Embedding of Sobolev Spaces on H\"older Domains

D. A. Labutin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Investigations in the theory of differentiable functions of many variables and its applications. Part 18, Tome 227 (1999), pp. 170-179

Résumé

It is well known that the embedding $W^1_p(\Omega)\hookrightarrow L_q(\Omega)$, $1\leq p$, is equivalent to certain isoperimetric or capacity inequalities for the subsets of $\Omega$. P. Hajłasz with P. Koskela and T. Kilpeläinen with J. Malý have proved in their recent works the inequalities of this type for a wide class of $s$+John domains. In the present paper, we prove the exact isoperimetric inequality and the embedding $W^1_p(\Omega)\hookrightarrow L_q(\Omega)$ with the best index $q$ for a narrower class of Hölder domains. A Hölder domain locally coincides with the epigraph of a function satisfying the Hölder condition. The improvement of the index $q$ as compared with the case considered in the aforementioned works is achieved due to the application of special coverings of the subsets of $\Omega$.