Abstract: | We make a contribution to the theory of embeddings of anisotropic Sobolev spaces into L
p
-spaces (Sobolev case) and spaces of H?lder continuous functions (Morrey case). In the case of bounded domains the generalized
embedding theorems published so far pose quite restrictive conditions on the domain’s geometry (in fact, the domain must be
“almost rectangular”). Motivated by the study of some evolutionary PDEs, we introduce the so-called “semirectangular setting”,
where the geometry of the domain is compatible with the vector of integrability exponents of the various partial derivatives,
and show that the validity of the embedding theorems can be extended to this case. Second, we discuss the a priori integrability
requirement of the Sobolev anisotropic embedding theorem and show that under a purely algebraic condition on the vector of
exponents, this requirement can be weakened. Lastly, we present a counterexample showing that for domains with general shapes
the embeddings indeed do not hold. |