Paper detail

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