English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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

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