Infinity Laplacian equation with strong absorptions

TL;DR

This paper investigates the regularity of solutions to reaction-diffusion equations governed by the infinity Laplacian, especially near regions where solutions vanish, providing geometric estimates on the boundary of these regions.

Contribution

It offers new sharp regularity estimates for solutions near plateau boundaries in infinity Laplacian reaction-diffusion models, including finiteness of the Hausdorff measure.

Findings

Finite (n-ε)-Hausdorff measure of plateau boundaries

Sharp geometric regularity estimates for solutions

Analysis focused on regions where solutions vanish

Abstract

We study regularity properties of solutions to reaction-diffusion equations ruled by the infinity laplacian operator. We focus our analysis in models presenting plateaus, i.e. regions where a non-negative solution vanishes identically. We obtain sharp geometric regularity estimates for solutions along the boundary of plateaus sets. In particular we show that the $(n-\epsilon)$-Hausdorff measure of the plateaus boundary is finite, for a universal number $\epsilon>0$.