Research on canonical forms to nonlinear elliptic boundary value problems and the global structure of all solutions including singular solutions

非线性椭圆边值问题的规范形式和包括奇异解在内的所有解的全局结构研究

基本信息

批准号：
12640225
负责人：
YOTSUTANI Shoji
金额：
$ 2.24万
依托单位：
Ryukoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

First, a head investigator S. Yotsutani have obtained canonical forms and structure theorems for radial solutions to semilinear elliptic problems with Y. Kabeya and E. Yanagida.Radial solutions of semilinear elliptic problems satisfy some boundary value problems for second order differential equations. It is seen that the boundary value problems can be reduced to a canonical form with the Dirichlet, Neumann or Robin boundary condition after suitable change of variables. We get structure theorems to canonical forms to equations with power nonlinearities and various boundary conditions. By using these theorems, it is possible to study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations.Second, it is possible through the canonical forms to convert results for one problem to that of others, and moreover, to find an original methods to investigate singular solutions. As a concrete example, S. Yotsutani clarified the structure of solutions including singular solutions in a unit ball with H. Myogahara and E. Yanagida.As related problems, we are investigating a limiting equation to a cross-diffusion equation that appears in mathematical biology. We showed that it has different kinds of singular solutions with Y. Lou and W.-M. Ni. This problem is a nonlocal nonlinear elliptic boundary problem, for which no method was known to solve it. We discovered a new method. There are a lot of interesting problems for which the method is applicable. A problem of the Ossen's spiral flow is one of them.

首先，首席研究员S. Yotsutani与Y. Kabeya和E. Yanagida一起获得了半线性椭圆问题径向解的规范形式和结构定理。半线性椭圆问题的径向解满足二阶微分方程的一些边值问题。可以看出，适当改变变量后，边值问题可以简化为具有Dirichlet、Neumann 或Robin 边界条件的规范形式。我们将结构定理转化为规范形式，再转化为具有幂非线性和各种边界条件的方程。利用这些定理，可以系统地研究半线性椭圆方程径向解的性质，明确各种方程的未知结构。其次，可以通过规范形式将一个问题的结果转换为另一个问题的结果。此外，还寻找一种原始方法来研究奇异解决方案。作为一个具体的例子，S. Yotsutani 与 H. Myogahara 和 E. Yanagida 一起阐明了包括单位球中的奇异解在内的解的结构。作为相关问题，我们正在研究数学生物学中出现的交叉扩散方程的极限方程。我们与 Y. Lou 和 W.-M. 一起证明了它具有不同类型的奇异解。尼。该问题是一个非局部非线性椭圆边界问题，目前尚无解决方法。我们发现了一种新方法。该方法适用于许多有趣的问题。奥森螺旋流的问题就是其中之一。