Research on the Regularity of Solutions for Geometric Variational Problems

黎曼流形间映射所定义的能量泛函的变分问题已被许多数学家所研究。也就是说，关于黎曼流形之间的调和映射，我们有很多深刻的结果。近年来，调和图的一些推广引起了许多研究者的兴趣。本文研究了调和映射的一些广义概念，得到了调和映射正则性的一些结果。2000年，我们用势处理调和映射，得到了调和映射的存在性和正则性结果。在2001年，我们考虑了更广义的调和映射的概念，即所谓的f -法莫天，并得到了它们的存在性和部分正则性结果。在2002年，我们将调和映射处理成Finsler极小集。芬斯勒流形是黎曼流形的自然推广。p . centore定义了芬斯勒流形之间映射的能量。我们使用了中心点的定义，并专门用于源流形是欧几里德空间的情况。我们计算了它的欧拉-拉格朗日方程，得到了汉尼克映射的方程。到一个芬斯勒流形的欧几里德空间R^m。此外，在m=3,4的情况下，我们得到了从R^m到Finsler流形的能量最小化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，能量最小化映射在Hausdorff维数小于m-2的奇异集外是Wilder连续的。

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.