## Abstract

Let Ω be a connected open subset of R^{d}. We analyse L_{1}-uniqueness of real second-order partial differential operators, and, on Ω where, and C(x) = (c_{kl}(x)) > 0 for all x ∈ Ω. Boundedness properties of the coefficients are expressed indirectly in terms of the balls B(r) associated with the Riemannian metric C^{-1} and their Lebesgue measure {pipe}B(r){pipe}. First, we establish that if the balls B(r) are bounded, the Täcklind condition ∫^{∞}_{R} dr r(log {pipe}B(r){pipe})^{-1} = ∞ is satisfied for all large R and H is Markov unique then H is L_{1}-unique. If, in addition, C(x) ≥ κ (c^{T} ⊗ c)(x)} for some κ > 0 and almost all x ∈ Ω div, is upper semi-bounded and c_{0} is lower semi-bounded, then K is also L_{1}-unique. Secondly, if the c_{kl} extend continuously to functions which are locally bounded on ∂Ω and if the balls B(r) are bounded, we characterize Markov uniqueness of H in terms of local capacity estimates and boundary capacity estimates. For example, H is Markov unique if and only if for each bounded subset A of, there exist, satisfying, where, and, for each, or if and only if cap(∂Ω) = 0.

Original language | English |
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Pages (from-to) | 229-250 |

Number of pages | 22 |

Journal | Journal of Evolution Equations |

Volume | 13 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2013 |