Just a quick question. Under what conditions does $L^2(\Omega)$ embed compactly into the dual Sobolev space $H^{1}(\Omega)$? Specifically, I'm looking for conditions on a bounded domain $\Omega\subset \mathbb{C}^n$. Any good references?
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5$\begingroup$ It's the adjoint of the embedding of $H^1_0(\Omega)$ into $L^2(\Omega)$, and an operator is compact iff its adjoint is. So this should follow from the usual Sobolev embedding theorems. $\endgroup$– Nate EldredgeCommented Mar 2, 2017 at 17:35

2$\begingroup$ This is "Schauder's theorem", right? $\endgroup$– NealCommented Mar 2, 2017 at 17:43

$\begingroup$ @Neal Yes it is. $\endgroup$– anonymousCommented Apr 9, 2017 at 18:26
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For the sake of completeness, taking Nate Eldredge's comment and elevating to an answer. This is known as Schauder's theorem; the statement (as Nate Eldredge says) is that an operator is compact iff its adjoint is compact. A proof sketch can be found in this MSE answer. In this context, $L^2(\Omega)$ will embed compactly into $H^{1}(\Omega)$ exactly when $H^1_0(\Omega)$ embeds compactly into $L^2(\Omega)$, which is the case when $\Omega$ is bounded by the RellichKondrachov theorem.