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Analysis of gradient flow equations and Lagrange equations of action integrals associated to quasiconvex functionals

与拟凸泛函相关的梯度流方程和作用积分拉格朗日方程的分析

基本信息

批准号：
14540202
负责人：
KIKUCHI Koji
金额：
$ 2.56万
依托单位：
Shizuoka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540202/
关键词：
partial differential equations variational problems quasiconvexity

项目摘要

This research was projected in order to investigate the following problems. 1.Constructing gradient flows associated to typical quasiconvex functionals, 2.Study in Lagrange equations of action integrals associated to typical quasiconvex functionals, 3.Discovering phenomena that show differences between convex and quasiconvex functions. During the term of the project the head investigator, Kikuchi, attended various conferences and discussed with specialists in related research areas. In the second year Workshop on spectral theory and differential operators was held at Fudan University, Shanghai, China, and the head investigator attended this conference, anounced his recent result and gathered information. Other investigators also attended various conferences held in Japan or abroad and gathered recent information. Thereby following research results are obtained. The most progresses are obtained in Problem 2. Linear application is investigated for a Lagrange equation of an action integra … More ls associated to a functional that corresponds a value of the integral of F(Du(x)) for a function u, and several results are obtained in case that F is quasiconvex and linear growth. Before obtaining this result, it is obtained for the same equation that a sequence of approximate solutions to this equation constructed by Rothe's method converges to a function and that, if it satisfies the energy conservation law, it is a weak solution in the space of BV functions. This is already established for convex cases, and now it is successfully established for quasiconvex cases. Related to Problem 3, the problem requires a different observation from that in convex cases. So far, energy inequality is obtained by the use of the convexity of the functional, and hence this method is not available in quasiconvex cases. Instead our constructiong approximate solutions elementwisely makes it possible to obtain energy inequality. This seems to be a large difference between convex and quasiconvex functions. In research related to Problem 1, although constructing a gradient flow is not complete, it is sucseeded to find an identity in the process of constructing approximate solutions, which should be a key for our destination. Less

本研究旨在调查以下问题。 1.构造与典型拟凸泛函相关的梯度流，2.研究与典型拟凸泛函相关的作用积分拉格朗日方程，3.发现凸函数和拟凸函数之间差异的现象。在项目期间，首席研究员菊池参加了各种会议并与相关研究领域的专家进行了讨论。第二年，谱理论和微分算子研讨会在中国上海复旦大学举行，首席研究员出席了这次会议，公布了他的最新成果并收集了信息。其他调查人员还参加了在日本或国外举行的各种会议并收集了最新信息。从而得到以下研究结果。最大的进展是在问题 2 中取得的。研究了与函数 u 对应的 F(Du(x)) 积分值相关的动作积分的拉格朗日方程的线性应用，并且在 F 是拟凸和线性增长的情况下获得了几个结果。在得到这个结果之前，对于同一个方程，通过Rothe方法构造的该方程的一系列近似解收敛于一个函数，并且如果它满足能量守恒定律，则它是BV函数空间中的弱解。这对于凸情况已经成立，现在对于拟凸情况也成功成立。与问题 3 相关，该问题需要与凸情况下不同的观察。到目前为止，能量不等式是通过使用泛函的凸性来获得的，因此该方法在拟凸情况下不可用。相反，我们的构造近似解使得获得能量不等式成为可能。这似乎是凸函数和拟凸函数之间的很大差异。在与问题1相关的研究中，虽然构建梯度流并不完整，但在构建近似解的过程中成功找到了恒等式，这应该是我们到达目的地的关键。较少的