View in EDS PDF Full Text

ON A CONJECTURE RELATIVE TO THE MAXIMA OF HARMONIC FUNCTIONS ON CONVEX DOMAINS.

Saved in:
Bibliographic Details
Title: ON A CONJECTURE RELATIVE TO THE MAXIMA OF HARMONIC FUNCTIONS ON CONVEX DOMAINS.
Authors: Berrone, Lucio R.1 berrone@unrctu.edu.ar
Source: SIAM Journal on Mathematical Analysis. 1999, Vol. 30 Issue 6, p1185-1207. 23p. 5 Diagrams.
Subjects: Harmonic functions, Fourier series, Harmonic analysis (Mathematics), Boundary value problems, Convex surfaces
Abstract: We consider a harmonic function u defined on a bounded domain Ω ⊂ R² and satisfying the mixed boundary conditions u|Γ0 = 0; (∂u/∂n)|Γ1 = 1, where Γ1 is composed by a finite number of arcs of ∂Ω and Γ0 = ∂Ω ∼ Γ1. In [Berrone, Subsistencia de Modelos Matematicos que Involucran a la Ecuacion del Calor-Difusion, Ph.D. thesis, Universidad Nacional de Rosario, Argentina, 1994] it was conjectured that if Ω is convex and the subset Γ1 is made to vary on ∂Ω so as to maintain its measure equal to a constant C > 0, then Γ1 ... u attains its maximum value when Γ1 is a certain connected arc of measure C. The present paper has evolved from attempts to prove this conjecture. When certain geometric restrictions are satisfied by the components of Γ1, the property stated by the conjecture is shown to hold for every regular domain Ω, convex or not, and every connected arc, provided that the measure |Γ1| is sufficiently small (see Theorem 5). However, convexity becomes a necessary condition in order that the full conjecture can be supportable (see section 2). In addition, some variations of the conjecture are proposed. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Mathematical Analysis is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
FullText Links:
  – Type: pdflink
Text:
  Availability: 0
Header DbId: egs
DbLabel: Engineering Source
An: 13208585
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: ON A CONJECTURE RELATIVE TO THE MAXIMA OF HARMONIC FUNCTIONS ON CONVEX DOMAINS.
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Berrone%2C+Lucio+R%2E%22">Berrone, Lucio R.</searchLink><relatesTo>1</relatesTo><i> berrone@unrctu.edu.ar</i>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Mathematical+Analysis%22">SIAM Journal on Mathematical Analysis</searchLink>. 1999, Vol. 30 Issue 6, p1185-1207. 23p. 5 Diagrams.
– Name: Subject
  Label: Subjects
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Harmonic+functions%22">Harmonic functions</searchLink><br /><searchLink fieldCode="DE" term="%22Fourier+series%22">Fourier series</searchLink><br /><searchLink fieldCode="DE" term="%22Harmonic+analysis+%28Mathematics%29%22">Harmonic analysis (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Boundary+value+problems%22">Boundary value problems</searchLink><br /><searchLink fieldCode="DE" term="%22Convex+surfaces%22">Convex surfaces</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: We consider a harmonic function u defined on a bounded domain Ω ⊂ R² and satisfying the mixed boundary conditions u|Γ0 = 0; (∂u/∂n)|Γ1 = 1, where Γ1 is composed by a finite number of arcs of ∂Ω and Γ0 = ∂Ω ∼ Γ1. In [Berrone, Subsistencia de Modelos Matematicos que Involucran a la Ecuacion del Calor-Difusion, Ph.D. thesis, Universidad Nacional de Rosario, Argentina, 1994] it was conjectured that if Ω is convex and the subset Γ1 is made to vary on ∂Ω so as to maintain its measure equal to a constant C > 0, then Γ1 ... u attains its maximum value when Γ1 is a certain connected arc of measure C. The present paper has evolved from attempts to prove this conjecture. When certain geometric restrictions are satisfied by the components of Γ1, the property stated by the conjecture is shown to hold for every regular domain Ω, convex or not, and every connected arc, provided that the measure |Γ1| is sufficiently small (see Theorem 5). However, convexity becomes a necessary condition in order that the full conjecture can be supportable (see section 2). In addition, some variations of the conjecture are proposed. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of SIAM Journal on Mathematical Analysis is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=13208585
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1137/S0036141098334973
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 23
        StartPage: 1185
    Subjects:
      – SubjectFull: Harmonic functions
        Type: general
      – SubjectFull: Fourier series
        Type: general
      – SubjectFull: Harmonic analysis (Mathematics)
        Type: general
      – SubjectFull: Boundary value problems
        Type: general
      – SubjectFull: Convex surfaces
        Type: general
    Titles:
      – TitleFull: ON A CONJECTURE RELATIVE TO THE MAXIMA OF HARMONIC FUNCTIONS ON CONVEX DOMAINS.
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Berrone, Lucio R.
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 07
              Text: 1999
              Type: published
              Y: 1999
          Identifiers:
            – Type: issn-print
              Value: 00361410
          Numbering:
            – Type: volume
              Value: 30
            – Type: issue
              Value: 6
          Titles:
            – TitleFull: SIAM Journal on Mathematical Analysis
              Type: main
ResultId 1