On self-adjointness of symmetric diffusion operators

Abstract

Let $Ω$ be a domain in $\Ri^d$ with boundary $Γ$ and let $d_Γ$ denote the Euclidean distance to $Γ$. Further let $H=-\divv(C\nabla)$ where $C=(\,c_{kl}\,)>0$ with $c_{kl}=c_{lk}$ are real, bounded, Lipschitz continuous functions and $D(H)=C_c^\infty(Ω)$. Assume also that there is a $δ\geq0$ such that $\|C/d_Γ^{\,δ}-aI\|\to 0$ as $d_Γ\to0$ with $δ\geq0$ where $a$ is a bounded Lipschitz continuous function with $a\geqμ>0$ on a boundary layer $Γ_{\!\!r}=\{x\inΩ: d_Γ(x) 2-(d-d_H)/2$ is sufficient for $H$ to be essentially self-adjoint as an operator on $L_2(Ω)$. In particular $δ>3/2$ suffices for $C^2$-domains. Finally we prove that $δ\geq 3/2$ is necessary in the $C^2$-case.