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

我们用黏度解的框架来表述障碍问题，特别是双边障碍问题，并得到了该解的接触子域的大小估计。该估计是对先前估计单侧障碍问题的接触子域大小的工作的扩展。晶体曲率控制多边形的运动是晶体在过冷条件下生长的一种模式。我们考虑了一个具有晶体能量的二维问题，其水平集是正n多边形，得到了这些解收敛于平均曲率流的唯一光滑解的结果。我们还研究了一个与表面张力自由能相关的偏微分方程组。该系统被提出作为两种金属合金相分离现象的模型。我们证明了解的全局存在性和唯一性。考虑了各种非线性常微分方程，得到了其解为振荡或非振荡的条件。这些研究不仅本身很有趣，而且对我们项目中处理的偏微分方程的近似解的构造也很有用。

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.