Under which circumstances is the Laplacian compact?

Question

I want to know when the Laplacian is a compact Operator. Do you know some good literature about this topic?

For instance, is the Laplacian compact on the Sobolev space $H^2(\Omega)$? Or maybe on the Hilbert space $C^{\infty}_c(\Omega)$ with the inner product $\langle f,g\rangle:=\int\limits_{\Omega}f\cdot g\text{ }dx$ ?

Thanks you for your answers.

Answer 1

Let $E:=\{u\in H^1_0(\Omega),\Delta u\in L^2(\Omega)\}$ endowed with the norm $||u||_E:=||u||_{H^1_0(\Omega)}$. We can see that $-\Delta\colon E\to L^2(\Omega)$ is an isomorphism. Let $T\colon L^2(\Omega)\to E$ its inverse. Then, using the fact that $E \hookrightarrow H^1_0(\Omega)$ is continuous and $H_0^1(\Omega)\hookrightarrow L^2(\Omega)$ is compact, we get that $T$ is compact, so $-\Delta$ cannot be compact on $E$. In particular, it cannot be compact in $H^2(\Omega)$.