First off, I apologize for asking a question which I'm sure has been studied to death, but I can't seem to find an answer with google.
I want to see a proof that the Laplace operator $\Delta$ with domain $W^{2,p}(\Omega) \cap W^{1,p}_0(\Omega)$ generates an analytic semigroup on $L^p(\Omega)$ where $\Omega$ is a reasonably nice domain and $1 < p < \infty$. I'm assuming this boils down to the applying the following theorem:
If $A$ is closed and densely defined then $A$ generates an analytic semigroup iff there exists $\omega \in R$ such that the half plane Re $\lambda > \omega$ is contained in the resolvent set of $A$ and there is a constant $C$ such that the resolvent $R_{\lambda}(A)$ satisfies $\|R_\lambda(A)\| \le C/|\lambda - \omega|$ for all Re $\lambda > \omega$.
That $\Delta$ is closed and densely defined is clear to me, but how do I prove the remaining 2 hypotheses?
Thanks in advance.