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

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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.