Research in evolution equations related to variational problems

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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