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

1. Kurata研究了以下问题：（1）一维非齐次Cahn-Hilliard能量极小解的单调性的破缺和Chern-Simons-Higgs理论中一类非线性椭圆方程非拓扑解的存在性。(2)第一狄利克雷本征值到薛定谔算子的优化问题中障碍物的最优位置. Jimbo研究了具有磁效应的Ginzburg-Landau方程在凸区域中稳定涡旋解的存在性和永久电流的不存在性。Tanaka构造了非齐次Allen-Cahn方程和非线性Schrodinger方程的复模式解。Murata研究了斜积型椭圆型方程正解的结构，并对Martin边界和Martin核进行了完整的分类. Mochizuki利用内部数据研究了Dirac算子和Sturm-Liouville算子的反谱问题。Sakai详细研究了初始区域有夹角时Hale-Shaw流动的动边界的渐近行为。

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.