Singularities of solutions for Monge-Ampere equations

物理学中的基本方程，特别是流体力学、电磁学和相对论中的基本方程，都是用非线性双曲方程表示的。关于这些方程的整体理论在今天还不完备。其中一个原因是经典解在大范围内不存在，也就是说在它们的解中出现了奇异性。此外，我们看到，“奇点”引起许多有趣的现象。我们研究的第一个目的是“描述经典解存在的区域”，第二个目的是“将解扩展到奇点之外”。在这个项目中，我们考虑了二阶非线性偏微分方程“Monge-Ampere方程”的上述问题。解决这些问题的方法正是世纪法国学派主要发展起来的“特征法”，特别是达布和古萨的方法。为了应用他们的方法，我们必须假设强条件 ...更多信息 在方程式上。由于目前还没有关于上述问题的结果，我们考虑了Darboux-Goursat型方程，并由此可以看出这些方程解的奇性结构。其次，我们将这个结果应用到曲面理论中，得到了关于双曲曲面奇点的一些结果。最后，我们进一步研究了在不考虑Darboux-Goursat可积性条件下的上述问题。从而得出了一类“一阶双曲型方程组”的可解性问题。虽然求解起来非常困难，但在某些非线性波动方程的情况下，我们可以得到系统的精确解和整体解。我们认为，由于我们的解决办法是具体的，我们的理由是可以接受的。在研究解的精确表示时，我们开始对弱解的定义产生疑问。现在，考虑到弱解的原始含义，我们研究如何引入弱解的概念。这是我们下一年要研究的主要课题。少

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