Study of Solutions to Partial Differential Equations, Variational Problems and Inverse Problems

偏微分方程、变分问题和反问题解的研究

• 批准号: 13640183
• 负责人: KURATA Kazuhiro
• 金额: $ 1.47万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640183/
• 关键词: Cahn-Hilliard energy; Chern-Simons-Higgs theory; optimization problem; first Dirichlet eigenvalue; Ginzburg-Landau equation; Martin boundary; inverse spectrum problem; Hale-Shaw flow; 変分問題; 楕円型境界値問題; Ginzburg-Landau方程式; 非線形Schrodinger方程式; スペクト

1. Kurata studied the following:(1) breakdown of the monotonicity of the minimizer to a one-dimensional Cahn-Hilliard energy with inhomogeneous weight and the existence of non-topological solution to a nonlinear elliptic equation arising from Chern-Simons-Higgs theory.(2) optimal location of a obstacle in an optimization problem for the first Dirichlet eigenvalue to Schrodinger operator.2. Jimbo studied the existence of stable vortex solutions and the non-existence of permanent current in a convex domain to Ginzburg-Landau equation with magnetic effect. Tanaka constructed solutions with complex patterns to inhomogeneous Allen-Cahn equation and nonlinear Schrodinger equation. Murata studied the structure of positive solutions to elliptic equation of skew-product type and classifies the Martin boundary and Martin kernel completely.3. Mochizuki studied the inverse spectrum problem for Dirac operator and Sturm-Liouville operator by interior datas. Sakai studied the asymptotic behavior of the moving boundary for Hale-Shaw flow when the initial region has an angle in details.

1. Kurata研究了以下内容：（1）极小值的单调性分解为具有非均匀权重的一维Cahn-Hilliard能量以及由Chern-Simons-Higgs理论产生的非线性椭圆方程非拓扑解的存在性(2)薛定谔算子的第一狄利克雷特征值优化问题中障碍物的最优位置。2． Jimbo 研究了具有磁效应的 Ginzburg-Landau 方程稳定涡解的存在性和凸域中永久电流的不存在性。田中构造了非齐次 Allen-Cahn 方程和非线性薛定谔方程的复杂模式解。 Murata研究了斜积型椭圆方程正解的结构，并对Martin边界和Martin核进行了完整的分类。 3．望月新一利用内部数据研究了Dirac算子和Sturm-Liouville算子的逆谱问题。 Sakai详细研究了当初始区域有角度时Hale-Shaw流移动边界的渐近行为。

项目成果

Mochizuki(with I.A.Shishmarev): "Large time asymptotics of small solutions to generalized Kolmogorov-Petrovskii-Piskunov equation"Funkcialai Ekvasioj. 44. 99-117 (2001)

Mochizuki（与 I.A.Shishmarev）：“广义 Kolmogorov-Petrovskii-Piskunov 方程小解的大时间渐近”Funkcialai Ekvasioj。

Kazuhiro Kurata: "Existence of non-topological solutions for a nonlinear elliptic equation arising from Chern-Simons-Higgs theory in a general background metric"Duff. Integral Equations. 14. 925-935 (2001)

Kazuhiro Kurata：“在一般背景度量中，由 Chern-Simons-Higgs 理论产生的非线性椭圆方程的非拓扑解的存在”Duff。

Kimie Nakashima: "Clustering layers and boundary layers in spatially inhomogeneous phase transition problems"Ann. Inst, H. Poincare Anal. Non Lineaire. 20. 107-143 (2003)

Kimie Nakashima：“空间非均匀相变问题中的聚类层和边界层”Ann。

Kunio Hidano: "Scattering and self-similar solutions for the nonlinear wave equation"Differential and Integral Equations. 15. 405-462 (2002)

Kunio Hidano：“非线性波动方程的散射和自相似解”微分和积分方程。

B.M.Harrell II(with P.Kroger, K.Kurata): "On the placement of an obstacle or a well so as to optimize the fumdamental eigenvalue"SIAM J Math. Ana.. 33. 240-259 (2001)

B.M.Harrell II（与 P.Kroger、K.Kurata）：“关于放置障碍物或井以优化基本特征值”SIAM J Math。