Stability estimates in $H_0^1$ for solutions of elliptic equations in varying domains

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

10 pages