Stability estimates in $H^1_0$ for solutions of elliptic equations in varying domains

TL;DR

This paper derives stability estimates for solutions of elliptic equations under domain perturbations, quantifying how solutions change with domain deformations using bi-Lipschitz maps and symmetric difference measures.

Contribution

It introduces new stability estimates in $H^1_0$ for elliptic solutions under domain variations, linking solution differences to geometric measures of domain perturbation.

Findings

Estimates of solution gradient differences in terms of domain deformation measures.

Sharpness of the derived estimates demonstrated through examples.

Applicability to domains with bi-Lipschitz boundary transformations.

Abstract

We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain $\Omega$ and on the domain $\phi(\Omega)$ resulting from $\Omega$ by means of a bi-Lipschitz map $\phi$. We consider the solutions $u$ and $\tilde u$ of the corresponding elliptic equations with the same right-hand side $f\in L^2(\Omega\cup\phi(\Omega))$. Under certain assumptions we estimate the difference $\|\nabla\tilde u-\nabla u\|_{L^2(\Omega\cup\phi(\Omega))}$ in terms of certain measure of vicinity of $\phi$ to the identity map. For domains within a certain class this provides estimates in terms of the Lebesgue measure of the symmetric difference of $\phi(\Omega)$ and $\Omega$, that is $|\phi(\Omega)\triangle \Omega|$. We provide an example which shows that the estimates obtained are in a certain sense sharp.