Uniform estimates for a family of Poisson problems: `rounding off the corners'

TL;DR

This paper establishes uniform solvability estimates for elliptic problems on domains approaching polygonal shapes, using conformal metric modifications to handle corner rounding in elliptic PDEs.

Contribution

It introduces a novel approach to obtain uniform estimates for elliptic problems on domains with corners by conformally modifying the metric to ensure bounded geometry.

Findings

Uniform estimates hold for elliptic problems on domains approaching polygons.

The method applies both in weighted and standard Sobolev spaces.

The technique effectively 'rounds off' corners via conformal metric adjustments.

Abstract

We prove \emph{uniform solvability estimates} for certain families of elliptic problems posed in a bounded family of domains (for example, a sequence that converges to another domain). We provide uniform estimates both in weighted and in usual Sobolev spaces. When the limit domain is a \emph{polygon} and the other domains are smooth, our results amount to rounding off'' the corners of the limit domain. The technique of proof is based on a suitable conformal modification of the metric, which makes the union of the domains a manifold with boundary and relative bounded geometry.