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The higher dimensional plateau problems

Chapter
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Part of the Grundlehren der mathematischen Wissenschaften book series (volume 130)

Abstract

Until recently, no general results had been obtained concerning the existence and/or differentiability of the solutions of parametric problems in more than two variables. The greatest single stumbling block was the non-existence of a useful generalization of a conformal map to higher dimensions. Now, by imitating the proof of the author’s old con-formal mapping theorem (Morrey [3]), one can prove that a “non-degenerate” Fréchet variety of the topological type of the v-ball (i.e. a Fréchet variety which possesses a representation on \( \overline {B\left( {0,1} \right)} \) in which no continuum is carried into a point) which possesses a representation of class H v 1 [B(0,1)] possesses such a representation which minimizes ∫|▽z| v dx among all such. However, one can not conclude that B(0.1) the value of this integral ≤C · L[z, B(0,1)] or even that L [z, B(0,1)] is given by the area integral for such a representation. So the methods which had been successful in the two dimensional problems did not lead to results in the higher dimensional cases.

Keywords

HAUSDORFF Measure Polyhedral Cone Simplicial Cone Disjoint Family Radial Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

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