Compact Embedding/ Sobolev's Theorem

Question

I have a domain given $\Omega =[0,1]^2$ and need a compact embedding in $C^1(\bar{\Omega})$ for my function in $H^3(\Omega)$. We have $N=2$, so i tried to get the embedding with the help of Sobolev's embedding theorem. However, for this $\partial \Omega$ must be in $C^1$. For $\Omega$ we have the square $[0,1]^2$, this has corners and therefore no smooth edge. Does a compact embedding still apply to this?

Answer 1

The embedding is true for Lipschitz domains and not just $C^1$ domains. This is because if you have a Lipschitz domain, you can extend a function in $H^3(\Omega)$ to a function in $H^3(\mathbb{R}^2)$ controlling the norm. This extension is due to Stein.