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Research on the Regularity of Solutions for Geometric Variational Problems

几何变分问题解的规律性研究

基本信息

批准号：
12640221
负责人：
TACHIKAWA Atsushi
金额：
$ 1.41万
依托单位：
Tokyo University of Science
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640221/
关键词：
Variattional Problems regularity Harmonic maps Finsler manifold energy functional

项目摘要

The variational problems for the energy functional defined for maps between Riemmannian manifolds have been studied by many mathematicians.Namely, about harmonic maps between Riemannian manifolds, we have many deep results.Recently, some generalisations of harmonic maps attract the interest of several reseachers.In this research, we considered some generalised notion of harmonic maps and got some results on their regularityIn the year 2000, we treated harmonic maps with potentials, and got their existence and regularity results.In the year 2001, we considered more generalised notion of harmonic maps so-called F-farmonic tnays and get their existence and partial regularity resultsIn the year 2002, we treated harmonic maps into Finsler manidolds.Finsler manifolds are natural generalization of Riemannian ones.P.Centore defined the energy of a map between Finsler manifolds.We used Centores definition and specialized it for the case that the source manifold is Eucldean space. We calculated the Euler-Lagrange equation of it and got the equation for hannonic maps from an. Euclidean space R^m to a Finsler manifold. Moreover, we got a partial regularity result for energy minimizing map from R^m to a Finsler manifold for the case that m=3,4.More precisely, we proved that for the above cases the enegy minimizing maps are Wilder continuous outside the singular set whose Hausdorff dimension is less that m-2.

定义在黎曼流形之间的映射的能量泛函的变分问题已经被许多数学家研究过。同样，关于黎曼流形之间的调和映射，我们得到了许多深刻的结果。最近，调和映射的一些推广引起了一些研究者的兴趣。在这项研究中，我们考虑了调和映射的一些广义概念，并得到了它们的正则性的一些结果。2000年，我们处理了具有位势的调和映射，得到了它们的存在性和正则性结果。2001年，我们考虑了更广义的调和映射的概念，即F-Far monic映射，得到了它们的存在性和部分正则性结果将调和映射转化为Finsler流形，Finsler流形是黎曼流形的自然推广.P.Centore定义了Finsler流形之间映射的能量.我们利用Centore定义，并将其推广到源流形为欧氏空间的情况.我们计算了它的欧拉-拉格朗日方程，从An得到了单调映射的方程。欧几里得空间R^m到Finsler流形。在m=3，4时，得到了从R^m到Finsler流形的能量极小映射的部分正则性结果。更准确地说，我们证明了对于上述情况，能量极小映射在Hausdorff维小于m-2的奇异集外是Wilder连续的。