Sobolev inequalities of exponential type
We give sufficient conditions for domains to satisfy Sobolev inequalities of single exponential type. Earlier work in this area imposed more stringent conditions on the domains and is thus contained in our results. Moreover, the class of functions considered is based onL n log an L witha<1−1/n, n being the dimension of the underlying space. The limiting casea=1−1/n gives rise to an inequality of double exponential type which is shown to be valid in a large class of irregular domains. This inequality is new even in smooth domains.
KeywordsSOBOLEV Inequality Orlicz Space Exponential Type Orlicz Function Irregular Domain
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- B. Bojarski,Remarks on Sobolev imbedding inequalities, Proceedings of the Conference on Complex Analysis, Joensuu, Lecture Notes in Mathematics1351, Springer-Verlag, Berlin, 1987, pp. 52–68.Google Scholar
- A. Cianchi,Some results in the theory of Orlicz spaces and applications to variational problems, inProceedings of the Spring School ‘Nonlinear Analysis, Function Spaces and Applications’, Prague, May 31–June 6, 1999, Olympia Press, Prague, 1999, pp. 50–92.Google Scholar
- W. Smith and D. Stegenga,Sobolev imbeddings and integrability of harmonic functions on Hölder domains, inPotential Theory (Nagoya, 1990), de Gruyter, Berlin, 1992, pp. 303–313.Google Scholar