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Zeitschrift für Analysis und ihre Anwendungen


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Volume 35, Issue 2, 2016, pp. 139–152
DOI: 10.4171/ZAA/1558

Published online: 2016-04-13

Sobolev Embedding Theorem for Irregular Domains and Discontinuity of $p \to p^*(p,n)$

Tomáš G. Roskovec[1]

(1) Charles University, Prague, Czech Republic

For a domain $\Omega\subset\mathbb{R}^n$ we denote $$\begin{aligned}q_{\Omega}(p):=\operatorname{sup}\big\{r\in[1,\infty];\text{ for all } f:\Omega\rightarrow\mathbb{R}:(f\in W^{1,p}(\Omega)\Rightarrow f\in L^{r}(\Omega))\big\}. \end{aligned}$$ Let $p_0 \!\in \! [2,\infty).$ We construct a domain $\Omega \! \subset \! \mathbb{R}^2$ such that $q_{\Omega}(p)$ is discontinuous at $p_0.$

Keywords: Sobolev space, Sobolev embedding

Roskovec Tomáš: Sobolev Embedding Theorem for Irregular Domains and Discontinuity of $p \to p^*(p,n)$. Z. Anal. Anwend. 35 (2016), 139-152. doi: 10.4171/ZAA/1558