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 $-{\Delta }_{p}u={\mu }_p$ 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.

Corrected some typos

References in corpus (1)

• Eigenvalue Problems in ${\mathrm{L}}^{\infty }$: Optimality Conditions, Duality, and Relations with Optimal Transport