Study of the structure of solutions to Variational Problems, Inverse Problems and Partial Differential Equations

变分问题、反问题和偏微分方程解的结构研究

基本信息

批准号：
15540177
负责人：
KURATA Kazuhiro
金额：
$ 2.18万
依托单位：
Tokyo Metropolitan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

1. Kurata studied the existence and qualitative properties of optimal solutions to several optimization problems for nonlinear elliptic boundary value problems arising in mathematical biology and nonlinear heat conduction phenomena. Kurata also proved the existence of multiple stable patterns in population growth model with Allee effect, symmetry breaking phenomena of the least energuy solution to nonlinear Schroedindger equation and asymptotic profile of radial solution with vortex to 2-dimensional nonlinear Schroedinger equation.2. Okada studied numerial simulation and numerical analysis of nonlinear paratial differential equations. Especially, he constructed boundary spline function by using Newton extrapolation polynomials.3. Sakai studied Hele-Shaw free boundary problem in the case that initial data has a cusp and found sufficient conditions to specify the typical pheneomena.4. Isozaki discovered the relationship between the hyperbolic geometry and inverse problem. He also studied the inverse conductivity problem with discontinuous inclusions and found a numerical algorithm to detect discontinuities.5. Jimbo continued his research on the study of solution structure of the Ginzburg-Landau equation arising in superconductivity under heterogeneous environments. He also studied the spectrum of elliptic operator associated with the Maxwell equation and proved characterization of eigenvalues and proved a perturbation formula by using weak forms.6. Tanaka studied concentration phenomena of solutions and clustered solutions for nonlinear elliptic singular perturbation problems. Especially, he constructed high frequency solution to nonlinear Schroedinger equations and multi-clustered high energySolutions to a phase transition prolem.

1. Kurata研究了数学生物学和非线性热传导现象中出现的非线性椭圆边值问题的几个优化问题的最优解的存在性和定性性质。 Kurata还利用Allee效应证明了人口增长模型中多重稳定模式的存在、非线性薛定谔方程最小能量解的对称破缺现象以及二维非线性薛定谔方程涡旋径向解的渐近轮廓。 2．冈田研究了非线性偏微分方程的数值模拟和数值分析。特别是利用牛顿外推多项式构造了边界样条函数。 3． Sakai研究了初始数据有尖点情况下的Hele-Shaw自由边界问题，找到了描述典型现象的充分条件。 4．矶崎新发现了双曲几何与反问题之间的关系。他还研究了不连续夹杂物的电导率反问题，并找到了检测不连续性的数值算法。5． Jimbo 继续研究异质环境下超导中的 Ginzburg-Landau 方程的解结构。他还研究了与麦克斯韦方程相关的椭圆算子谱，证明了特征值的表征，并用弱形式证明了微扰公式。 6．田中研究了非线性椭圆奇异摄动问题的解和聚类解的集中现象。特别是，他构造了非线性薛定谔方程的高频解和相变问题的多簇高能解。