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 .......................................................................................23 1.3. The Hilbert transformation .............................................................................................................................28 1.4. The boundary space Ḣ²(∂Ω)......................................................................................................................................31 1.5. The generalized Neumann-Poincaré operator ...............................................................................................36II. Quasisymmetric automorphisms of the unit circle...........................................................................................................41 2.1. The Douady-Earle extension ..........................................................................................................................42 2.2. On an approximation of the Hersch-Pfluger distortion function ......................................................................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 .........................................................................................................................................60III. The generalized harmonic conjugation operator............................................................................................................64 3.1. The generalized harmonic conjugation operator .............................................................................................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................................................................................................................................................................................124
1991 Mathematics Subject Classification: Primary 30C62; Secondary 30F10, 45C05, 41A25.
@book{bwmeta1.element.zamlynska-7660e1af-389e-4bf7-91a1-fb5f3b08165e, author = {Partyka Dariusz}, title = {The generalized Neumann-Poincar\'e operator and its spectrum}, series = {GDML\_Books}, publisher = {Instytut Matematyczny Polskiej Akademi Nauk}, address = {Warszawa}, year = {1997}, zbl = {0885.30014}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.zamlynska-7660e1af-389e-4bf7-91a1-fb5f3b08165e} }
Partyka Dariusz. The generalized Neumann-Poincaré operator and its spectrum. GDML_Books (1997), http://gdmltest.u-ga.fr/item/bwmeta1.element.zamlynska-7660e1af-389e-4bf7-91a1-fb5f3b08165e/