Weighted Korn inequality on John domains

TL;DR

This paper establishes a weighted Korn inequality on John domains with explicit constant estimates and extends the analysis to solve div(u)=f in weighted Sobolev spaces, highlighting geometric and boundary distance considerations.

Contribution

It introduces a weighted Korn inequality on John domains with explicit constant bounds and proves solvability of div(u)=f in weighted Sobolev spaces on these domains.

Findings

Weighted Korn inequality with explicit constants on John domains

Solvability of div(u)=f in weighted Sobolev spaces on John domains

Dependence of constants on geometric and weight parameters

Abstract

We show a weighted version of Korn inequality on bounded euclidean John domains, where the weights are nonnegative powers of the distance to the boundary. In this theorem, we also provide an estimate of the constant involved in the inequality which depends on the power that appears in the weight and a geometric condition that characterizes John domains. The proof uses a local-to-global argument based on a certain decomposition of functions. In addition, we prove the solvability in weighted Sobolev spaces of div(u)=f on the same class of domains. In this case, the weights are nonpositive powers of the distance to the boundary. The constant appearing in this problem is also estimated.