The inhomogeneous p-Laplacian equation with Neumann boundary conditions in the limit pto infty

Abstract

We investigate the limiting behavior of solutions to the inhomogeneous p-Laplacian equation -Delta _{p} u = mu _{p} subject to Neumann boundary conditions. For right-hand sides, which are arbitrary signed measures, we show that solutions converge to a Kantorovich potential associated with the geodesic Wasserstein-1 distance. In the regular case with continuous right-hand sides, we characterize the limit as viscosity solution to an infinity Laplacian / eikonal type equation.