Edgard A. Pimentel

Time-Dependent Mean-Field Games with Logarithmic Nonlinearities

In this paper, we prove the existence of classical solutions for time-dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument which combines Lipschitz regularity for the Hamilton--Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker--Planck equation and a concavity argument for the nonlinearity.

Regularity for Mean-Field Games Systems with Initial-Initial Boundary Conditions: The Subquadratic Case

In the present paper, we study forward-forward mean-field games with a power dependence on the measure and subquadratic Hamiltonians. These problems arise in the numerical approximation of stationary mean-field games. We prove the existence of smooth solutions under dimension and growth conditions for the Hamiltonian. To obtain the main result, we combine Sobolev regularity for solutions of the Hamilton-Jacobi equation (using Gagliardo-Nirenberg interpolation) with estimates of polynomial type for solutions of the Fokker-Planck equation.

A Priori Bounds for Stationary Models

We draw upon our earlier results to study stationary MFGs. Here, we illustrate various techniques in three models. First, we use the Bernstein estimates given in Theorem 3.11, to obtain Sobolev estimates for the value function. Next, we consider a congestion problem and show, through a remarkable identity, that m > 0. Finally, we examine an MFG with a logarithmic nonlinearity. This model presents substantial challenges since the logarithm is not bounded from below. However, a clever integration by parts argument gives the necessary bounds for its study.

A Priori Bounds for Time-Dependent Models

We continue our study of the regularity of MFGs by considering the time-dependent problem

Non-local Mean-Field Games: Existence

\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} -u_{t} + \frac{1} {\gamma } \left \vert Du\right \vert ^{\gamma } = \Delta u + m^{\alpha } \quad &\;\;\;\mbox{ in}\;\;\;\mathbb{T}^{d} \times [0,T], \\ m_{t} -\mathop{\mathrm{div}}\nolimits (\left \vert Du\right \vert ^{\gamma -1}m) = \Delta m\quad &\;\;\;\mbox{ in}\;\;\;\mathbb{T}^{d} \times [0,T], \end{array} \right. }

Estimates for the Transport and Fokker–Planck Equations

where 1 0. For γ 2, based on the nonlinear adjoint method. In the next chapter, we investigate two time-dependent problems with singularities—the logarithmic nonlinearity and the congestion problem—for which different methods are required.

Estimates for MFGs

The nonlinear adjoint method was introduced by L.C. Evans as a tool to study Hamilton–Jacobi equations.

A Priori Bounds for Models with Singularities

MFGs where the Hamilton–Jacobi equation depends on the distribution of players in a non-local way make up an important group of problems. In many examples, this dependence is given by regularizing convolution operators. We split the discussion of non-local problems into two cases. First, we consider first-order MFGs. Here, semiconcavity bounds and the optimal control characterization of the Hamilton–Jacobi equation are the main tools. Next, we examine second-order MFGs. Here, the regularizing effects of parabolic equations and the L2 stability of the Fokker–Planck equation are the main ingredients of the proof.