Cavity type problems ruled by infinity Laplacian operator

TL;DR

This paper investigates a singularly perturbed problem involving the infinity Laplacian, establishing regularity, growth, non-degeneracy, porosity of level surfaces, and measure properties of level sets, along with asymptotic analysis.

Contribution

It provides new regularity and geometric properties of solutions to infinity Laplacian problems, including Lipschitz continuity, growth behavior, and measure estimates of level sets.

Findings

Solutions are locally Lipschitz continuous.

Solutions grow linearly and are strongly non-degenerate.

Level surfaces are porous and have finite (n-1)-dimensional measure in some cases.

Abstract

We study a singularly perturbed problem related to infinity Laplacian operator with prescribed boundary values in a region. We prove that solutions are locally (uniformly) Lipschitz continuous, they grow as a linear function, are strongly non-degenerate and have porous level surfaces. Moreover, for some restricted cases we show the finiteness of the (n-1)-dimensional Hausdorff measure of level sets. The analysis of the asymptotic limits is carried out as well.