### Abstract

Let Ω be an open subset of R ^{d} and H _{Ω} - Σ ^{d} _{i,j 1} ∂ _{i}ij ∂ _{j} be a second-order partial differential operator on L _{2}(Ω) with domain C ^{∞} _{0}(Ω), where the coefficients c _{ij} ε 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.

Original language | English |
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Pages (from-to) | 683-710 |

Number of pages | 28 |

Journal | Annali della Scuola Normale - Classe di Scienze |

Volume | 10 |

Issue number | 3 |

Publication status | Published - 2011 |

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## Cite this

*Annali della Scuola Normale - Classe di Scienze*,

*10*(3), 683-710.