On self-adjointness of symmetric diffusion operators

Abstract

Let $$\Omega $$ be a domain in $$\mathbf{R}^d$$ with boundary $$\Gamma $$ and let $$d_\Gamma $$ denote the Euclidean distance to $$\Gamma $$ . Further let $$H=-\,\mathrm{div}(C\nabla )$$ where $$C=(\,c_{kl}\,)>0$$ with $$c_{kl}=c_{lk}$$ real, bounded, Lipschitz continuous functions and $$D(H)=C_c^\infty (\Omega )$$ . The matrix $$Cd_\Gamma ^{\,-\delta }$$ is assumed to converge uniformly to a diagonal matrix $$a\,I$$ as $$d_\Gamma \rightarrow 0$$ . Thus $$\delta \ge 0$$ measures the order of degeneracy of the operator and a, a positive Lipschitz function, gives the boundary profile of the operator. In addition we place a mild restriction on the order of degeneracy of the derivatives of the coefficients at the boundary. Then we derive sufficient conditions for H to be essentially self-adjoint as an operator on $$L_2(\Omega )$$ in three general cases. Specifically, if $$\Omega $$ is a $$C^2$$ -domain, or if $$\Omega =\mathbf{R}^d\backslash S$$ where S is a countable set of positively separated points, or if $$\Omega =\mathbf{R}^d\backslash \overline{\Pi }$$ with $$\Pi $$ a convex set whose boundary has Hausdorff dimension $$d_H\in \{1,\ldots , d-1\}$$ then the condition $$\delta >2-(d-d_H)/2$$ is sufficient for essential self-adjointness. 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.