哈密顿—雅克比方程的有效哈密顿函数
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摘要
本文考虑Hamilton-Jacobi方程H(x,P+Du)=λ的与有效Hamilton函数相关的一系列问题。
Lions, Papanicolaou和Varadhan首先证明了存在唯一的实数λ∈R使得Hamilton-Jacobi方程存在全局粘性解,记此唯一的与P有关的实数λ∈R为H(P),称为有效Hamilton函数。
有效Hamilton函数有着十分明确的物理意义：它表征了量子本征态所对应的能量值。并且它在Hamilton-Jacobi方程的均匀化理论、解的长时间渐近行为等的研究中都起着非常重要的作用,与弱KAM理论、Aubry-Mather理论等也有着十分密切的联系。
第一章首先介绍这一方面研究进展,既包括理论上的推广,也包括数值计算和应用方面的进展。然后给出必要的预备知识。
第二章首先给出Lions, Papanicolaou和Varadhan等人的原始证明,然后给出一个新的关于有效Hamilton函数存在唯一性的几何方法的证明。这种方法不仅可以证明有效Hamilton函数的存在唯一性,而且由此出发还可以讨论有效Hamilton函数的一些性质以及建立它与Aubry-Mather理论的密切联系。
第三章将对有效Hamilton函数进行刻画。首先是它的一些等价表示,这是由有效Hamilton函数的存在唯一性的几何证明出发而得到的关于H(P)的一些表达式,它们表征了有效Hamilton函数的一些极限特性。还有H(P)的两个变分表示,它们是后面数值计算的理论基础。之后讨论了有效Hamilton函数的一些基本性质,它们反映了H(P)的性质与Hamilton函数H的性质之间的密切关系。
第四章讨论有效Hamilton函数的计算问题。首先对于一些具体的Hamilton函数给出相应的有效Hamilton函数的解析表达式。由于一般而言只有对于特殊的Hamilton函数才能得到有效Hamilton函数的解析表达式,所以接下来讨论了相应的数值计算问题。数值计算的方法又分为两类：偏微分方程方法和变分方法,对每一类方法都进行了简单的分析讨论。
第五章给出有效Hamilton函数的一些应用。首先说明有效Hamilton函数的物理意义：它表征了本征态所对应的能量值,接着给出它在Hamilton-Jacobi方程的均匀化理论、解的长时间渐近行为等的研究中的应用,最后指出了有效Hamilton函数与弱KAM理论、Aubry-Mather理论的密切联系。
Consider the Hamilton-Jacobi equation H(x, P+Du) =λ, there exists one and only one real numberλ∈R such that the equation has a global viscosity solution. It is called the effective Hamiltonian and denoted by H(P).
The physical meaning of the effective Hamiltonian is that it represents the eigenvalue of an eigenstate. It is important in the study of asymptotic solutions of Hamilton-Jacobi equations and in the homogenization theory. It is also closely related to the Weak KAM theory and Aubry-Mather theory.
Some developments in the field is discussed in Chapter 1. It includes the generalization of theoretical analysis and also the developments in numerical calculations and in applications. Some preparations are also given.
Chapter 2 begins with the original proof of existence and uniqueness of effec-tive Hamiltonian given by Lions,Papanicolaou,Varadhan. Then it proceeds with a new geometric proof of the result. Using the geometric proof, some proper-ties of the effective Hamiltonian can be discussed and the close relation between effective Hamiltonian and Aubry-Mather theory can be found.
Some equivalent expressions of the effective Hamiltonian will be discussed in Chapter 3. These equivalent expressions comes from the new geometric proof and shows the limit property of the effective Hamiltonian. Two variational ex-pressions are also discussed, which are the theoretical foundation of numerical calculations. The second part of this chapter deals with the basic properties of the effective Hamiltonian which shows the relation between properties of the effective Hamiltonian and properties of the Hamiltonian function.
Chapter 4 is devoted to the calculation of the effective Hamiltonian. Some examples are given. Since usually it is impossible to give the explicit expressions of the effective Hamiltonian, numerical calculation methods are discussed. There are basically two kinds of numerical calculation methods:PDE methods and variational methods. Both are discussed briefly.
Some applications of the effective Hamiltonian are given in Chapter 5. First the physical meaning is discussed. Then it proceeds with the discussion of the role the effective Hamiltonian played in the study of asymptotic solutions of Hamilton-Jacobi equations and in the homogenization theory. At last, it points out the close relation among the effective Hamiltonian, the Weak KAM theory and Aubry-Mather theory.
Lions, Papanicolaou和Varadhan首先证明了存在唯一的实数λ∈R使得Hamilton-Jacobi方程存在全局粘性解,记此唯一的与P有关的实数λ∈R为H(P),称为有效Hamilton函数。
有效Hamilton函数有着十分明确的物理意义：它表征了量子本征态所对应的能量值。并且它在Hamilton-Jacobi方程的均匀化理论、解的长时间渐近行为等的研究中都起着非常重要的作用,与弱KAM理论、Aubry-Mather理论等也有着十分密切的联系。
第一章首先介绍这一方面研究进展,既包括理论上的推广,也包括数值计算和应用方面的进展。然后给出必要的预备知识。
第二章首先给出Lions, Papanicolaou和Varadhan等人的原始证明,然后给出一个新的关于有效Hamilton函数存在唯一性的几何方法的证明。这种方法不仅可以证明有效Hamilton函数的存在唯一性,而且由此出发还可以讨论有效Hamilton函数的一些性质以及建立它与Aubry-Mather理论的密切联系。
第三章将对有效Hamilton函数进行刻画。首先是它的一些等价表示,这是由有效Hamilton函数的存在唯一性的几何证明出发而得到的关于H(P)的一些表达式,它们表征了有效Hamilton函数的一些极限特性。还有H(P)的两个变分表示,它们是后面数值计算的理论基础。之后讨论了有效Hamilton函数的一些基本性质,它们反映了H(P)的性质与Hamilton函数H的性质之间的密切关系。
第四章讨论有效Hamilton函数的计算问题。首先对于一些具体的Hamilton函数给出相应的有效Hamilton函数的解析表达式。由于一般而言只有对于特殊的Hamilton函数才能得到有效Hamilton函数的解析表达式,所以接下来讨论了相应的数值计算问题。数值计算的方法又分为两类：偏微分方程方法和变分方法,对每一类方法都进行了简单的分析讨论。
第五章给出有效Hamilton函数的一些应用。首先说明有效Hamilton函数的物理意义：它表征了本征态所对应的能量值,接着给出它在Hamilton-Jacobi方程的均匀化理论、解的长时间渐近行为等的研究中的应用,最后指出了有效Hamilton函数与弱KAM理论、Aubry-Mather理论的密切联系。
Consider the Hamilton-Jacobi equation H(x, P+Du) =λ, there exists one and only one real numberλ∈R such that the equation has a global viscosity solution. It is called the effective Hamiltonian and denoted by H(P).
The physical meaning of the effective Hamiltonian is that it represents the eigenvalue of an eigenstate. It is important in the study of asymptotic solutions of Hamilton-Jacobi equations and in the homogenization theory. It is also closely related to the Weak KAM theory and Aubry-Mather theory.
Some developments in the field is discussed in Chapter 1. It includes the generalization of theoretical analysis and also the developments in numerical calculations and in applications. Some preparations are also given.
Chapter 2 begins with the original proof of existence and uniqueness of effec-tive Hamiltonian given by Lions,Papanicolaou,Varadhan. Then it proceeds with a new geometric proof of the result. Using the geometric proof, some proper-ties of the effective Hamiltonian can be discussed and the close relation between effective Hamiltonian and Aubry-Mather theory can be found.
Some equivalent expressions of the effective Hamiltonian will be discussed in Chapter 3. These equivalent expressions comes from the new geometric proof and shows the limit property of the effective Hamiltonian. Two variational ex-pressions are also discussed, which are the theoretical foundation of numerical calculations. The second part of this chapter deals with the basic properties of the effective Hamiltonian which shows the relation between properties of the effective Hamiltonian and properties of the Hamiltonian function.
Chapter 4 is devoted to the calculation of the effective Hamiltonian. Some examples are given. Since usually it is impossible to give the explicit expressions of the effective Hamiltonian, numerical calculation methods are discussed. There are basically two kinds of numerical calculation methods:PDE methods and variational methods. Both are discussed briefly.
Some applications of the effective Hamiltonian are given in Chapter 5. First the physical meaning is discussed. Then it proceeds with the discussion of the role the effective Hamiltonian played in the study of asymptotic solutions of Hamilton-Jacobi equations and in the homogenization theory. At last, it points out the close relation among the effective Hamiltonian, the Weak KAM theory and Aubry-Mather theory.
引文
[1]P.L.Lions,G.Papanicolaou,S.R.S.Varadhan. Homogenization of Hamilton-Jacobi equations, preprint
[2]A.Fathi. Theoreme KAM faible et theorie de Mather sur les systemes la-grangiens. C.R.Acad.Sci.Paris,1997,324:1043-1046
[3]A.Fathi. Solutions KAM faibles conjuguees et barrieres de Peierls. C.R.Acad.Sci.Paris,1997,325:649-652
[4]G. Contreras,R. Iturriaga,G.P. Paternain,M. Paternain. Lagrangian Graphs, Minimizing Measures And Mane's Critical Values. GAFA,Geom.funct.anal.,1998,8:788-809
[5]F.Rezakhanlou,J.Tarver. Homogenization for Stochastic Hamilton-Jacobi Equations. Arch.Rational.Mech.Anal.,2000,151:277-309
[6]A.Fathi,E.Maderna. Weak KAM theorem on non compact manifolds. Non-linear Differential Equations and Applications.2007,14:1-27
[7]L.C.Evans,D.Gomes. Effective Hamiltonians and Averaging for Hamiltonian Dynamics I. Arch.Rational.Mech.Anal.,2001,157:1-33
[8]L.C.Evans,D.Gomes. Effective Hamiltonians and Averaging for Hamiltonian Dynamics II. Arch.Rational.Mech.Anal.,2002,161:271-305
[9]P.L.Lions,P.E.Souganidis. Correctors for the Homogenization of Hamilton-Jacobi Equations in the Stationary Ergodic Setting. Communications on Pure and Applied Mathematics.,2003,56:1501-1524
[10]A.Fathi,A.Siconolfi. PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians. Calculus of Variations and PDE.2005,22:185-228
[11]F.Camilli,A.Siconolfi. Effective Hamiltonian and Homogenization of Measur-able Eikonal Equations. Arch.Rational.Mech.Anal.,2007,183:1-20
[12]N.Ichihara,H.Ishii. Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians. Communications in Partial Differential Equations.,2008,33:784-807
[13]H.Ishii. Asymptotic solutions for large time of Hamilton-Jacobi equations. Proceedings of the International Congress of Mathematicians,Madrid,Spain.,2006:213-227
[14]H.Ishii. Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space. Ann.I.Poincare.,2008,25:231-266
[15]P.Bernard,J.M.Roquejoffre. Convergence to Time-Periodic Solutions in Time-Periodic Hamilton-Jacobi Equations on the Circle. Communications in Partial Differential Equations.,2004,19:457-469
[16]G.Wolansky. Minimizers of Dirichlet functionals on the n-torus and the weak KAM theory. Ann.I.Poincare.,2009,26:521-545
[17]Qian Jianliang. Two Approximations for Effective Hamiltonians Aris-ing from Homogenization of Hamilton-Jacobi Equations. UCLA,Math Dept.,2003,preprint
[18]D.Gomes,A.Oberman. Computing the Effective Hamiltonian Using a Varia-tional Approach. SIAM J.Control Optim.,2004,43:792-812
[19]M.Falcone,M.Rorro. On a Variational Approximation of the Effective Hamil-tonian. Numerical mathematics and advanced applications:proceeding of ENUMATH 2007,719-726
[20]P.L.Lions. Generalized solutions of Hamilton-Jacobi equations. Boston:Pitman Advanced Publishing Program.,1982
[21]M.G.Crandall,L.C.Evans,P.L.Lions. Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. Trans.Amer.Math.Soc.,1984,282:487-502
[22]H.Ishii. Perron's Method for Hamilton-Jacobi Equations. Duke Mathemati-cal Journal.,1987,55:369-384
[23]G.Barles. Solutions de viscosite des equations de Hamilton-Jacobi. Paris:Springer-Verlag.,1994
[24]M.Bardi,I.Capuzzo-Dolcetta. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Boston:Birkhauser.,1997
[25]K.F.Siberg. The Principle of Least Action in Geometry and Dynamics. Berlin:Springer-Verlag.,2004
[26]D.Gomes. Aubry-Mather Theory for Multi-dimensional Maps.2002,preprint
[27]L.C.Evans. Weak KAM Theory and Partial Differential Equations. Calculus Variations and Partial Differential Equations,2008:123-154
[28]L.C.Evans. Towards a Quantum Analog of Weak KAM Theory. Com-mun.Math.Phys.,2004,244:311-334
[29]A.Fathi. Weak KAM Theorem in Lagrangian Dynamics. Cambridge:Cam-bridge University Press,2008
[30]J.N.Mather.Minimal measures. Comment.Math.Helvetici.,1989,64:375-394
[31]J.N.Mather. Action minimizing invariant measures for positive definite La-grgian systems. Math.Z.,1991,207:169-207
[32]J.N.Mather. Variational Construction of Connecting Orbits. Ann.Inst.Fourier.,1993,43:1349-1386
[33]V.Kaloushin. Mather theory, Weak KAM Theory,and Viscosity Solutions of Hamilton-Jacobi PDE's.2003,preprint
[34]L.C.Evans. A Survey of Partial Differential Equations Methods in Weak KAM Theory. Communications on Pure and Applied Mathematics.,2004,57:445-480
[2]A.Fathi. Theoreme KAM faible et theorie de Mather sur les systemes la-grangiens. C.R.Acad.Sci.Paris,1997,324:1043-1046
[3]A.Fathi. Solutions KAM faibles conjuguees et barrieres de Peierls. C.R.Acad.Sci.Paris,1997,325:649-652
[4]G. Contreras,R. Iturriaga,G.P. Paternain,M. Paternain. Lagrangian Graphs, Minimizing Measures And Mane's Critical Values. GAFA,Geom.funct.anal.,1998,8:788-809
[5]F.Rezakhanlou,J.Tarver. Homogenization for Stochastic Hamilton-Jacobi Equations. Arch.Rational.Mech.Anal.,2000,151:277-309
[6]A.Fathi,E.Maderna. Weak KAM theorem on non compact manifolds. Non-linear Differential Equations and Applications.2007,14:1-27
[7]L.C.Evans,D.Gomes. Effective Hamiltonians and Averaging for Hamiltonian Dynamics I. Arch.Rational.Mech.Anal.,2001,157:1-33
[8]L.C.Evans,D.Gomes. Effective Hamiltonians and Averaging for Hamiltonian Dynamics II. Arch.Rational.Mech.Anal.,2002,161:271-305
[9]P.L.Lions,P.E.Souganidis. Correctors for the Homogenization of Hamilton-Jacobi Equations in the Stationary Ergodic Setting. Communications on Pure and Applied Mathematics.,2003,56:1501-1524
[10]A.Fathi,A.Siconolfi. PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians. Calculus of Variations and PDE.2005,22:185-228
[11]F.Camilli,A.Siconolfi. Effective Hamiltonian and Homogenization of Measur-able Eikonal Equations. Arch.Rational.Mech.Anal.,2007,183:1-20
[12]N.Ichihara,H.Ishii. Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians. Communications in Partial Differential Equations.,2008,33:784-807
[13]H.Ishii. Asymptotic solutions for large time of Hamilton-Jacobi equations. Proceedings of the International Congress of Mathematicians,Madrid,Spain.,2006:213-227
[14]H.Ishii. Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space. Ann.I.Poincare.,2008,25:231-266
[15]P.Bernard,J.M.Roquejoffre. Convergence to Time-Periodic Solutions in Time-Periodic Hamilton-Jacobi Equations on the Circle. Communications in Partial Differential Equations.,2004,19:457-469
[16]G.Wolansky. Minimizers of Dirichlet functionals on the n-torus and the weak KAM theory. Ann.I.Poincare.,2009,26:521-545
[17]Qian Jianliang. Two Approximations for Effective Hamiltonians Aris-ing from Homogenization of Hamilton-Jacobi Equations. UCLA,Math Dept.,2003,preprint
[18]D.Gomes,A.Oberman. Computing the Effective Hamiltonian Using a Varia-tional Approach. SIAM J.Control Optim.,2004,43:792-812
[19]M.Falcone,M.Rorro. On a Variational Approximation of the Effective Hamil-tonian. Numerical mathematics and advanced applications:proceeding of ENUMATH 2007,719-726
[20]P.L.Lions. Generalized solutions of Hamilton-Jacobi equations. Boston:Pitman Advanced Publishing Program.,1982
[21]M.G.Crandall,L.C.Evans,P.L.Lions. Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. Trans.Amer.Math.Soc.,1984,282:487-502
[22]H.Ishii. Perron's Method for Hamilton-Jacobi Equations. Duke Mathemati-cal Journal.,1987,55:369-384
[23]G.Barles. Solutions de viscosite des equations de Hamilton-Jacobi. Paris:Springer-Verlag.,1994
[24]M.Bardi,I.Capuzzo-Dolcetta. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Boston:Birkhauser.,1997
[25]K.F.Siberg. The Principle of Least Action in Geometry and Dynamics. Berlin:Springer-Verlag.,2004
[26]D.Gomes. Aubry-Mather Theory for Multi-dimensional Maps.2002,preprint
[27]L.C.Evans. Weak KAM Theory and Partial Differential Equations. Calculus Variations and Partial Differential Equations,2008:123-154
[28]L.C.Evans. Towards a Quantum Analog of Weak KAM Theory. Com-mun.Math.Phys.,2004,244:311-334
[29]A.Fathi. Weak KAM Theorem in Lagrangian Dynamics. Cambridge:Cam-bridge University Press,2008
[30]J.N.Mather.Minimal measures. Comment.Math.Helvetici.,1989,64:375-394
[31]J.N.Mather. Action minimizing invariant measures for positive definite La-grgian systems. Math.Z.,1991,207:169-207
[32]J.N.Mather. Variational Construction of Connecting Orbits. Ann.Inst.Fourier.,1993,43:1349-1386
[33]V.Kaloushin. Mather theory, Weak KAM Theory,and Viscosity Solutions of Hamilton-Jacobi PDE's.2003,preprint
[34]L.C.Evans. A Survey of Partial Differential Equations Methods in Weak KAM Theory. Communications on Pure and Applied Mathematics.,2004,57:445-480