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manuscripta mathematica

, Volume 28, Issue 1–3, pp 81–88 | Cite as

Über eine Verallgemeinerung des Plateauschen Problems

  • Erhard Heinz
Article

Abstract

Let T be the domain in ℝN defined by the inequalities O < τ1 < ... < τN < +Π. Put τN+k = Π/2(1+k) (k=1,2,3), τN+41+2Π, and denote byF(τ) the set of functions x=x(u,v)=(x1(u,v),...,xp(u,v)), (p≥2) of class\(C^2 (B) \cap \cap C^0 (\bar B)\), where B is the unit disk u2+v2<1, which maps the circular arcs γk={w=ek<ϕ<τK+1} (k=1,..., N+3) into the straight lines containing the edges ak, ak+1 (aN+4=a1) of a polygon Γ⊂IRp. Then we show that the function Θ(τ)= inf x∈F(τ) D(x) is analytic in T. This generalizes and sharpens an unproved result of I. Marx and M. Shiffman (see [4]).

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Literatur

  1. [1]
    COURANT, R.: dirichlet's principle, conformal mapping, and minimal surfaces. Interscience publishers, New York 1950Google Scholar
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    HEINZ, E.: On surfaces of constant mean curvature with polygonal boundaries. Arch. Rational Mech. Anal.36, 335–347 (1970)Google Scholar
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    HEINZ, E.: Über die analytische Abhängigkeit der Lösungen eines linearen elliptischen Randwertproblems von Parametern. Nachr. Akad. Wiss. Göttingen, II. math. phys. Kl., erscheint demnächstGoogle Scholar
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    MARX, I.: On the classification of unstable minimal surfaces with polygonal boundaries. Comm. P. Appl. Math.8, 235–244 (1955)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Erhard Heinz
    • 1
  1. 1.Mathematisches Institut der UniversitätGöttingenBundesrepublik Deutschland

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