The inhomogeneous p-Laplacian equation with Neumann boundary conditions in the limit $p\to \infty $

Abstract

We investigate the limiting behavior of solutions to the inhomogeneous p-Laplacian equation $-\Delta _{p} u = \mu _{p}$ − Δ p u = μ 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.