Let $\Omega\subset \mathbb R^d$, $d\ge 2$, be a bounded Lipschitz domain, and consider the $L^2$-Dirichlet Laplacian $(\Delta_{2}^D,\mathscr{D}(\Delta_2^D))$ on $L^2(\Omega)$. Say that $\mathcal C\subset \mathscr{D}(\Delta_2^D))$ is a core for $(\Delta_{2}^D,\mathscr{D}(\Delta_2^D))$ if $(\Delta_{2}^D,\mathscr{D}(\Delta_2^D))$ is the unique $L^2$-self-adjoint extenion of $(\Delta_{2}^D,\mathcal C)$.
Question: What is the best regularity we can expect for functions in $\mathcal C$ ?
Precisely, can we expect that there exists a core $\mathcal C\subset \mathcal C^{k,\alpha}(\Omega)$ for either
$k=2$ and $\alpha=0$, having defined $\mathcal C^{k,0}(\Omega):= \mathcal C^k(\overline\Omega)$?
$k=2$ and some $\alpha>0$?
$k=2$ and $\alpha=1$?
$k\geq 3$?
More difficult question: What is the maximal boundary regularity we can expect from eigenfunctions of $(\Delta_{2}^D,\mathscr{D}(\Delta_2^D))$?
The second question is intimately related to the first, since, if all eigenfunctions have regularity $\mathcal C^{k,\alpha}(\Omega)$, then their linear span is a core of regularity $\mathcal C^{k,\alpha}(\Omega)$.
Note: of course on somewhat nice domains we can expect $\mathcal C^\infty(\overline\Omega)\cap \mathscr{D}(\Delta_2^D)$ to be a core. This is the case on smooth domains, or domains with nice spectral properties, e.g. rectangles, on which one can prove that eigenfunctions are smooth up to the boundary.