Uniqueness of diffusion on domains with rough boundaries

Abstract

Let Ω be a domain in Rd and h(φ)=∑k,l=1d(∂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−1dΓδI≤C≤adΓδI for dΓ≤1 where dΓ is the Euclidean distance to the boundary Γ of Ω.We assume that Γ is Ahlfors s-regular and if s, the Hausdorff dimension of Γ, is larger or equal to d−1 we also assume a mild uniformity property for Ω in the neighbourhood of one z∈Γ. 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 Γ 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.