Markov uniqueness of degenerate elliptic operators

Robinson, Derek W.¹  ; Sikora, Adam²

¹ Centre for Mathematics and its Applications Mathematical Sciences Institute Australian National University Canberra, ACT 0200, Australia
² Department of Mathematics Macquarie University Sydney, NSW 2109, Australia

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 683-710

Résumé

Let $\Omega$ be an open subset of ${ℝ}^d$ and $H_{\Omega }=-{\sum }_{i,j=1}^d{\partial }_i{c}_{ij}{\partial }_j$ be a second-order partial differential operator on $L_2\left(\Omega \right)$ with domain $C_c^{\infty }\left(\Omega \right)$, where the coefficients $c_{ij}\in W^{1,\infty }\left(\Omega \right)$ are real symmetric and $C=\left(c_{ij}\right)$ 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-adjoint extensions that generate submarkovian semigroups. Our main result states that $H_{\Omega }$ is Markov unique, i.e., it has a unique submarkovian extension, if and only if ${\mathrm{cap}}_{\Omega }\left(\partial \Omega \right)=0$ where ${\mathrm{cap}}_{\Omega }\left(\partial \Omega \right)$ is the capacity of the boundary of $\Omega$ measured with respect to $H_{\Omega }$. The second main result shows that Markov uniqueness of $H_{\Omega }$ is equivalent to the semigroup generated by the Friedrichs extension of $H_{\Omega }$ being conservative.

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Publié le : 2018-06-21

Classification : 47B25, 47D07, 35J70