The edge algebra structure of boundary value problems
type: Item
instance of: scholarly article
title: The edge algebra structure of boundary value problems
zbMATH Open document ID: 1024.58011
DOI: 10.1023/A:1019939316595
author: Yanyan Li
published in: Annals of Global Analysis and Geometry
publication date: 2003-04-27T00:00:00Z
review text: The aim of this paper is to treat boundary value problems in the frame of the edge pseudo-differential calculus with constant discrete asymptotics [cf. \textit{B.-W. Schulze}, `Boundary value problems and singular pseudo-differential operators', Wiley, Chichester (1998; Zbl 0907.35146)]. Basic question is the following. Let \(A\) be a given classical pseudo-differential operator on the double \(2X\) of a compact smooth manifold \(X\) with boundary \(Y\), and consider \(B= r^+Ae^+\) where \(r^+\), \(e^+\) are, respectively, the restriction to \(X\) and the extension by zero from \(X\) to \(2X\). Then, if \(A\) has not the transmission property, the operator \(B: C^\infty_0(\text{int }X)\to C^\infty(\text{int }X)\) has not an extension to the standard Sobolev spaces. Generally speaking, one should like to identify an algebra containing \(B\), with good continuity properties on suitable function spaces. This is achieved by the authors passing to the scale of the weighted Sobolev spaces is employed on manifolds with edges, \(Y=\partial X\) being the edge and \(\overline{\mathbb{R}}_+\) the inner normal, model cone of the wedge \(\overline{\mathbb{R}}_+\times Y\). In this frame, the authors obtain a symbolic calculus, extending preceding results in literature [see \textit{L. Boutet de Monvel}, Acta Math. 126, 11-51 (1971; Zbl 0206.39401), \textit{G. I. Eskin}, `Boundary value problems for elliptic pseudodifferential equations', Math. Monogr. 52, Am. Math. Soc., Providence, RI (1980; Zbl 0458.35002)].
reviewed by: Luigi Rodino
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35S15
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 1901457
zbMATH Keywords: boundary value problems
zbMATH Keywords: pseudodifferential operators
MaRDI profile type: Publication
Recommended article: Boundary value problems in edge representation
Recommended article: Q4341310
Recommended article: Q3540470
Recommended article: Mixed boundary value problems and parametrices in the edge calculus
Recommended article: Edge-degenerate boundary value problems on cones
Recommended article: Elliptic theory of differential edge operators II: boundary value problems
Recommended article: Q3342968
Recommended article: Q5286613
Recommended article: Edge-boundary problems with singular trace conditions
Recommended article: Boundary problems on a manifold with edge
description: scientific article; zbMATH DE number 1901457