The limit as $p\to \infty$ for the $p$-Laplacian with mixed boundary conditions and the mass transport problem through a given window

Abstract

In this paper we study the limit as $p\to \infty$ in a PDE problem involving the $p$-Laplacian with a right hand side, $-\mathrm{div}(|Du{|}^{p-2}Du)=f$, with mixed boundary conditions, $u=0$ on $\Gamma$ and $|Du{|}^{p-2}\frac{\partial u}{\partial \nu }=0$ on $\partial \Omega \hspace{0.17em}\backslash \hspace{0.17em}\Gamma$. 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 $\Gamma$.