Mosco convergence of Sobolev spaces and Sobolev inequalities for nonsmooth domains

TL;DR

This paper establishes broad conditions under which Sobolev spaces on nonsmooth domains converge in the Mosco sense and satisfy Sobolev inequalities uniformly, simplifying stability analysis in acoustic scattering problems.

Contribution

It introduces general criteria for Mosco convergence and Sobolev inequalities on nonsmooth domains, unifying and simplifying previous conditions, with applications to acoustic scattering stability.

Findings

Mosco convergence holds for a wide class of nonsmooth open sets.

Sobolev inequalities are valid with uniform constants under these conditions.

Results extend previous work to higher dimensions and simplify stability analysis.

Abstract

We find extremely general classes of nonsmooth open sets which guarantee Mosco convergence for corresponding Sobolev spaces and the validity of Sobolev inequalities with a uniform constant. An important feature of our results is that the conditions we impose on the open sets for Mosco convergence and for the Sobolev inequalities are of the same nature, therefore it is easy to check when both are satisfied. Our analysis is motivated, in particular, by the study of the stability of the direct acoustic scattering problem with respect to the scatterer, which we also discuss. Concerning Mosco convergence in dimension 3 or higher, our result extends all those previously known in the literature. Concerning Sobolev inequalities, our approach seems to be new and considerably simplifies the conditions previously required for the stability of acoustic direct scattering problems.