The Schwarz problem for analytic functions in torus-related domains

The Cauchy kernel is one of the two significant tools for solving the Riemann boundary value problem for analytic functions. For poly-domains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in poly-domains. This s...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Applicable analysis Jg. 85; H. 9; S. 1079 - 1101
Hauptverfasser: Begehr, Heinrich, Mohammed, Alip
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Taylor & Francis Group 01.09.2006
Schlagworte:
Holomorphic functions
Mathematics Subject Classifications: 32A26
Poly-domain
Schwarz problem
ISSN:0003-6811, 1563-504X
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:The Cauchy kernel is one of the two significant tools for solving the Riemann boundary value problem for analytic functions. For poly-domains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in poly-domains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. It is also decisive for the establishment of connection between the Riemann-Hilbert problem and the Riemann problem. Thus, not only the Schwarz problem for holomorphic functions in poly-domains is solved, but also the basis is established for solving some other problems. The boundary values of functions, holomorphic in poly-domains, are classified in the Wiener algebra. The general integral representation formulas for these functions, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary poly-domains is given. At the end, well-posed formulations of the torus-related problems are considered. †Dedicated to Professor Wei Lin on the occasion of his 70th birthday.
ISSN:0003-6811
1563-504X
DOI:10.1080/00036810600835128