The weighted Hardy inequality and self-adjointness of symmetric diffusion operators

Abstract

Let $\Omega$ be a domain in $\Ri^d$ with boundary $\Gamma$$,$ $d_{\Gamma }$ the Euclidean distance to the boundary and $H=-\divv(C\,\nabla)$ an elliptic operator with $C=(c_{kl})>0$ where $c_{kl}=c_{lk}$ are real, bounded, Lipschitz functions. We assume that $C\sim c{d}_{\Gamma }^{\delta }$ as $d_{\Gamma }\to 0$ in the sense of asymptotic analysis where $c$ is a strictly positive, bounded, Lipschitz function and $\delta \ge 0$. We also assume that there is an $r>0$ and a $b_{\delta ,r}>0$ such that the weighted Hardy inequality \[ \int_{Γ_{\!\!r}} d_Γ^{\,δ}\,|\nabla ψ|^2\geq b_{δ,r}^{\,2}\int_{Γ_{\!\!r}} d_Γ^{\,δ-2}\,| ψ|^2 \] is valid for all $\psi \in C_c^{\infty }({\Gamma }_r)$ where ${\Gamma }_r=\{x\in \Omega :d_{\Gamma }(x)<r\}$. We then prove that the condition $(2-\delta )/2<b_{\delta }$ is sufficient for the essential self-adjointness of $H$ on $C_c^{\infty }(\Omega )$ with $b_{\delta }$ the supremum over $r$ of all possible $b_{\delta ,r}$ in the Hardy inequality. This result extends all known results for domains with smooth boundaries and also gives information on self-adjointness for a large family of domains with rough, e.g.\ fractal, boundaries.