A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems

Jan Haskovec*, Christian Schmeiser

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

29 Scopus citations

Abstract

We make a contribution to the theory of embeddings of anisotropic Sobolev spaces into Lp-spaces (Sobolev case) and spaces of Hölder 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.

Original languageEnglish (US)
Pages (from-to)71-79
Number of pages9
JournalMonatshefte fur Mathematik
Volume158
Issue number1
DOIs
StatePublished - Jan 1 2009

Keywords

  • Anisotropic Sobolev spaces
  • Embedding theorems
  • Gagliardo-Nirenberg-Sobolev inequality
  • Morrey inequality

ASJC Scopus subject areas

  • Mathematics(all)

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