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

Let Omega be a domain in R-d with boundary Gamma, d(Gamma) the Euclidean distance to the boundary and H = -div(C del) an elliptic operator with C = (c(kl)) > 0 where c(kl) = c(lk) are real, bounded, Lipschitz functions. We assume that C similar to c d(Gamma)(delta)I as d(Gamma) -> 0 in the sense of asymptotic analysis where c is a strictly positive, bounded, Lipschitz function and delta >= 0. We also assume that there is an r > 0 and a b(delta,r) > 0 such that the weighted Hardy inequality integral(Gamma r) d(Gamma)(delta) vertical bar del psi vertical bar(2) >= b(delta,r)(2) integral(Gamma r) d(Gamma)(delta-2) vertical bar psi vertical bar(2) is valid for all psi is an element of C-c(infinity)(Gamma(r)) where Gamma(r) = {x is an element of Omega : d(Gamma)(x) < r}. We then prove that the condition (2 - delta)/2 < b(delta) is sufficient for the essential self-adjointness of H on C-c(infinity)(Omega) with b(delta) the supremum over r of all possible b(delta,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. (C) 2021 Elsevier Inc. All rights reserved.