The standard formulation of the PDE system of Mean Field Games (MFG) requires the differentiability of the Hamiltonians. However in many cases, the structure of the underlying optimal problem leads to a convex but non-differentiable Hamiltonian. For time-dependent MFG systems, we introduce a generalization of the problem as a Partial Differential Inclusions (PDI) by interpreting the derivative of the Hamiltonian in terms of the subdifferential set. In particular, we prove the existence and uniqueness of weak solutions to the resulting MFG PDI system under standard assumptions in the literature. We propose a monotone stabilized finite element discretization of the problem, using conforming affine elements in space and an implicit Euler discretization in time with mass-lumping. We prove the strong convergence in $L^2(H^1)$ of the value function approximations, and strong convergence in $L^p(L^2)$ of the density function approximations, together with strong $L^2$-convergence of the value function approximations at the initial time.

翻译：平均场博弈（MFG）偏微分方程组的标准形式要求Hamiltonian可微。然而在许多情况下，底层最优问题的结构会导致凸但非可微的Hamiltonian。针对时间依赖的MFG系统，我们通过将Hamiltonian的导数解释为次微分集，将问题推广为偏微分包含（PDI）。特别地，在文献中标准假设下，我们证明了所得到MFG PDI系统弱解的存在唯一性。我们提出了一种单调稳定化的有限元离散方法，空间上采用协调仿射单元，时间上采用含质量集中的隐式欧拉离散。我们证明了值函数逼近在$L^2(H^1)$中的强收敛性、密度函数逼近在$L^p(L^2)$中的强收敛性，以及初始时刻值函数逼近的强$L^2$收敛性。