$L_1$-uniqueness of degenerate elliptic operators

Abstract

Let $\Omega$ be an open subset of $\Ri^d$ with $0\in \Omega$. Further let $H_\Omega=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j$ be a second-order partial differential operator with domain $C_c^\infty(\Omega)$ where the coefficients $c_{ij}\in W^{1,\infty}_{\rm loc}(\bar\Omega)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $0<C(x)\leq c(|x|) I$ for all $x\in \Omega$. If \[ \int^\infty_0ds\,s^{d/2}\,e^{-\lambda\,\mu(s)^2}<\infty \] for some $\lambda>0$ where $\mu(s)=\int^s_0dt\,c(t)^{-1/2}$ then we establish that $H_\Omega$ 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 $\Omega$, measured with respect to $H_\Omega$, 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 $H_\Omega$ at $A$.