Research in evolution equations related to variational problems

本研究旨在探讨以下问题。1.构造非线性弹性力学中各种变分问题的梯度流，2.梯度流方程的分支现象，3.与弹性变形和面积泛函有关的双曲型方程，4.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Morse半流方法之间的关系.在这个项目的第一年，每四年举行一次的世界非线性分析师大会举行了，因此首席研究员菊口和另一位研究员太田出席了这次大会，并收集了一些与该项目有关的最新信息。第二年，捷克斯洛伐克举行了关于微分方程及其应用的国际会议，首席调查员出席了这次会议，宣布了他的最新成果并收集了信息。除了e…更多的ACH调查人员参加了在日本或国外举行的各种会议，宣布了每一次结果，并公布了最近的信息。从而取得了以下研究成果。问题1和问题3取得了最大的进展。与问题1相关的结果是，当拟凸泛函满足一定的胁迫条件时，可以构造一个梯度流。此外，虽然方程的形式是限制性的，但证明了对于某些拟凸泛函，即使它不满足这种强制性条件，也可以构造一个梯度流。与文献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(迹线消失)弱。这一结果是通过将直接变分方法的一个结果应用于发展方程理论而得到的，这是本研究项目的最大特点。还得到了与问题4有关的一些事实。它相信，与2和5相关的一些新理论也将被开发出来。但到目前为止，这些作品的框架还没有得到。这在未来应该是意料之中的。较少

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