Minimization Method In Metric Space And Higher Dimensional Frenkel-Kontorova Model

Variational method is a fundamental tool in studying dynamic systems and partial differential equations.Minimization method is the key idea of Aubry-Mather theory,which is based on 1-dimensional Frenkel-Kontorova model and monotone twist maps on cylinder.For Tonelli Lagrangian,J.Mather established a theory of variational method to analyze the form of minimal orbits of the system.A.Fathi developed the so-called weak KAM,which established the connection between Mather's theory and global viscosity solutions of the related Hamilton-Jacobi equation.If the Lagrangian is induced by a Riemannian metric,weak KAM theory is a powerful tool in studying minimal geodesics and viscosity solutions of Eikonal equation.In recent years,benefitting from the development of optimal transport,Eikon-al equation on metric space,especially on Wasserstein space,has received a lot of attention.On manifold a class of weak solutions,i.e.,viscosity solutions,has been extensively studied.A natural idea is to extend the definition of viscosity solution to metric space.Since lacking differential structure on metric space,the first question is how to define the viscosity solution on metric space.Notice that viscosity solution is equivalent to the fixed point of Lax-Oleinik operator in weak KAM theory.We use this idea to define viscosity solution on metric space.We also discuss some properties of viscosity solution on metric space and compare our definition with that in[13],[7].1-dimensional Frenkel-Kontorova model is one of the important sources of Aubry-Mather theory.A suitable method for studying higher dimensional Frenkel-Kontorova model is Moser-Bangert theory.In this thesis,we study the heteroclinic solutions in higher dimensional Frenkel-Kontorova model.This problem has been considered in[17]by the method of Bangert[4].Using the method of Rabinowitz and Stredulinsky,we prove that if the rotation vector of the configuration is rational and if there is an adjacent pair of periodic configurations,then there is a solution that is heteroclinic in one fixed direction.For example,if ?=0 we can construct a solution heteroclinic in ei and periodic in other orthogonal directions e2,…,en of e1.Furthermore,if the above heteroclinic solutions have an adjacent pair,then there is a solution that is heteroclinic in e1,e2 and periodic in e3,…,en.The procedure can be repeated to produce more complex solutions and hence we obtain a better classification for minimal and without self-intersections solutions.Using this approach,we plan to further study more complex solutions,such as homoclinic solutions and heteroclinic solutions crossing the gap between an adjacent pair.