Uniqueness of diffusion on domains with rough boundaries

Let ΩΩ be a domain in  Rd and h(φ)=∑k,l=1d(∂kφ,ckl∂lφ) a quadratic form on L2(Ω)L2(Ω) with domain Cc∞(Ω) where the cklckl are real symmetric L∞(Ω)L∞(Ω)-functions with C(x)=(ckl(x))>0C(x)=(ckl(x))>0 for almost all x∈Ωx∈Ω. Further assume there are a,δ>0a,δ>0 such that a−1dΓδI≤C≤adΓδI for dΓ≤1dΓ≤1 where dΓdΓ is the Euclidean distance to the boundary ΓΓ of ΩΩ.We assume that ΓΓ is Ahlfors ss-regular and if ss, the Hausdorff dimension of ΓΓ, is larger or equal to d−1d−1 we also assume a mild uniformity property for ΩΩ in the neighbourhood of one z∈Γz∈Γ. Then we establish that hh is Markov unique, i.e. it has a unique Dirichlet form extension, if and only if δ≥1+(s−(d−1))δ≥1+(s−(d−1)). The result applies to forms on Lipschitz domains or on a wide class of domains with ΓΓ 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.