Strong uniqueness of finite dimensional Dirichlet operators with singular drifts

Abstract

We show the $L^r({\mathbb{R}}^d,\mu )$-uniqueness for any $r\in (1,2]$ and the essential self-adjointness of a Dirichlet operator $Lf=\Delta f+⟨\frac{1}{\rho }\nabla \rho ,\nabla f⟩$, $f\in C_0^{\infty }({\mathbb{R}}^d)$ with $d\ge 3$ and $\mu =\rho dx$. In particular, $\nabla \rho$ is allowed to be in $L_{loc}^d({\mathbb{R}}^d,{\mathbb{R}}^d)$ or in $L_{loc}^{2+\epsilon }({\mathbb{R}}^d,{\mathbb{R}}^d)$ for some $\epsilon >0$, while $\rho$ is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-divergence type operators and basic properties of Dirichlet forms and their resolvents.

arXiv comment15 pages, minor corrections on typos and omissions