A theorem on the potential of a double layer
type: Item
instance of: scholarly article
title: A theorem on the potential of a double layer
zbMATH Open document ID: 0755.31008
DOI: 10.1016/0022-247X(91)90352-Z
author: Gabor G. Groh
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: It is well known that the double layer potential \(D\mu(P)=\int_ S\mu(q)\cdot(\partial/\partial n_ q)(1/| P-q|)dS_ q\) is discontinuous at the boundary \(S\) crossing: \(\lim_{\varepsilon\to 0}\mu(p\pm\varepsilon n_ p)=D\mu(p)\pm 2\pi\mu(p)\), \((\varepsilon>0)\). The author establishes a similar discontinuous behaviour for the surface gradients in the case of closed surfaces \(S\): \[ \overline\nabla(D\mu)\pm(p)=\oint_ S[\mu(q)-\mu(p)]\overline\nabla_ pK(q,p) dS_ q\pm 2\pi\overline\nabla\mu(p). \] The following notations are used: \(P\) --- a point in \(R_ +\) or \(R_ -\); \(R_ +,R_ -\) --- the interior and exterior domains, respectively, defined by \(S\); \(S\) --- a Lyapunov surface in \(R^ 3\); \(p,q\) --- points on \(S\); \(\mu(q)\in C'(S)\) --- the density of the double layer potential; \(n_ q\) --- the inner unit normal at \(q\); \(K(p,q)=(\partial/\partial n_ q)\) \((1/| P-q|)\) --- the kernel of the integral in the double layer potential definition. The author compares the presented jump relation with other ones, established by other authors, and shows its advantages.
Mathematics Subject Classification ID: 31B25
Mathematics Subject Classification ID: 31B10
zbMATH DE Number: 57732
zbMATH Keywords: jump relations in the double layer potential
zbMATH Keywords: jump relations for the gradient of double layer potential
reviewed by: Vasile Ionescu
MaRDI profile type: Publication
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full work available at URL: 0022-247x(91)90352-z
OpenAlex ID: W2084827118
description: scientific article; zbMATH DE number 57732