The weighted Hardy inequality and self-adjointness of symmetric diffusion operators

Abstract

Let Ω be a domain in Rd with boundary Γ, dΓ the Euclidean distance to the boundary and H=−div(C∇) an elliptic operator with C=(ckl)>0 where ckl=clk are real, bounded, Lipschitz functions. We assume that C∼cdΓδI as dΓ→0 in the sense of asymptotic analysis where c is a strictly positive, bounded, Lipschitz function and δ≥0. We also assume that there is an r>0 and a bδ,r>0 such that the weighted Hardy inequality∫ΓrdΓδ|∇ψ|2≥bδ,r2∫ΓrdΓδ−2|ψ|2 is valid for all ψ∈Cc∞(Γr) where Γr={x∈Ω:dΓ(x)<r}. We then prove that the condition (2−δ)/2<bδ is sufficient for the essential self-adjointness of H on Cc∞(Ω) with bδ the supremum over r of all possible bδ,r in the Hardy inequality. This result extends all known results for domains with smooth boundaries and also gives information on self-adjointness for a large family of domains with rough, e.g. fractal, boundaries.