# Markov uniqueness of degenerate elliptic operators - Mathematics > Analysis of PDEs

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Abstract: Let $\Omega$ be an open subset of $\Ri^d$ and$H \Omega=-\sum^d {i,j=1}\partial i c {ij} \partial j$ a second-order partialdifferential operator on $L 2\Omega$ with domain $C c^\infty\Omega$ wherethe 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-adjointextensions which generate submarkovian semigroups. Our main result establishesthat $H \Omega$ is Markov unique, i.e. it has a unique submarkovian extension,if and only if $\capp \Omega\partial\Omega=0$ where$\capp \Omega\partial\Omega$ is the capacity of the boundary of $\Omega$measured with respect to $H \Omega$. The second main result establishes thatMarkov uniqueness of $H \Omega$ is equivalent to the semigroup generated by theFriedrichs extension of $H \Omega$ being conservative.

Author: ** Derek W. Robinson, Adam Sikora**

Source: https://arxiv.org/