The Existence Of Quasi-periodic Solutions For Two Classes Of Differential Equations

In this paper,we are interested in the existence of quasi-periodic solutions for a class of quasi-periodically forced reversible harmonic oscillators and a class of nonlinear elliptic equations.A lot of nonlinear vibration problems in physics,mechanics,and engineer-ing applications are characterized by the quasi-periodically forced harmonic os-cillators.A classical question,which has been posed explicitly by Stoker,asks for solutions that are quasi-periodic with the same frequency as the forcing:so called response solutions.This question is nowadays called 'Stoker's prob-lem.' There are lots of existing results on this problem.Among them KAM(Kolmogorov-Arnold-Moser)theory is a powerful tool for studying the quasi-periodical solutions of almost integrable conservative systems.In the first part,our proof is based on a finite dimensional KAM theory for quasi-periodically forced reversible systems with multi-dimensional Liouvillean frequency.As we know,the results existing in the literature deal with two-dimensional frequency.The key is to exploiting the theory of continued fractions to control the influ-ence of small divisors and construct two KAM schemes to realize the reduction of the quasi-periodic linear systems.In this part,we extend the analysis to higher dimensional frequency and impose a non-resonance condition weaker than the Brjuno condition,so allowing a class of Liouvillean frequencies.The main idea in the classical KAM theory is to perform a normal form reduc-tion under some non-resonance conditions,and then impose further Melnikov conditions on the small divisors.In this way one can obtain that most of the invariant torus of the unperturbed system are persisted by sacrificing some parameters.Thus we obtain the existence of the solutions.The overall strat-egy of this paper comes from the literature[22],but the method is still in the spirit of[38].In the Hamiltonian systems,the symplectic transformation can maintain the Hamiltonian structure in each step of KAM iteration.In this paper,the reversible system requires that the coordinate transformation can commute with some involution to preserve reversible structure.This makes our proof more complicated.In the second part,we study the existence of solutions for a class of non-linear elliptic equations and a class of the elliptic-type evolution equations.In this part,we give simple proofs and extensions of the results of the existing literature[76].We know that the small-divisor difficulty appears in studying nonlinear elliptic equations.In fact,the KAM technique and CWB(Craig-Wayne-Bourgain)methods are powerful tools to overcome the small-divisor d-ifficulty.The paper[76]uses the CWB methods to construct analytic solutions of an elliptic problem involving parameters.In this paper,by constructing a suitable space,we use several classical methods(freezing of coefficients for non-linear elliptic equations and center manifold theorem for elliptic type evolution equations)to obtain results on the same models.We relax some assumptions on the structure of the nonlinearity and cover more parameters.Moreover,we also relax the regularity assumption of the nonlinearity.Thus,we can not only get the analytic solutions but also the finite regularity solutions.The specific arrangement of this paper is as follows:In the first chapter,we give the preliminary knowledge that will be used in the following paper,such as some definitions,lemmas,propositions,etc.,and briefly introduce the main definitions and properties of Hamiltonian systems and reversible systems.Then,we introduce the classical KAM theory.Finally,we introduce the background and current status of our research,and give the main work we have done in this article.In Chapter 2,we give a detailed construction of the response solution-s for a class of quasi-periodically forced reversible harmonic oscillators with multi-dimensional Liouvillean frequency.Specifically,we give an abstract fi-nite dimensional KAM theorem for reversible system to prove our main results.In Chapter 3,we introduce in detail how to use the classical freezing coefficient method to solve the existence of solutions for a class of nonlinear elliptic equations and apply the time dependent center manifold theorem to construct solutions for a class of nonlinear elliptic type evolution equations.