Analysis on the nonlinear elliptic eigenvalue problems

非线性椭圆特征值问题分析

• 批准号: 14540207
• 负责人: SHIBATA Tetsutaro
• 金额: $ 2.18万
• 依托单位: HIROSHIMA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540207/
• 关键词: Nonlinear; Eigenvalue; Asymptotic analysis; Variational methods; Singular perturbation; Boundary layer; Simple pendulum; Interior behavior; 変文法

We studied nonlinear elliptic eigenvalue problems with several parameters. Our main concern was to clarify the asymptotic properties of eigenvalue parameters and associated eigenfunctions by using variational methods and singular perturbation approaches when a parameter was very large.We studied two-parameter ordinary differential equations with two pure power nonlinear terms. We established several asymptotic formulas for eigenvalues by using several variational approaches. We also studied the two-parameter problems which were related to the simple pendulum problems. We established the precise asymptotic formulas for the solutions with boundary layers when a parameter was very large under the Dirichlet boundary conditions. The formulas obtained here were totally different from those for the associated one-parameter simple pendulum problems.For problems with one-parameter, we studied the problems which were related to the simple pendulum problems. We first studied ordinary differential equations and established precise asymptotic formulas for the boundary layers when a parameter was large under the Dirichlet boundary conditions. We next extended this result to the nonlinear elliptic eigenvalue problems in a smooth bounded domain. We studied nonlinear elliptic eigenvalue problems with pure power nonlinearity and established the asymptotic formula for the eigenvalues in L-2 framework. We also treated the perturbed simple pendulum problems in a bounded domain. It is known that if parameter is large, then an associated solution is nearly flat inside the domain. We have succeeded to establish the precise asymptotic formulas for the interior behavior of the solutions to understand precisely how flat the solutions were inside a domain when a parameter was very large. The formulas obtained here were exactly represented by using the nonlinear term.

研究了具有多个参数的非线性椭圆型特征值问题。我们的主要工作是利用变分方法和奇异摄动方法来阐明当参数很大时特征值参数及其相关特征函数的渐近性质。我们研究了具有两个纯幂非线性项的两参数常微分方程解。我们用几种变分方法建立了几个特征值的渐近公式。我们还研究了与单摆问题相关的两参数问题。在Dirichlet边界条件下，建立了参数很大时边界层解的精确渐近公式。所得到的公式与相关的单参数单摆问题的公式完全不同。对于单参数问题，我们研究了与单参数单摆问题相关的问题。我们首先研究了常微分方程组，在Dirichlet边界条件下建立了参数较大时边界层的精确渐近公式。接下来，我们将这一结果推广到光滑有界域上的非线性椭圆型特征值问题。研究了具有纯幂非线性的非线性椭圆型特征值问题，在L-2框架下建立了特征值的渐近公式。我们还处理了有界域上的摄动单摆问题。已知，如果参数较大，则相关解在域内几乎是平坦的。我们已经成功地建立了解的内部行为的精确渐近公式，以便准确地了解当参数很大时，解在区域内是多么平坦。文中所得公式用非线性项精确表示。

项目成果

Asymptotic expansion of the variational eigencurve for two-parameter simple pendulum type equations

二参数单摆型方程变分特征曲线的渐近展开

• 发表时间: 2004
• 期刊: Nonlinear Analysis 58
• 作者: T.Kobayashi;G.Hector;T.Shibata;T.Shibata;T.Shibata;T.Shibata;A.Bi's;T.Kobayashi;T.Shibata;T.Shibata;T.Shibata
• 通讯作者: T.Shibata

Tetsutaro Shibata: "Asymptotic expansion of the boundary layers of the perturbed simple pendulum problems"Journal of Mathematical Analysis and Applications. 283. 431-439 (2003)

Tetsutaro Shibata：“扰动单摆问题边界层的渐近展开”数学分析与应用杂志。

Kiyoshi Yoshida: "Self-similar solutions to a parabolic system modeling chemotaxis"journal of Differential Equations. 184・2. 386-421 (2002)

Kiyoshi Yoshida：“抛物线系统模拟趋化性的自相似解”微分方程杂志184・2（2002）。

Tetsutaro Shibata: "The effect of the variational framework on the spectral asymptotics for two-parameter nonlinear eigenvalue problems"Mathematische Nachrichten. 248-249. 168-184 (2003)

Tetsutaro Shibata：“变分框架对二参数非线性特征值问题谱渐进的影响”Mathematische Nachrichten。

Three-term spectral asymptotics for the perturbed simple pendulum problems

扰动单摆问题的三项谱渐近

• 发表时间: 2004
• 期刊: Differential Integral Equations 17
• 作者: T.Kobayashi;G.Hector;T.Shibata;T.Shibata;T.Shibata;T.Shibata;A.Bi's;T.Kobayashi;T.Shibata;T.Shibata
• 通讯作者: T.Shibata