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

TL;DR

This paper studies the behavior of solutions to the inhomogeneous p-Laplacian equation with Neumann boundary conditions as p approaches infinity, revealing convergence to a Kantorovich potential and characterizing the limit as a viscosity solution.

Contribution

It establishes the convergence of solutions to a Kantorovich potential for signed measures and characterizes the limit as a viscosity solution in the regular case.

Findings

Solutions converge to a Kantorovich potential for Wasserstein-1 distance.

In the regular case, the limit solves an infinity Laplacian/eikonal equation.

The results connect PDE solutions with optimal transport theory.

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.