Boundary behavior of Robin problems in non-smooth domains

TL;DR

This paper investigates the boundary positivity of solutions to Robin boundary value problems in non-smooth domains, providing geometric conditions and quantitative estimates for solutions of elliptic operators of p-Laplacian type.

Contribution

It introduces geometric criteria ensuring boundary positivity of Robin problem solutions in irregular domains, extending previous harmonic function results to more general elliptic operators.

Findings

Established geometric conditions for boundary positivity.

Provided quantitative estimates for solutions near the boundary.

Extended boundary behavior analysis to non-smooth domains.

Abstract

We analyze strict positivity at the boundary for nonnegative solutions of Robin problems in general (non-smooth) domains, e.g. open sets with rectifiable topological boundaries having finite Hausdorff measure. This question was raised by Bass, Burdzy and Chen in 2008 for harmonic functions, in a probabilistic context. We give geometric conditions such that the solutions of Robin problems associated to general elliptic operators of $p$-Laplacian type, with a positive right hand side, are globally or locally bounded away from zero at the boundary. Our method, of variational type, relies on the analysis of an isoperimetric profile of the set and provides quantitative estimates as well.