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On the Problem of Plateau / Subharmonic Functions

  • Book
  • © 1971

Overview

Authors:
  1. Tibor Radó
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Part of the book series: Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge (MATHE1, volume 2)

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Table of contents (14 chapters)

  1. Front Matter

    Pages I-XVII
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  2. Introduction

    • Tibor Radó
    Pages 1-1

  3. Curves and surfaces

    • Tibor Radó
    Pages 2-18

  4. Minimal surfaces in the small

    • Tibor Radó
    Pages 19-30

  5. Minimal surfaces in the large

    • Tibor Radó
    Pages 31-49

  6. The non-parametric problem

    • Tibor Radó
    Pages 49-68

  7. The problem of Plateau in the parametric form

    • Tibor Radó
    Pages 68-90

  8. The simultaneous problem in the parametric form. Generalizations

    • Tibor Radó
    Pages 90-109

  9. Definition and preliminary discussion of subharmonic functions

    • Tibor Radó
    Pages 111-116

  10. Integral means of subharmonic functions

    • Tibor Radó
    Pages 117-122

  11. Criteria and constructions for subharmonic functions

    • Tibor Radó
    Pages 122-132

  12. Examples of subharmonic functions

    • Tibor Radó
    Pages 132-141

  13. Harmonic majorants of subharmonic functions

    • Tibor Radó
    Pages 141-149

  14. Representation of subharmonic functions in terms of potentials

    • Tibor Radó
    Pages 150-155

  15. Analogies between harmonic and subharmonic functions

    • Tibor Radó
    Pages 156-163

  16. Back Matter

    Pages 164-166
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Keywords

About this book

A convex function f may be called sublinear in the following sense; if a linear function l is ::=: j at the boundary points of an interval, then l:> j in the interior of that interval also. If we replace the terms interval and linear junction by the terms domain and harmonic function, we obtain a statement which expresses the characteristic property of subharmonic functions of two or more variables. This ge­ neralization, formulated and developed by F. RIEsz, immediately at­ tracted the attention of many mathematicians, both on account of its intrinsic interest and on account of the wide range of its applications. If f (z) is an analytic function of the complex variable z = x + i y. then If (z) I is subharmonic. The potential of a negative mass-distribu­ tion is subharmonic. In differential geometry, surfaces of negative curvature and minimal surfaces can be characterized in terms of sub­ harmonic functions. The idea of a subharmonic function leads to significant applications and interpretations in the fields just referred to, and· conversely, every one of these fields is an apparently in­ exhaustible source of new theorems on subharmonic functions, either by analogy or by direct implication.

Bibliographic Information

  • Book Title: On the Problem of Plateau / Subharmonic Functions

  • Authors: Tibor Radó

  • Series Title: Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge

  • DOI: https://doi.org/10.1007/978-3-642-65236-3

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1971

  • Softcover ISBN: 978-3-540-05479-5Published: 04 January 1971

  • eBook ISBN: 978-3-642-65236-3Published: 29 June 2013

  • Edition Number: 1

  • Number of Pages: XI, 109

  • Additional Information: Originally published as Band 2, Heft 2 and Band 5, Heft 1 of the series: Ergebnisse der Mathematik und ihrer Grenzgebiete

  • Topics: Analysis

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