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 HΩ=-i,j=1dicijj be a second-order partial differential operator on L2(Ω) with domain Cc(Ω), where the coefficients cijW1,(Ω) are real symmetric and C=(cij) is a strictly positive-definite matrix over Ω. In particular, HΩ is locally strongly elliptic. We analyze the submarkovian extensions of HΩ, i.e., the self-adjoint extensions that generate submarkovian semigroups. Our main result states that HΩ is Markov unique, i.e., it has a unique submarkovian extension, if and only if capΩ(Ω)=0 where capΩ(Ω) is the capacity of the boundary of Ω measured with respect to HΩ. The second main result shows that Markov uniqueness of HΩ is equivalent to the semigroup generated by the Friedrichs extension of HΩ 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://www.numdam.org/item/ASNSP_2011_5_10_3_683_0/

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