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

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640225/
• 关键词: elliptic equation; radial solution; canonical form; cross-diffusion; reaction diffusion; nonlocal; 半線形楕円形方程式; スカラー曲率方程式; 不完全分岐; 反応拡張方程式; 進行波解

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其他问题的结果，并找到一种研究奇异解的原始方法。作为一个具体例子，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，目前尚无解决该问题的方法。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。

项目成果

S.Jimbo, Y.Morita: "Ginzburg-Landau equation with magnetic effect in a thin domain"Calc.Var.Partial Differential Equations. 15・3. 325-352 (2002)

S.Jimbo，Y.Morita：“薄域中的磁效应的金茨堡-朗道方程”Calc.Var.偏微分方程 15・3（2002）。

D.Hilhorst,M.Iida,M.Mimura and H.Ninomiya: "A competition-diffusion system approximation to the classical two-phase Stefan problem"JJIAM. (to appear).

D.Hilhorst、M.Iida、M.Mimura 和 H.Ninomiya：“近似经典两相 Stefan 问题的竞争扩散系统”JJIAM。

S.Jinbo, Y.Morita: "Ginzburg-Landau Equation with magnetic effect in a thin domain"Calc.Var.Partial Differential Equations. 15. 325-352 (2002)

S.Jinbo，Y.Morita：“薄域中具有磁效应的 Ginzburg-Landau 方程”Calc.Var.偏微分方程。

H.Morishita,E.Yanagida and S.Yotsutani: "Structural change of solutions for a scalar curvature equation"Differential and Integral Equations. 14・3(to appear). (2001)

H.Morishita、E.Yanagida 和 S.Yotsutani：“标量曲率方程解的结构变化”微分方程和积分方程 14・3（待出版）。

Y.Kabeya, E.Yanagida, S.Yotsutani: "Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems"Communication on Pure and Applied Analysis. 1. 85-102 (2002)

Y.Kabeya、E.Yanagida、S.Yotsutani：“半线性椭圆问题径向解的规范形式和结构定理”纯粹与应用分析交流。