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 $p_0.$

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

Tomáš G. Roskovec

Abstract

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