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

本研究旨在探讨以下问题。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拉格朗日作用积分方程；3.发现凸函数与拟凸函数之间存在差异的现象。在项目期间，首席调查员菊内出席了各种会议，并与相关研究领域的专家进行了讨论。第二年，谱理论与微分算符研讨会在上海复旦大学举行，中国主任参加了这次会议，宣布了他的最新成果，收集了信息。其他调查人员也参加了在日本或国外举行的各种会议，收集了最新的信息。从而获得了以下研究成果。在问题2中取得了最大的进展。研究了作用积分…的拉格朗日方程的线性应用更多的L对应于函数u的F(Du(X))的积分值，并且在F是拟凸和线性增长的情况下得到了几个结果。在得到这一结果之前，对于同一方程，用Rothe方法构造的近似解序列收敛到一个函数，并且如果它满足能量守恒定律，则它是BV函数空间中的一个弱解。这对于凸情形已经建立了，现在对于拟凸情形也成功地建立了。与问题3相关，该问题需要不同于凸情况下的观察。到目前为止，能量不等式是利用泛函的凸性得到的，因此这种方法在拟凸的情况下是不可行的。取而代之的是，我们的近似解的构造使得获得能量不等成为可能。这似乎是凸函数和拟凸函数之间的一个很大的区别。在与问题1相关的研究中，虽然构造梯度流是不完全的，但在构造近似解的过程中找到一个恒等式是成功的，这应该是我们到达目的地的关键。较少

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