## Abstract

Let Ω be a domain in R^{d} and h(φ)=(Formula presented.)(∂_{k}φ,c_{kl}∂_{l}φ) a quadratic form on L_{2}(Ω) with domain C_{c}^{∞}(Ω) where the c_{kl} are real symmetric L_{∞}(Ω)-functions with C(x)=(c_{kl}(x)) > 0 for almost all x ∈ Ω. Further assume there are a,δ > 0 such that a^{-1}d_{F}^{δ} I ≤ C ≤ a d_{F}^{δ} I for d_{F} ≤ 1 where d_{F} is the Euclidean distance to the boundary F of Ω. We assume that F is Ahlfors s-regular and if s, the Hausdorff dimension of F, is larger or equal to d - 1 we also assume a mild uniformity property for Ω in the neighbourhood of one z ∈ F. Then we establish that h is Markov unique, i.e. it has a unique Dirichlet form extension, if and only if δ ≥ 1+(s-(d-1)). The result applies to forms on Lipschitz domains or on a wide class of domains with F a self-similar fractal. In particular it applies to the interior or exterior of the von Koch snowflake curve in R^{2} or the complement of a uniformly disconnected set in R^{d}.

Original language | English |
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Pages (from-to) | 60-80 |

Number of pages | 21 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 131 |

DOIs | |

Publication status | Published - Jan 2016 |