# On the Problem of Plateau / Subharmonic Functions

• Authors
Book

Part of the Ergebnisse der Mathematik und Ihrer Grenƶgebiete book series (MATHE1, volume 2)

1. Front Matter
Pages I-XVII
Pages 1-1
Pages 2-18
Pages 19-30
Pages 31-49
Pages 49-68
Pages 68-90
Pages 90-109
Pages 111-116
Pages 117-122
Pages 122-132
Pages 132-141
Pages 141-149
Pages 150-155
Pages 156-163
16. Back Matter
Pages 164-166

### Introduction

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.

### Keywords

Functions Plateausches Problem Problem of Plateau Subharmonische Funktion function minimum subharmonic function

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-65236-3
• Copyright Information Springer-Verlag Berlin Heidelberg 1971
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-540-05479-5
• Online ISBN 978-3-642-65236-3
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