This paper studies a mean field game inspired by crowd motion in which agents evolve in a compact domain and want to reach its boundary minimizing the sum of their travel time and a given boundary cost. Interactions between agents occur through their dynamic, which depends on the distribution of all agents.We start by considering the associated optimal control problem, showing that semi-concavity in space of the corresponding value function can be obtained by requiring as time regularity only a lower Lipschitz bound on the dynamics. We also prove differentiability of the value function along optimal trajectories under extra regularity assumptions.We then provide a Lagrangian formulation for our mean field game and use classical techniques to prove existence of equilibria, which are shown to satisfy a MFG system. Our main result, which relies on the semi-concavity of the value function, states that an absolutely continuous initial distribution of agents with an L p density gives rise to an absolutely continuous distribution of agents at all positive times with a uniform bound on its L p norm. This is also used to prove existence of equilibria under fewer regularity assumptions on the dynamics thanks to a limit argument.

We show that a solution to a variant of the Beckmann problem can be obtained by studying the limit of some weighted p − p- Laplacian problems. More precisely, we find a solution to the following minimization problem: min { ∫ Ω k d | w | + ∫ ∂ Ω g − d ν − − ∫ ∂ Ω g + d ν + : w ∈ M d ( Ω ) , ν ∈ M ( ∂ Ω ) , − ∇ ⋅ w = f + ν } , \begin{equation*} \min \bigg \{\int _\Omega k \,\mathrm {d}|w| + \int _{\partial \Omega } g^-\,\mathrm {d}\nu ^- - \int _{\partial \Omega } g^+\,\mathrm {d}\nu ^+\,:\,w \in \mathcal {M}^d(\Omega ),\,\nu \in \mathcal {M}(\partial \Omega ),\,-\nabla \cdot w =f + \nu \bigg \}, \end{equation*} where f , k f,\,k , and g ± g^\pm are given. In addition, we connect this problem to a formulation with Kantorovich potentials with Dirichlet boundary conditions.