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Research in evolution equations related to variational problems

与变分问题相关的演化方程研究

基本信息

批准号：
12640205
负责人：
KIKUCHI Koji
金额：
$ 2.3万
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：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640205/
关键词：
variational problems nonlinear partial differential equations geometric measure theory

项目摘要

This research was projected in order to investigate the following problems. 1. Constructing gradient flows for various variational problems in, for example, nonlinear elasticity, 2. Bifurcation phenomena for gradient flow equations, 3. Hyperbolic equations related to deformation of elasticity and to area functional, 4. Application of the method of discrete Morse semiflow to the theory of Schrodinger equations, 5. Relation between blowup solutions and the method of discrete Morse semiflow. In the first year of this project World Congress of Nonlinear Analysts which is held once in each four years was held and hence the head investigator, Kikuchi, and another investigator, Ohta, attended this congress and gathered some recent information related to this project. In the second year Czechoslovak International Conference on Differential Equations and Their Applications was held and the head investigator attended this conference, anounced his recent result and gathered information. Besides e … More ach investigators attended various conferences held in Japan or abroad, announced each results and gattered recent information. Thereby following research results have been obtained. The most progresses are obtained in problems 1 and 3. The result related to 1 is that a gradient flow can be consructed when a quasiconvex functional satisfies some coersiveness condition. Furthermore, though the form of equation is restrictive, it turns out that a gradient flow for some quasiconvex functional can be constructed even if it does not satisfy such a coersiveness condition. The result related to 3 is that Dirichle condition for the equation of motion of vibrating membrane should be weaker than the usual weak formulation (that the trace vanishes). This result is obtined by applying a result in direct variational method to the theory of evolution equations, what is the most feature of this research project. Some facts related to Problem 4 are also obtained. It is confident that some new theories related to 2 and 5 will also be developed. But by now frames of these works have not yet been obtained. It should be expected in the future. Less

本研究旨在探讨以下问题。1. 构造各种变分问题的梯度流，例如非线性弹性，2。3.梯度流动方程的分岔现象。与弹性变形和面积泛函有关的双曲方程；离散Morse半流方法在薛定谔方程理论中的应用，5。爆破解与离散莫尔斯半流方法的关系。在该项目的第一年，每四年举行一次的世界非线性分析学家大会举行了，因此首席研究员菊池和另一位研究员Ohta参加了这次大会，并收集了与该项目有关的一些最新信息。第二年召开了捷克斯洛伐克国际微分方程及其应用会议，首席研究员出席了会议，宣布了他的最新成果并收集了资料。此外，更多的研究人员参加了在日本或国外举行的各种会议，公布了每个结果并收集了最新信息。从而得到以下研究结果。在问题1和问题3中取得了最大的进展。与1相关的结果是，当拟凸泛函满足某些强制条件时，可以构造梯度流。此外，尽管方程的形式是有限制的，但对于某些拟凸泛函，即使不满足强制条件，也可以构造出梯度流。与3相关的结果是振动膜运动方程的Dirichle条件应该比通常的弱公式（即迹消失）弱。该结果是将直接变分法的结果应用于进化方程理论，这是本研究项目的最大特点。与问题4相关的一些事实也得到了。相信与2和5相关的一些新理论也会发展出来。但到目前为止，这些作品的框架还没有得到。这应该是未来的预期。少