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The generalized Neumann-Poincaré operator and its spectrum

Seria
Rozprawy Matematyczne tom/nr w serii: 366 wydano: 1997
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Warianty tytułu
Abstrakty
EN
CONTENTS
Introduction..........................................................................................................................................................................5
Preliminaries. Complex harmonic functions..........................................................................................................................7
I. Spectral values and eigenvalues of a Jordan curve........................................................................................................19
 1.1. On a boundary integral..............................................................................................................................................20
 1.2. The generalized Cauchy singular integral operator $C_𝕍$.......................................................................................23
 1.3. The Hilbert transformation $T_Ω$.............................................................................................................................28
 1.4. The boundary space Ḣ²(∂Ω)......................................................................................................................................31
 1.5. The generalized Neumann-Poincaré operator $N_𝕍$...............................................................................................36
II. 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 $B_γ$.........................................................................................................................................60
III. 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............84
IV. 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....................................................................................................108
Appendix. The inner completion of pseudo-normed spaces............................................................................................114
References......................................................................................................................................................................117
List of symbols.................................................................................................................................................................122
Index................................................................................................................................................................................124
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 366
Liczba stron
125
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXVI
Daty
wydano
1997
otrzymano
1995-10-03
poprawiono
1996-07-06
poprawiono
1997-03-24
Twórcy
  • Institute of Mathematics, The Catholic University of Lublin, P.O. Box 129, Al. Racławickie 14, 20-950 Lublin, Poland, partyka@adam.kul.lublin.pl
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Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: Primary 30C62; Secondary 30F10, 45C05, 41A25.
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0012-3862
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