When does $\{u\in H^1_0: \Delta_{\mu}u\in L^2\}=H_0^1\cap H^2$.

Question

Let $(M,\mu)$ be a weighted Riemannian manifold. In Grigor’yan's book, he proves that the Dirichlet Laplacian $\Delta_{\mu}$ is self-adjoint on the set $u\in\{u\in H^1_0(M): \Delta_{\mu}u\in L^2\}$. It's well known that for bounded regions $\Omega$ in $R^n$ with smooth boundary (say), that

$ \{u\in H^1_0(\Omega): \Delta_{\mu}u\in L^2(\Omega)\}=H_0^1(\Omega)\cap H^2(\Omega)$

since for $u\in H_0^1(\Omega)\cap H^2(\Omega)$ we have $||u||_{H^2}\le C(||u||_{L^2}+||\Delta u||_{L^2})$. I'm curious to know if there exist examples (elementary ones preferred) where we do not have equality of these two sets, or (more ambitiously) a classification of when they're equal?

Answer 1

Renardy-Rogers, Theorem 9.53 states that $C^2$ boundary is sufficient. Example 9.52 before shows that this cannot be heavily relaxed.