Research on structures of solutions for geometric variational problems

几何变分问题解的结构研究

基本信息

  • 批准号:
    15540214
  • 负责人:
  • 金额:
    $ 2.11万
  • 依托单位:
  • 依托单位国家:
    日本
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
  • 财政年份:
    2003
  • 资助国家:
    日本
  • 起止时间:
    2003 至 2005
  • 项目状态:
    已结题

项目摘要

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.
本研究计画的目的是探讨几何变分问题解的结构。在研究Finsler流形上的调和映射的同时,产生了研究奇异变分泛函的必要性。因此,在本研究计划中,我们认为,作为重要的点,具有奇异性的变分问题或具有奇异系数的偏微分方程的弱解的解的正则性。为此,我们与卡塔尼亚大学(意大利)的Maria Alessandra拉古萨教授合作,研究了具有VMO(消失平均振荡)系数的泛函极小元的部分正则性.作为与M.A.拉古萨合作的结果,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是某个VMO系数泛函的极小元,则u满足“u是保持器连续的,除了m-2-ε维Hausdorff测度为0的区域的子集之外,其中m是区域的维数”。T.Nagasawa研究了人体红细胞形状变换理论的数学模型之一Helfrich变分问题。对于任意的初始数据,局部地证明了伴随梯度流的存在性,而在球附近则证明了伴随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。他表明,唯一存在的解决方案,当地的时间,即使最初的表面和底部是不均匀的。

项目成果

期刊论文数量(30)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Periodic vortical flows of an incompressible ideal fluid with free boundary
具有自由边界的不可压缩理想流体的周期性涡流
On the existence of solutions of the Helfrich flow and its center manifold near spheres
  • DOI:
    10.57262/die/1356050521
  • 发表时间:
    2006-01
  • 期刊:
  • 影响因子:
    1.4
  • 作者:
    Y. Kohsaka;Takeyuki Nagasawa
  • 通讯作者:
    Y. Kohsaka;Takeyuki Nagasawa
Free surface motion of an incompressible ideal fluid
不可压缩理想流体的自由表面运动
On continuity of minimizers for certain quadratic growth functionals
关于某些二次增长泛函的最小化器的连续性
Partial regularity of the miniraizers of quadratic functionals with VMO coefficients
具有VMO系数的二次泛函极小化器的偏正则性
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TACHIKAWA Atsushi其他文献

TACHIKAWA Atsushi的其他文献

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{{ truncateString('TACHIKAWA Atsushi', 18)}}的其他基金

Research on the regularity of solutions for nonlinear partial differential equations related to variational problems
与变分问题有关的非线性偏微分方程解的规律性研究
  • 批准号:
    22540207
  • 财政年份:
    2010
  • 资助金额:
    $ 2.11万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Research on the Regularity of Solutions for Geometric Variational Problems
几何变分问题解的规律性研究
  • 批准号:
    12640221
  • 财政年份:
    2000
  • 资助金额:
    $ 2.11万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Research of Nonlinear Partial Differential Equations Using Variational Methods and Time-discretization Schemes (1999)
使用变分方法和时间离散方案的非线性偏微分方程研究(1999)
  • 批准号:
    09640177
  • 财政年份:
    1997
  • 资助金额:
    $ 2.11万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)

相似海外基金

Regularity and Partial Regularity for Monge-Ampere-Type Equations, with Applications to Numerics
Monge-Ampere 型方程的正则性和偏正则性及其在数值中的应用
  • 批准号:
    1700094
  • 财政年份:
    2017
  • 资助金额:
    $ 2.11万
  • 项目类别:
    Continuing Grant
Partial regularity and rigidity problems associated to geometric elliptic systems
与几何椭圆系统相关的部分正则性和刚性问题
  • 批准号:
    1104592
  • 财政年份:
    2011
  • 资助金额:
    $ 2.11万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Partial Regularity in the Calculus ofVariations
数学科学:变分演算中的部分正则性
  • 批准号:
    8704111
  • 财政年份:
    1987
  • 资助金额:
    $ 2.11万
  • 项目类别:
    Continuing Grant
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