| Abstrakt: |
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 ]> in which no continuum is carried into a point) which possesses a representation of class Hv1[B(0,1)] possesses such a representation which minimizes ʃ|∇z|vdx 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. [ABSTRACT FROM AUTHOR] |
