Embedding and compactness theorems for irregular and unbounded domains in weighted Sobolev spaces (Q1108547)

The classical embedding and compactness theorems for Sobolev spaces hold essentially only in the case of a bounded domain with the cone property. In the case of domains failing to have these properties, many authors have pointed out the rôle played by weights in counter-balancing geometrical irregularities, either in the case of unboundedness, or in presence of cusps: weighted Sobolev spaces seem to be unavoidable if one wishes to obtain general theorems with classical embedding exponents. In this direction, the authors of this paper investigate deeply the link between the weight functions and the geometric structure of the underlying domain. In sections 1 and 2, general embedding and compactness theorems for Sobolev spaces are stated and proved; in section 3 a compact embedding theorem is proved for a class of bounded domains, the weights being related to their shapes; finally, in section 4 examples of concrete domains and weights are given.