The Spike-type Solutions For 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.In chapter 2, with knowledge of singular perturbation and contrast structure, the existence of spile-like contrast structure for Variational problems had been given. By being used 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 simple example.In chapter 3, by showing some special Variational problems, we extended the foregoing problems forward for the readers to think about and work it out.