\(L_p\)-theory of boundary integral equations on a contour with outward peak
type: Item
instance of: scholarly article
title: \(L_p\)-theory of boundary integral equations on a contour with outward peak
zbMATH Open document ID: 0910.31001
DOI: 10.1007/BF01193508
published in: Integral Equations and Operator Theory
publication date: 1998-12-02T00:00:00Z
review text: The authors consider the Dirichlet problem \(\Delta u=0\) in \(D\), \(u=f\) on \(\Gamma\setminus \{0\}\), where \(D\) is a plane simply connected bounded domain having boundary \(\Gamma\) with a peak \((0,0)\). This means that \(\Gamma\setminus \{0\}\) belongs to the class \(C^2\) and \(D\) is given near 0 by the inequalities \[ \kappa_-(x)< y<\kappa_+(x), \quad 0<x<\delta, \] where \(x^{-\mu-1} \kappa_\pm(x)\in C^2[0,\delta]\), \(\lim_{x\to+0} x^{-\mu-1} \kappa_\pm(x)= \alpha_\pm\) with \(\mu>0\) and \(\alpha_+> \alpha_-\). The Dirichlet problem is reduced to the integral equation \[ (\pi I-T)\sigma=-f \quad\text{on }\Gamma\setminus \{0\}, \tag{1} \] where \(T\sigma\) is the double layer potential. The Fredholm theory cannot be directly applied to (1) considered in the space of continuous functions. The authors introduce a pair of functional spaces such that the operator \(\pi I-T\) maps one of them onto another. These spaces are introduced as spaces of absolutely continuous functions on \(\Gamma\setminus \{0\}\) with special weights. \(\text{Ker}(\pi I-T)\) is described. The exterior Dirichlet and Neumann problems are also studied.
Mathematics Subject Classification ID: 31A10
Mathematics Subject Classification ID: 45A05
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 1213756
zbMATH Keywords: boundary integral equations
zbMATH Keywords: outward peak
zbMATH Keywords: Laplace equation
zbMATH Keywords: cuspidal domains
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: simply connected bounded domain
zbMATH Keywords: double layer potential
zbMATH Keywords: absolutely continuous functions
zbMATH Keywords: exterior Dirichlet and Neumann problems
author: V. Maz'ya
author: Alexander A. Solov'ev
reviewed by: Vladimir Mityushev
MaRDI profile type: Publication
cites work: Boundary integral equations of the logarithmic potential theory for domains with peaks
cites work: Note on Singular Integrals
cites work: Weighted Norm Inequalities for the Hardy Maximal Function
cites work: On the Derivatives of Harmonic Functions on the Boundary
full work available at URL: https://doi.org/10.1007/bf01193508
OpenAlex ID: W3213477049
description: scientific article; zbMATH DE number 1213756