Study of evolution equations from the aspects of the theory of minimizing movements

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Minimizing movement is proposed by De Giorgi which is based on theories used in the studies of partial differential equations and mean curvature flows. This research was projected in order to investigate the problems which appear in the following studies: (A) Study of minimizing movements in mathematical physics and geometry, (B) Study of minimizing movements associated with second order quasilinear hyperbolic partial differential equations. In the first year Seminar on Partial Differential Equations and Its Applications was held at Pukyong National University, Pusan, Korea, and, head investigator attended this conference, announced his recent result and gathered information. Besides, during the term of the project the head investigator and other investigators attended vari-ous conferences and discussed with specialists in related research areas. Thereby following research results are obtained. The most progresses are obtained in Study (B). It has been expected that ap-plication by minimizing movement method is equivalent to Yosida approximation, however the head investigator presents an example and shows that they are different from each other. Furthermore this example is also an example of a minimizing movement which does not satisfy energy conser-vation law. The equation which appears in the example does not satisfy uniqueness of a solution and hence it also turns out that uniqueness is not important in existence of a minimizing movement. Namely, this research has obtained important but negative facts. Probably structures of minimiz-ing movements associated with second order quasilinear hyperbolic partial differential equations are much more complicated than one expects at first. Some facts related to Study (A) are also obtained. However, the results seem to be the halfway stage and the future investigations are expected.