Uniqueness of diffusion on domains with rough boundaries. (January 2016) - Lehrbäck, Juha; Robinson, Derek W.
Abstract: Let Ω be a domain in R d and h ( φ ) = ∑ k, l = 1 d ( ∂ k φ, c k l ∂ l φ ) a quadratic form on L 2 ( Ω ) with domain C c ∞ ( Ω ) where the c k l are real symmetric L ∞ ( Ω ) -functions with C ( x ) = ( c k l ( x ) ) > 0 for almost all x ∈ Ω . Further assume there are a, δ > 0 such that a − 1 d Γ δ I ≤ C ≤ a d Γ δ I for d Γ ≤ 1 where d Γ 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 R 2 or the complement of a uniformly disconnected set in R d .