Stability estimates in H_0~1 for solutions of elliptic equations in varying domains

摘要

We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain ? and on the domain φ(?) resulting from ? 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 ∈L~2(? ∪ φ (?)). Under certain assumptions, we estimate the difference ||△?-△u||_L~2(?∪φ(?)) 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 (?) and ?, that is, |φ (?) △?|.We provide an example that shows that the estimates obtained are in a certain sense sharp.