Let $\Omega$ be a domain in $\mathbb{R}^d$ (not necessarily bounded, no regularity assumption) and $K \subset \Omega$ a compact.
Is it true that the embedding $H^1_0(\Omega) \rightarrow L^2_K(\Omega)$ is compact, $L^2_K(\Omega)$ being the set of $L^2$ functions that are supported in $K$?
Thanks for any response.