Functional calculus of pseudo-differential boundary problems
type: Item
instance of: scholarly article
title: Functional calculus of pseudo-differential boundary problems
zbMATH Open document ID: 0622.35001
author: Gerd Grubb
published in: Progress in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let \(\Omega\) be an open subset of \({\mathbb{R}}^ n\) with smooth boundary \(\partial \Omega\). Let us consider the boundary value problem \[ (1)\quad \Delta u=f\quad in\quad \Omega,\quad Tu=\phi \quad at\quad \partial \Omega, \] where \(\Delta\) is the Laplacian and T a trace operator (e.g. the usual trace on \(\partial \Omega)\). If R, K are the operators solving (1) with \(\phi =0\), resp. \(f=0\), then the operator \(\left( \begin{matrix} \Delta \\ T\end{matrix} \right)\) is inverted by (R,K). Actually R is equal to \(Q_{\Omega}+G\), where \(Q_{\Omega}\) is a restriction to \(\Omega\) of an inverse Q of \(\Delta\), and G is a special term adapted to the boundary condition and called a singular Green operator. The consideration of Problem (1) above therefore motivates the study of general operators of the form \[ (2)\quad \left( \begin{matrix} P_{\Omega}+G\\ T\end{matrix} \begin{matrix} K\\ S\end{matrix} \right): C({\bar \Omega})^ N\times C^{\infty}(\partial \Omega)^ M\quad \to \quad C^{\infty}({\bar \Omega})^{N'}\times C^{\infty}(\partial \Omega)^{M'}\quad, \] where P and S are pseudo-differential operators, G is a singular Green operator and K is a so-called Poisson operator. Operators of type (2) were introduced and studied by Boutet de Monvel. The main purpose of the book under review is to set up an operational calculus for operators of type (2) defined from differential and pseudo- differential boundary value problems via a resolvent construction, and to present some applications of this to evolution problems, fractional powers, spectral theory and singular perturbation problems. Here is a quick (and superficial) description of the contents of this book. In Chapter 1, the theory of boundary problems of type (2) is surveyed, with special attention devoted to the case of the so-called normal boundary conditions. Operators of type (2) depending on a parameter is introduced. For them a symbolic calculus is developed in Chapter 2, and a parametrix and resolven…
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35S15
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 47Gxx
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 4008910
zbMATH Keywords: boundary value problem
zbMATH Keywords: Laplacian
zbMATH Keywords: trace operator
zbMATH Keywords: singular Green operator
zbMATH Keywords: Poisson operator
zbMATH Keywords: operational calculus
zbMATH Keywords: resolvent construction
zbMATH Keywords: evolution problems
zbMATH Keywords: fractional powers
zbMATH Keywords: spectral theory
zbMATH Keywords: singular perturbation
zbMATH Keywords: normal boundary conditions
zbMATH Keywords: parametrix
zbMATH Keywords: solvability
zbMATH Keywords: parabolic problems
zbMATH Keywords: trace formula
zbMATH Keywords: index formulas
zbMATH Keywords: Complex powers
zbMATH Keywords: eigenvalues
reviewed by: Paul Godin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4008910