Sobolev embeddings into Orlicz spaces and isocapacitary inequalities

Abstract

Sobolev embeddings into Orlicz spaces on domains in the Euclidean space or, more generally, on Riemannian manifolds are considered. Highly irregular domains where the optimal degree of integrability of a function may be lower than the one of its gradient are focused. A necessary and sufficient condition for the validity of the relevant embeddings is established in terms of the isocapacitary function of the domain. Compact embeddings are discussed as well. Sufficient conditions involving the isoperimetric function of the domain are derived as a by-product.