$L_1$-uniqueness of degenerate elliptic operators

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

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