4
$\begingroup$

Let $\Omega\subset\mathbb{R}^n$ a open bounded set. The Dirichlet laplacian can be defined via it's closed semi-bounded form on $H^1_0(\Omega)$. The fact that it's spectrum is discrete is as far as I can tell proven by that fact that the embedding $H_0^1(\Omega)\rightarrow L^2(\Omega)$ is compact and that the spectrum is discrete if and only if the embedding $H_0^1(\Omega)=(D(q),\lVert\cdot\lVert_q)\rightarrow L^2(\Omega)$ is compact. Where $q$ is the associated form and $D(q)$ is the form domain.

I search for quite some time but didn't find a proof for the second claim. I'd appreciate hints on the proof itself and references very much!

  • 0
    Where did you find this equivalence: spectrum is discrete if and only if the embedding $H_0^1(\Omega)=(D(q),\lVert\cdot\lVert_q)\rightarrow L^2(\Omega)$ is compact2012-12-29
  • 1
    I never saw the second claim phrased as "if and only if". The embedding of $H_0^1(\Omega)$ into $L^2(\Omega)$ is compact for any bounded domain. An accessible proof that the spectrum of the Dirichlet Laplacian is discrete (on any bounded domain) can be found in the book [Spectral theory and differential operators](http://books.google.com/books/about/Spectral_Theory_and_Differential_Operato.html?id=j5L_xSkLjW8C) by E. B. Davies.2012-12-29
  • 0
    the if and only if is claimed in "Schmüdgen, Unbounded Operators on Hilbert Space, Chapter 10, Prop. 10.6"2012-12-29
  • 0
    There is a demonstration there of this result. Did you understood it?2012-12-29

1 Answers 1

1

I know how to prove one part of the assertion. Let $f\in L^2$ and consider the problem $$ \left\{ \begin{array}{rl} -\Delta u=f &\mbox{in $\Omega$} \\ u=0 &\mbox{in $\partial\Omega$ } \end{array} \right. $$

We know that for each $f$ there exist a unique weak solution $u\in H_0^1$ of the previous problem, i.e. $$\int_\Omega \nabla u\nabla v=\int_\Omega fv,\ \forall\ v\in H_0^1$$

Define $T:L^2\rightarrow H_0^1$ by $Tf=u$, where $u$ is the weak solution. Moreover, you can conclude from the characterization of weak solution that $\|u\|_{H_0^1}\leq C\|f\|_2$ and $T$ is self-adjoint. Because $H_0^1$ is compactly embedded in $L^2$, you have that $T:L^2\rightarrow L^2$ is a self-adjoint compact operator. Now you can conclude.