Markov uniqueness of degenerate elliptic operators
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 3, p. 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.

Published online : 2018-06-21
Classification:  47B25,  47D07,  35J70
@article{ASNSP_2011_5_10_3_683_0,
     author = {Robinson, Derek W. and Sikora, Adam},
     title = {Markov uniqueness of degenerate elliptic operators},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {3},
     year = {2011},
     pages = {683-710},
     zbl = {1259.47026},
     mrnumber = {2905383},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2011_5_10_3_683_0}
}
Robinson, Derek W.; Sikora, Adam. Markov uniqueness of degenerate elliptic operators. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 3, pp. 683-710. http://www.numdam.org/item/ASNSP_2011_5_10_3_683_0/

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