$L_1$-uniqueness of degenerate elliptic operators

Abstract

Let $\hat{I}\text{c◯}$ be an open subset of $\Ri^d$ with $0\in \hat{I}\text{c◯}$. Further let $H_{\hat{I}}\text{c◯}=-{\sum }_{i,j=1}^d{\partial }_i{c}_{ij}{\partial }_j$ be a second-order partial differential operator with domain $C_c^{\infty }(\hat{I}\text{c◯})$ where the coefficients $c_{ij}\in W_{\mathrm{loc}}^{1,\infty }(\bar{\hat{I}}\text{c◯})$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $0<C(x)\le c(|x|)I$ for all $x\in \hat{I}\text{c◯}$. If \[ \int^\infty_0ds\,s^{d/2}\,e^{-λ\,μ(s)^2}<\infty \] for some $\hat{I}»>0$ where $\hat{I}¼(s)={\int }_0^{s}dtc(t{)}^{-1/2}$ then we establish that $H_{\hat{I}}\text{c◯}$ is $L_1$-unique, i.e.\ it has a unique $L_1$-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e.\ it has a unique $L_2$-extension which generates a submarkovian semigroup. Moreover these uniqueness conditions are equivalent with the capacity of the boundary of $\hat{I}\text{c◯}$, measured with respect to $H_{\hat{I}}\text{c◯}$, being zero. We also demonstrate that the capacity depends on two gross features, the Hausdorff dimension of subsets $A$ of the boundary the set and the order of degeneracy of $H_{\hat{I}}\text{c◯}$ at $A$.

21 pages