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

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 $fin L^2(Omegacupphi(Omega))$. Under certain assumptions we estimate the difference $| ablatilde u- abla u|_{L^2(Omegacupphi(Omega))}$ 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.