Analysis of partial differential equations arising in Material Science

(1)研究了具有磁效应的Ginzburg-Landau方程的非平凡状态解。特别地，在非均匀的薄三维域中，构造了模式（Jimbo和Morita）。在二维凸域上，证明了不存在模式形成（Jimbo with P. Sternberg）。（ii）研究了非平稳Ginzburg-Landan方程（不含磁效应）中的涡旋运动。本文以综合形式改写了F.H. Lin和gerard - soner得到的涡运动的简化ode。研究了Neumann bc案例的动力学（Jimno和Morita）。（iii）研究了具有不连续系数(或可变系数（或可变系数和穿孔区域）的椭圆算子的特征值问题的摄动（Jimbo with Kosugi）。（iv）研究了Allen-Cahn方程（小扩散系数）中出现的相变边界。研究了自由边界的正则性和几何性质（Tonegawa）。(v)研究了曲率各向异性效应驱动的表面演化方程（解的存在性和性质）（Tonegawa）。（六）研究了具有自由边界的极小曲面问题。研究了具有自由边界的双曲演化方程（Omata）。并进行了数值分析。(7)用计算方法（Omata）研究了双曲型金兹堡-朗道方程中涡的运动。

(I) Non-trivial state solutions to the Ginzburg-Landau equation with magnetic effect are studied. Particularly, in a non-uniform thin 3-d domain, pattern is constructed (Jimbo with Morita). In 2-d convex domain, it is proved that no pattern formation exists (Jimbo with P. Sternberg).(ii) Vortex motion in nonstationary Ginzburg-Landan equation (without magnetic effect) is studied. The reduced ODEs of vortex motion obtained by F.H. Lin and Jerrard-Soner are rewrittened in comprehensive form. The dynamics in the Neumann B.C. case is studied (Jimno and Morita).(iii) The perturbation of eigenvalue problem of elliptic operator with discontinuous coefficients (or vaiable coefficients (or vaiable coefficients and perforated domain) is studied (Jimbo with Kosugi).(iv) The phase transition boundary arising in the Allen-Cahn equation (with small diffusion coefficients) is studied. The regularity and the geometic property of the free boundaries are investigated (Tonegawa).(v) The surface evolution equation driven by anisotropic effect of curvature (existence of solution and properties) is studied (Tonegawa).(vi) Minimal surface problem with free boundary is studied. The hyperbolic evolution equation with free boundary is studied (Omata).Numerical analysis are also done.(vii) The vortex motion arising in the hyperbolic Ginzburg-Landau equation is studied by computational method (Omata).