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
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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 -