Analysis of differential operators associated with boundary value problems
type: Item
instance of: scholarly article
title: Analysis of differential operators associated with boundary value problems
zbMATH Open document ID: 1214.47038
author: M. A. Antonets
published in: Journal of Mathematical Sciences (New York)
publication date: 2010-12-21T00:00:00Z
review text: In general, boundary value problems (BVP) for differential equations in vector bundles are studied by means of reducing the BVP to equations of traces of the solution on the boundary. The paper considers BVP for differential operators with transmission boundary conditions and impedance boundary conditions in a domain of the \(n\)-dimensional Euclidean space obtained by eliminating from \(\mathbb R^n\) the boundary \(\mathfrak S\). On both sides of the boundary \(\mathfrak S\), without the use of limit passages, operators of trace computation are constructed. Traces on the boundary, the Hilbert space of traces and the projection operators are defined first. Based on the projection operators and on the operators that define the transmission and impedance type boundary conditions, operators on the trace space are constructed. The author proves the invertibility of these operators, which further implies the unique solvability of the corresponding BVP. For the resolvents, Krein-type representations are obtained. Several simple examples of the application of the above results are shown. Further, after presenting some results on traces and von Neumann expansions, the author shows that, for a symmetric differential operator, the constructed resolvents of BVP correspond to closed von Neumann extensions of this operator.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 5828854
zbMATH Keywords: boundary value problems
zbMATH Keywords: differential operators
zbMATH Keywords: traces
zbMATH Keywords: projection operators
zbMATH Keywords: resolvent
zbMATH Keywords: von Neumann expansions
MaRDI profile type: Publication
full work available at URL: s10958-009-9424-0
OpenAlex ID: W2137628800
cites work: Q3843833
cites work: Singular Integrals and Boundary Value Problems
cites work: Q4500905
cites work: Q4328095
DOI: 10.1007/S10958-009-9424-0
description: scientific article; zbMATH DE number 5828854