Research On Several Kinds Of Singularly Perturbed Variational Problems

In this paper,two kinds of variational problems are studied by using singular perturbation theory and method.In recent years,there have many new results on variational problems by applying singular perturbation theory.In this paper,a new kind of singularly perturbed variational problem is studied by means of boundary layer function method on the basis of the existed papers,and we obtained some new results.This paper is divided into four chapters,the specific contents and research results are summarized as follows.Chapter 1 is mainly focus on the history and present situation of singular perturbation theory and variational method,and introduce the research work of this paper.In Chapter 2,we study a class of singularly perturbed variational problems with movable boundaries.The formal asymptotic solutions of this problem are constructed by using boundary layer function method and qualitative analysis method.The uniformly valid of the asymptotic solutions is proved by differential inequality method.In Chapter 3,a class of singularly perturbed variational problems with corner points is studied.The formal asymptotic solution of the problem is constructed by using boundary layer function method and qualitative analysis method.The uniformly valid of the asymptotic solution is proved by differential inequality method.Chapter 4 summarizes and prospects this paper.