Stability estimates in $H_0^1$ 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 $\mathrm{\Omega }$ and on the domain $\varphi (\mathrm{\Omega })$ resulting from $\mathrm{\Omega }$ 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 $f\in L^2(\Omega\cup\phi(\Omega))$. Under certain assumptions we estimate the difference $\mathrm{\parallel }\mathrm{\nabla }\tilde{u}-\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega }\cup \varphi (\mathrm{\Omega }))}$ 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 $\varphi (\mathrm{\Omega })$ and $\mathrm{\Omega }$, that is $\mathrm{\mid }\varphi (\mathrm{\Omega })\mathrm{△}\mathrm{\Omega }\mathrm{\mid }$. We provide an example which shows that the estimates obtained are in a certain sense sharp.