Asymptotics of the solutions of nonlinear partial differential equations and their applications to phase transition

利用粘性解的框架对障碍问题，特别是双边障碍问题进行了研究，得到了解的接触子域的大小估计。这一估计是对以往单边障碍物问题接触子域大小估计的推广。晶体曲率控制的多边形运动是过冷条件下晶体生长的模型。本文考虑了一个水平集为正n-多边形的二维晶体能量问题，得到了这些解收敛于平均曲率流的唯一光滑解的结果，并研究了一个与表面张力自由能有关的偏微分方程组.该系统被提出作为两种金属合金中相分离现象的模型。我们证明了解的整体存在唯一性，并考虑了各种非线性常微分方程，得到了其解振动或非振动的条件。这些调查不仅本身是有趣的，但也有用的构造在我们的项目中处理偏微分方程的近似解。

We formulated obstacle problems especially bilateral obstacle problems by the framework of viscosity solutions, and obtained an estimate of the size of touching subdomains of the solution. This estimate is an extension of the previous works that estimated the size of touching subdomains for unilateral obstacle problems.The motion of polygons governed by crystalline curvature is a model of crystal growth under the super cooling. We consider a two-dimensional problem with a crystalline energy whose level sets are regular n-polygons and got an result of the convergence of these solutions to the unique smooth solution of the mean curvature flow.We also investigate a system of partial differential equations associated with the free energy of surface tension. This system is proposed as a model of phase separation phenomena in alloy of two metals. We proved the global existence and the uniqueness of the solution.We considered various nonlinear ordinary differential equations and got conditions that the solutions to be oscillatory or nonoscillatory. These investigations are not only interesting in itself but also useful to construct approximate solutions in partial differential equations treated in our project.