The generalized Neumann-Poincaré operator and its spectrum

Partyka Dariusz

  • Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1997

Abstract

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CONTENTSIntroduction..........................................................................................................................................................................5Preliminaries. Complex harmonic functions..........................................................................................................................7I. Spectral values and eigenvalues of a Jordan curve........................................................................................................19 1.1. On a boundary integral..............................................................................................................................................20 1.2. The generalized Cauchy singular integral operator C .......................................................................................23 1.3. The Hilbert transformation T Ω .............................................................................................................................28 1.4. The boundary space Ḣ²(∂Ω)......................................................................................................................................31 1.5. The generalized Neumann-Poincaré operator N ...............................................................................................36II. Quasisymmetric automorphisms of the unit circle...........................................................................................................41 2.1. The Douady-Earle extension E γ ..........................................................................................................................42 2.2. On an approximation of the Hersch-Pfluger distortion function Φ K ......................................................................46 2.3. On the maximal dilatation of the Douady-Earle extension..........................................................................................48 2.4. The Hilbert space H...................................................................................................................................................54 2.5. The linear operator B γ .........................................................................................................................................60III. The generalized harmonic conjugation operator............................................................................................................64 3.1. The generalized harmonic conjugation operator A γ .............................................................................................64 3.2. Spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle..............................................73 3.3. The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle..............................................80 3.4. Limiting properties of spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle............84IV. Spectral values of a quasicircle.....................................................................................................................................90 4.1. Characterizations of the boundary space Ḣ²(∂Ω).......................................................................................................91 4.2. Spaces symmetric with respect to a Jordan curve.....................................................................................................93 4.3. Plemelj’s formula for a quasicircle..............................................................................................................................96 4.4. The main spectral theorem for quasicircles.............................................................................................................103 4.5. Spectral values and eigenvalues of a quasicircle....................................................................................................108Appendix. The inner completion of pseudo-normed spaces............................................................................................114References......................................................................................................................................................................117List of symbols.................................................................................................................................................................122Index................................................................................................................................................................................1241991 Mathematics Subject Classification: Primary 30C62; Secondary 30F10, 45C05, 41A25.

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Partyka Dariusz. The generalized Neumann-Poincaré operator and its spectrum. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1997. <http://eudml.org/doc/271129>.

@book{PartykaDariusz1997,
abstract = {CONTENTSIntroduction..........................................................................................................................................................................5Preliminaries. Complex harmonic functions..........................................................................................................................7I. Spectral values and eigenvalues of a Jordan curve........................................................................................................19 1.1. On a boundary integral..............................................................................................................................................20 1.2. The generalized Cauchy singular integral operator $C_$.......................................................................................23 1.3. The Hilbert transformation $T_Ω$.............................................................................................................................28 1.4. The boundary space Ḣ²(∂Ω)......................................................................................................................................31 1.5. The generalized Neumann-Poincaré operator $N_$...............................................................................................36II. Quasisymmetric automorphisms of the unit circle...........................................................................................................41 2.1. The Douady-Earle extension $E_γ$..........................................................................................................................42 2.2. On an approximation of the Hersch-Pfluger distortion function $Φ_K$......................................................................46 2.3. On the maximal dilatation of the Douady-Earle extension..........................................................................................48 2.4. The Hilbert space H...................................................................................................................................................54 2.5. The linear operator $B_γ$.........................................................................................................................................60III. The generalized harmonic conjugation operator............................................................................................................64 3.1. The generalized harmonic conjugation operator $A_γ$.............................................................................................64 3.2. Spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle..............................................73 3.3. The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle..............................................80 3.4. Limiting properties of spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle............84IV. Spectral values of a quasicircle.....................................................................................................................................90 4.1. Characterizations of the boundary space Ḣ²(∂Ω).......................................................................................................91 4.2. Spaces symmetric with respect to a Jordan curve.....................................................................................................93 4.3. Plemelj’s formula for a quasicircle..............................................................................................................................96 4.4. The main spectral theorem for quasicircles.............................................................................................................103 4.5. Spectral values and eigenvalues of a quasicircle....................................................................................................108Appendix. The inner completion of pseudo-normed spaces............................................................................................114References......................................................................................................................................................................117List of symbols.................................................................................................................................................................122Index................................................................................................................................................................................1241991 Mathematics Subject Classification: Primary 30C62; Secondary 30F10, 45C05, 41A25.},
author = {Partyka Dariusz},
keywords = {boundary limiting values of harmonic functions; Cauchy integral; Cauchy singular integral; completion of pseudo-normed spaces; Dirichlet integral; Douady-Earle extension; eigenvalues and spectral values of a linear operator; extremal quasiconformal mappings; Grunsky inequality; Grunsky matrixes; harmonic conjugation operator; harmonic functions; Hersch-Pfluger distortion function; Hilbert transformation; Neumann-Poincaré kernel; Neumann-Poincaré operator; Plemelj's formula; Poisson integral; quasiconformal mappings in the plane; quasisymmetric automorphisms; quasisymmetric functions; special functions; Teichmüller mappings; univalent functions; universal Teichmüller space; welding homeomorphism; quasiconformal maps in the plane},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {The generalized Neumann-Poincaré operator and its spectrum},
url = {http://eudml.org/doc/271129},
year = {1997},
}

TY - BOOK
AU - Partyka Dariusz
TI - The generalized Neumann-Poincaré operator and its spectrum
PY - 1997
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - CONTENTSIntroduction..........................................................................................................................................................................5Preliminaries. Complex harmonic functions..........................................................................................................................7I. Spectral values and eigenvalues of a Jordan curve........................................................................................................19 1.1. On a boundary integral..............................................................................................................................................20 1.2. The generalized Cauchy singular integral operator $C_$.......................................................................................23 1.3. The Hilbert transformation $T_Ω$.............................................................................................................................28 1.4. The boundary space Ḣ²(∂Ω)......................................................................................................................................31 1.5. The generalized Neumann-Poincaré operator $N_$...............................................................................................36II. Quasisymmetric automorphisms of the unit circle...........................................................................................................41 2.1. The Douady-Earle extension $E_γ$..........................................................................................................................42 2.2. On an approximation of the Hersch-Pfluger distortion function $Φ_K$......................................................................46 2.3. On the maximal dilatation of the Douady-Earle extension..........................................................................................48 2.4. The Hilbert space H...................................................................................................................................................54 2.5. The linear operator $B_γ$.........................................................................................................................................60III. The generalized harmonic conjugation operator............................................................................................................64 3.1. The generalized harmonic conjugation operator $A_γ$.............................................................................................64 3.2. Spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle..............................................73 3.3. The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle..............................................80 3.4. Limiting properties of spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle............84IV. Spectral values of a quasicircle.....................................................................................................................................90 4.1. Characterizations of the boundary space Ḣ²(∂Ω).......................................................................................................91 4.2. Spaces symmetric with respect to a Jordan curve.....................................................................................................93 4.3. Plemelj’s formula for a quasicircle..............................................................................................................................96 4.4. The main spectral theorem for quasicircles.............................................................................................................103 4.5. Spectral values and eigenvalues of a quasicircle....................................................................................................108Appendix. The inner completion of pseudo-normed spaces............................................................................................114References......................................................................................................................................................................117List of symbols.................................................................................................................................................................122Index................................................................................................................................................................................1241991 Mathematics Subject Classification: Primary 30C62; Secondary 30F10, 45C05, 41A25.
LA - eng
KW - boundary limiting values of harmonic functions; Cauchy integral; Cauchy singular integral; completion of pseudo-normed spaces; Dirichlet integral; Douady-Earle extension; eigenvalues and spectral values of a linear operator; extremal quasiconformal mappings; Grunsky inequality; Grunsky matrixes; harmonic conjugation operator; harmonic functions; Hersch-Pfluger distortion function; Hilbert transformation; Neumann-Poincaré kernel; Neumann-Poincaré operator; Plemelj's formula; Poisson integral; quasiconformal mappings in the plane; quasisymmetric automorphisms; quasisymmetric functions; special functions; Teichmüller mappings; univalent functions; universal Teichmüller space; welding homeomorphism; quasiconformal maps in the plane
UR - http://eudml.org/doc/271129
ER -

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