On self-adjointness of symmetric diffusion operators

Abstract

Let Ω be a domain in R d with boundary Γ and let d Γ denote the Euclidean distance to Γ. Further let H = − div(C∇) where C = ( c kl ) > 0 with c kl = c lk are real, bounded, Lipschitz continuous functions and D(H) = C ∞ c (Ω). Assume also that there is a δ ≥ 0 suchto be bounded on Γ r . Then we prove that if Ω is a C 2 -domain, or if Ω = R d \S where S is a countable set of positively separated points, or if Ω = R d \Π with Π a convex set whose boundary has Hausdorff dimension d H ∈ {1, . . . , d − 1} then the condition δ > 2 − (d − d H )/2 is sufficient for H to be essentially self-adjoint as an operator on L 2 (Ω). In particular δ > 3/2 suffices for C 2 -domains. Finally we prove that δ ≥ 3/2 is necessary in the C 2 -case.