Weakly coupled mean-field game systems
aa r X i v : . [ m a t h . A P ] O c t WEAKLY COUPLED MEAN-FIELD GAME SYSTEMS
DIOGO A. GOMES AND STEFANIA PATRIZI
Abstract.
Here, we prove the existence of solutions to first-order mean-fieldgames (MFGs) arising in optimal switching. First, we use the penalizationmethod to construct approximate solutions. Then, we prove uniform estimatesfor the penalized problem. Finally, by a limiting procedure, we obtain solutionsto the MFG problem. Introduction
The mean-field game (MFG) framework [34, 35, 36, 37, 38, 39] is a class ofmethods used to study large populations of rational, non-cooperative agents. MFGshave been the focus of intense research, see, for example, the surveys [28, 31]. Here,we investigate MFGs that arise in optimal switching. These games are given bya weakly coupled system of Hamilton-Jacobi equations of the obstacle type and acorresponding system of transport equations.To simplify the presentation, we use periodic boundary conditions. Thus, thespatial domain is the N -dimensional flat torus, T N . Our MFG is determined bya value function, u : T N → R d , a probability density, θ : T N → ( R + ) d , anda switching current, ν, that together satisfy the following system of variationalinequalities:(1.1) max (cid:18) H i ( Du i , x ) + u i − g ( θ i ) , max j (cid:0) u i − u j − ψ ij (cid:1)(cid:19) = 0coupled with the system(1.2) − div( D p H i ( Du i , x ) θ i ) + θ i + X j = i (cid:0) ν ij − ν ji (cid:1) = 1 . Moreover, for 1 ≤ i, j ≤ d , ν ij is a non-negative measure on T N supported in theset u i − u j − ψ ij = 0.This system models a stationary population of agents. Each agent moves in T N and can switch between different modes that are given by the index i . Their actionsseek to minimize a certain cost. Agents can change their state by continuouslymodifying their spatial position, x , and by switching between different modes, i to j , at a cost ψ ji . The function u i ( x ) is the value function for an agent whose spatiallocation is x and whose mode is i . The function θ i ( x ) is the density of the agentson T N × { , . . . , d } . Thus, we require that θ i ( x ) ≥
0. We note that θ i is not a Date : July 30, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Mean Field Games; Weakly coupled systems; Optimal Switching.D. Gomes was partially supported by KAUST baseline and start-up funds. S. Patrizi waspartially supported by NSF grant DMS-1262411 “Regularity and stability results in variationalproblems”. probability measure on T N × { , . . . , d } because the source term in the right-handside of (1.2) is not normalized.In Section 2, we discuss detailed assumptions on the Hamiltonians H i , on thenonlinearity g , and on the switching costs ψ ij . A concrete example that satisfiesthose is(1.3) H i ( x, p ) = | p | V i ( x ) , g ( θ ) = ln θ, and ψ ij ( x ) = η, with V i : T N → R being a C ∞ function and η being a positive real number.Another case of interest is the polynomial nonlinearity, g ( m ) = m α for α > H i ( Du i , x ) + u i + X j = i β ǫ ( u i − u j − ψ ij ) = g ( θ i )(1.5) − div( D p H i ( Du i , x ) θ i )+ θ i + X j = i β ′ ǫ ( u i − u j − ψ ij ) θ i − β ′ ǫ ( u j − u i − ψ ji ) θ j = 1 , where the penalty function, β ǫ , is an increasing C ∞ function and ǫ > . We assumethat, as ǫ → β ǫ ( s ) → ∞ for s > β ǫ ( s ) = 0 for s ≤
0, see Assumption 8.The study of optimal switching has a long history that predates viscosity solutionsand, certainly, MFGs, see, for example [2, 6, 7, 17]. In those references, the use of apenalty to approximate a non-smooth Hamilton-Jacobi equation is a fundamentaltool. The penalty in (1.4) is similar to the ones in the aforementioned references.More recently, several authors have investigated weakly coupled Hamilton-Jacobiequations [44], the corresponding extension of the weak KAM and Aubry-Mathertheories [5, 14, 40], the asymptotic behavior of solutions [4, 3, 42, 41, 45], andhomogenization [43]. In these references, the state of the system has differentmodes, and a random process drives the switching between them. In contrast,here, the switches occur at deterministic times. Thus, our models are the MFGcounterpart of the Hamilton-Jacobi systems considered in [18, 32]. MFGs withdifferent populations [12, 13] are a limit case of (1.1)-(1.2). This can be seen bytaking the limit ψ ij → + ∞ ; that is, the case where agents are not allowed to changetheir state.The development of the existence and regularity theory for MFGs has seen sub-stantial progress in recent years. Uniformly elliptic and parabolic MFGs are nowwell understood, and the existence of smooth and weak solutions has been estab-lished in a broad range of problems, see, respectively, [23, 24, 25, 30, 29, 27, 26]and [10, 46, 47]. However, the regularity theory for first-order MFGs is less devel-oped and, in general, only weak solutions are known to exist [8, 9, 11]. Variationalinequality methods are at the heart of a new class of techniques to establish theexistence of weak solutions, both for first- and second-order problems [19] and fortheir numerical approximation [1]. Some MFGs that arise in applications, suchas congestion [20, 33] or obstacle-type problems [22], feature singularities. Thus,there is keen interest in developing methods for their analysis. To the best of ourknowledge, this paper is the first to address MFGs arising in optimal switching. EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 3
Moreover, our techniques contribute to better understanding of the regularity offirst-order MFGs.Our main result is the following theorem.
Theorem 1.1.
Suppose that Assumptions 1-4 (see Section 2) hold and that either - Assumption 5 L or - Assumptions 5 P- N , 6 and 7hold. Then, there exists a solution, ( u, θ ) , of (1.1) - (1.2) , with u ∈ ( W , ( T N )) d ∩ ( C γ ( T N )) d for some γ ∈ (0 , and θ ∈ ( W , ( T N )) d . As mentioned before, to prove the existence of solutions for (1.1)-(1.2), we firstexamine the existence of solutions for (1.4)-(1.5), prove ǫ independent bounds and,subsequently, consider the limit ǫ →
0. On the existence of solutions, our mainresult is the following theorem.
Theorem 1.2.
Suppose that Assumptions 1-4 (see Section 2) and 8 hold, andeither - Assumption 5 L or - Assumptions 5 P- N , 6 and 7hold. Then, there exists a unique solution, ( u, θ ) ∈ ( C ∞ ( T N )) d × ( C ∞ ( T N )) d , of (1.4) - (1.5) with θ i ≥ θ > for some constant θ that does not depend on ǫ . To prove Theorem 1.1, we establish the existence of solutions for (1.4)-(1.5) inTheorem 1.2 and prove ǫ independent bounds. The analysis of (1.4)-(1.5) beginsin Section 3 where we examine various a priori estimates. Next, in Section 4, weconsider separately the two different nonlinearities, g ( m ) = ln m and g ( m ) = m α .In these two sections, our estimates are uniform in ǫ . In contrast, in Section 5,we prove L ∞ estimates for θ and Lipschitz bounds for u that depend on ǫ . Theseare crucial in the proof of Theorem 1.2 that we present in Section 6. This proofcombines the a priori estimates with the continuation method. The paper endswith the proof of Theorem 1.1 in Section 7 and a brief discussion of convergenceand uniqueness in Section 8. 2. Main assumptions
We begin by discussing the assumptions on H i , g, and ψ used in the study of(1.1)-(1.2). On the Hamiltonian, H i , we assume standard hypotheses that hold ina large class of problems. In particular, they are satisfied by the example (1.3). Tosimplify the presentation, we select assumptions compatible with quadratic growthof the Hamiltonian, see Remark 2.3 below. Regarding the dependence on the mea-sure: for every coordinate, i , we have the same nonlinearity, g , evaluated at thecoordinate θ i . Some of our estimates are valid without substantial changes in thecorresponding proofs if g is replaced by a function, g i , depending on all coordinatesof θ or even on x . Naturally, Assumption 2 must be modified in a suitable way.Finally, we work with positive switching costs, ψ ij . The positivity condition isnatural in optimal switching because it prevents the occurrence of infinitely manyswitches. These conditions and the assumptions that follow are unlikely to give themost general case under which our techniques hold. Our choice reflects a balancebetween generality and simplicity of the proofs. DIOGO A. GOMES AND STEFANIA PATRIZI
Assumption 1.
The Hamiltonian, H i , the nonlinearity, g, and the switching cost, ψ ij , satisfy: (1) For ≤ i ≤ d , H i : T N × R N → R is C ∞ and positive. (2) g : R + → R is C ∞ and strictly increasing; that is, g ′ > . (3) For ≤ i, j ≤ d , the function ψ ij : T N → R is of class C ∞ ( T N ) . Further-more, for x ∈ T N , ψ ij ( x ) > . As usual, we identify whenever convenient, functions in T N as Z N -periodic func-tions in R N . Assumption 2.
The function g satisfies the following. (1) For any C > , there exists C such that (2.1) Z T N θg ( θ ) dx ≥ − C for any non-negative θ ∈ L ( T N ) with R T N θdx ≤ C . (2) There exists
C > such that, for any θ > , (2.2) g ( θ ) ≤ θg ( θ ) + C. Remark 2.1.
The functions g ( θ ) = ln θ and g ( θ ) = θ α , for α > , satisfy thepreceding assumption. Assumption 3.
There exist constants, c, C ≥ , such that (2.3) H i ( p, x ) − D p H i ( p, x ) · p ≤ − cH i ( p, x ) + C for all p ∈ R N , x ∈ T N , and ≤ i ≤ d . Remark 2.2.
Consider the Lagrangian, L i , associated with the Hamiltonian H i given by L i ( x, v ) = sup p ∈ R N − p · v − H i ( p, x ) . Because the supremum is achieved for v = − D p H i ( p, x ) , L i ( x, v ) = D p H i ( p, x ) · p − H i ( p, x ) . Accordingly, the preceding hypothesis gives a lower bound on L i . Assumption 4.
There exists γ > such that (2.4) H ip k p j ( p, x ) ξ k ξ j ≥ γ | ξ | for all x ∈ T N and p, ξ ∈ R N .There exist C, c > such that | D pp H i | ≤ C, | D xp H i | ≤ C (1 + | p | ) , | D xx H i | ≤ C (1 + | p | ) . (2.5) Remark 2.3.
The preceding assumption implies that there exists
C > such that (2.6) γ | p | − C ≤ H i ( p, x ) ≤ C | p | + C, EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 5 and | D p H i ( p, x ) | ≤ C (1 + | p | ) , | D x H i ( p, x ) | ≤ C (1 + | p | )(2.7) for all p ∈ R N and x ∈ T N . Assumption 5.
There exist constants, C, e C > , and α ≥ such that (2.8) Cθ α − ≤ g ′ ( θ ) ≤ e Cθ α − + e C for any θ ≥ . Two specific cases of interest are L - g ( θ ) = ln θ ; P - g ( θ ) = θ α , ≤ α ≤ .In the P case, the additional constraint, α < N , is denoted by P- N . The next two assumptions are of a technical nature and are used in the studyof the P- N case. Assumption 6 is employed in Proposition 4.3 to obtain a lowerbound for θ i . Assumption 7 is fundamental in the proof of Proposition 5.1. Assumption 6.
For ≤ i ≤ d , we have (2.9) D px H i (0 , x ) = D x H i (0 , x ) D p H i (0 , x ) = 0 for any x ∈ T N and (2.10) max x ∈ T N H i (0 , x ) < . Remark 2.4.
The preceding assumption is used to prove lower bounds for θ i . Be-cause H i ≥ , the bound (2.10) gives an oscillation condition for H i (0 , x ) . Thisoscillation condition is natural in light of the example considered in [28] , Chapter 3.In that reference and also in [21] , various examples of first-order MFGs are shownto have a vanishing density. The oscillation of H (0 , x ) plays an essential role inthese examples. Remark 2.5.
The number on the right-hand side of (2.10) corresponds to thesource term in the Fokker-Planck equation (1.5) . Suppose that we modify (1.5) andconsider a source, υ > ; that is, − div ( D p H i ( Du i , x ) θ i ) + θ i + X j = i β ′ ǫ ( u i − u j − ψ ij ) θ i − β ′ ǫ ( u j − u i − ψ ji ) θ j = υ. Then, (2.10) becomes max x ∈ T N H i (0 , x ) < υ. Assumption 7.
The value α in Assumption 5 satisfies α ∈ [0 , α ) , where α solves (2.11) 2 α = ( α + 1) β ( β − , with β = r ∗ , if N > , and α = ∞ if N ≤ . Remark 2.6.
In the N ≤ case, the value α determined by (2.11) is larger than N . Whereas, if N > the opposite inequality holds. Our last assumption is required in the study of the penalized problem. For ǫ > β ǫ , satisfying the following assumption. DIOGO A. GOMES AND STEFANIA PATRIZI
Assumption 8. β ǫ : R → R , smooth, with β ′ ǫ ≥ , β ′′ ǫ ≥ with (2.12) β ǫ ( s ) = 0 for s ≤ , β ′ ǫ ( s ) ≤ Cβ ′′ ǫ ( s ) for s > , and β ǫ ( s ) → ∞ as ǫ → for s > . Remark 2.7.
From the preceding assumption, we get (2.13) β ǫ ( s ) − sβ ′ ǫ ( s ) ≤ for s ∈ R . The preceding assumption is standard in the setting of variational inequalitiesand optimal switching. In the context of MFGs, a similar penalty was used in [22]to study the obstacle problem.3.
A priori estimates
Here, we prove a priori estimates for classical solutions of (1.4)-(1.5). The pur-pose of these estimates is twofold: first, to obtain the existence of solutions; second,to take the limit ǫ →
0. For that, we seek to prove bounds that are uniform in ǫ .We begin with a simple consequence of the maximum principle for weakly coupledsystems. Proposition 3.1.
Suppose that Assumptions 1 and 8 hold. Let ( u, θ ) be a C ∞ solution of (1.4) - (1.5) . Then, for i = 1 , . . . , d , we have θ i ≥ .Proof. The proof of this Lemma is a straightforward application of the maximumprinciple to (1.5), see [5] for a similar proof. (cid:3)
As is standard in MFG problems, we can get several estimates by multiplying(1.4)-(1.5) by 1, θ i or u i , adding or subtracting, and integrating by parts. We recordthese in the next lemma. Lemma 3.2.
Suppose that Assumptions 1-3 and 8 hold. Let ( u, θ ) be a C ∞ solu-tion of (1.4) - (1.5) . Then, there exists a constant, C, that does not depend on theparticular solution nor on ǫ , such that, for i = 1 . . . d , (3.1) 0 ≤ Z T N θ i dx ≤ C, (3.2) (cid:12)(cid:12)(cid:12)(cid:12)Z T N θ i g ( θ i ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C, (3.3) (cid:12)(cid:12)(cid:12)(cid:12)Z T N u i dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C, (3.4) Z T N | Du i | θ i dx ≤ C, (3.5) d X i,j =1 ,i = j Z T N β ′ ǫ ( u i − u j − ψ ij ) ψ ij θ i dx ≤ C, and (3.6) Z T N | Du i | dx ≤ C. EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 7
Proof.
By summing over i the equations in (1.5), we gather that d X i =1 − div( D p H i ( Du i , x ) θ i ) + θ i = d. Hence, integrating on T N , we get Z T N θ i dx ≤ d X i =1 Z T N θ i dx = d for any i = 1 , ..., d . Thus, (3.1) holds. Due to Assumptions 1 and 8, H i and β ǫ arenon-negative. Consequently, we infer that(3.7) Z T N u i dx ≤ Z T N g ( θ i ) dx. Next, we multiply (1.4) by θ i , sum over i, and integrate. Accordingly, we gatherthe identity d X i =1 Z T N H i ( Du i , x ) θ i + u i θ i + X j = i β ǫ ( u i − u j − ψ ij ) θ i dx = d X i =1 Z T N θ i g ( θ i ) dx. (3.8)Next, we multiply (1.5) by u i , add over i, and integrate by parts to conclude that d X i =1 Z T N h D p H i ( Du i , x ) · Du i θ i + u i θ i + X j = i β ′ ǫ ( u i − u j − ψ ij ) θ i u i − β ′ ǫ ( u j − u i − ψ ji ) θ j u i i dx = d X i =1 Z T N u i dx. (3.9)Subtracting equations (3.8) and (3.9), we get d X i =1 Z T N θ i g ( θ i ) dx = d X i =1 Z T N H i ( Du i , x ) θ i + u i θ i + X j = i β ǫ ( u i − u j − ψ ij ) θ i dx = d X i =1 Z T N ( H i ( Du i , x ) − D p H i ( Du i , x ) · Du i ) θ i + u i dx + d X i,j =1 ,i = j Z T N β ǫ ( u i − u j − ψ ij ) θ i dx + d X i,j =1 ,i = j Z T N − β ′ ǫ ( u i − u j − ψ ij ) θ i u i + β ′ ǫ ( u j − u i − ψ ji ) θ j u i dx. DIOGO A. GOMES AND STEFANIA PATRIZI
According to Assumption 3, we have d X i =1 Z T N ( H i ( Du i , x ) − D p H i ( Du i , x ) · Du i ) θ i dx ≤ − c d X i =1 Z T N H i ( Du i , x ) θ i dx + C using (3.1). Moreover, we have d X i,j =1 ,i = j Z T N β ǫ ( u i − u j − ψ ij ) θ i dx + d X i,j =1 ,i = j Z T N − β ′ ǫ ( u i − u j − ψ ij ) θ i u i + β ′ ǫ ( u j − u i − ψ ji ) θ j u i dx = d X i,j =1 ,i = j Z T N [ β ǫ ( u i − u j − ψ ij ) − β ′ ǫ ( u i − u j − ψ ij )( u i − u j − ψ ij )] θ i dx − d X i,j =1 ,i = j Z T N β ′ ǫ ( u i − u j − ψ ij ) ψ ij θ i dx ≤ − d X i,j =1 ,i = j Z T N β ′ ǫ ( u i − u j − ψ ij ) ψ ij θ i dx by (2.13) in Remark 2.7. Gathering the previous estimates, we conclude that d X i =1 Z T N θ i g ( θ i ) + cH i ( Du i , x ) θ i dx + d X i,j =1 ,i = j Z T N β ′ ǫ ( u i − u j − ψ ij ) ψ ij θ i dx ≤ d X i =1 Z T N u i dx + C ≤ d X i =1 Z T N g ( θ i ) dx + C (3.10)using (3.7). Then, Assumption 2 implies Z T N θ i g ( θ i ) dx ≤ C. On the other hand, (2.1) in Assumption 2 and (3.1) give Z T N θ i g ( θ i ) dx ≥ − C. Therefore, (3.2) holds. Using (3.2) and the bound (2.2) from Assumption 2, we get(3.3). In addition, for any i = 1 , ..., d , Z T N H i ( Du i , x ) θ i dx ≤ C. The last estimate combined with (2.4) implies (3.4). A similar argument yields(3.5).
EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 9
Finally, the bound (3.6) follows from (1.4) by combining (2.4), the non-negativityof β ǫ , and the previous results with the estimate Z T N | Du i | dx ≤ C + Z T N ( g ( θ i ) − u i ) dx ≤ C. (cid:3) Lemma 3.3.
Suppose that Assumptions 1-5 and 8 hold. Let ( u, θ ) be a C ∞ solu-tion of (1.4) - (1.5) . Then, there exists a constant, C, that does not depend on theparticular solution nor on ǫ , such that, for i = 1 , . . . , d , (3.11) Z T N | D u i | θ i dx ≤ C, (3.12) Z T N g ′ ( θ i ) | Dθ i | dx ≤ C, and (3.13) k ( θ i ) α +12 k W , ( T N ) ≤ C. Proof.
We begin by differentiating (1.4) twice with respect to x k and then summingover k . In this way, we get D p H i · D (∆ u i ) + H ix k x k + 2 H ix k p l u ix l x k + H ip l p m u ix l x k u ix m x k + ∆ u i + X j = i β ′ ǫ ( u i − u j − ψ ij )∆( u i − u j − ψ ij ) + β ′′ ǫ ( u i − u j − ψ ij ) | D ( u i − u j − ψ ij ) | = ∆( g ( θ i )) . Next, we multiply the previous equation by θ i , add in the index i, and integrate byparts to conclude that d X i =1 Z T N ∆( g ( θ i )) θ i dx = d X i =1 Z T N ( D p H i · D (∆ u i ) + ∆ u i + X j = i β ′ ǫ ( u i − u j − ψ ij )∆ u i ) θ i dx − d X i =1 X j = i Z T N β ′ ǫ ( u i − u j − ψ ij )∆( u j + ψ ij ) θ i dx + Z T N ( H ix k x k + 2 H ix k p l u ix l x k + H ip l p m u ix l x k u ix m x k ) θ i dx + d X i =1 X j = i Z T N β ′′ ǫ ( u i − u j − ψ ij ) | D ( u i − u j − ψ ij ) | θ i dx. (3.14) Multiplying (1.5) by ∆ u i and integrating by parts results in d X i =1 Z T N ( D p H i · D (∆ u i ) + ∆ u i + X j = i β ′ ǫ ( u i − u j − ψ ij )∆ u i ) θ i dx = d X i =1 X j = i Z T N ( β ′ ǫ ( u j − u i − ψ ji )∆ u i θ j + ∆ u i dx = d X i =1 X j = i Z T N ( β ′ ǫ ( u i − u j − ψ ij )∆ u j θ i dx. Using the previous identity in (3.14) gives d X i =1 Z T N ∆( g ( θ i )) θ i dx = − d X i =1 X j = i Z T N β ′ ǫ ( u i − u j − ψ ij )∆ ψ ij θ i dx + Z T N ( H ix k x k + 2 H ix k p l u ix l x k + H ip l p m u ix l x k u ix m x k ) θ i dx + d X i =1 X j = i Z T N β ′′ ǫ ( u i − u j − ψ ij ) | D ( u i − u j − ψ ij ) | θ i dx. Taking into account that ∆ ψ ij is bounded and ψ ij >
0, estimate (3.5) implies that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i,j =1 ,i = j Z T N β ′ ǫ ( u i − u j − ψ ij )∆ ψ ij θ i dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C. Because β ′′ ǫ ≥ d X i =1 Z T N g ′ ( θ i ) | Dθ i | dx + C d X i =1 Z T N | D u i | θ i dx ≤ C d X i =1 Z T N (1 + | Du i | ) θ i dx ≤ C. (3.15)Hence, we have (3.11) and (3.12). Moreover, from (3.12) and (2.8), we infer that Z T N ( θ i ) α − | Dθ i | dx ≤ C Z T N g ′ ( θ i ) | Dθ i | dx ≤ C ;that is, | D ( θ i ) α +12 | ∈ L ( T N ). By (3.1), θ i ∈ L ( T N ). Thus, the previous inequalityand the Poincar´e inequality imply (3.13). (cid:3) Further a priori estimates
In this section, we prove additional a priori estimates for logarithmic (Assump-tion 5 L ) and power-like nonlinearities (Assumptions 5 P and P- N ). These twocases are examined separately. Nevertheless, for both the logarithmic nonlinearityand for the power case, if α < N (Assumption 5 P- N ), we obtain similar, ǫ -independent lower bounds on θ , on k u k W , ( T N ) , and on k u k W ,p ( T N ) for any p ≥ EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 11
The logarithmic case.
Here, we consider the logarithmic nonlinearity g ( θ ) =ln θ . Proposition 4.1.
Suppose that Assumptions 1-4, 5 L and 8 hold. Let ( u, θ ) be a C ∞ solution of (1.4) - (1.5) . Then, there exist constants C, C p , θ > that do notdepend on the particular solution nor on ǫ , such that, for i = 1 , . . . , d and for any p ∈ [1 , + ∞ ) , (4.1) k u i k W ,p ( T N ) ≤ C p . Moreover, for any γ ∈ (0 , , (4.2) k u i k C γ ( T N ) ≤ C. In addition, (4.3) θ i ≥ θ in T N , (4.4) k θ i k W , ( T N ) ≤ C, and (4.5) k u i k W , ( T N ) ≤ C. Proof.
In what follows, we use C and C p to denote any of several constants, possiblydepending on p but independent of ǫ . We remark that, for any p ≥
1, there existsa constant, C p > , such that log( θ i ) ≤ ( θ i ) p + C p . Therefore, from (1.4), using (2.6) in Remark 2.3 and the positivity of β ǫ , we inferthat C | Du i | ≤ ( θ i ) p + C p − u i = ( θ i ) p + C p − (cid:18) u i − Z T N u i dx (cid:19) − Z T N u i dx. Combining the previous inequality with (3.3) yields | Du i | p ≤ Cθ i + C (cid:12)(cid:12)(cid:12)(cid:12) u i − Z T N u i dx (cid:12)(cid:12)(cid:12)(cid:12) p + C p . Then, integrating, using (3.1) and the Poincar´e inequality, we get Z T N | Du i | p dx ≤ C Z T N θ i dx + C Z T N (cid:12)(cid:12)(cid:12)(cid:12) u i − Z T N u i dx (cid:12)(cid:12)(cid:12)(cid:12) p dx + C p ≤ C p Z T N | Du i | p dx + C p ≤ Z T N | Du i | p dx + C p . We conclude that, for any p ≥ Z T N | Du i | p dx ≤ C. This bound, together with (3.3) and the Poincar´e inequality, gives (4.1). TheSobolev Embedding Theorem then implies (4.2). In particular, we have k u i k L ∞ ( T N ) ≤ C. From (1.4), the previous estimate and the positivity of H and β , we infer thatlog( θ i ) ≥ − C, from which (4.3) follows.Estimate (4.4) is a consequence of (3.1), (3.12), (4.3) and the Poincar´e inequality.Finally, estimate (4.5) is a consequence of (3.3), (3.6), (3.11), (4.3) and thePoincar´e inequality. (cid:3) Power case.
We devote this section to the study of power nonlinearities. Webegin by examining the general case, Assumption 5 P . Then, we obtain additionalresults by considering Assumption 5 P- N . As in the previous section, our estimatesare uniform in ǫ . Proposition 4.2.
Suppose that Assumptions 1-4, 5 P and 8 hold. Let ( u, θ ) be a C ∞ solution of (1.4) - (1.5) . Then, there exist constants, C > and γ ∈ (0 , , thatdo not depend on the particular solution nor on ǫ , such that, for i = 1 , . . . , d, (4.6) k u i k W , α ( T N ) ≤ C and (4.7) Z T N | D (( θ i ) α +12 Du i ) | α +1 dx ≤ C. If, in addition, Assumption 5 P- N holds, then there exists γ = γ ( α ) such that (4.8) k u i k C γ ( T N ) ≤ C. Proof.
In what follows, we denote by C several constants that are independent of ǫ and δ . From (1.4), (2.6), and β ǫ ≥
0, we infer that C | Du i | ≤ ( θ i ) α + C − u i = ( θ i ) α + C − (cid:18) u i − Z T N u i dx (cid:19) − Z T N u i dx. Consequently, from (3.3), | Du i | α ≤ Cθ i + C (cid:12)(cid:12)(cid:12)(cid:12) u i − Z T N u i dx (cid:12)(cid:12)(cid:12)(cid:12) α + C. Then, integrating and using (3.1) and the Poincar´e inequality, we get Z T N | Du i | α dx ≤ C Z T N θ i dx + C Z T N (cid:12)(cid:12)(cid:12)(cid:12) u i − Z T N u i dx (cid:12)(cid:12)(cid:12)(cid:12) α dx + C ≤ C Z T N | Du i | α dx + C ≤ Z T N | Du i | α dx + C. Therefore, Z T N | Du i | α dx ≤ C, which gives, together with (3.3) and the Poincar´e inequality, the bound (4.6). If α ∈ (cid:0) , N (cid:1) , then α > N . Therefore, estimate (4.8) is a consequence of (4.6)combined with the Sobolev Embedding Theorem. EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 13
Next, to prove (4.7), we compute(4.9) D (( θ i ) α +12 Du i ) = α + 12 θ α − Du i ⊗ Dθ i + θ α +12 D u i . Now, using H¨older’s inequality, we have Z T N h θ α − | Dθ i || Du i | i α +1 dx = Z T N ( θ i ) α − α +1 | Dθ i | α +1 | Du i | α +1 dx ≤ (cid:20)Z T N h ( θ i ) α − α +1 | Dθ i | α +1 i α +1 dx (cid:21) α +1 (cid:20)Z T N | Du i | α +1 ( α +1) ′ dx (cid:21) α +1) ′ = (cid:20)Z T N ( θ i ) α − | Dθ i | dx (cid:21) α +1 (cid:20)Z T N | Du i | α dx (cid:21) αα +1 . From (3.13) and (4.6), we infer that(4.10) Z T N h θ α − | Dθ i || Du i | i α +1 dx ≤ C. Next, using H¨older’s inequality again, we have Z T N h | D u i | ( θ i ) α +12 i α +1 dx = Z T N | D u i | α +1 θ i dx = Z T N (cid:2) | D u i | θ i (cid:3) α +1 ( θ i ) αα +1 dx ≤ (cid:20)Z T N | D u i | θ i dx (cid:21) α +1 (cid:20)Z T N θ i dx (cid:21) αα +1 . From (3.1) and (3.11), we gather the bound(4.11) Z T N h | D u i | ( θ i ) α +12 i α +1 dx ≤ C. Estimate (4.7) is then a consequence of (4.9), (4.10) and (4.11). (cid:3)
Proposition 4.3.
Suppose that Assumptions 1-4, 5 P- N , 6, and 8 hold. Let ( u, θ ) be a C ∞ solution of (1.4) - (1.5) . Then, for i = 1 , . . . , d and any x ∈ T N , we have (4.12) θ i ( x ) ≥ − max x ∈ T Nj =1 ,...,d H j (0 , x ) α . Moreover, there exists
C > that does not depend on the particular solution noron ǫ , such that (4.13) k θ i k W , ( T N ) ≤ C and (4.14) k u i k W , ( T N ) ≤ C. Proof.
We begin the proof by establishing a lower bound on u . Let i ∈ { , . . . , d } and x ∈ T N be such that u i ( x ) = min j =1 ,...,dx ∈ T N u j ( x ) . Then, we have(4.15) Du i ( x ) = 0 , D u i ( x ) ≥ , and u i ( x ) ≤ u j ( x ) for any j = 1 , . . . , d. In particular, the last inequality implies(4.16) β ǫ ( u i ( x ) − u j ( x ) − ψ ij ( x )) = β ′ ǫ ( u i ( x ) − u j ( x ) − ψ ij ( x )) = 0 for any j = 1 , . . . , d. From (1.4), (4.15), and (4.16), we infer that(4.17) H i (0 , x ) + u i ( x ) = g ( θ i ( x )) . We can substitute θ i = g − H i ( Du i , x ) + u i + X j = i β ǫ ( u i − u j − ψ ij ) in (1.5) to get − θ i H ip k p j u ix j x k − θ i H ip k x k − g ′ ( θ i ) (cid:2) H ip k H ip j u ix j x k + H ix k H ip k + H ip k u ix k + X j = i β ′ ǫ ( u i − u j − ψ ij )( u i − u j − ψ ij ) x k H ip k (cid:3) + θ i + X j = i β ′ ǫ ( u i − u j − ψ ij ) θ i = X j = i β ′ ǫ ( u j − u i − ψ ji ) θ j + 1 . Evaluating at x = x and using (2.4), (4.15) and (4.16), we obtain − θ i H ip k x k (0 , x ) − g ′ ( θ i ( x )) H ix k (0 , x ) H ip k (0 , x ) + θ i ( x ) ≥ . Since H i satisfies (2.9), the preceding inequality can be rewritten as θ i ( x ) ≥ . Then, (4.17) and the last estimate imply(4.18) u i ( x ) ≥ g − (cid:0) − H i (0 , x ) (cid:1) . Now, from (1.4), (4.18), (2.9), (2.10) and the positivity of H j and β ǫ , we infer that,for any x ∈ T N and j = 1 , . . . , d , g ( θ j ( x )) ≥ H j ( Du j , x ) + u j ( x ) ≥ u i ( x ) ≥ − H i (0 , x ) . Thus, (4.12) follows from the preceding inequality.Estimate (4.13) follows by combining (3.1), (3.12), (4.12) and the Poincar´e in-equality. Finally, (4.14) is a consequence of (3.3), (3.6), (3.11), (4.12), and thePoincar´e inequality. (cid:3)
EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 15 Lipschitz bounds
In this section, we prove the Lipschitz continuity of u for any solution ( u, θ )of (1.4)-(1.5). These bounds are used to establish the existence of solutions bythe continuation method. In contrast to the results in the preceding sections, theestimates here depend on ǫ and are not valid for (1.1)-(1.2). Lemma 5.1.
Suppose that Assumptions 1-5 and 8 hold, and that either - Assumption 5 L or - Assumptions 5 P- N , 6 and 7hold. Let ( u, θ ) be a C ∞ solution of (1.4) - (1.5) . Then, there exists C ǫ > thatdoes not depend on the particular solution, such that, for any i = 1 , . . . , d , (5.1) k θ i k L ∞ ( T N ) ≤ C ǫ and (5.2) k u i k W , ∞ ( T N ) ≤ C ǫ . Proof.
First, note that (5.2) is an immediate consequence of (5.1). Indeed, bycombining (5.1) with (2.6), by the positivity of β ǫ , by the boundedness of u i (c.f.Proposition 4.1 and Proposition 4.2), and by (1.4), we get C | Du i | ≤ H i ( Du i ) + C ≤ g ( θ i ) − u i + C ≤ C. Consequently, we only need to prove (5.1).If Assumption 5 L holds or if Assumptions 5 P- N and 6 hold by, respectively,Propositions 4.1 and 4.3, then there exists θ >
0, such that, for any i = 1 , . . . , d and for any x ∈ T N , (5.3) θ i ( x ) ≥ θ > . For any fixed ǫ , by (4.2) and (4.8), there exists a constant, C, depending on ǫ ,such that, for any i, j = 1 , . . . , d ,(5.4) β ′ ǫ ( u i − u j − ψ ij ) ≤ C. To prove (5.1), we use a technique introduced in [16] and used in [22] to studya mean-field-game obstacle problem. For p >
0, we multiply the equation (1.5) by div(( θ i ) p D p H i ( Du i , x )) and integrate by parts. Accordingly, we get Z T N [ θ i + X j = i β ′ ǫ ( u i − u j − ψ ij ) θ i − X j = i β ′ ǫ ( u j − u i − ψ ji ) θ j ]div(( θ i ) p D p H i ) dx = Z T N ( θ i H ip k ) x k (( θ i ) p H ip j ) x j dx = Z T N ( θ i H ip k ) x j (( θ i ) p H ip j ) x k dx = Z T N ( θ i ( H ip k ) x j + θ ix j H ip k )(( θ i ) p ( H ip j ) x k + p ( θ i ) p − θ ix k H ip j ) dx = Z T N ( θ i ) p +1 ( H ip k ) x j ( H ip j ) x k + p ( θ i ) p − H ip k θ ix k H ip j θ ix j + ( p + 1)( θ i ) p θ ix k H ip j ( H ip k ) x j dx =: Z T N I + I + I dx. (5.5)Using (2.5) in Assumption 4, we get I = ( θ i ) p +1 ( H ip k p l u x l x j + H ip k x j )( H ip j p m u x m x k + H ip j x k ) ≥ ( θ i ) p +1 [ γ | D u i | − C (1 + | Du i | ) | D u i | − C (1 + | Du i | )] ≥ ( θ i ) p +1 ˜ γ | D u i | − C ( θ i ) p +1 (1 + | Du i | )for some ˜ γ >
0. Clearly, I = p ( θ i ) p − | D p H i · Dθ i | . Next, we estimate I from below. From equation (1.4), we gather that(5.6) H ip j u ix j x l = g ′ ( θ i ) θ ix l − H ix l − u ix l − X j = i β ′ ǫ ( u i − u j − ψ ij )( u i − u j − ψ ij ) x l . The estimate (2.8) in Assumption 4 and the lower bound (5.3) on θ i imply theexistence of a positive constant, C (depending on ǫ ), such that(5.7) g ′ ( θ i ) θ i ≥ C > . Then, using (2.4), (2.5), (5.4), (5.6), (5.7), and the Cauchy-Schwarz inequality, weget I = ( p + 1)( θ i ) p θ ix k H ip j ( H ip k p l u ix l x j + H ip k x j )= ( p + 1) g ′ ( θ i )( θ i ) p H ip k p l θ ix k θ ix l + ( p + 1)( θ i ) p θ ix k ( H ip j H ip k x j − H ip k p l H ix l ) − ( p + 1)( θ i ) p H ip k p l θ ix k u ix l + X j = i β ′ ǫ ( u i − u j − ψ ij )( u i − u j − ψ ij ) x l ≥ ( p + 1) γC ( θ i ) p − | Dθ i | − C ( p + 1)( θ i ) p | Dθ i | (1 + | Du i | ) − C ( p + 1)( θ i ) p | Dθ i | (1 + | Du i | ) − X j = i C ( p + 1)( θ i ) p | Dθ i | (1 + | Du j | ) ≥ C ( p + 1)( θ i ) p − | Dθ i | − C ( p + 1)( θ i ) p +1 | Du i | + X j = i | Du j | . EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 17
Next, we bound the left-hand side of (5.5) from above. Using (2.5), (5.4), andthe Cauchy-Schwarz inequality, we obtain( θ i + X j = i β ′ ǫ ( u i − u j − ψ ij ) θ i )div(( θ i ) p D p H i )= [ θ i + X j = i β ′ ǫ ( u i − u j − ψ ij ) θ i ][ p ( θ i ) p − D p H i · Dθ i + ( θ i ) p H ip k p j u ix j x k + ( θ i ) p H ip k x k ] ≤ Cθ i [ p ( θ i ) p − | D p H i · Dθ i | + ( θ i ) p | H ip k p j u ix j x k | + ( θ i ) p | H ip k x k | ] ≤ p θ i ) p − | D p H i · Dθ i | + ˜ γ θ i ) p +1 | D u i | + Cp ( θ i ) p +1 (1 + | Du i | ) . Similarly, − X j = i β ′ ǫ ( u j − u i − ψ ji ) θ j div(( θ i ) p D p H i ) ≤ X j = i Cθ j (cid:12)(cid:12)(cid:12) p ( θ i ) p − | D p H i · Dθ i | + ( θ i ) p | H ip k p j u ix j x k | + ( θ i ) p | H ip k x k | (cid:12)(cid:12)(cid:12) ≤ X j = i Cθ j h p ( θ i ) p − ( θ i ) p − (cid:12)(cid:12) D p H i · Dθ i (cid:12)(cid:12) + ( θ i ) p − ( θ i ) p +12 (cid:12)(cid:12)(cid:12) H ip k p j u ix j x k (cid:12)(cid:12)(cid:12) + ( θ i ) p − ( θ i ) p +12 (cid:12)(cid:12) H ip k x k (cid:12)(cid:12)i ≤ X j = i Cp ( θ j ) ( θ i ) p − + p θ i ) p − | D p H i · Dθ i | + ˜ γ θ i ) p +1 | D u i | + Cp ( θ i ) p +1 (1 + | Du i | ) . From the preceding estimates, we conclude that C ( p + 1) Z T N ( θ i ) p − | Dθ i | dx − C ( p + 1) Z T N ( θ i ) p +1 (1 + | Du i | + X j = i | Du j | ) dx + p Z T N ( θ i ) p − | D p H i · Dθ i | dx + ˜ γ Z T N ( θ i ) p +1 | D u i | − C Z T N ( θ i ) p +1 (1 + | Du i | ) dx ≤ Z T N I + I + I dx = Z T N [ θ i + X j = i β ′ ǫ ( u i − u j − ψ ij ) θ i − X j = i β ′ ǫ ( u j − u i − ψ ji ) θ j ]div(( θ i ) p D p H i ) dx ≤ p Z T N ( θ i ) p − | D p H i · Dθ i | dx + ˜ γ Z T N ( θ i ) p +1 | D u i | dx + Cp Z T N ( θ i ) p +1 (1 + | Du i | ) dx + X j = i Cp Z T N ( θ j ) ( θ i ) p − . Consequently, Z T N ( θ i ) p − | Dθ i | dx ≤ Z T N ( θ i ) p +1 (1 + | Du i | + X j = i | Du j | ) dx + X j = i C Z T N ( θ j ) ( θ i ) p − dx. Applying Young’s inequality, we gather( θ j ) ( θ i ) p − ≤ p + 1 ( θ j ) p +1 + p − p + 1 ( θ i ) p +1 . Therefore, Z T N ( θ i ) p − | Dθ i | dx ≤ Z T N ( θ i ) p +1 (1 + | Du i | + X j = i | Du j | ) dx + X j = i C Z T N ( θ j ) p +1 dx. (5.8)If N = 1, (5.1) is a consequence of Morrey’s Theorem. If N = 2, then θ i ∈ L p for all p . Moreover, in this case, the argument that follows can be modified by replacingthe Sobolev exponent, 2 ∗ , by any arbitrarily large number, M . Therefore, weassume that N >
2. Accordingly, by (3.13), we have θ i ∈ L ∗ (1+ α )2 . In addition,Sobolev’s inequality provides the bound (cid:18)Z T N ( θ i ) p +12 ∗ dx (cid:19) ∗ ≤ C Z T N ( θ i ) p +1 dx + C Z T N | D (( θ i ) p +12 ) | dx = C Z T N ( θ i ) p +1 dx + C ( p + 1) Z T N ( θ i ) p − | Dθ i | dx. (5.9)Let β := q ∗ = q NN − >
1. Consider first the P case. Then, Assumption 7implies that 2 α ≤ ( α + 1) β β − β . The previous inequality together with (2.6) implies that | Du i | ≤ C ( g ( θ i )) + C ≤ C ( θ i ) α + C ≤ C (1 + ( θ i ) ( α +1) β β − β ) . The same inequality holds in the logarithmic case with α = 0: | Du i | ≤ C ( g ( θ i )) + C ≤ C (log( θ i )) + C ≤ C (1 + ( θ i ) β β − β ) . Therefore, from H¨older’s inequality, we get Z T N ( θ i ) p +1 (1 + | Du i | ) dx ≤ C Z T N ( θ i ) p +1 (1 + ( θ i ) ( α +1) β β − β ) dx ≤ C Z T N ( θ i ) p +1 dx + C (cid:18)Z T N ( θ i ) ( p +1) β dx (cid:19) β (cid:18)Z T N ( θ i ) ( α +1) β dx (cid:19) β − β ≤ C Z T N ( θ i ) p +1 dx + C (cid:18)Z T N ( θ i ) ( p +1) β dx (cid:19) β ≤ C (cid:18)Z T N ( θ i ) ( p +1) β dx (cid:19) β . Similarly, Z T N ( θ i ) p +1 (1 + | Du j | ) dx ≤ C Z T N ( θ i ) p +1 (1 + ( θ j ) ( α +1) β β − β ) dx ≤ C Z T N ( θ i ) p +1 dx + C (cid:18)Z T N ( θ i ) ( p +1) β dx (cid:19) β (cid:18)Z T N ( θ j ) ( α +1) β dx (cid:19) β − β ≤ C Z T N ( θ i ) p +1 dx + C (cid:18)Z T N ( θ i ) ( p +1) β dx (cid:19) β ≤ C (cid:18)Z T N ( θ i ) ( p +1) β dx (cid:19) β . EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 19
The last two inequalities, combined with (5.8) and (5.9) give the bound (cid:18)Z T N ( θ i ) ( p +1) β dx (cid:19) β ≤ Cp (cid:18)Z T N ( θ i ) ( p +1) β dx (cid:19) β + Cp X j = i (cid:18)Z T N ( θ j ) ( p +1) β dx (cid:19) β for i = 1 , . . . , d . Summing on i , we finally obtain(5.10) d X i =1 (cid:18)Z T N ( θ i ) ( p +1) β dx (cid:19) β ≤ Cp d X i =1 (cid:18)Z T N ( θ i ) ( p +1) β dx (cid:19) β . Arguing as in [16], we get d X i =1 k θ i k L ∞ ( T N ) ≤ C and, hence, (5.1). (cid:3) Corollary 5.2.
Suppose that Assumptions 1-5 and 8 hold, and either - Assumption 5 L or - Assumptions 5 P- N , 6 and 7hold. Let ( u, θ ) be a C ∞ solution of (1.4) - (1.5) . Then, for any k ∈ N , there exists C ǫ,k > that does not depend on the particular solution, such that (5.11) k u i k W k, ∞ ( T N ) + k θ i k W k, ∞ ( T N ) ≤ C ǫ,k , for any i = 1 , . . . , d .Proof. If Assumption 5 L holds or if Assumptions 5 P- N and 6 hold by, respectively,Propositions 4.1, and 4.3, then there exists θ >
0, such that, for any i = 1 , . . . , d and any x ∈ T N ,(5.12) θ i ( x ) ≥ θ > . Thus, we use θ i = g − H i ( Du i , x ) + u i + X j = i β ǫ ( u i − u j − ψ ij ) in (1.5) to get − θ i H ip k p j u ix j x k − θ i H ip k x k − g ′ ( θ i ) (cid:2) H ip k H ip j u ix j x k + H ix k H ip k + H ip k u ix k + X j = i β ′ ǫ ( u i − u j − ψ ij )( u i − u j − ψ ij ) x k H ip k (cid:3) + θ i + X j = i β ′ ǫ ( u i − u j − ψ ij ) θ i = X j = i β ′ ǫ ( u j − u i − ψ ji ) θ j + 1 . From estimates (5.1), (5.2), and (5.12), we have that the previous equation is auniformly elliptic equation for each i . Therefore, from the elliptic regularity theory,we infer that k u i k W ,p ( T N ) + k θ i k W ,p ( T N ) ≤ C ǫ for any 1 < p < ∞ . Repeated differentiation and a bootstrapping argument give(5.11). (cid:3) Proof of Theorem 1.2
In this section, we show the existence and uniqueness of a classical solution of(1.4)-(1.5). In the proof of existence, we use the continuation method. In theproof of uniqueness, we rely on a monotonicity argument. Here, we work underAssumptions 1-4 and either 5 L or 5 P- N together with Assumptions 6 and 7.6.1. Existence.
To prove the existence of a classical solution of (1.4)-(1.5) usingthe continuation method, we define H iλ ( p, x ) := λH i ( p, x ) + (1 − λ ) | p | ≤ λ ≤
1, ( p, x ) ∈ R N × T N , and i = 1 , . . . , d . We introduce the mean-fieldgame(6.1) H iλ ( Du iλ , x ) + u iλ + X j = i β ǫ ( u iλ − u jλ − ψ ij ) = g ( θ iλ ) , (6.2) − div( D p H iλ ( Du iλ , x ) θ iλ )+ θ iλ + X j = i β ′ ǫ ( u iλ − u jλ − ψ ij ) θ iλ − β ′ ǫ ( u jλ − u iλ − ψ ji ) θ jλ = 1for x ∈ T N and i = 1 , . . . , d .Next, for k ∈ N , we set E k := ( H k ( T N )) d , E := ( L ( T N )) d . If k > N , E k isan algebra. Moreover, E k ⊂ (cid:0) C γ ( T N ) (cid:1) d for any 0 < γ < − Nk . Given θ > k > N , we set E kθ := { θ ∈ E k | θ i ≥ θ , i = 1 , . . . , d } . Finally, for k > N , we define F : [0 , × E k +2 × E k +1 θ → E k × E k +1 by F ( λ, u, θ ) := div( D p H iλ ( Du iλ , x ) θ iλ ) − θ iλ − P j = i ( β ′ ǫ,λ ( u iλ − u jλ − ψ ij ) θ iλ − β ′ ǫ,λ ( u jλ − u iλ − ψ ji ) θ jλ ) + 1 H iλ ( Du iλ , x ) + u iλ + P j = i β ǫ ( u iλ − u jλ − ψ ij ) − g ( θ iλ ) ! . Then, (6.1)-(6.2) is equivalent to F ( λ, u λ , θ λ ) = 0 . Let Λ := { λ ∈ [0 , | (6.1)-(6.2) has a classical solution ( u λ , θ λ ) } . Next, we show that Λ = [0 , . We divide the proof of this identity into the three following claims.
Claim 1: ∈ Λ. Indeed, for λ = 0 , we have the explicit solution: u i = g (1) , θ i = 1 , i = 1 , . . . , d. Claim 2: Λ is closed. To prove this claim, we show that, for any sequence,( λ k ) k ⊂ Λ , such that λ k → λ as k → + ∞ , we have that λ ∈ Λ. Accordingly,let ( u λ k , θ λ k ) be a classical solution of (6.1)-(6.2) for λ = λ k . Recall that H λ satisfies Assumptions 1-4 and Assumption 6 uniformly in 0 ≤ λ ≤
1. Therefore,by (5.11), we can bound the derivatives of any order of ( u λ k , θ λ k ) by a constantthat is independent of k . Consequently, we can extract a subsequence of smooth EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 21 solutions converging to a limit function ( u, θ ) that solves (6.1)-(6.2) with λ = λ .Thus, λ ∈ Λ. Claim 3: Λ is open. To prove this last claim, we need to check that, for any λ ∈ Λ , there exists a neighborhood of λ contained in Λ. To do so, we use theimplicit function theorem. To simplify the notation, for h = β, β ′ , β ′′ , we set h ǫ,λ ( i, j ) := h ǫ ( u iλ − u jλ − ψ ij ) . For λ ∈ Λ, we consider the Fr´echet derivative, L λ : E k +2 × E k +1 → E k × E k +1 ,of ( u, θ ) → F ( λ , u, θ ) at ( u λ , θ λ ). We have L λ ( v, f )= ( H iλ ,p k p j ( Du iλ , x ) v ix j θ iλ + H iλ ,p k ( Du iλ , x ) f i ) x k − f i − P j = i [( β ′′ ǫ,λ ( i, j ) θ iλ + β ′′ ǫ,λ ( j, i ) θ jλ )( v i − v j ) + β ′ ǫ,λ ( i, j ) f i − β ′ ǫ,λ ( j, i ) f j ] D p H iλ ( Du iλ , x ) · Dv i + v i + P j = i ( β ′ ǫ,λ ( i, j )( v i − v j )) − g ′ ( θ iλ ) f i . (6.3)Because of the a priori bounds for smooth solutions (5.11) and either estimate(4.3), in the L case, or (4.12), in the P- N case, the operator, L λ , is well definedin E k +2 × E k +1 for any k ≥
0. Next, we prove that L λ is an isomorphism from E k +2 × E k +1 to E k × E k +1 for any k ≥ w = ( v, f ) ∈ E × E . Define the bilinear form, B λ [ w , w ] : E × E → R , by(6.4) B λ [ w , w ] := d X i =1 B iλ [ w , w ] , where B iλ [ w , w ] := Z T N − H iλ ,p k p j ( Du iλ , x ) v i ,x j v i ,x k θ iλ − D p H iλ ( Du iλ , x ) · Dv i f i − f i v i dx − Z T N h X j = i ( β ′′ ǫ,λ ( i, j ) θ iλ + β ′′ ǫ,λ ( j, i ) θ jλ )( v i − v j ) + β ′ ǫ,λ ( i, j ) f i − β ′ ǫ,λ ( j, i ) f j i v i dx + Z T N [ D p H iλ ( Du iλ , x ) · Dv i + v i + X j = i ( β ′ ǫ,λ ( i, j )( v i − v j )) − g ′ ( θ iλ ) f i ] f i dx. If w ∈ E k +2 × E k +1 with k ≥
0, then B λ [ w , w ] = Z T N L λ ( w ) · w dx. The following lemma is a straightforward consequence of estimate (5.11) combinedwith either (4.3) or (4.12).
Lemma 6.1.
Let B be the bilinear form given by (6.4) . Then, there exists C > such that | B λ [ w , w ] | ≤ C k w k E × E k w k E × E for any w , w ∈ E × E . Thus, in view of Riesz’s representation theorem for Hilbert spaces, there existsa continuous linear mapping, A : E × E → E × E , such that(6.5) B λ [ w , w ] = ( Aw , w ) E × E . Lemma 6.2.
The operator, A, defined in (6.5) is injective.Proof. Let w = ( v, f ). Then, we have B iλ [ w, w ] = Z T N − H iλ ,p k p j ( Du iλ , x ) v ix j v ix k θ iλ − Z T N X j = i ( β ′′ ǫ,λ ( i, j ) θ iλ + β ′′ ǫ,λ ( j, i ) θ jλ )( v i − v j ) v i + g ′ ( θ iλ )( f i ) dx. Summing the preceding expression on i , using the identity X i X j = i ( β ′′ ǫ,λ ( i, j ) θ iλ + β ′′ ǫ,λ ( j, i ) θ jλ )( v i − v j ) v i = X i X j = i β ′′ ǫ,λ ( i, j ) θ iλ ( v i − v j ) , the convexity property (2.4) from Assumption 4 and either (4.3), in the L case, or(4.12), in the P- N case, we get B λ [ w, w ] = d X i =1 B iλ [ w, w ] ≤ − d X i =1 Z T N H iλ ,p k p j ( Du iλ , x ) v ix j v ix k θ iλ + g ′ ( θ iλ )( f i ) dx − X i X j = i Z T N β ′′ ǫ,λ ( i, j ) θ iλ ( v i − v j ) ≤ − C λ Z T N k Dv k + k f k dx − θ X i X j = i Z T N β ′′ ǫ,λ ( i, j )( v i − v j ) dx. (6.6)According to the previous inequality, if Aw = 0, we have w = ( µ,
0) for some µ ∈ R d . Next, by computing0 = ( Aw, (0 , µ )) = B [( µ, , (0 , µ )] = d X i =1 Z T N v i f i dx = | µ | , we conclude that µ = 0. (cid:3) Lemma 6.3.
The operator, A, given by (6.5) is surjective.Proof. First, we prove that the range of A is closed in E × E . For that, take aCauchy sequence, ( z n ) n , in the range of A ; that is, z n = Aw n for some sequence( w n ) n in E × E . We claim that ( w n ) n is a Cauchy sequence. Let w n = ( v n , f n ). EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 23
Then, according to (6.6), we have( z n − z m , w n − w m ) E × E = ( A ( w n − w m ) , w n − w m ) E × E = B [ w n − w m , w n − w m ] ≤ − C ( k D ( v n − v m ) k E + k f n − f m k E ) − X i X j = i Z T N β ′′ ǫ,λ ( i, j )(( v in − v im ) − ( v jn − v jm )) dx. Moreover, we have | ( z n − z m ,w n − w m ) E × E | ≤ k z n − z m k E × E k w n − w m k E × E = k z n − z m k E × E ( k v n − v m k E + k Dv n − Dv m k E + k f n − f m k E ) ≤ δ ( k Dv n − Dv m k E + k f n − f m k E ) + C δ k z n − z m k E × E + k z n − z m k E × E k v n − v m k E . Let µ be a positive constant to be chosen later. By selecting a suitably small δ andcombining the inequalities above, we get k Dv n − Dv m k E + k f n − f m k E + X i X j = i Z T N ∩{ u i − u j − ψ ij > } β ′′ ǫ,λ ( i, j )(( v in − v im ) − ( v jn − v jm )) dx ≤ C k z n − z m k E × E + k z n − z m k E × E k v n − v m k E ≤ C µ k z n − z m k E × E + µ k v n − v m k E . (6.7)Next, we have B [ w n − w m , (0 , v n − v m )] = Z T N d X i =1 (cid:2) D p H iλ ( Du iλ , x ) · (cid:0) D ( v in − v im ) (cid:1) ( v in − v im ) + ( v in − v im ) (cid:3) dx + Z T N d X i =1 X i = j β ′ ǫ,λ ( i, j )[( v in − v im ) − ( v jn − v jm )]( v in − v im ) dx − Z T N d X i =1 g ′ ( θ iλ )( f in − f in )( v in − v im ) dx. Set M = d X i =1 Z T N X i = j β ′ ǫ,λ ( i, j )[( v in − v im ) − ( v jn − v jm )] / . Then, using (5.1), (5.2), H¨older’s inequality, and Young’s inequality, we get B [ w n − w m , (0 , v n − v m )] ≥ k v n − v m k E − C ( k Dv n − Dv m k E + k v n − v m k E k f n − f m k E + M k v n − v m k E ) ≥ C k v n − v m k E − C ( k Dv n − Dv m k E + k f n − f m k E + M ) . (6.8) On the other hand, (6.5) implies that B [ w n − w m , (0 , v n − v m )] ≤ C k z n − z m k E × E k v n − v m k E . Therefore, C k v n − v m k E − C ( k Dv n − Dv m k E + k f n − f m k E + M ) ≤ C k z n − z m k E × E k v n − v m k E ≤ C k z n − z m k E × E + µ k v n − v m k E for some µ to be selected later. Next, taking into account that M ≤ C X i X j = i Z T N β ′′ ǫ,λ ( i, j )[( v in − v im ) − ( v jn − v jm )] dx and using (6.7) and (6.8), we obtain k v n − v m k E ≤ C k z n − z m k E × E . Consequently, v n is a Cauchy sequence in L . Finally, from (6.7) we infer that w n is a Cauchy sequence in E × E because of the bound k w n − w m k E × E ≤ C k z n − z m k E × E . The last inequality and the continuity of A imply that R ( A ) is closed.Finally, we prove that R ( A ) = E × E . Suppose that R ( A ) = E × E . Since R ( A ) is closed, there exists z ∈ R ( A ) ⊥ with z = 0, such that B λ [ z, z ] = 0. Theargument in the proof of Lemma 6.2 implies that z = 0, which is a contradiction. (cid:3) Lemma 6.4.
The operator, L λ : E k +2 × E k +1 → E k × E k +1 , given by (6.3) is anisomorphism for all k ∈ N .Proof. By Lemma 6.2, L λ is injective. Therefore, it suffices to prove that L λ issurjective. To do so, we fix w ∈ E × E with w = ( v , f ). We claim that thereexists a solution, w ∈ E × E , to L λ w = w .Consider the bounded linear functional, w → ( w , w ) ( E ) , in E × E . Ac-cording to the Riesz representation theorem, there exists e w ∈ E × E , such that( w , w ) ( E ) = ( e w, w ) E × E for any w ∈ E × E . In light of Lemmas 6.2 and 6.3,the operator A is invertible in E × E . We define w := A − e w . Then, for any w ∈ E × E , ( Aw , w ) E × E = ( w , w ) E × E ;that is, w = ( v, f ) is a weak solution of(6.9) ( H iλ ,p k p j ( Du iλ , x ) v ix j θ iλ + H iλ ,p k ( Du iλ , x ) f i ) x k − f i − P j = i [( β ′′ ǫ,λ ( i, j ) θ iλ + β ′′ ǫ,λ ( j, i ) θ jλ )( v i − v j ) + β ′ ǫ,λ ( i, j ) f i − β ′ ǫ,λ ( j, i ) f j ] = v i D p H iλ ( Du iλ , x ) · Dv i + v i + P j = i ( β ′ ǫ,λ ( i, j )( v i − v j )) − g ′ ( θ iλ ) f i = f i for x ∈ T N and i = 1 , . . . , d . From the second equation in (6.9), we obtain(6.10) f i = ( g ′ ( θ iλ )) − ( D p H iλ ( Du iλ , x ) · Dv i + v i + X j = i ( β ′ ǫ,λ ( i, j )( v i − v j )) − f i ) . Using (6.10) in the first equation of (6.9), we see that v i is a weak solution to EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 25 [ H iλ ,p k p j ( Du iλ , x ) v ix j θ iλ + H iλ ,p k ( Du iλ , x ) H iλ ,p j ( Du iλ , x ) v ix j ( g ′ ( θ iλ )) − ] x k = − [( g ′ ( θ iλ )) − ( v i + X j = i ( β ′ ǫ,λ ( i, j ) v i − β ′ ǫ,λ ( i, j ) v j ) − f i )] x k + f i + X j = i [( β ′′ ǫ,λ ( i, j ) θ iλ + β ′′ ǫ,λ ( j, i ) θ jλ )( v i − v j ) + β ′ ǫ,λ ( i, j ) f i − β ′ ǫ,λ ( j, i ) f j ] + v i . For any i = 1 , . . . , d , the right-hand side of the previous equation is in L ( T N ).Thus, the elliptic regularity theory implies that v ∈ E . Consequently, (6.10) gives f ∈ E . By induction, if w = ( v , f ) ∈ E k × E k +1 , then w = ( v, f ) ∈ E k +2 × E k +1 .This concludes the proof of the lemma. (cid:3) Claim 3 is now a straightforward consequence of Lemma 6.4 combined with theimplicit function theorem in Banach spaces, see, for instance, [15].Finally, we gather the previous results to prove the existence claim in Theorem1.2.
Proof of Theorem 1.2 - existence.
Claims 1-3 imply that Λ = [0 , ∈ Λ, i.e., there exists a C ∞ solution of (1.4)-(1.5). (cid:3) Uniqueness.
Here, we complete the proof of Theorem 1.2 by proving theuniqueness of solutions for (1.4)-(1.5) using the monotonicity method.
Proof of Theorem 1.2 - uniqueness.
Let ( u , θ ) and ( u , θ ) be classical solutionsof (1.4)-(1.5). First, we take (1.4) with ( u, θ ) = ( u , θ ) and ( u, θ ) = ( u , θ ) andsubtract the corresponding equations. Next, we multiply by θ i − θ i and integrate.Accordingly, we obtain Z T N [ H i ( Du i , x ) − H i ( Du i , x )]( θ i − θ i ) + ( u i − u i )( θ i − θ i ) dx + X j = i Z T N ( β ǫ ( u i − u j − ψ ij ) − β ǫ ( u i − u j − ψ ij ))( θ i − θ i ) dx = Z T N ( g ( θ i ) − g ( θ i ))( θ i − θ i ) dx. Now, we take (1.5) with ( u, θ ) = ( u , θ ) and ( u, θ ) = ( u , θ ). Next, we subtract thecorresponding equations, multiply by u i − u i , and integrate by parts. Accordingly,we get0 = Z T N ( D p H i ( Du i , x ) θ i − D p H i ( Du i , x ) θ i ) D ( u i − u i ) + ( u i − u i )( θ i − θ i ) dx + X j = i Z T N h β ′ ǫ ( u i − u j − ψ ij ) θ i − β ′ ǫ ( u j − u i − ψ ji ) θ j − (cid:16) β ′ ǫ ( u i − u j − ψ ij ) θ i − β ′ ǫ ( u j − u i − ψ ji ) θ j (cid:17)i ( u i − u i ) dx. Finally, we subtract the two previous identities, sum on i, and use the monotonicityof g to conclude that0 ≤ d X i =1 Z T N ( g ( θ i ) − g ( θ i ))( θ i − θ i ) dx = d X i =1 Z T N [( H i ( Du i , x ) − H i ( Du i , x ))( θ i − θ i ) − ( D p H i ( Du i , x ) θ − D p H i ( Du i , x ) θ ) D ( u i − u i )] dx + d X i =1 X j = i Z T N ( β ǫ ( u i − u j − ψ ij ) − β ǫ ( u i − u j − ψ ij ))( θ i − θ i ) dx − d X i =1 X j = i Z T N h β ′ ǫ ( u i − u j − ψ ij ) θ i − β ′ ǫ ( u j − u i − ψ ji ) θ j − (cid:16) β ′ ǫ ( u i − u j − ψ ij ) θ i − β ′ ǫ ( u j − u i − ψ ji ) θ j (cid:17)i ( u i − u i ) dx. Now, from the convexity of β ǫ , we infer that d X i =1 X j = i Z T N ( β ǫ ( u i − u j − ψ ij ) − β ǫ ( u i − u j − ψ ij ))( θ i − θ i ) dx − d X i =1 X j = i Z T N [ β ′ ǫ ( u i − u j − ψ ij ) θ i − β ′ ǫ ( u j − u i − ψ ji ) θ j − ( β ′ ǫ ( u i − u j − ψ ij ) θ i − β ′ ǫ ( u j − u i − ψ ji ) θ j )]( u i − u i ) dx = d X i =1 X j = i Z T N ( β ǫ ( u i − u j − ψ ij ) − β ǫ ( u i − u j − ψ ij ))( θ i − θ i ) dx − d X i =1 X j = i Z T N [ β ′ ǫ ( u i − u j − ψ ij ) θ i − β ′ ǫ ( u i − u j − ψ ij ) θ i ]( u i − u i − u j + u j ) dx = − d X i =1 X j = i Z T N [ β ǫ ( u i − u j − ψ ij ) − β ǫ ( u i − u j − ψ ij ) − β ′ ǫ ( u i − u j − ψ ij )( u i − u j − u i + u j )] θ i dx − d X i =1 X j = i Z T N [ β ǫ ( u i − u j − ψ ij ) − β ǫ ( u i − u j − ψ ij ) − β ′ ǫ ( u i − u j − ψ ij )( u i − u j − u i + u j )] θ i dx ≤ . EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 27
Moreover, using (2.4) of Assumption 4, we get d X i =1 Z T N [( H i ( Du i , x ) − H i ( Du i , x ))( θ i − θ i ) − ( D p H i ( Du i , x ) θ − D p H i ( Du i , x ) θ ) D ( u i − u i )] dx = − d X i =1 Z T N [ H i ( Du i , x ) − H i ( Du i , x ) − D p H i ( Du i , x ) D ( u i − u i )] θ i dx − d X i =1 Z T N [ H i ( Du i , x ) − H i ( Du i , x ) − D p H i ( Du i , x ) D ( u i − u i )] θ i dx ≤ − C d X i =1 Z T N | D ( u i − u i ) | dx. By combining the last two inequalities, we conclude that0 ≤ − C d X i =1 Z T N | D ( u i − u i ) | dx. Thus, we infer that u i = u i for any i = 1 , . . . , d . Consequently, θ i = g − H i ( Du i , x ) + u i + X j = i β ǫ ( u i − u j − ψ ij ) = g − H i ( Du i , x ) + u i + X j = i β ǫ ( u i − u j − ψ ij ) = θ i . This concludes the proof of the uniqueness of the solution of (1.4)-(1.5). (cid:3) Proof of Theorem 1.1
Now, we use Theorem 1.2 and a limiting procedure to prove Theorem 1.1.
Proof of Theorem 1.1.
Let ( u ǫ , θ ǫ ) be the classical solution of (1.4)-(1.5), whoseexistence is guaranteed by Theorem 1.2. By estimates (4.5), (4.2), and (4.4) ifAssumption 5 L holds, or by estimates (4.8), (4.14) and (4.13) if Assumptions5 P- N and 6 hold, there exist γ ∈ (0 , u ∈ ( W , ( T N )) d ∩ ( C γ ( T N )) d , and θ ∈ ( W , ( T N )) d , such that, up to extracting a subsequence, as ǫ → u ǫ → u in ( L ∞ ( T N )) d ,Du ǫ → Du, θ ǫ → θ in ( L ( T N )) d ,D u ǫ ⇀ D u, in ( L ( T N )) d , and, for any i = 1 , . . . , d and j = i,u i − u j − ψ ij ≤ T N . The uniform convergence of u iǫ to u i implies that, if x ∈ T N is such that u i ( x ) − u j ( x ) − ψ ij ( x ) <
0, then, for a small enough ǫ , we have u iǫ ( x ) − u jǫ ( x ) − ψ ij ( x ) < β ǫ ( u iǫ ( x ) − u jǫ ( x ) − ψ ij ( x )) = β ′ ǫ ( u iǫ ( x ) − u jǫ ( x ) − ψ ij ( x )) = 0 . We deduce that the limit, ( u, θ ) , is a weak solution of H i ( Du i , x ) + u i ≤ g ( θ i ) in T N ,H i ( Du i , x ) + u i = g ( θ i ) in ∩ j = i { u i − u j − ψ ij < } . Next, for j = i , we introduce the measures ν ijǫ = β ′ ǫ ( u iǫ ( x ) − u jǫ ( x ) − ψ ij ( x )) θ j . By (3.5), we have that R T N ν ijǫ dx ≤ C for some constant, C , independent of ǫ .Thus, there exist non-negative measures, ν ij , such that − div( D p H i ( Du i , x ) θ i ) + θ i + X j = i (cid:0) ν ij − ν ji (cid:1) = 1 . Moreover, ν ij is supported in the set u i − u j − ψ ij = 0. (cid:3) Uniqueness of the limit
In this last section, we discuss the uniqueness of the limit, ( u, θ ), in the proof ofTheorem 1.1.
Proposition 8.1.
Suppose that Assumptions 1-4 and 8 hold, and that either - Assumption 5 L or - Assumptions 5 P- N , 6 and 7hold. For ǫ > , let ( u ǫ , θ ǫ ) be the solution of (1.4) - (1.5) . Then, the limit, ( u, θ ) ,as ǫ → of the family ( u ǫ , θ ǫ ) exists; that is, ( u, θ ) is independent of the choice ofsubsequence.Proof. For k = 0 ,
1, consider a sequence, ǫ kn , converging to 0. Let ( u k , θ k ) =lim n →∞ ( u ǫ kn , θ ǫ kn ). By (1.1), we have H i ( Du ik , x ) + u ik − g ( θ ik ) ≤ u ik − u jk − ψ ij ≤ . For k = 0 ,
1, let ˜ k = 1 − k . Taking into account the preceding inequalities and theuniform convexity of H i , we have0 ≥ H i ( Du ik , x ) − g ( θ ik ) + u ik + X j β ǫ ˜ kn ( u ik − u jk − ψ ij ) ≥ H i ( Du iǫ ˜ kn , x ) − g ( θ iǫ ˜ kn ) + u iǫ ˜ kn + X j β ǫ ˜ kn ( u iǫ ˜ kn − u jǫ ˜ kn − ψ ij )+ u ik − u iǫ ˜ kn + D p H i ( Du iǫ ˜ kn , x ) · D ( u ik − u iǫ ˜ kn )+ X j = i β ′ ǫ ˜ kn ( u iǫ ˜ kn − u jǫ ˜ kn − ψ ij )(( u ik − u iǫ ˜ kn ) − ( u jk − u jǫ ˜ kn ))+ cγ | D ( u ik − u iǫ ˜ kn ) | + g ( θ iǫ ˜ kn ) − g ( θ ik ) EAKLY COUPLED MEAN-FIELD GAME SYSTEMS 29 for some c >
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King Abdullah University of Science and Technology (KAUST), CEMSEDivision , Thuwal 23955-6900. Saudi Arabia.
E-mail address : [email protected] (S. Patrizi) University of Texas at Austin, Austin, Texas, USA
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