Krein resolvent formulas for elliptic boundary problems in nonsmooth domains

TL;DR

This paper develops Krein resolvent formulas and M-functions for elliptic boundary problems, extending classical results from smooth to nonsmooth domains using advanced pseudodifferential calculus.

Contribution

It introduces a framework for elliptic boundary problems in nonsmooth domains, expanding the applicability of resolvent formulas with new pseudodifferential techniques.

Findings

Krein resolvent formulas are extended to nonsmooth domains.

Construction of M-functions for elliptic boundary problems.

Application of pseudodifferential calculus to nonsmooth boundary operators.

Abstract

The paper reports on a recent construction of M-functions and Krein resolvent formulas for general closed extensions of an adjoint pair, and their implementation to boundary value problems for second-order strongly elliptic operators on smooth domains. The results are then extended to domains with $C^{1,1}$ H\"older smoothness, by use of a recently developed calculus of pseudodifferential boundary operators with nonsmooth symbols.