$L_1$-uniqueness of degenerate elliptic operators

TL;DR

This paper establishes conditions under which degenerate elliptic operators are uniquely extendable in $L_1$ and $L_2$ spaces, linking these to boundary capacity and degeneracy features.

Contribution

It provides a characterization of $L_1$-uniqueness for degenerate elliptic operators via boundary capacity and degeneracy order, extending previous results.

Findings

$L_1$-uniqueness is equivalent to Markov uniqueness under certain integrability conditions.

Boundary capacity with respect to the operator determines uniqueness.

Degeneracy and boundary Hausdorff dimension influence the capacity and thus the uniqueness.

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 $A$.