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

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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