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

Mosco convergence of Sobolev spaces and Sobolev inequalities for nonsmooth domains