Classical and weak solutions to local first-order mean field games through elliptic regularity

We study the regularity and well-posedness of the local, first-order forward-backward mean field games system, assuming a polynomially growing cost function and a Hamiltonian of quadratic growth. We consider systems and terminal data that are strictly monotone in the density and study two different regimes depending on whether there exists a lower bound for the running cost function. The work relies on a transformation due to P.-L. Lions, which gives rise to an elliptic partial differential equation with oblique boundary conditions, that is strictly elliptic when the coupling is unbounded from below. In this case, we prove that the solution is smooth. When the problem is degenerate elliptic, we obtain existence and uniqueness of weak solutions analogous to those obtained by P. Cardaliaguet and P.J. Graber for the case of a terminal condition that is independent of the density. The weak solutions are shown to arise as viscous limits of classical solutions to strictly elliptic problems.


Introduction
The purpose of this paper is to study the well-posedness of the first-order mean field games system (MFG for short) with a local coupling: where H : T d × R d → R is a strictly convex Hamiltonian of quadratic growth, f, g : are strictly increasing in their second variable m, f has polynomial growth in m, and m 0 is a strictly positive probability density. As is standard, we work on the flat d-dimensional torus T d = R d /Z d to avoid additional technicalities with spatial boundary conditions. MFG were introduced by Lasry and Lions [14,18], and at the same time, in a particular setting, by Huang, Malhamé, and Caines [13]. They are non-cooperative differential games with infinitely many players, in which the players find an optimal strategy by observing the distribution of the others. When the game is completely deterministic, such games are typically modeled by the system (MFG), which has been successfully studied in the case where the function g is independent of the density m, and a complete theory of weak solutions has been obtained through variational methods by Cardaliaguet, Graber, Porretta, and Tonon [2,3,4,12].
Our two main contributions are, proving well-posedness when the terminal condition is strictly increasing with respect to m, and the attainment of classical solutions under the additional assumption that f (·, 0) ≡ −∞. When the latter blowup condition does not hold, we obtain weak solutions that are in line with the variational theory, and they are shown to enjoy higher regularity than in the case g m ≡ 0, by virtue of the strict monotonicity of g.
The precise statements of our main results are as follows. We refer to Section 2 for the exact assumptions (M), (H), (F), (G), (SE), and (DE), to Section 5 for the definition of a weak solution, and to the notation subsection for the meaning of the function spaces mentioned in the theorems below. (ii) The solution (u, m) is the a.e. limit, as ǫ → 0, of solutions (u ǫ , m ǫ ) ∈ C 3,α (Q T ) × C 2,α (Q T ) to MFG systems satisfying (SE). Furthermore, (u ǫ (·, T ), m ǫ (·, T )) → (u(·, T ), m(·, T )) a.e. in T d .
Despite the connections with the variational theory, we do not use variational methods. Instead, we follow the ideas of Lions and his work on the so-called planning problem, where the initial and terminal densities m(·, 0) and m(·, T ) are prescribed [14,18]. It was first observed by Lions that if, for each fixed x ∈ T d , f −1 (x, ·) is the inverse function of f (x, ·), it is possible to formally eliminate the variable m from the system. This transforms the problem into a second order quasilinear elliptic equation with a non-linear oblique boundary condition which, in the special case where D x H, D x f, D x g ≡ 0, may be written as follows (see Section 2 for the general setting): where the function χ(w) is defined by We emphasize the fact that, while (1.1) is an elliptic second order problem, the original system (MFG) is of first order and, in particular, it models a game with no diffusion. Our approach to obtain classical solutions when (1.1) is strictly elliptic was developed by Lions, who applied, in his lectures at Collège de France, the following strategy for finding regular solutions to the planning problem. Viewed as a quasilinear elliptic equation with a non-linear boundary condition, the problem can be tackled with classical methods from the field of a priori estimates: specifically, maximum principle techniques and the Bernstein method to obtain bounds on the solution and its gradient, the application of classical estimates to bound the Hölder norm of the gradient up to the boundary, and soft functional analytic tools to attain the classical solutions.
In order to study the general MFG system, even when (1.1) happens to be degenerate elliptic and the solutions are expected to be discontinuous, our strategy is to first obtain smooth solutions in the strictly elliptic case, and to subsequently find the weak solution as a viscous limit of strictly elliptic problems. The success of this approach is based on the fact that, once smooth solutions are known to exist, every a priori estimate that is independent of the ellipticity constant can be used as a source of compactness and regularity for the limit. Our a priori estimates are supplemented by energy computations based on the Lasry-Lions monotonicity procedure, which is the canonical method for obtaining integral bounds and proving uniqueness in MFG systems.
To identify and motivate the condition that determines the strict or degenerate ellipticity of the system, we remark that the determinant corresponding to the elliptic equation in (1.1) becomes zero precisely as χ = mf m → 0. This is in accordance with the heuristic principle that the regularity of u is lost in regions where there are few to no players (no information), as well as when the cost fails to be strictly monotone (concentration blowup). Because, as is standard, f is assumed to grow at least logarithmically as m → ∞, this degeneracy can only happen as m → 0. In the absence of diffusion, for the strict positivity of m to be preserved, we expect to have a very strong incentive for the players to navigate through regions of low density. With these considerations in place, we will classify the system (MFG) as being strictly elliptic precisely when f has a singularity at m = 0, and as degenerate elliptic otherwise.
It should be noted that, for the stationary problem, classical solutions were obtained in [6], for the case where f = log m, and in [11], for the case where H(x, p) , under a small-oscillation assumption. For second order systems with a (possibly) degenerate diffusion and a density-independent terminal condition, the variational theory was extended in [4], where it was shown (compare with Theorem 1.2) that the weak solutions to the first order problem arise as viscous limits of weak solutions to second order MFG systems. Finally, the most general result for weak solutions to the second order problem is due to Porretta [21], and, unlike [4], it does not use variational methods.
The content and structure of the paper are described as follows. Section 2 explains the general setting and assumptions that will be used, followed by the statements of the preliminary results from the classical literature on quasilinear elliptic equations and oblique derivative problems that will be used to prove existence of classical solutions.
In Section 3, we obtain all the necessary a priori estimates for the strictly elliptic problem. The main results, which deal with the system in full generality, are summarized in Theorem 3.9. We also establish a minor variant, in the special case when the x dependence has a simple structure, that is, when Theorem 3.10. This result states that, with this structural assumption, it is not necessary to require f to grow at most polynomially in m, allowing for examples such as f (m) = e m + log m. Section 3.1 contains the L ∞ -bounds on the solution u, as well as two-sided bounds for the terminal density m(·, T ), obtained through maximum principle methods. These methods exploit the fact that the strict monotonicity property of g is equivalent to the linearized version of the problem (1.1) having an oblique boundary condition, which is of Robin type in the upper component of ∂Q T . In Section 3.2, the gradient bound is obtained by means of the Bernstein method. To deal with the asymmetry between the space and time derivatives in (1.1), it is necessary to first get a precise bound for u t in terms of the space gradient, utilizing the a priori lower bound on m(·, T ) and the maximum principle. This, in turn, provides a "conditional" a priori lower bound for m, namely a lower bound that holds exclusively at points (x, t) where the function H(x, D x u) is close to its maximum value. The conditional nature of this bound, as well as the structure of (1.1) in its fully general form, require a non-conventional choice of an auxiliary function of the space-time gradient.
Section 4 deals with the existence of classical solutions for the strictly elliptic problem, including the proof of Theorem 1.1. The corresponding variant for the case of a fast-growing f is presented in Theorem 4.3. It is first explained how a classical result from the theory of oblique derivative problems, due to G.M. Lieberman [16], immediately yields an a priori Hölder estimate for Du up to the boundary in terms of the L ∞ -bounds on u and Du. Existence is then proved through an application of the non-linear method of continuity, the classical Schauder estimates for the linear oblique derivative problem, and a variant of a convergence theorem of R. Fiorenza [7,8,17].
In Section 5, we develop the weak theory for the degenerate elliptic problem, and obtain the proof of Theorem 1.2. It is first established that, for strictly elliptic problems, there exists an upper bound for the density that is independent of any lower bounds on m(·, T ). After deriving some necessary energy estimates and defining an ǫ-perturbation of the coupling f that makes the problem strictly elliptic, the solution is obtained as the limit when ǫ → 0 of the corresponding smooth solutions. It is also proved, in Theorem 5.5, that when the data is independent of the space variable, the value function u and the terminal density m(·, T ) are globally Lipschitz continuous.
Remark 1.3. We mention here some related work that was released after this paper. In [20], the author showed existence of classical solutions for the so-called extended MFG, a generalization of (MFG) introduced by Lions and Souganidis [19], having a fully general continuity equation, and a non-separated Hamiltonian, namely H = H(x, p, m), with arbitrary superlinear growth. In particular, classical solutions were obtained for first order MFG with congestion. As for weak solutions to (MFG), the most general result to date was obtained by Cardaliaguet and Porretta [5], where the solution is obtained as a vanishing viscosity limit to the weak solutions from [21].

Notation
Let n, k ∈ N. Given x, y ∈ R n , x and y will always be understood to be row vectors, and their scalar product xy T will be denoted by x · y. For any bounded set Ω, with Ω ⊂ Q T , Ω ⊂ T d , or Ω ⊂ [0, T ], and 0 ≤ α < 1, C k,α (Ω), refers to the space of k times differentiable real-valued functions with α-Hölder continuous kth order derivatives, and, for u ∈ C 0,α (Ω), the Hölder semi-norm of u will be denoted by the conventions x ≡ (x, t) and q ≡ (p, s) will always be in place. The notation Du, DΦ will always refer to the full gradient in all variables, so that, for instance Du = Dxu = (D x u, u t ), and DΦ = (DxΦ, Φ z , D q Φ). For (x, t) ∈ ∂Q T , ν(x, t) = ±(0, 0, . . . , 1) denotes the outward pointing unit normal vector. We write C = C(K 1 , K 2 , . . . , K M ) for a positive constant C depending monotonically on the non-negative quantities K 1 , . . . , K M . We also define, for K > 0, and any set V, V K = {(y, z, q) ∈ V × R×R d+1 : |z| + |q| ≤ K}. We write C k (Q T ) * for the dual space of C k (Q T ). In particular, C 0 (Q T ) * is the space of finite signed Borel measures on Q T , and C ∞ (Q T ) * is the space of distributions. Moreover, BV(Q T ) is the space of functions of bounded variation, that is, the space of L 1 (Q T ) functions such that their distributional derivatives are elements of C 0 (Q T ) * , and L ∞ + (Q T ) consists of the functions m ∈ L ∞ (Q T ) such that m ≥ 0 almost everywhere (a.e. for short) in Q T . Finally, for m ∈ L ∞ + (Q T ), L 2 m (Q T ) consists of the functions v such that |v| 2 m ∈ L 1 (Q T ).

The MFG system as an elliptic problem
We now present the general elliptic formulation of the MFG system. As explained in the previous section, it is an equivalent problem satisfied by u, whenever the pair (u, m) is a classical solution to (MFG). It is obtained after eliminating m from the system, and it consists of a quasilinear elliptic equation with a non-linear oblique boundary condition, where, for all (x, t, z, p, s) ∈ Q T × R × R d+1 , with the function χ(x, w) being defined by We remark that the matrix A is clearly non-negative, and since det(A) = χ d det D 2 pp H, the condition for degeneracy is χ = mf m = 0. For future use, we set h(x, w) = χ(x, w).

Assumptions
We now state the main assumptions (M), (H), (F), (G), and (E), that will be in place throughout the paper, except when explicitly stated. Assumption (E) on the ellipticity of the system contains the mutually exclusive possibilities (SE) and (DE), and it will always be made clear which of the two is in place. For the theory of weak solutions, the differentiability assumptions on the data can naturally be weakened through standard approximation arguments, but in the interest of clarity such matters will not be considered. Throughout the assumptions, the quantities C 0 > 0 and 0 ≤ τ < 1 are fixed constants.
for all (x, p) ∈ T d × R d . The space oscillation of H is at most subquadratic in p, namely is four times continuously differentiable on T d × (0, ∞) and strictly increasing in the second variable, with f m > 0. f grows polynomially as m → ∞, in the sense that its growth is at least of degree zero, namely and its derivative f m satisfies a polynomial bound |mf mm | ≤ C 0 f m , which can be equivalently expressed in terms of χ(x, w) as The space derivative of f satisfies the same polynomial bound, as well as the control is four times continuously differentiable on T d × (0, ∞) and strictly increasing in the second variable, with g m > 0. The control required for its space oscillation is that, for each x ∈ T d , (E) (Ellipticity of the system) One of the following conditions holds: In the case of (DE), since the density is not expected to be strictly positive, we assume that g(·, 0) > −∞.
A few comments should be made about the assumptions on the spatial oscillation. First, we remark that the subquadratic growth assumption (HX) can be interpreted as requiring that the purely quadratic part of H is independent of x. Condition (FX2), on the other hand, can be interpreted as being dual to (HX). Indeed, heuristically, since f is assumed to have polynomial growth, mf m ≈ f , and f = −u t + H ≈ |p| 2 , so both conditions impose the same polynomial growth bound in the variable |p|. We consider now the assumption (GX) on the x-oscillation of g. When g is bounded, the first (resp. second) condition in (GX) corresponds to a purely qualitative control on |D x g| that becomes stricter as m → ∞ (resp. m → 0 + ). From the modeling point of view, it can be interpreted as the requirement that extremely crowded regions (resp. nearly empty regions) have roughly the same terminal value for the players.
Remark 2.1. For simplicity of the presentation, we observe that, up to increasing the value of C 0 , the following inequalities are trivial consequences of (H1), (HX), and (SE), and they will be used freely when pertinent.

Preliminary results
This subsection includes the classical results that will be required in Section 4 to obtain the higher regularity from a priori C 1 bounds. In this subsection only, it will not be assumed that the problem (Q0) is explicitly given by (Q1), (Q2), and (B1), but instead (Q, N ) will be a general pair of an elliptic quasilinear operator and a fully non-linear boundary operator. In particular, A and b will not necessarily be assumed to be independent of t and u.
Suppose that ||u|| C 1 (QT ) ≤ K, and that the constants λ K , µ K satisfy, in Q T,K , Then, for any V ⊂⊂ Q T , there exist constants Next is the following local boundary Hölder estimate for the gradient in oblique problems, due to Lieberman [16,Lem. 2.3]. In Theorem 2.3, the following definitions are in place: Then there are constants C and γ depending only on K and For the next theorem, which is the basic Schauder estimate for linear oblique problems [9, Thm. 6.30], we recall that ν(x, t) = ±(0, 0, . . . , 1) denotes the outward pointing normal vector at (x, t) ∈ ∂Q T .
Then there exists The last result of this subsection is a variant of a convergence theorem of Fiorenza, which is a basic tool for using the method of continuity without the need of a priori second derivative estimates [17, Lem. 2, Cor. 1].

Estimates for the solution and the terminal density
We first obtain a priori bounds for the C 0 norm of the solution u. As a corollary, positive, two-sided bounds for the terminal density are established.

Lemma 3.1. Assume that (SE) holds. Then, there exists a constant
Proof. The goal here is to modify u into a function that necessarily achieves its maximum at {t = T }, which is the region of the boundary where, by the strict monotonicity of g, the boundary condition of (Q0) provides information about u. This requires some estimates for the terms in (Q2). By (2.3) and (F2), Moreover, by (FX2), Now, given u, define the linear, uniformly elliptic operator Q u by be a function to be chosen later, and define where the constant C increases in each line. Now, set C 1 = 2C and fix C 1 , still allowing C to increase at each step. We choose and, consequently, at any interior maximum point ( Hence, taking k > max x∈T d f (x, m 0 (x))+||H(·, 0)|| C 0 (T d ) , it follows that v attains its maximum value at t = T . At this point, , and, as before, Thus, since u(x, T ) = v(x, T ), taking into account the surjectivity of f (x, ·), The lower estimate follows from a completely symmetrical argument.
Corollary 3.2. Assume (SE), and let C be the constant from Lemma 3.1. Then, for every x ∈ T d ,

4)
Proof. From the first inequality in (3.1), for each x ∈ T d , and thus, by definition of g 1 , Observe that the application of g −1 1 on both sides of (3.5) is possible because, by (GX), the functions g 0 and g 1 have the same range. This yields the first inequality in (3.4). The second inequality is obtained through the same reasoning. Remark 3.3. A minor modification of the proof of Lemma 3.1 shows that, when H, f , and g are independent of x, the following sharper estimates hold:

Estimates for the space-time gradient
Given the operator Q from (Q0), we recall that its linearization at u ∈ C 2 (Q T ) is the linear, uniformly elliptic operator The gradient estimate will be obtained through Bernstein's method. Specifically, we will bound ||Du|| C 0 (QT ) by evaluating the linearization L u (v) at appropriately chosen functions v( is convex in q, exploiting the fact that, roughly speaking, convex functions of the gradient are expected to be subsolutions. For this purpose, we first obtain an explicit form for the terms in (3.6), as well as a general expression for the linearization applied to such functions v.
Du(x, t)). Then, for each q = (p, s) ∈ R d+1 , and for each x = (x, t) ∈ Q T , Proof. Using (Q1), which shows (3.7). Equation (3.8) is an immediate consequence of (Q2). From the definition of v, it follows that Thus, differentiating the equation Qu = 0 and taking the inner product with D q Φ yields Using the fact that A and b are independent of t, as well as (3.6), we obtain Tr(A xi D 2 u)Φ pi + DxΦ · D q Tr(AD 2 u), which proves (3.9). By Corollary 3.2, this result reduces the problem to estimating ||D x u|| C 0 , but it is also a key ingredient for obtaining that bound, particularly due to the fact that the term ||H(·, D x u)|| C 0 (QT ) has coefficient 1 in (3.10). We now begin to simplify the quantity (3.9) for the specific Φ that will be used in the proof of the gradient estimate, bounding one of the dominant signed terms by a simpler expression, using matrix algebra. Lemma 3.6. Assume that (SE) holds. For each (x, t, p, s) ∈ Q T × R d+1 , set H(x, t, p, s) = H(x, p), and define the matrixĨ = (δ ij (1 − δ i,d+1 )) d+1 i,j=1 . Then, for every u ∈ C 2 (Q T ),

be a solution to (MFG), and set
Proof. By (H1), thus, since the matrix D 2 uAD 2 u is non-negative, multiplying both sides of the inequality by this matrix and taking the trace of both sides gives Now, by (Q1) and (H1), Tr(D 2 pp HD 2 uAD 2 u) The next Lemma continues to simplify the linearizations. Since one of the dominant signed terms will later be shown to be of order |D x u| 4 , the goal will be to bound everything else by (4 − ǫ)th powers of |D x u|, (2 − ǫ)th powers of u t (dealing with these through Corollary 3.5), and second derivative terms that can be dealt with using the other dominant term (3.11). The usage of Φ(x, t, D x u, u t ) = H(x, D x u), as opposed to a more standard choice such as |D x u| 2 or |Du| 2 , is crucial in the next two results, in order to produce structural cancellation of terms that can not be otherwise estimated, as well as to be able to use (3.10) without gaining any constant factors in the process.
Proof. As was mentioned, the proof will proceed through Bernstein's method. By Corollary 3.5, it is sufficient to bound the space gradient. Since the estimate will be up to the boundary, as in [18], we linearize the HJ equation that holds at the extremal times: We now normalize u to have a prescribed sign at the initial and terminal times. That is, we set where 0 < c 1 ≤ 1 is a constant to be chosen later. Let (x 0 , t 0 ) ∈ Q T be a point where v achieves its maximum value. We will distinguish three cases: (2.2), and the HJ equation in (MFG), together with the fact that m(·, T ) = g −1 (·, u(·, T )), yield Thus, by (2.1), This implies −c 1ũ (H(x 0 , D x u)) ≤ C(1 + |D x u|), and so, we conclude once more that In order to make use of Lemma 3.7, it is necessary to eliminate the (x 0 , t 0 ) dependence of the "constant" C(x 0 , t 0 ) from the Lemma, which amounts to establishing an a priori upper bound on the quantities 1/χ and |h w | at the point (x 0 , t 0 ). By (F1) and (F2), 1/χ and |h w | = |χ w /2 √ χ| are both bounded above as w → ∞, so it is enough to establish a lower bound for w = f (x 0 , m(x 0 , t 0 )). By Corollary 3.5, there exists a point (x 1 , t 1 ) ∈ ∂Q T where u t achieves its maximum value. Then, since (x 0 , t 0 ) is a maximum point for v, and the initial and terminal densities are both bounded below a priori, This estimate, together with (H2) and (2.1), allows us to identify the dominant power of |D x u| in the linearization, (3.31) Now, because of the form of the estimate in Lemma 3.7, it is also necessary to be able to compare powers of |f | with powers of |D x u|. By Corollary 3.5 and (2.1), With these preliminaries done, we now apply Lemma 3.7, obtaining Applying (3.31) and (3.32) yields where C is as in the previous line. This gives which may be rearranged as This finally implies that either of which yields |H(x 0 , D x u(x 0 , t 0 ))| ≤ C.
We now summarize all of the a priori bounds obtained in this section.
Theorem 3.9. Assume that (SE) holds, let (u, m) ∈ C 3 (Q T ) × C 2 (Q T ) be a solution to (MFG), and let β be defined by (3.29). Then there exist constants L, L 1 , K, K 1 , with , L 1 = L 1 (C 0 , T ), such that Proof. This result follows simply by the successive application of Lemma 3.1, Corollary 3.2, and Lemma 3.8.
The following variation of Theorem 3.9 shows that, in the standard case where H(x, p) = H(p) − V (x) and f (x, m) = f (m), the condition (F2) which requires f to grow at most polynomially may be significantly relaxed.
Theorem 3.10. The conclusion of Theorem 3.9 still holds if condition (F2) is replaced by: Proof. We simply address all of the instances in which condition (F2) has been used so far. In the proofs of Lemma 3.1, Corollary 3.2, and Lemma 3.7, (F2) was exclusively used to estimate either space derivatives D x f, D x H, or terms that involve mixed derivatives D 2 xm f, D 2 xp H. With (HFX*) in place, such terms are, respectively, either bounded in C 1 norm or trivially zero. Condition (F2) was also used in the proof of Lemma 3.8 in order to obtain a bound for |h w | as w → ∞, but this bound exists here by assumption.
We note that the condition that (HFX*) imposes on h may be equivalently rewritten, in terms of f , as lim sup

Classical solutions
To obtain classical solutions, it is necessary to have Hölder estimates for the gradient of the solution in terms of the C 1 norm. The following Lemma, which is merely a restatement of Theorems 2.2 and 2.3 in the context of the MFG system, provides such an estimate.
Lemma 4.1. Let (u, m) ∈ C 3 (Q T ) × C 2 (Q T ) be a solution to (Q0), and set K = ||u|| C 1 (QT ) . Let µ K , λ K > 0 be such that (2.5) holds in T d K , and the conditions (2.6) and hold in ∂Q T,K . There exist constants C > 0, 0 < γ < 1, with Proof. The only thing to remark is that in order to apply Theorem 2.3, it is necessary to verify that λ K can be chosen to satisfy (4.1), or, in other words, that N is indeed an oblique boundary operator. This follows directly from (B1), since Therefore, the result follows by applying Theorems 2.2 and 2.3 locally, and extracting a finite subcover of Q T . The use of Theorem 2.3 is particularly straighforward since the boundary of Q T is already flat.
The strategy to prove existence will be to use the nonlinear method of continuity, by constructing an explicit homotopy (Q θ , N θ ) θ∈[0,1] between (Q0) and an elliptic problem that comes from a much simpler MFG system, and trivially has a smooth solution. For each θ ∈ [0, 1] and each (x, p, m) and consider the family of MFG systems We observe that, when θ = 0, the unique solution is (u, m) ≡ (1, 1). Let (Q θ u , N θ u) be the operators for the corresponding elliptic problem associated to (MFG θ ), and let A θ , b θ , and B θ be their coefficients. The following straightforward Lemma is a version of Theorem 3.9, tailored to the family (MFG θ ), that also includes the Hölder estimates of Lemma 4.1, and provides a priori bounds that hold uniformly in θ.

Now, by definition,
Therefore, and similarly, On the other hand, the following inequalities hold: Indeed, by (4.4), which shows the first inequality in (4.7), with the second one following in the same fashion. Now, (4.7) yields . (4.8) Thus, (4.5), (4.6), and (4.8) yield and Next, we obtain the gradient Hölder estimate with the help of Lemma 4.1. We remark that the operator (Q θ , N θ ) is clearly elliptic and oblique, because it comes from (MFG θ ). Moreover, since A θ , b θ , and B θ and their derivatives are, respectively, continuous functions of (x, t, z, p, s, θ) on the compact sets Q T,K × [0, 1] and ∂Q T,K × [0, 1], it follows that there exist constants µ L+K > 0, λ L+K > 0, independent of θ, satisfying (2.5) in (T d ) L+K , and (2.6), (4.1) in ∂Q T,L+K , when the operators (Q, N ) are replaced by (Q θ , N θ ). Lemma 4.1 then yields constants C > 0, 0 < γ < 1, independent of θ, such that With the help of this uniform estimate, the main theorem for the strictly elliptic problem may now be proved.
Proof of Theorem 1.1. The uniqueness part of the statement is an immediate consequence of the standard Lasry-Lions monotonicity method, and will be omitted. We define the Banach spaces The partial Fréchet derivative of S with respect to the variable u at the point (u, θ) is the corresponding linearization, for fixed θ, of the differential operator (Q θ , N θ ), namely (L 1 (u,θ) , L 2 (u,θ) ), where For fixed (u, θ) ∈ E × [0, 1], the linear operator L 1 (u,θ) is uniformly elliptic and the linear boundary operator L 2 (u,θ) is oblique. Moreover, the homogeneous problem (L 1 (u,θ) w, L 2 (u,θ) w) = (0, 0) has the form whereB · ν > 0,c ≥ 0 andc ≡ 0, which implies that it has only the trivial solution in C 3,α (Q T ). Hence, by the standard Fredholm alternative for linear oblique problems (see [9]), the operator (L 1 (u,θ) , L 2 (u,θ) ) is invertible in C 3,α (Q T ). The infinite-dimensional implicit function theorem then implies that the set The next step is to show that D is also closed. Let {θ n } ⊂ D be a sequence such that θ n → θ ∈ [0, 1], and let {u n } ⊂ E be the corresponding sequence of solutions to S(u n , θ n ) = (0, 0). By Lemma 4.2, there exist numbers C > 0, 0 < γ < 1, independent of n, such that The Arzelà-Ascoli Theorem implies that, up to a subsequence, there exists u ∈ C 1,γ (Q T ) such that u n → u in C 1 (Q T ). By Theorem 2.5, it follows that u ∈ C 2,α (Q T ), u n → u in C 2,α (Q T ), and S(u, θ) = 0. In particular, the u n are uniformly bounded in C 2,α (Q T ). Now, given i ∈ {1, . . . , d}, differentiating the equation (Q θn (u n ), N θn (u n )) = (0, 0) yields, for w = D xi u n , Therefore, by Theorem 2.4, there exists C > 0, independent of n, such that implying that D x u n is bounded in C 2,α (Q T ). In particular, u n | ∂QT is bounded in C 3,α (∂Q T ), and by the standard Schauder theory for the Dirichlet problem, u n is therefore bounded in C 3,α (Q T ). Consequently, u ∈ C 3,α (Q T ) and θ ∈ D, proving that D is closed. Since 0 ∈ D, it follows that D = [0, 1], which completes the proof.
The next theorem is the corresponding variant of Theorem 1.2 for the case of a fast-growing f , which follows from the estimates in Theorem 3.10. Proof. All of the results in this section follow in this case by simply replacing the use of Theorem 3.9 by Theorem 3.10.

Weak solutions
In this section we develop the theory of weak solutions, for the case where the strict ellipticity condition (SE) fails to hold. We begin by stating the definition of weak solution that will be used, which is in direct analogy with the one used in [2,3,4] to study the degenerate case in which g m ≡ 0.
is called a weak solution to (MFG) if the following conditions hold: (ii) u satisfies the HJ inequality in the distributional sense, with u(·, T ) = g(·, m(·, T )) in the sense of traces.
(iii) m satisfies the continuity equation in the distributional sense, with m(·, 0) = m 0 in H −1 (T d ).
The only missing ingredient necessary to obtain a solution as the limit when ǫ → 0 is the following minor modification of Lemma 3.8, which provides a global, a priori upper bound for the density that is independent of the size of 1 min(m(T )) .
Proof. The argument is a simple variant of the proof of Lemma 3.8. Let v andũ have the same meaning as in said proof, with 0 < c 1 < 1 once more being a free parameter, setṽ = −u t + v = f + c1 2ũ 2 , and let (x 0 , t 0 ) be a point whereṽ achieves its maximum value.
Once more, as in Lemma 3.8, fix c 1 such that c 1 < 1 4C0C(1+C0) , where the constant C is as in the previous line, obtaining The left-and right-hand sides have, respectively, degree 4 and degree 3 + τ < 4 in the non-negative variables (|D x u|, √ f ), thus |D x u| 2 + f ≤ C, and, in particular, it follows that v(x 0 , t 0 ) ≤ C.
We now obtain several a priori bounds for (u ǫ , m ǫ ) that are independent of ǫ.
Therefore, by (5.7) we conclude that ||m ǫ || C 0 (QT ) ≤ C for large enough C and small enough ǫ. The lower bound for u ǫ t is simply a consequence of (5.7), the relation −u ǫ t + H = f ǫ , and the fact that H is bounded below. This completes the proof of (5.4). Now, integrating the HJ equation for u ǫ yields D x u ǫ → D x u in L 2 m (Q T ) and a.e. in {m > 0}, m ǫ |D x u ǫ | 2 → m|D x u| 2 in L 1 (Q T ) and a.e. in Q T , (5.12) ǫ log m ǫ → 0 in L 1 m (Q T ).
We finally note that, in the case d = 1, since there exists an a priori lower bound for the density m in terms of its boundary values (obtained in [1,10,15]), the solutions are seen to be smooth. However, when d > 1, even in the special case of Theorem 5.5, where an a priori bound for the gradient was obtained, we do not know whether the solution to the degenerate elliptic problem enjoys higher regularity.