The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates

TL;DR

This paper derives a Krein resolvent formula for mixed boundary value problems of elliptic operators, leading to new spectral asymptotics that relate the resolvent difference to boundary geometry.

Contribution

It introduces a Krein resolvent formula for mixed boundary problems and establishes new Weyl-type spectral asymptotics for the resolvent difference.

Findings

Krein resolvent formula expressed via boundary operators

New spectral asymptotics proportional to boundary area

Upper estimates previously known, now with precise asymptotics

Abstract

For a second-order symmetric strongly elliptic operator A on a smooth bounded open set \Omega in R^n with boundary \Sigma, the mixed problem is defined by a Neumann-type condition on a part Sigma_+ of the boundary and a Dirichlet condition on the other part Sigma_-. We show a Krein resolvent formula, where the difference between its resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Sigma_+. This is used to obtain a new Weyl-type spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely s_j j^{2/(n-1)}\to C_{0,+}^{2/(n-1)}, where C_{0,+} is proportional to the area of Sigma_+, in the case where A is principally equal to the Laplacian.