Singularly Perturbed Variational Problems With Discontinuous Integrands

Based on the variational method and the singularly perturbation theory,we study the variational problem with small parameters for the discontinuous integrand,and give the proof of the existence of spatial contrast structure solution and the uniformly valid asymptotic solution.At present,there are many researches on spatial contrast structure,and have obtained a very deep scientific research result correspondingly,which provides a strong basis for the theory of spatial contrast structure solution of variational problem.The first chapter mainly introduces the development background and current achievements of singular perturbation and variational method,reviews the development process and relevant conclusions of the variational method and the comparison structure of singular perturbation space,and introduces some basic definitions and lemmas related to this paper,which provides a strong theoretical support for the research of singular perturbation variational problem.In the second chapter,we discuss and study the variational problem of the discontinuous integrand function.Based on the theory of variational method,we can get the necessary conditions of extremum,that is,Euler equation,then we study the Euler equation deeply and find that it is a second order differential equation in essence.Combined with singular perturbation theory,in this paper,we prove the existence of the solution of step space contrast structure and construct the corresponding formal asymptotic solution.Especially in the proof of the existence of the solution,we spend a lot of time to prove the existence of the solution of step space contrast structure.For the structural asymptotic solution,we use the direct expansion method and the boundary layer function method to transform the original problem into more simple variational problems,and then,the uniformly valid asymptotic solution is obtained.Finally,an example is given to verify the accuracy of the conclusions,which makes the conclusions more full,vivid and practical.In the third chapter,we continue to study step space contrast structure solution for the high dimensional singular perturbation variational problem.Similarly,we can get the second order differential equations from the necessary conditions of the extremum,which is complicated,but the basic idea is very similar to that of the singularly perturbed quantitative variational problem,especially in the construction of the formal asymptotic solution,the direct expansion method and the boundary layer function method are also used to construct the formal asymptotic solution.Finally,the corresponding example is used to show the validity and application of the conclusion.