Singular Perturbation In Variational Problems

In this paper, a class of variational problems with small parameters was considered, the existence of solutions for variational problem was obtained, and we constructed the uniformly valid asymptotic solutions, the remainder estimation was done also.In chapter 1, at first we simply introduced the development in singular perturbation, next we introduced some developments in singular perturbation of variational problem, listed the preparing knowledge and introduce the work had been down.With knowledge of singular perturbation we discussed the existence of the solutions of this problem. As the method of boundary layer function developing in 1970s, the method of direct scheme which based on boundary layer functions was applied to solve optimal control problems and variational problems. In chapter 2, we constructed the uniformly valid asymptotic solutions of this problem, and the remainder estimation was done, finally an example was raised to support our point.It was late 90's last century to begin study contrast structure, which contain spike-like and step-like contrast structure, and it is the mainstream direct to the research of singularly perturbed recently. The contrast structure in variational problems developed also, the same as singular perturbation in critical case, variational problem in critical case was considered also. In chapter 3, the existence of step-like contrast structures for the variational problems bad been gave in critical case. With the method of direct scheme and optimality condition, the uniformly valid asymptotic solutions and transition points of this problem are obtained. At last we showed it by an example.In chapter 4, by showing some special variational problems, we extended the foregoing problems forward for the readers to think about and work it out.