Singularities of solutions for Monge-Ampere equations

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Fundamental equations appearing in physics, especially in fluid mechanics, electro-magnetics and theory of relativity, are written in the form of nonlinear hyperbolic equations. The global theory concerning these equations is not complete at today's point. One of the reasons is that classical solutions do not exist in the large, that is to say that singularities appear in their solutions. Moreover we see that "singularities" cause many interesting phenomena. The first aim of our research is "to describe the domain where classical solutions exist", and the second one is "to extend the solutions beyond the singularities". In this project, we have considered the above problems for "Monge-Ampere equations" which are nonlinear partial differential equations of second order. The method to solve these exactly is "characteristic method" principally developed by French school in the nighteen century, especially by G.Darboux and E.Goursat. To apply their method, we must assume strong conditions … More on the equations. As we do not have any result on the abobe subjects at today's point, we considered the equations of Darboux-Goursat type and could see the structure of singularities of solutions to these equations. Next we applied this result to the theory of surfaces and we could get some results on the singularities of hyperbolic surfaces. Finally we advanced to the subject such that we study the above problems without the integrability condition of Darboux-Goursat. As the result, we arrived at the problem on the solvability of certain "hyperbolic system of first order". It was very difficult to solve it. But we could get exact and global solutions of the system in the case of certain nonlinear wave equations. We believe that, as our solutions are concrete, our reasong is acceptable. Studying the exact representation of solutions, we began to have some question on the definition of weak solutions. Now, considering the original meaning of weak solutions, we investigate how to introduce the notion of weak solutions. This is the principal subject which we would like to study in the following year. Less