On self-adjointness of symmetric diffusion operators

Let $\mathrm{\Omega }$ be a domain in ${\text{Ri}}^d$ with boundary $\mathrm{\Gamma }$ and let $d_{\mathrm{\Gamma }}$ denote the Euclidean distance to $\mathrm{\Gamma }$. Further let $H=-\text{divv}(C\mathrm{\nabla })$ where $C=(c_{kl})>0$ with $c_{kl}=c_{lk}$ are real, bounded, Lipschitz continuous functions and $D(H)=C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. Assume also that there is a $\delta \ge 0$ such that $\mathrm{\parallel }C\mathrm{/}{d}_{\mathrm{\Gamma }}^{\delta }-aI\mathrm{\parallel }\to 0$ as $d_{\mathrm{\Gamma }}\to 0$ with $\delta \ge 0$ where $a$ is a bounded Lipschitz continuous function with $a\ge \mu >0$ on a boundary layer $\Gamma_{!!r}={x\in\Omega: d_\Gamma(x)<r}$\mathrm{.}Finallywerequire$|(\divv C).(\nabla d_\Gamma)|d_\Gamma^{,-\delta+1}$tobeboundedon$\Gamma_{!!r}$. Then we prove that if $\mathrm{\Omega }$ is a $C^2$-domain, or if $\mathrm{\Omega }={\text{Ri}}^d\mathrm{\backslash }S$ where $S$ is a countable set of positively separated points, or if $\mathrm{\Omega }={\text{Ri}}^d\mathrm{\backslash }\bar{\mathrm{\Pi }}$ with $\mathrm{\Pi }$ a convex set whose boundary has Hausdorff dimension $d_H\in \{1,\dots ,d-1\}$ then the condition $\delta >2-(d-d_H)\mathrm{/}2$ is sufficient for $H$ to be essentially self-adjoint as an operator on $L_2(\mathrm{\Omega })$. In particular $\delta >3\mathrm{/}2$ suffices for $C^2$-domains. Finally we prove that $\delta \ge 3\mathrm{/}2$ is necessary in the $C^2$-case.