Research on structures of solutions for geometric variational problems

本研究项目的目的是研究几何变分问题解的结构。当我们研究Finsler流形上的调和映射时，有必要研究具有奇异性的变分泛函。因此，在本研究项目中，我们把奇异变分问题或奇异系数偏微分方程解的正则性作为重点来考虑。为此，我们与意大利卡塔尼亚大学的Maria Alessandra Ragusa教授合作，研究了具有VMO(消失平均振荡)系数的泛函的极小元的部分正则性。作为与M.A.Ragusa合作的结果，我们得到了极小元的部分正则性的一些结果。也就是说，如果u是具有vmo系数的泛函的极小元，则u满足“u是Holder连续的，但m-2-ε维Hausdorff测度为0的区域的子集除外”。另一方面，每个研究人员都研究了自己的问题：T.Nagawa研究了人类红细胞形状变换理论的数学模型之一Helfrich变分问题。对于任意初始数据，局部证明了相关梯度流的存在性，并在球面附近证明了相关梯度流的全局存在性。M.Ogawa研究了不可压缩理想流体流动的自由边界问题。他证明了解的唯一存在，在时间的局部，即使初始表面和底部是不平坦的。

The aim of this research project is to investigate structures of solutions for geometric variational problems. While we were studying harmonic maps into Finsler manifolds, the necessity of the research on the variational functionals with singularities occurred. So, in this research project, we considered, as important points, regularity of the solutions for variational problems with singularities or weak solutions of partial differential equations with singular coefficients. For this purpose, we investigated partial regularity of minimizers for functionals with VMO (Vanishing Mean Oscillation)-coefficients in cooperation with Prof.Maria Alessandra Ragusa (Universita di Catania (Italy)). As results of cooperation with M.A.Ragusa, we got some results on partial regularity of minimizers. Namely, we proved that if u is a minimizer of certain functional with VMO-coefficients then u satisfies "u is Holder continuous except a subset of the domain whose m-2-ε dimensional Hausdorff measure is 0, where m is the dimension of the domain".On the other hand, each researchers investigated their own problems : T.Nagasawa studied Helfrich variational problem which is one of mathematical models for shape transformation theory of human red blood cells. The existence of associated gradient flow was proved locally for arbitrary initial data, and globally near spheres. M.Ogawa studied free boundary problems for flows of an incompressible ideal fluid. He showed the unique existence of the solution, locally in time, even if the initial surface and the bottom are uneven.