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

Let $Ω$ be a domain in $\Ri^d$ with boundary $Γ$${\!,}$ $d_Γ$ 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 $C\sim c\,d_Γ^{\,δ}$ as $d_Γ\to0$ in the sense of asymptotic analysis where $c$ is a strictly positive, bounded, Lipschitz function and $δ\geq0$. We also assume that there is an $r>0$ and a $b_{δ,r}>0$ 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 $ψ\in C_c^\infty(Γ_{\!\!r})$ where $Γ_{\!\!r}=\{x\inΩ: d_Γ(x)<r\}$. We then prove that the condition $(2-δ)/2<b_δ$ is sufficient for the essential self-adjointness of $H$ on $C_c^\infty(Ω)$ with $b_δ$ the supremum over $r$ of all possible $b_{δ,r}$ 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.

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