Uniqueness of diffusion on domains with rough boundaries

Abstract

Let Ω be a domain in Rd and h(φ)=(Formula presented.)(∂kφ,ckl∂lφ) a quadratic form on L2(Ω) with domain Cc ∞(Ω) where the ckl are real symmetric L∞(Ω)-functions with C(x)=(ckl(x)) > 0 for almost all x ∈ Ω. Further assume there are a,δ > 0 such that a-1dF δ I ≤ C ≤ a dF δ I for dF ≤ 1 where dF 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 R2 or the complement of a uniformly disconnected set in Rd.