A Monotone Operator Approach to Separable Mean-Field Games with Mixed Boundary Conditions

We study a class of local, first-order, stationary mean-field games (MFGs) on bounded domains with nonstandard mixed boundary conditions: prescribed inflow on $\Gamma_N$ and a relaxed Signorini-type exit condition on $\Gamma_D$ (complementarity between exit flux and boundary value). For separable Hamiltonians, we overcome the lack of coercivity and the boundary complementarity constraints by introducing a monotone operator on a convex domain, augmented with an auxiliary nonnegative boundary variable $h$ encoding exit flux. To address a constant-shift degeneracy in the value function $u$ (the transport equation depends only on $Du$), we employ a quotient-space formulation that restores coercivity. Using the Browder--Minty theorem, we prove existence for a penalized operator $A_\epsilon$ on a convex domain and pass to the limit as $\epsilon \to 0^+$. We obtain weak solutions $(m,u,h)$ solving the associated variational inequality, with $m \in L^{\beta+1}(\Omega)$, $u \in W^{1,\gamma}(\Omega)$, and $h$ in the dual trace space on $\Gamma_D$.