Essential norms and localization moduli of Sobolev embeddings for general domains

We introduce new measures of non-compactness for the embedding operator E-p,E-q(Omega) : L-p(1)(Omega)-> L-q(Omega) and describe their relations with the essential norm of E-p,E-q(Omega), 'local' isoperimetric and isocapacitary constants. An explicit formula for the essential norm of E-p,E-q(Omega) is obtained for domains with a power cusp on the boundary and bounded C-1 domains. The Neumann problem for a particular Schrodinger operator is discussed on domains with a power cusp.