Mayer-Vietoris type formula for determinants of elliptic differential operators
type: Item
instance of: scholarly article
title: Mayer-Vietoris type formula for determinants of elliptic differential operators
zbMATH Open document ID: 0759.58043
DOI: 10.1016/0022-1236(92)90099-5
author: Leonid Friedlander
author: Dan Burghelea
author: T. Kappeler
published in: Journal of Functional Analysis
publication date: 1993-01-16T00:00:00Z
review text: The paper consists of four sections and an appendix. Section 1 is an introduction. In Section 2 the definitions and auxiliary results are given. The main results are formulated and proved in Section 3: For a closed codimension one submanifold \(\Gamma\) of a compact manifold \(M\), let \(M\) be the manifold with boundary obtained by cutting \(M\) along \(\Gamma\). Let \(A\) be an elliptic differential operator on \(M\) and \(B\) and \(C\) be two complementary boundary conditions on \(\Gamma\). If \((A,B)\) is an elliptic boundary value problem on \(M_ \Gamma\), then one defines an elliptic pseudodifferential operator \(R\) of Neumann type on \(\Gamma\) and proves the following factorization formula for the \(\zeta\)-regularized determinants: \(\text{Det }A/\text{Det}(A,B)=K\text{ Det }R\), with \(K\) a local quantity depending only on the jets of the symbols of \(A\), \(B\) and \(C\) along \(\Gamma\). In Section 4 the particular case when \(M\) has dimension 2, \(A\) is the Laplace-Beltrami operator, and \(B\) resp. \(C\) is the Dirichlet resp. Neumann boundary condition is considered. In the appendix the asymptotics of determinants of elliptic pseudodifferential operators with parameter is considered.
reviewed by: Sattor Baĭzaev
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 58J52
Mathematics Subject Classification ID: 35S15
zbMATH DE Number: 90035
zbMATH Keywords: elliptic differential operator
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: determinants
zbMATH Keywords: elliptic pseudodifferential operators
MaRDI profile type: Publication
full work available at URL: 0022-1236(92)90099-5
OpenAlex ID: W2054137042
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description: scientific article; zbMATH DE number 90035