weak solutions to Problem follows from Schaefer s fixed-point Theorem. We state next the version of this result that we use here, whose proof is a straightforward adaptation of the proof of Theorem 4, Section 9.., in 15]. Theorem 6.. Let X be a convex and closed subset of a Banach space with the property that λw X whenever w X and λ 0,1]. Assume that A : X X is a continuous and compact mapping such that the set { w X w = λaw] for some λ 0,1] } is bounded. Then, A has a fixed point. Proposition 6.3. Consider Problem, let A be the mapping defined in (6.1), and suppose that Assumptions 1 5 hold for some γ > 1. Then, there exists a unique weak solution, (m,u) H k () H k (), to Problem with h = 0 in the sense of Definition 1.3. Proof. (Existence) Fix λ 0,1], and let (m λ,u λ ) à Hk 1 () be such that ] ] mλ mλ = λa. u λ u λ If λ = 0, then (m λ,u λ ) = (0,0). Assume that 0 < λ 1; then, by the definition of A, Proposition 4.1, Corollary 4.3, and Proposition 5.1, we have m λ λ A, u λ λ H0 k (), and λ ( u λ H(x,Du λ )+g(m λ ) V ) (w m λ )dx + ǫm λ (w m λ )+ǫ α m λ ( α w α m λ )dx 0, λ ( m λ div ( m λ D p H(x,Du λ ) ) φ ) v + ǫ (u λ v + ] dx ) α u λ α v +ǫ(ξ + k ξ)v ] dx = 0,

13 1 RITA FERREIRA, DIOGO GOMES, AND TERUO TADA for all w A and v H0 k (). Hence, arguing as in Corollary 3., we have λm λ g(m λ )+m λ Du λ γ +φ Du λ γ ] dx +ǫ m λ +u λ + ( ( α m λ ) +( α u λ ) )] dx C, where C is a positive constant independent of λ. Consequently, by Assumption 5 and the positivity of m λ and φ, we have ǫ m λ +u λ + ( ( α m λ ) +( α u λ ) )] dx C where C is another positive constant independent of λ. Invoking the Gagliardo Nirenberg interpolationinequality, weconcludethat(m λ,u λ )isuniformlyboundedinh k () H k () with respect to λ. This fact and Proposition 6.1 allow us to use Theorem 6. to conclude that A has a fixed point, (m,u) Ã Hk 1 (). Finally, as before, using the definition of A, Proposition 4.1, Corollary 4.3, and Proposition 5.1, we conclude that (m, u) belongs to H k () H k () and satisfies (E1) (E3) with h = 0. (Uniqueness) Assume that there are two fixed points, (m 1,u 1 ) and (m,u ). Taking w = m in (E) for (u 1,m 1 ) and w = m 1 in (E) for (u,m ), and then summing the resulting inequalities, we have u1 u +H(x,Du 1 ) H(x,Du ) (g(m 1 ) g(m )) ] (m 1 m )dx ǫ(m 1 m ) +ǫ ( α m 1 α m ) ] dx 0. (6.6) Because u 1 u H0 k (), choosing v = u 1 u in (E3) for (u 1,m 1 ) and (u,m ), and then subtracting the resulting equalities, we obtain ( m 1 m div ( m 1 D p H(x,Du 1 ) m D p H(x,Du ) )) (u 1 u )dx + ǫ(u 1 u ) +ǫ ( α u 1 α u ) ] (6.7) = 0. Subtracting (6.6) from (6.7) and using Assumption 7, we have 0 ǫ(m 1 m ) +ǫ ( α m 1 α m ) +ǫ(u 1 u ) +ǫ ( α u 1 α u ) ] dx + F ] m1 F u 1 ] m, u ] m1 u 1 m ] 0. u Invoking Assumption 7 once more, we conclude that the integral in the preceding estimate is equal to zero, from which we the identity (m 1,u 1 ) = (m,u ) follows. Proof of Theorem 1.5. Define, for x and p R d, Ĥ(x,p) := H(x,p+Dh(x)), V(x) := V(x)+h(x), and ξ(x) := ξ(x)+h(x). Note that H satisfies Assumptions 1 3 for some γ > 1 if and only if Ĥ satisfies Assumptions 1 3 for the same γ. Moreover, (u,m) H k () H k () satisfies (E1) (E3) if and only if (û,m) := (u h,m) H k () H k () satisfies (E1) (E3) with h = 0 and with H, V, and ξ replaced by Ĥ, V, and ξ, respectively. To conclude, it suffices to invoke Proposition 6.3, which gives existence and uniqueness of a pair (û,m) H k () H k () satisfying (E1) (E3) with h = 0 and with H, V, and ξ replaced by Ĥ, V, and ξ, respectively.

14 MEAN-FIELD GAMES WITH DIRICHLET CONDITIONS Proof of Theorem 1. Here, we prove Theorem 1.. First, we study a compactness property of the unique weak solution to Problem. Then, we introduce a linear functional, F ǫ, corresponding to the equations (1.4) in Problem and address its monotonicity. Finally, using Minty s method, we prove the existence of a weak solution to Problem 1. Lemma 7.1. Consider Problem and suppose that Assumptions 1 6 hold for some γ > 1. Let (m ǫ,u ǫ ) H k () H k () be the unique weak solution to Problem. Then, there exists (m,u) L 1 () W 1,γ () such that m 0, u = h on in the sense of traces, and (m ǫ,u ǫ ) converges to (m,u) weakly in L 1 () W 1,γ () as ǫ 0, extracting a subsequence if necessary. Proof. The existence of u W 1,γ () as stated follows from the fact that u ǫ = h on in the sense of traces and, by Corollary 3.3, u ǫ is uniformly bounded in W 1,γ () with respect to ǫ. On the other hand, by Corollary 3. and the positivity of m ǫ and φ, we have sup ǫ (0,1) m ǫ g(m ǫ )dx <. Therefore, by Assumption 6, there exists m L 1 () such that m ǫ m in L 1 () as ǫ 0, extracting a subsequence if necessary. Because m ǫ 0, we have m 0. Fix (η,v) H k () H k (), let Fη, be the functional introduced in (1.3), and let F ǫ η, : H k () H k () R be the linear functional given by ] ] ] η w1 η w1 F ǫ, := F, + (ǫηw v w w 1 +ǫ α η α w 1 )dx + ǫ(v +ξ)w +ǫ (7.1) α (v +ξ) α w ]dx. Next, we prove the monotonicity of F ǫ over A Hh k (), where, we recall, A and Hk h () are given by (1.1) (1.). Lemma 7.. Let H,g,V,φ,ξ, and h be as in Problem, let F ǫ be given by (7.1), and suppose that Assumption 7 holds. Then, for any (η 1,v 1 ), (η,v ) A Hh k (), we have ] ] ] ] η1 η η1 η F ǫ F v ǫ, 0. 1 v v 1 v Proof. Let(η 1,v 1 ), (η,v ) A Hh k(). Then, v 1 v H0 k (); thus, using Assumption 7 and integrating by parts, we obtain ] ] ] ] η1 η η1 η F ǫ F v ǫ, 1 v v 1 v ǫ (η 1 η ) +(v 1 v ) + ( ( α η 1 α η ) +( α v 1 α v ) )] dx 0. Proof of Theorem 1.. Let (m ǫ,u ǫ ) H k () H k () be the unique weak solution to Problem in the sense of Definition 1.3. Fix (η,v) A Hh k (). By (E) and (E3), we have ] ] mǫ η mǫ F ǫ, 0. u ǫ u ǫ Thus, by Lemma 7., ] ] η mǫ 0 F ǫ F v ǫ, u ǫ ] ] ] η mǫ η η mǫ η η mǫ F u ǫ, = F, ǫ v u ǫ u ǫ ] +c ǫ, (7.)

15 14 RITA FERREIRA, DIOGO GOMES, AND TERUO TADA where c ǫ := (ǫη(η m ǫ )+ǫ + ǫ(v +ξ)(v u ǫ )+ǫ ) α η α (η m ǫ ) dx By Hölder s inequality and Corollary 3.4, we conclude that ] α (v +ξ) α (v u ǫ ) dx. lim ǫ 0 c ǫ = 0. (7.3) On the other hand, by Lemma 7.1, there exists (m,u) L 1 () W 1,γ () satisfying (D1) and such that (m ǫ,u ǫ ) converges to (m,u) weakly in L 1 () W 1,γ () as ǫ 0, extracting a subsequence if necessary. Then, using the definition of Fη, (see (1.3)), we get lim ǫ 0 F η, η mǫ ] = u ǫ F η, From (7.), (7.3), and (7.4), we conclude that η η m F, 0; u] η m u]. (7.4) that is, (m,u) also satisfies (D). Hence, (m,u) is a weak solution of Problem 1 in the sense of Definition 1.1. References 1] Y. Achdou. Finite difference methods for mean field games. In Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, pages Springer, 013. ] N. Almayouf, E. Bachini, A. Chapouto, R. Ferreira, D. Gomes, D. Jordão, D. Evangelista, A. Karagulyan, J. Monasterio, L. Nurbekyan, et al. Existence of positive solutions for an approximation of stationary mean-field games. Involve, a Journal of Mathematics, 10(3): , ] N. Almulla, R. Ferreira, and D. Gomes. Two numerical approaches to stationary mean-field games. Dynamic Games and Applications, pages 1 6, ] L. Boccardo, L. Orsina, and A. Porretta. Strongly coupled elliptic equations related to mean-field games systems. J. Differential Equations, 61(3): , ] L.M. Briceño Arias, D. Kalise, and F. J. Silva. Proximal methods for stationary mean field games with local couplings. Preprint, ] P. Cardaliaguet. Notes on Mean Field Games: from P.-L. Lions lectures at Collège de France. Lecture Notes given at Tor Vergata, ] P. Cardaliaguet, P. Garber, A. Porretta, and D. Tonon. Second order mean field games with degenerate diffusion and local coupling. NoDEA Nonlinear Differential Equations Appl., (5): , ] M. Cirant. A generalization of the Hopf Cole transformation for stationary Mean-Field Games systems. C. R. Math. Acad. Sci. Paris, 353(9): , ] M. Cirant. Stationary focusing mean-field games. Comm. Partial Differential Equations, 41(8): , ] M. Cirant, D. Gomes, E. Pimentel, and H. Sánchez-Morgado. Singular mean-field games. ArXiv e-prints, November ] M. Cirant and G. Verzini. Bifurcation and segregation in quadratic two-populations mean field games systems. Preprint, ] D. Evangelista, R. Ferreira, D. Gomes, L. Nurbekyan, and V. Voskanyan. First-order, stationary meanfield games with congestion. To appear in Nonlinear Analysis, ] D. Evangelista and D. Gomes. On the existence of solutions for stationary mean-field games with congestion. Journal of Dynamics and Differential Equations, pages 1 4, ] D. Evangelista, D. Gomes, and L. Nurbekyan. Radially symmetric mean-field games with congestion. In 017 IEEE 56th Annual Conference on Decision and Control (CDC), pages , Dec ] L. C. Evans. Partial Differential Equations. Graduate Studies in Mathematics. American Mathematical Society, ] R. Ferreira and D. Gomes. Existence of weak solutions for stationary mean-field games through variational inequalities. To appear in SIAM J. Math. Anal., ] I. Fonseca and G. Leoni. Modern methods in the calculus of variations: L p spaces. Springer Monographs in Mathematics. Springer, New York, ] D. Gomes and H. Mitake. Existence for stationary mean-field games with congestion and quadratic Hamiltonians. NoDEA Nonlinear Differential Equations Appl., (6): , ] D. Gomes, L. Nurbekyan, and M. Prazeres. Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion. 016 IEEE 55th Conference on Decision and Control, CDC 016, pages , 016.

16 MEAN-FIELD GAMES WITH DIRICHLET CONDITIONS 15 0] D. Gomes, L. Nurbekyan, and M. Prazeres. One-dimensional stationary mean-field games with local coupling. Dyn. Games and Applications, ] D. Gomes, S. Patrizi, and V. Voskanyan. On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal., 99:49 79, 014. ] D. Gomes, E. Pimentel, and V. Voskanyan. Regularity theory for mean-field game systems. Springer- Briefs in Mathematics. Springer, Cham], ] D. Gomes, G. E. Pires, and H. Sánchez-Morgado. A-priori estimates for stationary mean-field games. Netw. Heterog. Media, 7(): , 01. 4] D. Gomes and J. Saúde. Mean field games models a brief survey. Dyn. Games Appl., 4(): , ] M. Huang, P. E. Caines, and R. P. Malhamé. Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ǫ-nash equilibria. IEEE Trans. Automat. Control, 5(9): , ] M. Huang, R. P. Malhamé, and P. E. Caines. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst., 6(3):1 51, ] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris, 343(9):619 65, ] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris, 343(10): , ] J.-M. Lasry and P.-L. Lions. Mean field games. Jpn. J. Math., (1):9 60, ] P. L. Lions. Collége de France course on mean-field games ] A.R. Mészáros and F.J. Silva. A variational approach to second order mean field games with density constraints: The stationary case. J. Math. Pures Appl. (9), 104(6): , ] E. Pimentel and V. Voskanyan. Regularity for second-order stationary mean-field games. Indiana Univ. Math. J., 66(1):1, 017. (R. Ferreira) King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal , Saudi Arabia. address: rita.ferreira@kaust.edu.sa (D. Gomes) King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal , Saudi Arabia. address: diogo.gomes@kaust.edu.sa (T. Tada) King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal , Saudi Arabia. address: teruo.tada@kaust.edu.sa