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

1.仓田研究了以下问题：（1）具有非光滑位势的磁薛定谔算子基本解的二阶和三阶导数的估计以及某些算子的Calderon-Zygmund性质。(2)磁薛定谔算子热核的估计。(3)讨论了第一Dirichlet特征值的几个优化问题的最优配置的存在性及其性质。特别地，我们发现了某些对称域的最优构形的破缺现象。我们还研究了与最优构型相关的自由边界的正则性. Jimbo研究了具有磁效应的Ginzburg-Landau方程稳定的非常数解的不存在性. Tanaka研究了非线性椭圆型方程-Δu+u= u ^p.4在扰动下解的不连续现象.村田研究了非紧黎曼流形上二阶椭圆型和抛物型方程正解的结构. Mochizuki利用内点法研究了KPP方程解的爆破和渐近性态以及Sturm-Liouville算子的逆谱问题.石井证明了高斯曲率流几何逼近方法的收敛性. Sakai研究了在二维情况下，在多边形外有一个细支集且势相等的测度存在的条件。

1. Kurata studied the following :(1) estimates of the second and third derivatives of fundamental solutions to magnetic Schrodinger operators with non-smooth potentials and the Calderon-Zygmund property of certain operators.(2) estimates of the heat kernel of magnetic Schrodinger operators.(3) exiestence and further properties of the optimal configuration to several optimization problems for the first Dirichlet eigenvale. Especially, we find a symmetry-breaking pheneomena of the optimal configuration for certain symmetric domains. We also studied the regularity of the free boundary associated with the optimal configuration.2. Jimbo studied the non-existence of stable non-constant solution to Ginzburg-Landau equation with magnetic effect.3. Tanaka studied discontinuous phenomena for solutions under the perturbation to nonlinear ellitic equation -Δu+u=u^p.4. Murata studied the structure of positive solutions to elliptic and parabolic equations of second order on non-compact Riemannian manifolds.5. Mochizuki studied the blow-up and the asymptotic behavior of solutions to KPP equation and inverse spectrum problem for Sturm-Liouville operator by interior datas.6. Ishii showed the convergence of geometric approximation method for the Gauss curvature flow.7. Sakai studied the condition of the existence of a measure which has a fine support and makes the same potential outside the polygon in two-dimensional case.