Analysis of partial differential equations arising in Material Science

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(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).