Spectral Asymptotics for Nonsmooth Singular Green Operators

Abstract

Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω ⊂ ℝ n , and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order −t on a bounded domain, its eigenvalues or s-numbers have the behavior (*)s j (G) ∼ cj −t/(n−1) for j → ∞, governed by the boundary dimension n − 1. In some nonsmooth cases, upper estimates (**)s j (G) ≤ Cj −t/(n−1) are known. We show that (*) holds when G is a general selfadjoint nonnegative singular Green operator with symbol merely Hölder continuous in x. We also show (*) with t = 2 for the boundary term in the Krein resolvent formula comparing the Dirichlet and a Neumann-type problem for a strongly elliptic second-order differential operator (not necessarily selfadjoint) with coefficients in for some q > n.