Existences And Regularity Of Periodic Solutions To The Generalized Swift-Hohenberg Equation

In this paper,we study the existences of periodic solutions for a generalized Swift-Hohenberg equation which derived the symmetry breaking model,and the regularity of solutions for a class of fourth order elliptic equations.By the mountain pass lemma and periodic continuation method,the direct method of calculus of variations,Morrey theorem respectively,we obtain the existences of mountain pass solution and minimal energy solution for generalized SwiftHohenberg equation,the regularity of weak solutions for fourth order elliptic equations.This paper is mainly divided into three chapters.In chapter one,we introduce the background of the symmetry breaking model,and the derivation process of generalized Swift-Hohenberg equations,then we illustrate the history of classical Swift-Hohenberg equation.Finally,the main results of this paper are given.The second chapter we give some preliminary knowledge,the existences theorem of generalized Swift-Hohenberg equation for mountain pass solution and minimum energy solution are given by mountain pass lemma and the direct method of calculus of variations respectively.In chapter three,we review Morrey theorem,and some basic properties and conclusions of Lorentz space,then we give the proof of the regularity of weak solutions for fourth order elliptic equations.