On self-adjointness of symmetric diffusion operators

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=(ckl)>0 with c_kl= c_lk real, bounded, Lipschitz continuous functions and D(H)=Cc∞(Ω). The matrix CdΓ-δ is assumed to converge uniformly to a diagonal matrix aI as d_Γ→ 0. Thus δ≥ 0 measures the order of degeneracy of the operator and a, a positive Lipschitz function, gives the boundary profile of the operator. In addition we place a mild restriction on the order of degeneracy of the derivatives of the coefficients at the boundary. Then we derive sufficient conditions for H to be essentially self-adjoint as an operator on L₂(Ω) in three general cases. Specifically, if Ω is a C²-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 essential self-adjointness. In particular δ> 3 / 2 suffices for C²-domains. Finally we prove that δ≥ 3 / 2 is necessary in the C²-case.