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!
Abstract 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.
AbstractList 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.
Author Begehr, Heinrich
Mohammed, Alip
Author_xml – sequence: 1
  givenname: Heinrich
  surname: Begehr
  fullname: Begehr, Heinrich
  email: begehr@math.fu-berlin.de
  organization: I. Math. Institut, Freie Universität Berlin
– sequence: 2
  givenname: Alip
  surname: Mohammed
  fullname: Mohammed, Alip
  organization: I. Math. Institut, Freie Universität Berlin
BookMark eNqNkM9KAzEQh4NUsK0-gLe8wOpkd5NNwYsUtULBgxW8LbNJlkZ2k5Kk1Pr0btGTRfQ0h_l98-ebkJHzzhByyeCKgYRrACiEZCAAZMFZLk_ImHFRZBzK1xEZH_rZEGBnZBLjG8AQ4WJM5qu1oc9qvcPwQTfBN53paesDRYfdPllF261TyXoXqXU0-bCNWTAdJqOp9j1aF8_JaYtdNBffdUpe7u9W80W2fHp4nN8uM5ULlrJSGqObpqy4BFSlwJlBpXOuoaxguJVLNKXUqsRKiQaaRjRYoNDAMTdypospYV9zVfAxBtPWm2B7DPuaQX2wUB9ZGJjqB6NswsM_KaDt_kNaN_jocedDp-uE-86HNqBTNh5TdXpPA3nzJ1n8vvgTsDCIcA
CitedBy_id crossref_primary_10_1007_s11785_017_0705_1
crossref_primary_10_1016_j_jmaa_2006_03_011
crossref_primary_10_1080_17476930903276233
crossref_primary_10_1016_j_jmaa_2008_01_053
crossref_primary_10_1524_anly_2010_0932
crossref_primary_10_1007_s11868_017_0212_1
Cites_doi 10.1007/BF01193741
10.1142/2162
ContentType Journal Article
Copyright Copyright Taylor & Francis Group, LLC 2006
Copyright_xml – notice: Copyright Taylor & Francis Group, LLC 2006
DBID AAYXX
CITATION
DOI 10.1080/00036810600835128
DatabaseName CrossRef
DatabaseTitle CrossRef
DatabaseTitleList
DeliveryMethod fulltext_linktorsrc
Discipline Applied Sciences
Mathematics
EISSN 1563-504X
EndPage 1101
ExternalDocumentID 10_1080_00036810600835128
183449
GroupedDBID --Z
-~X
.7F
.QJ
0BK
0R~
23M
2DF
30N
4.4
5GY
5VS
AAENE
AAGDL
AAHIA
AAJMT
AALDU
AAMIU
AAPUL
AAQRR
ABCCY
ABFIM
ABHAV
ABJNI
ABLIJ
ABPAQ
ABPEM
ABTAI
ABUFD
ABXUL
ABXYU
ACAGQ
ACGEJ
ACGFS
ACGOD
ACIWK
ACTIO
ADCVX
ADGTB
ADXPE
AEISY
AENEX
AEOZL
AEPSL
AEYOC
AFKVX
AFRVT
AGDLA
AGMYJ
AGROQ
AHDZW
AHMOU
AIJEM
AIYEW
AJWEG
AKBVH
AKOOK
ALCKM
ALMA_UNASSIGNED_HOLDINGS
ALQZU
AMEWO
AQRUH
AQTUD
AVBZW
AWYRJ
BLEHA
CAG
CCCUG
CE4
COF
CRFIH
CS3
DGEBU
DKSSO
DMQIW
EBS
EJD
E~A
E~B
F5P
GTTXZ
H13
HZ~
H~P
IPNFZ
J.P
KYCEM
M4Z
NA5
NY~
O9-
P2P
PQQKQ
QCRFL
RIG
RNANH
ROSJB
RTWRZ
S-T
SNACF
TASJS
TBQAZ
TDBHL
TEJ
TFL
TFT
TFW
TN5
TOXWX
TTHFI
TUROJ
TWF
UPT
UT5
UU3
WH7
ZGOLN
~S~
07G
1TA
6TJ
AAIKQ
AAKBW
AAYXX
ACGEE
ACTCW
AEUMN
AFFNX
AGCQS
AGLEN
AMVHM
AMXXU
BCCOT
BPLKW
C06
CITATION
DWIFK
HF~
H~9
IVXBP
LJTGL
NUSFT
TAQ
TFMCV
UB9
UU8
V3K
V4Q
ZY4
ID FETCH-LOGICAL-c261t-48eedbb47580ac46a9eacd25d047015658ae48dc4a7c6b0bb6ba3a6d05a2e89d3
IEDL.DBID TFW
ISSN 0003-6811
IngestDate Sat Nov 29 03:58:57 EST 2025
Tue Nov 18 22:28:02 EST 2025
Mon May 13 12:10:01 EDT 2019
Mon Oct 20 23:46:21 EDT 2025
IsPeerReviewed true
IsScholarly true
Issue 9
Language English
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c261t-48eedbb47580ac46a9eacd25d047015658ae48dc4a7c6b0bb6ba3a6d05a2e89d3
PageCount 23
ParticipantIDs crossref_citationtrail_10_1080_00036810600835128
informaworld_taylorfrancis_310_1080_00036810600835128
crossref_primary_10_1080_00036810600835128
PublicationCentury 2000
PublicationDate 9/1/2006
PublicationDateYYYYMMDD 2006-09-01
PublicationDate_xml – month: 09
  year: 2006
  text: 9/1/2006
  day: 01
PublicationDecade 2000
PublicationTitle Applicable analysis
PublicationYear 2006
Publisher Taylor & Francis Group
Publisher_xml – name: Taylor & Francis Group
References Begehr H (CIT0002) 1994
Kufner A (CIT0007) 1971
Kumar A (CIT0008) 1994; 62
Begehr H (CIT0003) 1996
Vladimirov VS (CIT0011) 1969; 72
Krantz SG (CIT0005) 1990
Rudin W (CIT0010) 1969
Begehr H (CIT0001) 1997; 88
Maz’ya VG (CIT0004) 1991
Krantz S (CIT0006) 1992
CIT0009
References_xml – volume: 88
  volume-title: An introduction to Several Complex Variables and Partial Differential Equations
  year: 1997
  ident: CIT0001
– volume-title: Function Theory in Polydiscs
  year: 1969
  ident: CIT0010
– volume: 72
  start-page: 203
  year: 1969
  ident: CIT0011
  publication-title: Transactions of the American Mathematical Society
– volume-title: Complex Analysis: The Geometric Viewpoint
  year: 1990
  ident: CIT0005
– volume: 62
  start-page: 531
  year: 1994
  ident: CIT0008
  publication-title: Archiv der Mathematik
  doi: 10.1007/BF01193741
– volume-title: Complex Analytic Methods for Partial Differential Equations – An Introductory Text
  year: 1994
  ident: CIT0002
  doi: 10.1142/2162
– ident: CIT0009
– volume-title: Nonlinear Elliptic Boundary Value Problems and their Applications
  year: 1996
  ident: CIT0003
– volume-title: Analysis IV. Encyclopedia of Mathematical Sciences
  year: 1991
  ident: CIT0004
– volume-title: Fourier Series
  year: 1971
  ident: CIT0007
– volume-title: Function Theory of Several Complex Variables
  year: 1992
  ident: CIT0006
SSID ssj0012856
Score 1.6900431
Snippet 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...
SourceID crossref
informaworld
SourceType Enrichment Source
Index Database
Publisher
StartPage 1079
SubjectTerms Holomorphic functions
Mathematics Subject Classifications: 32A26
Poly-domain
Schwarz problem
Title The Schwarz problem for analytic functions in torus-related domains
URI https://www.tandfonline.com/doi/abs/10.1080/00036810600835128
Volume 85
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
journalDatabaseRights – providerCode: PRVAWR
  databaseName: Taylor & Francis Online Journals
  customDbUrl:
  eissn: 1563-504X
  dateEnd: 99991231
  omitProxy: false
  ssIdentifier: ssj0012856
  issn: 0003-6811
  databaseCode: TFW
  dateStart: 19710401
  isFulltext: true
  titleUrlDefault: https://www.tandfonline.com
  providerName: Taylor & Francis
link http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV07T8MwELZQxQADhQKivOSBCSkiDyexR1RRMUDFUES36GwnohKkqEkB8es5J27U8ugAc86OZZ_v7rPP3xFyJkOWiQi4wzKlEKAw4YDIhGO42iKBEYWuHwrfxIMBH43Enc3NKWxapcHQWU0UUdlqs7lBFvOMuIuKRIUjlqniBzSwaIHR7ZttOew_NHcIPq9qtxppB8W9-Z3mTz0seaUlztIFb9Nv_3Oc22TLhpn0staLHbKW5h3StiEntRu66JDN24a2tdglPVQa_Pj4BtMPakvNUBwkBUNdgjLUuMFKU-k4pwjXZ4VTvYbBTvXkGcZ5sUfu-1fD3rVjqyw4CtFT6TCOblJKhsDBBcUiEGiLtR9ql8XmnXXIIWVcKwaxiqQrZSQhgEi7IfgpFzrYJ618kqcHhLpcc5AgueQxUzqT3AdAzBTEnhdlvtcl7nyWE2UpyE0ljKfEa5hKv0xZl5w3TV5q_o1Vwu7i0iVldeiR1RVKvosn5XvZJeGKJsGvvzr8Y7sjslEf4pgstWPSKqez9ISsq9dyXExPKwX-BAiJ64A
linkProvider Taylor & Francis
linkToHtml http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LT8MwDLZgIAEHBgPEeObACamij7RNj2hiGmKbOAyxW5VHKyZBh9YOEL8ep-0qxmMHONdJo8SJ_TnOZ4Az4dI48DgzaCwlAhQaGDyIA0NztXkBehSqeCjc9ft9NhwGt2XALS3TKjWGjguiiPys1ptbB6NnKXEXOYsKQzCTOxB4wi7Diot2VnPnD9r31S2CzfLqrVrcQHlrdqv5UxdzdmmOtfSTvWnX_zvSLdgsPU1yWajGNixFSQPqpddJyj2dNmCjVzG3pjvQQr3Bjw-vfPJOymozBEdJuGYvQRmiLWGurGSUEETs09TIH8Rgp2r8xEdJugt37atBq2OUhRYMiQAqMyhDSykERexgckk9HuBxrGxXmdTXT61dxiPKlKTcl54whfAEd7inTJfbEQuUswe1ZJxE-0BMphgXXDDBfCpVLJjNOcImx7csL7atJpizaQ5lyUKui2E8hlZFVvplyppwXjV5Lig4Fgmbn9cuzPK4R1wUKfkuHmZvWRPcBU2cX3918Md2p7DWGfS6Yfe6f3MI60VMRyetHUEtm0yjY1iVL9konZzk2vwB_0Hvqg
linkToPdf http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LT8MwDLZgIAQHBgPEeObACami7dI2PaLBBGJMOwyxW5VHKyZBN60dIH49TttVG48d4FwnjRIn9pfYnwHOhEMj3-XMoJGUCFCob3A_8g3N1eb66FGoPFG47XU6rN_3u0VsTlKEVWoMHeVEEdlZrTf3SEXTiLiLjESFIZbJ_Ac8YJdhBd1mVyt4r_VYPiLYLCveqsUNlLemj5o_dTFnluZIS2fMTav6z4FuwWbhZ5LLXDG2YSmMa1AtfE5S7OikBhv3JW9rsgNN1Br8-PTGxx-kqDVDcJCEa-4SlCHaDmaqSgYxQbw-SYwsHQY7VcMXPoiTXXhoXfeaN0ZRZsGQCJ9SgzK0k0JQRA4ml9TlPh7GynaUST2daO0wHlKmJOWedIUphCt4g7vKdLgdMl819qASD-NwH4jJFOOCCyaYR6WKBLM5R9DU8CzLjWyrDuZ0lgNZcJDrUhjPgVVSlX6Zsjqcl01GOQHHImFzdumCNLv1iPISJd_Fg_Q9rYOzoEnj118d_LHdKax1r1pB-7Zzdwjr-YWOjlg7gko6noTHsCpf00EyPsl0-RMy8u5c
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Schwarz+problem+for+analytic+functions+in+torus-related+domains&rft.jtitle=Applicable+analysis&rft.au=Begehr%2C+Heinrich&rft.au=Mohammed%2C+Alip&rft.date=2006-09-01&rft.issn=0003-6811&rft.eissn=1563-504X&rft.volume=85&rft.issue=9&rft.spage=1079&rft.epage=1101&rft_id=info:doi/10.1080%2F00036810600835128&rft.externalDBID=n%2Fa&rft.externalDocID=10_1080_00036810600835128
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0003-6811&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0003-6811&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0003-6811&client=summon