Spectral asymptotics for nonsmooth singular Green operators

TL;DR

This paper establishes spectral asymptotic behavior for nonsmooth singular Green operators, extending known results to cases with less regularity in the symbol, and applies this to boundary terms in elliptic boundary value problems.

Contribution

It proves that the eigenvalue asymptotics hold for nonsmooth singular Green operators with Hölder continuous symbols and extends results to boundary terms in elliptic resolvent formulas with less regular coefficients.

Findings

Eigenvalue asymptotics hold for nonsmooth singular Green operators.

Asymptotics are established for boundary terms with Hölder continuous symbols.

Results apply to boundary correction terms in elliptic boundary value problems.

Abstract

Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain \Omega \subset 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) \sim c j^{-t/(n-1)} for j\to \infty, governed by the boundary dimension n-1. In some nonsmooth cases, upper estimates (**) s_j(G) \le Cj^{-t/(n-1)} are known. We show that (*) holds when G is a general selfadjoint nonnegative singular Green operator with symbol merely H\"older 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 W^1_p(\Omega) for some p>n.