\(L_p\)-theory of boundary integral equations on a contour with inward peak
type: Item
instance of: scholarly article
title: \(L_p\)-theory of boundary integral equations on a contour with inward peak
zbMATH Open document ID: 0910.31002
DOI: 10.4171/ZAA/843
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1999-01-11T00:00:00Z
review text: The unique solvability of the boundary integral equations for the Dirichlet problem in \(\mathbb{R}^2\): \[ \Delta u=0 \quad\text{in }\Omega, \qquad u|_\Gamma= \varphi \] and for the Neumann problem: \[ \Delta u=0 \quad\text{in }\Omega, \qquad (\partial u/\partial n)_\Gamma=\psi \] are studied in the paper. \(\Omega\) is a simply connected domain in the complex plane and \(\Gamma\) its boundary with an inward peak in \(z=0\). The problem was studied by the authors more than ten years ago, and they published their results in several papers quoted in the bibliography. In the present paper they analyse the integral operators corresponding to the two boundary value problems to which a pair of function spaces are associated. The integral operators map one space isomorphically onto another. The principal results of the paper concern their continuity, their range, their character (Fredholm or not) and their injectivity. In the final part the exterior Dirichlet problem and Neumann problem are studied, with outward peak.
Mathematics Subject Classification ID: 31A10
Mathematics Subject Classification ID: 45A05
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 1211545
zbMATH Keywords: cuspidal domains
zbMATH Keywords: existence
zbMATH Keywords: Fredholm character
zbMATH Keywords: unique solvability
zbMATH Keywords: boundary integral equations
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Neumann problem
zbMATH Keywords: integral operators
zbMATH Keywords: continuity
zbMATH Keywords: range
zbMATH Keywords: injectivity
zbMATH Keywords: exterior Dirichlet problem
zbMATH Keywords: outward peak
author: V. Maz'ya
author: Alexander A. Solov'ev
reviewed by: Vasile Ionescu
MaRDI profile type: Publication
cites work: Weighted Norm Inequalities for Hilbert Transforms and Conjugate Functions of Even and Odd Functions
cites work: Weighted Norm Inequalities for the Conjugate Function and Hilbert Transform
cites work: On the Derivatives of Harmonic Functions on the Boundary
cites work: ON AN INTEGRAL EQUATION FOR THE DIRICHLET PROBLEM IN A PLANE DOMAIN WITH CUSPS ON THE BOUNDARY
cites work: Boundary integral equations of the logarithmic potential theory for domains with peaks
cites work: Lp-theory of a boundary intelgral equation on a cuspidal contour
cites work: Weighted Norm Inequalities for the Hardy Maximal Function
description: scientific article; zbMATH DE number 1211545