Markov uniqueness of degenerate elliptic operators

Abstract

Let $\Omega$ be an open subset of $\Ri^d$ and $H_{\Omega }=-{\sum }_{i,j=1}^d{\partial }_i{c}_{ij}{\partial }_j$ a second-order partial differential operator on $L_2(\Omega )$ with domain $C_c^{\infty }(\Omega )$ where the coefficients $c_{ij}\in W^{1,\infty }(\Omega )$ are real symmetric and $C=(c_{ij})$ is a strictly positive-definite matrix over $\Omega$. In particular, $H_{\Omega }$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_{\Omega }$, i.e. the self-adjoint extensions which generate submarkovian semigroups. Our main result establishes that $H_{\Omega }$ is Markov unique, i.e. it has a unique submarkovian extension, if and only if $\capp_Ω(\partialΩ)=0$ where $\capp_Ω(\partialΩ)$ is the capacity of the boundary of $\Omega$ measured with respect to $H_{\Omega }$. The second main result establishes that Markov uniqueness of $H_{\Omega }$ is equivalent to the semigroup generated by the Friedrichs extension of $H_{\Omega }$ being conservative.

24 pages

References in corpus (1)

• Degenerate elliptic operators: capacity, flux and separation

Cited by in corpus (1)

• $L_1$-uniqueness of degenerate elliptic operators