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 = ( c kl ) > 0 with c kl = c lk real, bounded, Lipschitz continuous functions and D ( H ) = C c ∞ ( Ω ) . The matrix C d Γ - δ is assumed to converge uniformly to a diagonal matrix a I 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 L 2 ( Ω ) in three general cases. Specifically, 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 essential self-adjointness. In particular δ > 3 / 2 suffices for C 2 -domains. Finally we prove that δ ≥ 3 / 2 is necessary in the C 2 -case.