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

TL;DR

This paper establishes conditions under which certain elliptic diffusion operators are essentially self-adjoint, extending known results to domains with irregular, fractal-like boundaries using a weighted Hardy inequality.

Contribution

It provides a new criterion involving a weighted Hardy inequality for the self-adjointness of elliptic operators on irregular domains, including fractal boundaries.

Findings

Condition (2-δ)/2 < b_δ guarantees self-adjointness.

Extends results to domains with rough, fractal boundaries.

Provides a framework for analyzing self-adjointness via weighted Hardy inequalities.

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 $d_\Gamma\to0$ in the sense of asymptotic analysis where $c$ is a strictly positive, bounded, Lipschitz function and $\delta\geq0$. We also assume that there is an $r>0$ and a $ b_{\delta,r}>0$ such that the weighted Hardy inequality \[ \int_{\Gamma_{\!\!r}} d_\Gamma^{\,\delta}\,|\nabla \psi|^2\geq b_{\delta,r}^{\,2}\int_{\Gamma_{\!\!r}} d_\Gamma^{\,\delta-2}\,| \psi|^2 \] 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.