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On weak solutions to first-order discount mean field games

In this paper, we establish the existence and uniqueness of weak solutions to first-order discount mean field games and a stability result to give the existence for the ergodic problem. We show an example to illustrate the multiplicity of weak solutions to the ergodic problem. With this motivation, we address a selection condition, which is a necessary condition that any limit of solutions under subsequence satisfies. As an application, we show a nontrivial example to get the convergence of weak solutions.


introduction
In this paper, we consider the stationary first-order discount mean field game systems with a local coupling Mean field game (MFG) systems have been introduced simultaneously by Lasry and Lions [18] and by Huang, Caines and Malhamé [15], which describe the player's optimal strategies of the agents and their macroscopic distribution. These games are often determined by a system of a Hamilton-Jacobi equation coupled with a transport or Fokker-Planck equation. In this paper, we focus on deterministic games, and therefore we consider the first order Hamilton-Jacobi equation coupled with a transport equation. From a perspective of mathematical analysis, they are not monotone systems, and not uniformly elliptic. For first order MFG systems in which the coupling f is of nonlocal nature, the existence and uniqueness of solutions are well-understood (see [19]). On the other hand, in the case of local couplings, in general, we cannot expect the solvability in the classical sense, and therefore it is reasonable to study by introducing the notion of weak solutions. Recently the framework of weak solutions has been developed. See [1,4,5,8,13,24].
Our main interest of this paper is to establish the well-posedness of weak solutions to (1.1), and to study the asymptotic behavior of the solution (u ε , m ε ) as the discount factor ε → 0. We call this asymptotic problem the vanishing discount problem. As an analogy of the study of the vanishing discount problem for Hamilton-Jacobi equations (see [7,17,22]) and also for discount MFG systems (see [11]) we can naturally expect that the limit problem of (1.1), which is called the ergodic problem, is described by where a triple of u : T d → R, m : T d → [0, ∞) and λ ∈ R is unknown. We call λ an ergodic constant. Ergodic problem (1.2) appears in many context of the asymptotic problem of MFG systems. We refer to [3,5] for the study of the long time average of solutions. The discount problem naturally arises in optimal control theory and differential game theory, where ε is a discount factor. In recent years, there has been much interest and progress on the vanishing discount problem for Hamilton-Jacobi equations. Ergodic problem for Hamilton-Jacobi equations is given by where a pair of v : T d → R and c ∈ R is unknown. One of standard ways to establish the existence of viscosity solutions to (1.3) is to consider the solution v ε ∈ Lip(T d ) to the discount problem εv ε + H(x, Dv ε ) = 0 in T d , and pass to the limit. Under the coercivity assumption on Hamiltonians, we can easily get a priori estimate on Dv ε L ∞ (T d ) , and by the Arzelá-Ascoli theorem, we can prove that there exists a subsequence {ε j } j∈N with ε j → 0 as j → ∞ such that for some (v, c) ∈ Lip (T d ) × R, Here, it is worth emphasizing that due to the lack of uniqueness of viscosity solutions to (1.3), it is nontrivial whether the whole convergence of v ε − min T d v ε holds. Recently, the convergence to a unique limit and its characterization has been established by [7,17,22] using weak KAM theory, the nonlinear adjoint method, and the duality method, respectively. We also refer [20,28] and the references therein for further development.
For second-order MFG systems, that is, the systems are uniformly elliptic, the vanishing discount problem is studied in [6,21]. In first-order MFG systems, in [11], the authors study the existence of classical solutions under the specific Hamiltonian H(x, p) = 1 2 |p| 2 + V (x) with a small oscillatory potential. In this setting, since the ergodic problem has the uniqueness up to constants, it is rather easily proved that the uniform convergence holds. We also point out that in the argument in [11], the specific form of Hamiltonian is crucial. However, if one considers weak solutions, then the multiplicity of weak solutions to (1.2) rather naturally appears. In [11,Section 2.2], the authors consider a weak solution which is introduced in [8], and show an example to illustrate the non-uniqueness issue (see also [12]). This multiplicity of weak solutions makes the asymptotic problem harder and more interesting.
Main feature of this paper, compared particularly to [11], is to study the vanishing discount problem with a different framework of weak solutions which are introduced in [4,5,13]. In [8], they construct weak solutions based on variational inequality technics and Minty's method, and on the other hand, in [4,5,13], they introduce weak solutions by using variational structures and the Fenchel type of duality theorem. For a comparison between these two notions of weak solutions, we refer the readers to [8].
Main contribution of this paper is firstly to obtain the unique weak solution in the sense of [4,5,13] to discount MFG system (1.1). Next, we get a stability result for (u ε , m ε ) as ε → 0, which is a new way to prove the existence of weak solutions to ergodic problem (1.2). Moreover, we prove that the limit function satisfies a viscosity supersolution property, which will be clearly explained in Section 3. We also show an example to illustrate the multiplicity of weak solutions to the ergodic problem. With this motivation, we address a selection condition, which shows a necessary condition that any limit of solutions under subsequence satisfies. By using this condition, we show a nontrivial example to get the convergence result. In connection with the non-uniqueness issue on weak solutions to ergodic problem (1.2), we give several uniqueness set results. We explain the main results in the next sections in more details.
1.1. Assumptions. Throughout the paper we assume the following conditions: increasing with respect to the second variable, and there exists q > 1 and C > 0 such that By replacing H, f byH(x, p) For later purposes, we define F : It follows that F is continuous on T d × (0, ∞), differentiable and strictly convex in m and satisfies, for some C > 0, Let F * be the convex conjugate of F with respect to the second variable, i.e., Then, F * satisfies that, for some C > 0, where p is the conjugate exponent of q, that is, 1 p + 1 q = 1. We denote by H * the convex conjugate of H with respect to the second variable. Note that H * satisfies We give the definition of weak solutions to (1.1) following to the works in [4,5,13].
(ii) the first equation of (1.1) holds in the following sense: (1.5) and (iii) the second equation of (1.1) holds in the sense of distribution.
We notice here that, if pr > d, then u ε ∈ C 0,γ (T d ) with γ = 1− d pr , and u ε is differentiable almost everywhere.  Next, we obtain a weak compactness of (u ε , m ε ) and a stability result. (1.9) Here, we set We emphasize here that as in Proposition 4.1 we have the uniqueness of (m, λ), where (u, m, λ) ∈ W 1,pr (T d ) × L q (T d ) × R is a weak solution to (1.2). However, we have the multiplicity of u in general. We show an example to illustrate the multiplicity of weak solutions to (1.2) in Section 4.2. Therefore, it is not clear whether u ε converges or not. In the next main theorem, we give a condition which any limit of u εn satisfies. Let (u ε , m ε ) be the weak solution to (1.1). Assume that u ε ⇀ū weakly in W 1,pr (T d ), m ε ⇀ m weakly in L q (T d ) and T d εu ε dx → −λ as ε → 0. Then,ū is a minimizer of where we set As an application of Theorem 1.3, we present a nontrivial example that we get the whole convergence of u ε .
We also study the uniqueness issue of ergodic problem (1.2). We define Z ⊂ T d by We give a comparison principle on Z for ergodic problem (1.2).
Furthermore, we consider the case where m ∈ C(T d ). (1.12) Under assumption (1.12), we have an equivalence between weak solutions and viscosity solutions to (1.2) in the following sense.
We give sufficient conditions to have (1.12) in Section 3.2.
This paper is organized as follows. In Section 2, we prove Theorem 1.1, that is, we establish the existence and uniqueness of weak solutions to (1.1). In Section 3, we investigate weak compactness of (u ε , m ε ) and construct a weak solution to the ergodic problem (1.2) as a limit, which implies Theorem 1.2. Then, we observe lack of uniqueness of weak solutions for ergodic problem (1.2) in Section 4. We prove Theorem 1.3 and show a nontrivial example to get a convergence of u ε in Section 5. In Section 6, we prove Theorems 1.4, 1.5.

Well-posedness of discount problem
In this section, we establish the well-posedness result for (1.1) by following the arguments in [4,5,13] with a careful modification to the discount problem.
2.1. Optimization problems. We consider two optimization problems corresponding to We first notice that φ → A ε (φ) is convex due to the convexities of F * (x, ·) and H(x, ·).
We next define B : where if m(x) = 0, then we set First, we check that functionals A ε and B are weakly lower semi-continuous. Then, A ε and B are weakly lower semi-continuous in W 1,pr (T d ) and L q (T d )×L for all x ∈ T d and n ∈ N, because F * (x, ·) is nondecreasing and p > 1. It follows from Fatou's lemma that which is a contradiction. Hence, (2.1) is lower semi-continuous. Similarly, we can prove that B is weakly lower semi-continuous.
Here, we consider two optimization problems. At first, we study the one corresponding to the first equation of (1.1).
As a preliminary, we check that T d φ n dx is bounded. Note that F * (x, a) is nondecreasing in a. By Jensen's inequality, we have by using the Poincare-Wirtinger inequality and the Young inequality with arbitrary δ > 0 in the last inequality. By taking a small δ > 0, we obtain Dφ n L pr (T d ) C.
It follows from the Poincare-Wirtinger inequality that φ n is bounded in W 1,pr (T d ). Thus, we can choose a subsequence such that φ n converges to some φ weakly in W 1,pr (T d ) as n → ∞. In light of Lemma 2.1, φ is a minimizer of (2.4).
On the other hands, set Letφ n be the viscosity solution to By comparison, it holdsφ n φ n . Noting that Thus, we have Then, we can choose a subsequence such thatφ n converges to someφ weakly in W 1,pr (T d ). In light of Lemma 2.1,φ is a minimizer of (2.4). If pr > d, by the Rellich-Kondrachov compact embedding theorem, it holds thatφ n →φ uniformly on T d as n → ∞. Because εφ n + H(x, Dφ n (x)) f (x, 0) in T d holds in the viscosity sense, due to the stability of viscosity solutions,φ satisfies (2.5).
Next, we study the dual optimization problems, which corresponds to the second equation of (1.1).
Proof. Take a sequence (m n , w n ) ∈ K ε such that B(m n , w n ) → inf (m,w)∈Kε B(m, w) as n → ∞. Then, noting that (1, 0) ∈ K ε , for sufficiently large n ∈ N, we have In particular, m n L q is bounded. Note that w n = 0 a.e. in {m n = 0} because B(m n , w n ) < +∞. By the Hölder inequality, we have In light of Lemma 2.1, (m, w) is a minimizer of (2.9). Since (m, w) → B(m, w) is strictly convex, m is unique and so is w m in {m > 0}. As w = 0 in {m = 0}, uniqueness of w follows as well, which implies the uniqueness of the minimizer of (2.9).
Finally, we prove that the two optimization problems are in duality.
Indeed, because F (x, m) = +∞ for m < 0, it suffices to pay attention to the case (m, w) ∈ X does not satisfy (2.2). Then, there existsφ ∈ C 1 (T d ) satisfying Setφ n := nφ, changing the signature if necessary, to yield which can not be the infimum. Hence, this infimum is attained by (m, w) ∈ K ε obtained by Proposition 2.3. By Sion's min-max theorem, it holds that By using the interchange of minimization and integration (see [27,Theorem 14.60, p. 677]), we obtain An easy computation shows that

2.2.
Existence and uniqueness of weak solutions to discount problem. In this section, we prove that a pair of minimizers of optimization problems (2.4) and (2.9) is a weak solution to (1.1).
Proof. Let φ and (m, w) be a minimizer of (2.4) and (2.9), respectively. In view of the duality, Proposition 2.4, we have By convexities of F (x, ·) and H(x, ·), we get We now use the fact that (m, w) satisfies (2.2), Dφ ∈ L pr (T d ) and w ∈ L r ′ q r ′ +q−1 (T d ) = (L pr (T d )) ′ , where we denote by X ′ the dual space of a Banach space X. They imply Hence, equalities hold in estimate (2.12). In particular, combined with dualities of convexities of F (x, ·) and H(x, ·), we have Since F (x, ·) is strictly convex and smooth on (0, +∞), we get By the duality of H(x, ·), combined with (m, w) ∈ K ε , we obtain that, for ψ ∈ C 1 (T d ) which yields (1.7).
Here, we give a proof of Theorem 1.1.

weak compactness and stability
In this section we prove Theorem 1.2. We first recall the definition of weak solutions to (1.2) introduced by [5]. in the sense of distribution.
3.1. Discounted aproximations. The existence of weak solutions to (1.2) is firstly proved by [5]. In [5], they directly prove the existence by considering the optimization problem as in Section 2. In this paper, we prove it by considering weak compactness and stability for the vanishing discount problem.
be the weak solution to (1.1), and w ε = −m ε D p H(x, Du ε ). There exists a constant C > 0 independent of ε > 0 such that

4)
Proof. At first, we can see that |ε u ε dx| is bounded as (2.6). Next, as in (2.7), we get the bound for Du ε L pr . It follows from the Poincare-Wirtinger inequality that u ε W 1,pr is bounded uniformly in ε.
Finally, we study the stability of weak solutions using Γ-convergence type arguments. To prove Theorem 1.2, we introduce several notations. We define A : such that m 0 a.e. in T d , T d m = 1, and satisfies div(w) = 0, in the sense of distribution. Next, we prove the following two lemma.
Taking the infimum on (φ, c) yields the conclusion. Proof. We only need to consider the case where inf K B < +∞. Take any (m n , w n ) ∈ K so that B(m n , w n ) → inf K B as n → ∞. Consider Since m n ∈ L q (T d ), we have v n ∈ W 2,q (T d ). In particular, v n satisfies Set m ε n := ε + (1 − ε)m n , and w ε n := w n − ε(ε − 1)Dv n .
Note that if q ≥ d then we can easily check that (m ε n , w ε n ) ∈ L q (T d ) × L r ′ q r ′ +q−1 (T d ). If q < d then since we assume r ′ qd d−q , we have By the Sobolev inequality we get w ε n ∈ L r ′ q r ′ +q−1 (T d ). Moreover, it is easy to check that By a similar argument to Proposition 2.3 and using assumption r ′ qd d−q , we can see that the right hand side of (3.6) is in L 1 (T d ).
By the Fatou lemma, we obtain Sending n → ∞ yields the conclusion.
We are ready to prove Theorem 1.2.
Proof of Theorem Thus, equalities hold in the above estimates, which implies Let (m ε , w ε ) be the minimizer of (2.9). By Proposition 3.2, with choosing subese- which implies that (m, w) minimizes B among K.
Finally, if we assume pr > d, by the Rellich-Kondrachov compact embedding theorem, u ε → u uniformly on T d . In view of the stability of viscosity solutions and (1.8), u satisfies (1.9) in the viscosity sense.

3.2.
Continuity of density m. In this section, we give some conditions to obtain the continuity of m. (3.7) Then, we have m ∈ H 1 (T) ⊂ C(T). (3.8) Furthermore, assume that H * and F are twice differentiable in the first variable and H * xx (x, p) C(|p| r ′ + 1) and F xx (x, m) C(|m| q + 1), for all (x, p) ∈ T × R and m 0. Then, we have for some C ≥ 0, which is independent of ε.
Proof. The continuity of m, that is ( Remark 1. We point out here that due to (3.9) and the uniqueness of m we have m ε → m in C(T) as ε → 0, which is independently of interest.
In a higher dimension, we give another type of results. Proposition 3.6. Let c ∈ R and assume that (c, m, λ) is a weak solution to ergodic problem (1.2). Then, m ∈ C(T d ).

Proof. From (3.1), it holds that
Hence, we can explicitly get For instance, letting H(x, p) := 1 2 |p| 2 + V (x) for some V ∈ C(T d ), we can easily check that (0, m, λ) is a weak solution to ergodic problem (1.2).

Uniqueness issue of the ergodic problem
Here, we study uniqueness and non-uniqueness of solutions to the ergodic problem (1.2). We prove that (m, λ) ∈ L q (T d ) × R, which is a part of weak solutions to (1.2), is unique. On the other hand, we give an example which shows the multiplicity of u ∈ W 1,pr (T d ).
Proof. As in Proposition 2.3, inf (m,w)∈K B(m, w) has the unique minimizer (m, w) ∈ K. As in the proof of Proposition 2.5, we have w = −mD p H(x, Du i ), and combining with (3.1), for i = 1, 2. Then, we have Therefore, integrating over T d , we get since we have div(w) = 0 in the sense of distribution, which implies λ 1 = λ 2 . Set λ := λ 1 = λ 2 . Next, Suppose that In light of the strict convexity of H(x, ·), by a similar argument to that of the proof of Proposition 2.5, we have where we use the fact that (m, w) satisfies (3.3). Hence, (u 2 , λ) does not minimize A among Then, (1.2) becomes In this case, p = q = r = 2. Thus u ∈ W 1,4 (T) ⊂ C 0, 3 4 (T). Set λ = 0 and m(x) = max{W (x), 0}. Then, (0, m, λ) is a weak solution to (4.1). Indeed, clearly (3.1) and (3.2) hold and also we can easily check that for all ψ ∈ C 1 (T), We consider another type weak solution satisfying 1 2 |u x | 2 + W (x) 0 in T in the sense of viscosity solutions, (4.2) which corresponds to (1.9). In particular, due to (4.2), u x can not jump from negative value to positive value in {m = 0}. For any θ ∈ [ 1 8 , where χ A is the characteristic function for A ⊂ T, that is, with any C ∈ R. We can easily check u θ ∈ W 1,4 (T) ⊂ C 0, 3 4 (T) and (u θ , m, 0) is a weak solution to (4.1) and satisfies (4.2). Also, we emphasize that all weak solutions satisfying (4.2) are characterized by u θ for θ ∈ [ 1 8 , 3 8 ].   [10,11,22].
Proof. By (1.5), (3.2), for almost every x ∈ {m ε > 0}, we have Multiplying this by m ε and integrating on T d , we get Similarly, by (1.6), (3.1), for almost every x ∈ {m > 0}, we have Multiplying this by m and integrating over T d yields Combining (5.1) and (5.2) together, we get Therefore, dividing this by ε > 0, and sending ε = ε n → 0, we obtain which finishes the proof. where u θ is the function defined by (4.4). Because u θ is invariant with respect to adding constants, we can assume that u θ (0) = 0 without loss of generality. Moreover, because u θ and m are symmetric with respect to x = 1 2 , it suffices to consider the interval [0, 1 2 ]. Recall that m(x) = max{W (x), 0} and Then, we consider the following two cases.
Proposition 5.1. Let W be the function defined in Example 1, and let (u ε , m ε ) be the weak solution to Letũ be defined by (4.4) with θ = 1 4 and choosing C ∈ R such that T dũ dx = 0. Then, u ε →ũ as ε → 0 uniformly on T.

uniqueness set
In this section, we consider a uniqueness set for weak solutions to the ergodic problem (1.2). We call Z ⊂ T d a uniqueness set if two weak solutions (u 1 , m, λ), (u 2 , m, λ) ∈ W 1,pr (T d ) × L q (T d ) × R with (1.9) satisfy u 1 = u 2 on Z, then u 1 = u 2 on T d . 6.1. Classical comparison principle. We prove that Z given by (1.11) is a uniqueness set of the ergodic problem (1.2). We first notice that m may not be continuous in general, so the notion of viscosity solutions may not work in T d . However, if Z c := T d \ Z = ∅, then u satisfies H(x, Du) f (x, 0) + λ a.e. in Z c .
Then, u ∈ W 1,∞ (Z c ). Due to the convexity of Hamiltonian u satisfies H(x, Du) f (x, 0) + λ in Z c , in the viscosity sense.
Therefore, we can use a standard technique in the theory of viscosity solutions. Indeed, by a simple adaptation of the argument in [16], we can easily prove the following comparison result. We give it for the completeness for the paper.
Proof of Theorem 1.4. Let 0 < µ < 1. Suppose that and this maximum is attained at some x 0 ∈ Z c . For δ > 0 we define Ψ : Take (x δ , y δ ) so that Ψ(x δ , y δ ) = max T d ×T d Ψ. By a standard argument, we can prove x δ → x 0 and y δ → x 0 as δ → 0.
Let U x 0 be an open neighborhood of x 0 with U x 0 ⊂ Z c . Then, by (3.2), u is a viscosity subsolution to Taking δ > 0 so small that x δ , y δ ∈ U x 0 , by the definition of viscosity solutions, we get and where p δ := x δ −y δ δ . By the coercivity of H(x, ·), inequality (6.1) implies that |p δ | R for some R > 0 independent of δ. Subtracting (6.1) from (6.2) yields By passing to a subsequence if necessary, we have p δ j → p ∈ B(0, R).

Example 2.
Let H(x, p) = 1 2 |p| 2 + V (x) for V ∈ C(T d ) and f ∈ C([0, ∞)). Then, Then, for all c ∈ R, (c, m, λ) is a weak solution, where m is uniquely determined by m(x) := max{f −1 (−λ + V (x)), 0} and we choose λ ∈ R such that T d m(x) = 1. Thus, (1.11) becomes Combining Proposition 4.1 and Theorem 1.4, we get a sufficient condition to obtain the whole convergence of u ε . Corollary 6.1. Let c ∈ R and assume that (c, m, λ) is a weak solution to ergodic problem (1.2). Furthermore assume that the set defined by is connected and satisfies Then, weak solution u ∈ W 1,∞ (T d ) to (1.2) with (1.9) is unique up to constants.
Moreover, if pr > d, then u ε uniformly converges to a unique limit on T d as ε → 0.
Proof. By Proposition 3.6, we have m ∈ C(T d ), which implies u ∈ W 1,∞ (T d ). Because (c, m, λ) is a weak solution, it holds that Under assumption pr > d, u ε uniformly converges to u, which is a unique (up to constants) weak solution to (1.2) and satisfies (1.9), along subsequences. Because T d u ε dx = 0, one has u ε →ū uniformly as the whole sequence, whereū satisfies T dū dx = 0. 6.2. Comparison in terms of Mather measures. In this section, we first prove Theorem 1.5, and as a corollary, we give a comparison principle in terms of Mather measures.
Proof of Theorem 1.5. We first prove (i). We notice that under the continuity assumption on m we can easily see that λ coincides with the ergodic constant for the Hamilton-Jacobi equation. Thus, (1.13) admits at least one viscosity solution v ∈ Lip(T d ). Note that v satisfies H(x, Dv) = f (x, m) + λ a.e. in T d . Thus, we obtain Therefore, (v, λ) is a minimizer of inf (φ,c)∈W 1pr ×R A(φ, c). As in Proposition 2.5, (v, m, λ) is a weak solution to (1.2). Because f (x, ·) is increasing, v satisfies (1.9) in the sense of viscosity solutions.
Next, we prove (ii). In light of (3.2) and m ∈ C(T d ), we have u ∈ W 1,∞ (T d ). Due to the convexity of H(x, ·), we can easily check u is a viscosity subsolution to (1.2). We prove that u is a viscosity supersolution by contradiction. Suppose that there exists x 0 ∈ T d