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Journal of the London Mathematical Society Advance Access published online on June 23, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn035
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© 2008 London Mathematical Society

Essential norms and localization moduli of Sobolev embeddings for general domains

J. Lang

Department of Mathematics
The Ohio State University
231 West 18th Avenue
Columbus, OH 43210
USA

V. Maz'ya

Department of Mathematics
The Ohio State University
231 West 18th Avenue
Columbus, OH 43210
USA
vlmaz@math.ohio-state.edu
Department of Mathematical Sciences
University of Liverpool
M&O Building
Liverpool
L69 3BX
United Kingdom
and
Department of Mathematics
Linköping University
Linköping SE-581 83
Sweden

We introduce new measures of non-compactness for the embedding operator Ep,q({Omega}):Lp1({Omega}) -> Lq({Omega}) and describe their relations with the essential norm of Ep, q({Omega}), ‘local’ isoperimetric and isocapacitary constants. An explicit formula for the essential norm of Ep, q({Omega}) is obtained for domains with a power cusp on the boundary and bounded C1 domains. The Neumann problem for a particular Schrödinger operator is discussed on domains with a power cusp.

2000 Mathematics Subject Classification 46E35 (primary), 35P15, 54D45 (secondary).

Received December 11, 2007;
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