Analysis of partial differential equations arising in Material Science

(I)研究了具有磁效应的Ginzburg-Landau方程的非平凡态解。特别地，在非均匀薄3-d域中，构造图案（Jimbo与Morita）。在2-D凸域中，证明了不存在模式形成（Jimbo与P.斯滕贝格）。(ii)研究了非定常Ginzburg-Landan方程（无磁效应）中的涡旋运动。用F.H. Lin和Jerrard-Soner以综合形式改写。研究了Neumann B.C.情况下的动力学（Jimno和Morita）。(iii)研究了具有间断系数（或变系数（或变系数和穿孔区域）的椭圆算子特征值问题的扰动（Jimbo与Kosugi）. (iv)研究了小扩散系数Allen-Cahn方程中的相变边界。研究了自由边界的规则性和几何性质（利根川）。(v)研究了曲率各向异性效应驱动的曲面演化方程（解的存在性和性质）（利根川）。(vi)研究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型发展方程（Omata），并进行了数值分析. (vii)本文用计算方法（Omata）研究了双曲型Ginzburg-Landau方程中的涡旋运动。

(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).