Quadruple and octuple layer potentials in two dimensions I: Analytical apparatus

Petter Kolm; Shidong Jiang; Vladimir Rokhlin

doi:10.1016/S1063-5203(03)00004-6

Quadruple and octuple layer potentials in two dimensions I: Analytical apparatus

Petter Kolm
, Shidong Jiang
, Vladimir Rokhlin

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

Abstract

A detailed analysis is presented of all pseudo-differential operators of orders up to 2 encountered in classical potential theory in two dimensions. Each of the operators under investigation turns out to be a sum of one or more of standard operators (second derivative, derivative of the Hilbert transform, etc.), and an integral operator with smooth kernel. This classification leads to an extremely simple analysis of spectra of such operators, and simplifies the design of procedures for their numerical evaluation. In a sequel to this paper, the obtained apparatus will be used to construct stable discretizations of arbitrarily high order for a variety of boundary value problems for elliptic partial differential equations.

Original language	English (US)
Pages (from-to)	47-74
Number of pages	28
Journal	Applied and Computational Harmonic Analysis
Volume	14
Issue number	1
DOIs	https://doi.org/10.1016/S1063-5203(03)00004-6
State	Published - Jan 2003
Externally published	Yes

All Science Journal Classification (ASJC) codes

Applied Mathematics

Keywords

Hypersingular integral equations
Laplace equation
Potential theory
Pseudo-differential operators

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10.1016/S1063-5203(03)00004-6

Cite this

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title = "Quadruple and octuple layer potentials in two dimensions I: Analytical apparatus",

abstract = "A detailed analysis is presented of all pseudo-differential operators of orders up to 2 encountered in classical potential theory in two dimensions. Each of the operators under investigation turns out to be a sum of one or more of standard operators (second derivative, derivative of the Hilbert transform, etc.), and an integral operator with smooth kernel. This classification leads to an extremely simple analysis of spectra of such operators, and simplifies the design of procedures for their numerical evaluation. In a sequel to this paper, the obtained apparatus will be used to construct stable discretizations of arbitrarily high order for a variety of boundary value problems for elliptic partial differential equations.",

keywords = "Hypersingular integral equations, Laplace equation, Potential theory, Pseudo-differential operators",

author = "Petter Kolm and Shidong Jiang and Vladimir Rokhlin",

note = "Funding Information: * Corresponding author. E-mail address: [email protected] (V. Rokhlin). 1 Supported in part by DARPA/AFOSR under Contract F49620-97-1-0011. 2 Supported in part by DARPA under Grant MDA972-00-1-0033, and in part by ONR under Grant N0014-01-1-0364. 3 Supported in part by DARPA/AFOSR under Contract F49620-97-1-0011, in part by ONR under Grant N00014-96-1-0188, and in part by AFOSR under Contract F49620-97-C-0052.",

year = "2003",

month = jan,

doi = "10.1016/S1063-5203(03)00004-6",

language = "English (US)",

volume = "14",

pages = "47--74",

journal = "Applied and Computational Harmonic Analysis",

issn = "1063-5203",

publisher = "Academic Press Inc.",

number = "1",

}

TY - JOUR

T1 - Quadruple and octuple layer potentials in two dimensions I

T2 - Analytical apparatus

AU - Kolm, Petter

AU - Jiang, Shidong

AU - Rokhlin, Vladimir

N1 - Funding Information: * Corresponding author. E-mail address: [email protected] (V. Rokhlin). 1 Supported in part by DARPA/AFOSR under Contract F49620-97-1-0011. 2 Supported in part by DARPA under Grant MDA972-00-1-0033, and in part by ONR under Grant N0014-01-1-0364. 3 Supported in part by DARPA/AFOSR under Contract F49620-97-1-0011, in part by ONR under Grant N00014-96-1-0188, and in part by AFOSR under Contract F49620-97-C-0052.

PY - 2003/1

Y1 - 2003/1

N2 - A detailed analysis is presented of all pseudo-differential operators of orders up to 2 encountered in classical potential theory in two dimensions. Each of the operators under investigation turns out to be a sum of one or more of standard operators (second derivative, derivative of the Hilbert transform, etc.), and an integral operator with smooth kernel. This classification leads to an extremely simple analysis of spectra of such operators, and simplifies the design of procedures for their numerical evaluation. In a sequel to this paper, the obtained apparatus will be used to construct stable discretizations of arbitrarily high order for a variety of boundary value problems for elliptic partial differential equations.

AB - A detailed analysis is presented of all pseudo-differential operators of orders up to 2 encountered in classical potential theory in two dimensions. Each of the operators under investigation turns out to be a sum of one or more of standard operators (second derivative, derivative of the Hilbert transform, etc.), and an integral operator with smooth kernel. This classification leads to an extremely simple analysis of spectra of such operators, and simplifies the design of procedures for their numerical evaluation. In a sequel to this paper, the obtained apparatus will be used to construct stable discretizations of arbitrarily high order for a variety of boundary value problems for elliptic partial differential equations.

KW - Hypersingular integral equations

KW - Laplace equation

KW - Potential theory

KW - Pseudo-differential operators

UR - https://www.scopus.com/pages/publications/0242573575

UR - https://www.scopus.com/pages/publications/0242573575#tab=citedBy

U2 - 10.1016/S1063-5203(03)00004-6

DO - 10.1016/S1063-5203(03)00004-6

M3 - Article

AN - SCOPUS:0242573575

SN - 1063-5203

VL - 14

SP - 47

EP - 74

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

IS - 1

ER -

Quadruple and octuple layer potentials in two dimensions I: Analytical apparatus

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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