Stability estimates in for solutions of elliptic equations in varying domains
José M. Arrieta, Gerassimos Barbatis · Mathematical Methods in the Applied Sciences · 2013
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 of the corresponding elliptic equations with the same right‐hand side f ∈ L2(Ω ∪ ϕ(Ω)). Under certain assumptions, we estimate the difference 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) △ Ω | . We provide an example that shows that the estimates obtained are in a certain sense sharp. Copyright © 2013 John Wiley & Sons, Ltd.