5
$\begingroup$

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.

  • 0
    I don't know whether there are great result in the smooth boundary case. Are you sure that $C_c^{\infty}$ is complete for the norm which comes from this inner product?2012-02-29

1 Answers 1

5

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 $H^2(\Omega)$.

  • 0
    How exactly do you conclude the compactness of $T$? I've [asked a related question](http://math.stackexchange.com/questions/2086343/%ce%94-with-domain-u%e2%88%88h-01%ce%94u%e2%88%88l2-admits-an-orthonormal-basis-of-l2-consi); maybe you can help.2017-01-07