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

Let Ω be an open subset of d and H Ω =- i,j=1 d i c ij j be a second-order partial differential operator on L 2 (Ω) with domain C c (Ω), where the coefficients c ij W 1, (Ω) are real symmetric and C=(c ij ) is a strictly positive-definite matrix over Ω. In particular, H Ω is locally strongly elliptic. We analyze the submarkovian extensions of H Ω , i.e., the self-adjoint extensions that generate submarkovian semigroups. Our main result states that H Ω is Markov unique, i.e., it has a unique submarkovian extension, if and only if cap Ω (Ω)=0 where cap Ω (Ω) is the capacity of the boundary of Ω measured with respect to H Ω . The second main result shows that Markov uniqueness of H Ω is equivalent to the semigroup generated by the Friedrichs extension of H Ω being conservative.

Publié le :
Classification : 47B25, 47D07, 35J70
Robinson, Derek W. 1 ; Sikora, Adam 2

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
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Robinson, Derek W.; Sikora, Adam. Markov uniqueness of degenerate elliptic operators. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 683-710. http://archive.numdam.org/item/ASNSP_2011_5_10_3_683_0/

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