Calculus of Variations and Geometric Measure Theory

L. Bungert

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

created by bungert on 08 Jul 2022
modified on 17 Feb 2023


Published Paper

Inserted: 8 jul 2022
Last Updated: 17 feb 2023

Journal: Advances in Continuous and Discrete Models
Volume: 2023
Number: 1
Pages: 1-17
Year: 2023
Doi: 10.1186/s13662-023-03754-8

ArXiv: 2112.07401 PDF
Links: Journal version


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.

Keywords: Optimal transport, Wasserstein distance, p-Laplacian, viscosity solution, Infinity Laplacian