On self-adjointness of symmetric diffusion operators

Abstract

Let $\Omega$ be a domain in $\Ri^d$ with boundary $\Gamma$ and let $d_\Gamma$ denote the Euclidean distance to $\Gamma$. Further let $H=-\divv(C\nabla)$ where $C=(\,c_{kl}\,)>0$ with $c_{kl}=c_{lk}$ are real, bounded, Lipschitz continuous functions and $D(H)=C_c^\infty(\Omega)$. Assume also that there is a $\delta\geq0$ such that $\|C/d_\Gamma^{\,\delta}-aI\|\to 0$ as $d_\Gamma\to0$ with $\delta\geq0$ where $a$ is a bounded Lipschitz continuous function with $a\geq\mu>0$ on a boundary layer $\Gamma_{\!\!r}=\{x\in\Omega: d_\Gamma(x)<r\}$. Finally we require $|(\divv C).(\nabla d_\Gamma)|d_\Gamma^{\,-\delta+1}$ to be bounded on~$\Gamma_{\!\!r}$. Then we prove that if $\Omega$ is a $C^2$-domain, or if $\Omega=\Ri^d\backslash S$ where $S$ is a countable set of positively separated points, or if $\Omega=\Ri^d\backslash \overline \Pi$ with $\Pi$ a convex set whose boundary has Hausdorff dimension $d_H\in \{1,\ldots, d-1\}$ then the condition $\delta>2-(d-d_H)/2$ is sufficient for $H$ to be essentially self-adjoint as an operator on $L_2(\Omega)$. In particular $\delta>3/2$ suffices for $C^2$-domains. Finally we prove that $\delta\geq 3/2$ is necessary in the $C^2$-case.