The limit as p → ∞ for the p-Laplacian with mixed boundary conditions and the mass transport problem through a given window

In this paper we study the limit as p → ∞ in a PDE problem involving the p-Laplacian with a right hand side, − div(|Du|p − 2Du) = f, with mixed boundary conditions, u = 0 on Γ and |Du|p − 2 ∂u⁄∂ν = 0 on ∂Ω \ Γ. We find that this limit is related to an optimal mass transport problem, where the total mass given by f is transported outside the domain through a given window on the boundary Γ.