Let $\Omega \subseteq \mathbb R^d$ a bounded domain. My professor told me that there is a compact embedding $H_0^1(\Omega) \hookrightarrow L^2(\Omega)$. I am currently looking for some good sources for this theorem and its proof. So I would really appreciate some scripts or literature which deals with this embedding. Thank you in advance :)
2026-03-29 16:33:08.1774801988
Compact Sobolev Embedding
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As @Siddhant suggest, Evans book is a good resources. Here I want to recommend Leoni's book. It has more details in calculations and more lemmas.
Also, another good resources would be Adams book. It gives more information for higher order sobolev space and also deal with irregular domains.
Those two, plus Evans, should be very enough.