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 $\delta \ge 0$ such that $\Vert C/d_{\Gamma }^{\delta }-aI\Vert \to 0$ as $d_{\Gamma }\to 0$ with $\delta \ge 0$ where $a$ is a bounded Lipschitz continuous function with $a\ge \mu >0$ on a boundary layer ${\Gamma }_r=\{x\in \Omega :d_{\Gamma }(x)<r\}$. Finally we require $|(\divv C).(\nabla d_Γ)|d_Γ^{\,-δ+1}$ to be bounded on~${\Gamma }_r$. Then we prove that if $\Omega$ is a $C^2$-domain, or if $Ω=\Ri^d\backslash S$ where $S$ is a countable set of positively separated points, or if $Ω=\Ri^d\backslash \overline Π$ with $\Pi$ a convex set whose boundary has Hausdorff dimension $d_H\in \{1,\dots ,d-1\}$ 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 $\delta \ge 3/2$ is necessary in the $C^2$-case.