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，主管S. Yotsutani已获得了用于径向溶液的规范形式和结构定理，用于与Y. Kabeya和E. Yanagida的半线性椭圆形问题。半椭圆形问题的极限溶液满足了二阶微分方程的某些边界值问题。可以看出，在适当更改变量后，可以将边界价值问题与Dirichlet，Neumann或Robin边界条件还原为规范形式。我们将结构定理与具有功率非线性和各种边界条件的方程式的规范形式。通过使用这些定理，可以以系统的方式研究半线性椭圆方程的径向溶液的性能，并制定各种方程式的明确未知结构。第二，可以通过规范的形式将结果转换为一种问题，以便将其转换为其他问题的结果，并且可以找到一种原始方法来调查单个奇异的解决方案。作为一个具体的例子，Yotsutani阐明了与H. Myogahara和E. Yanagida的单位球中的奇异溶液的结构。与相关的问题，我们正在研究数学生物学中出现的交叉扩散方程的限制方程。我们证明它具有与Y. Lou和W.-M的不同种类的单数解决方案。 ni。这个问题是一个非局部非线性椭圆边界问题，为此，没有任何方法可以解决它。我们发现了一种新方法。该方法适用很多有趣的问题。 Ossen螺旋流的问题是其中之一。