## Abstract

Let Ω be an open subset of R^{d} with 0 ε Ω. Furthermore, let H_{Ω} = - Σ^{d} _{1,j=1} δ_{i}c_{ij}δ_{j} be a second-order partial differential operator with domain C^{∞} _{c}(Ω) where the coefficients C_{ij} ε W ^{1, infin;} _{loc}(Ω̄) are real, C_{ij} = C_{ji} and the coefficient matrix C = (c_{ij}) satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ε ω. If ∫^{∞} _{0} ds s_{d/2e-λμ(s)} ^{2}< ∞ for some λ > 0 where μ(s) = ∫^{s} _{0}dtc(t)^{-1/2} then we establish that H_{ω} is L_{1}-unique, i.e. it has a unique L_{2}-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique 1,2-extension which generates a submarkovian semigroup. Moreover these uniqueness conditions are equivalent to the capacity of the boundary of Ω, measured with respect to H _{ω}, being zero. We also demonstrate that the capacity depends on two gross features, the Hausdorff dimension of subsets A of the boundary of the set and the order of degeneracy of H_{ω} at A.

Original language | English |
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Pages (from-to) | 79-103 |

Number of pages | 25 |

Journal | Studia Mathematica |

Volume | 203 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2011 |