Aubry-Mather理论中的若干问题
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摘要
本文研究Aubry-Mather理论中的若干问题.全文共分四章,第一章是绪论,第二章至第四章为论文主体部分.
在第二章,我们首先将法国数学家A.Fathi建立的弱KAM理论(与Mather理论密切相关)参数化,对于一般形式的Hamilton函数得到低维作用极小测度的存在性定理.然后,将该定理局部地应用于近可积Hamilton系统,从而对动力系统理论的创始人,著名的法国博学家H.Poincar(?)提出的可积系统的稳定性机制有多少在小扰动下保持下来这一动力学基本问题作出进一步的回答.在第三章,我们研究了一类特殊的平面Lagrange系统极小解集的构造.Lagrange系统的极小解对应着相应Hamilton系统Poincar(?)映射的Mather集.我们给出了连接相邻的以0为旋转数的周期极小解的极小异宿解的一种新的变分构造.在第四章,我们研究了相应于具有变扩散系数的受控Markov扩散的Mather理论,讨论了随机作用极小测度的性质,给出了随机作用极小测度的变分构造,定义了相应的Mather函数,并且讨论了它们的可微性.
The study on the dynamical behaviour of area-preserving mappings is always on the cutting edge of the study on Hamiltonian systems and symplectic geometry. The class of area-preserving mappings with monotone twist property has been paid more attention by mathematicians. So-called Aubry-Mather theory of the mappings mentioned above is one of the most fascinating developments in this field.
Area-preserving mappings were first considered by H. Poincare in his work on the restricted three-body problem. The celebrated "last geometric theorem of Poincare" states that an area-preserving homeomorphism of the annulus, which rotates the two boundary circles in opposite directions, must have at least two fixed points. It was conjectured by H. Poincare (1912) from a consideration of the three-body problem in celestial mechanics and proved by G. Birkhoff (1913), who generalized the result in 1925. Poincar(?)-Birkhoff geometric theorem implies the existence of periodic orbits of the monotone twist mappings. Afterwards, in the field of area-preserving mappings we must mention the famous work on stability theory of A. Kolmogorov [90]-V. Arnold [12]-J. Moser [124]-KAM theory. Moser's twist mapping theorem [124] guarantees the existence of closed invariant curves of nearly integrable monotone twist homeomorphisms. Obviously, there should be a new kind of solutions between periodic orbits and closed invariant curves. Aubry-Mather theory confirmed the assertion.
Physicists S. Aubry and P. Le Daeron published their theory of minimum energy states for Frenkel-Kontorova models on《Physica D: Nonlinear Phenomena》in 1983. The theory is a beautiful body of work that has greatly enhanced understanding of how structure is determined in the solid state and played a key role in the interplay between condensed matter physics and dynamical systems theory and in our careers. They proved that the set of recurrent minimum energy states of mean spacing is either a curve or a Cantor set. Some of these results were also proved by J. Mather (starting independently of Aubry). Mathematician J. Mather obtained the existence of quasi-periodic orbits of monotone twist mappings of the annulus in his work, which was published on《Topology》in 1982. He proved that for each given numberαin the twist interval, there exists an invariant Mather set of rotation numberα. Mather sets are closed and invariant. For the smallest Mather set there are only three cases: when the rotation number is rational, it is an order-preserving periodic orbit, which is also called Birkhoff periodic orbit; when the rotation number is irrational, it is either a closed invariant curve or a Denjoy Cantor set. The works by S. Aubry and J. Mather were begun independently and with different motivations but led to similar results by different methods. So the subject is often called Aubry-Mather theory. Afterwards, many people generalized the theory in various directions. In particular, J. Mather [113] proposed a generalization of Aubry-Mather theory to periodic Hamiltonian systems in more degrees of freedom.
In this thesis, we make attempt to study several problems in Aubry-Mather theory. In Chapter 2, we parametrize weak KAM theory (closely related to Mather theory [113]), developed by A. Fathi [60, 61, 62, 63], using a variational approach, and then by applying the parametrized weak KAM theory to nearly integrable Hamiltonian systems locally, we obtain the existence of lower dimensional action minimizing measures, which gives a definite answer to the basic problem of dynamics raised by H. Poincar(?). In Chapter 3, we study the structure of the set of minimal solutions to a plane Lagrangian system. Minimal solutions of Lagrangian systems correspond to Mather sets of Poincare map of Hamiltonian systems. We give a constructive proof of the existence of minimal heteroclinic solutions connecting two adjacent periodic minimal solutions of rotation number 0. In Chapter 4, we study the stochastic Mather theory with respect to the non-uniform controlled Markov diffusion. More precisely, we discuss the properties of stochastic action minimizing measures and construct a stochastic action minimizing measure by using a variational approach. At last, we define the stochastic versions ofMather's functions and discuss the differentiability of them. Now let us introduce the main results of the thesis.
Lower dimensional action minimizing measures
LetΩ= T~n×R~n×X×X, where X is a real Hilbert space. Assume that Hamiltonian H :Ω→R~1, H＝H(x,p,α,β), is smooth, and satisfies the following two conditions:
(a) uniformly strict convexity in p: There exists a constantθ> 0, such thatfor each x∈T~n, p,ξ∈R~n,α,β∈X;
(b) growth bounds: There exists a constant C > 0, such thatfor each x∈T~n, p∈R~n,α,β∈X.
For given P∈R~n, (?), we obtain a Borel probability measureμonΩby using a variational approach. Setν= proj_(TT~n)μ, where TT~n= T~n×R~n. We have the following result:
Theorem 1 Probability measure v is Euler-Lagrange flow invariant (lower dimension), i.e.,for allφ∈C~1(T~n). Furthermore, v is an action minimizing measure on TT~n of rotation vector∫_(TT~n) qdv.
Consider the nearly integrable Hamiltonian systemH(x,y) = H_0(y) +εP(x,y), (1)where y∈G (?) R~n, x∈T~n, H_0 and P are C~3 functions defined on the closed bounded region G and T~n×G, respectively. Assume that H_0 is strictly convex. P is a perturbation andε> 0 is a small parameter.
For a given rank m_0 subgroup g of Z~n, set m = n -m_0 andis called a g-resonant surface.
Applying Theorem 1 to the nearly integrable Hamiltonian system (1), we have:
Theorem 2 For each y∈O(g,G), there exist at least m_0 + 1 action minimizing measures {μ_y~i}_(m=1)~(m_0+1) on T~m×K~m for the nearly integrable Hamiltonian system (1).
It is well known that most of the nonresonant tori of the integrable system persist under a small perturbation, and the resonant tori are destroyed. In the late 60's of the last century, V. Melnikov [123] used the then relatively new ideas of KAM theory to find lower dimensional invariant tori residing in the resonant surfaces of analytic, nearly integrable Hamiltonian systems. Since then, such tori have become a crucial element in attempts to establish a rigorous foundation for "Arnold diffusion" (an instability mechanism whose explication is considered by many to be the outstanding problem in the theory of nearly integrable Hamiltonian systems). Partly motivated by applications to the problem of Arnold diffusion, a number of authors (D. Bernstein and A. Katok [22], C. Cheng [37], L. Chierchia and G. Gallavotti [39], R. de la Llave and E. Wayne [46], L. Eliasson [51], S. Graff [78], J. P(?)schel [134], M. Rudnev and S. Wiggins [139], D. Treschev [146], and others) have given more refined results on the persistence of lower dimensional invariant tori. For the most part, these results are restricted to the multiplicity one resonant case and require hyperbolicity conditions in addition to the usual nondegeneracy conditions. For the multiplicity m_0 > 1 resonant case, in [40] the authors prove the persistence of lower dimensional tori in such systems under standard nondegeneracy conditions and without the requirement of hyperbolicity. More precisely, they proved that for most frequencies on the resonant surface (a Cantor subset of positive measure), the resonanttorus foliated by nonresonant lower dimensional tori is not destroyed completely and that there are some lower dimensional tori which survive the perturbation. Then what happens to the points outside the Cantor set in O(g,G)? Theorem 2 tells us that for each y∈O(g,G), there exist at least m_0 +1 action minimizing measures on T~m×R~m, which support certain either quasi-periodic motions or weak quasi-periodic motions. Therefore, Theorem 2 gives a definite answer to the basic problem of dynamics put forward by H. Poincar(?).
Minimal heteroclinic solutons
Next let us discuss the structure of the set of minimal solutions to plane Lagrangian systems. First we give the concept of minimal solutions. Given a Lagrangian L : R~3→R~1, the corresponding Euler-Lagrange equation reads
Definition 1 A C~1 function q : R~1→R~1 is called a minimal solution of (LS), if for each [a, b] andξ∈C_0~1([a, b]).The concept was given by M. Giaquinta and E. Guisti in [70].
Consider the standard Lagrangian L(t, q, (?)) = (?) + U(t, q) in classical mechanics. Assume that U satisfies the following three conditions: (U1) U∈C~2(R~1×R~1); (U2) U is 1-periodic t, q; (U3) U is even in t
The Euler-Lagrange equation is Newton equation(?). (2)LetΞ_0 = {q∈W_(loc)~(1,2)(R~1) : q is 1-periodic in t}.Consider a minimization problemFrom a well known result of Moser [125], we know that there exist minimizers of the problem above, and each minimizer q∈Ξ_0 is a C~2 solution to (2).
Set c_0 = inf{I_0(g) : q∈Ξ_0 }. It is easy to see that c_0∈R~1. Then the set of periodic minimal solutions of rotation number 0 of (2) is M(0) = {q∈Ξ_0 : I_0(q) = c_0}. Furthermore, M(0) is a nonempty ordered set. Consider the following condition:
(*) gap condition: There exist adjacent p_0, r_0∈M(0) with p_0 < r_0.
For i∈Z, set
Theorem 3 Let U satisfy (U1)-(U3) and gap condition (*) hold. Then there exists a classical solution Q of (2), such thatτ_iQ (?) p_0 and (?)r_0 in W~(1,2)([0,1]) as i→∞. Furthermore, Q is a minimal solution of (2).
Since minimal solutions of Lagrangian systems correspond to Mather sets of Poincare map of Hamiltonian systems, studying the structure of the set of minimal solutions is important for understanding of Aubry-Mather theory. Theorem 3 tells us that there exists a minimal heteroclinic solution connecting two adjacent periodic minimal solutions of rotation number 0. And the proof of Theorem 3 is constructive.
Stochastic Mather theory: non-uniform diffusion case
D. Gomes [74] generlized the work of J. Mather [113] on the action minimizing measures of positive definite Lagrangian systems to the stochastic case. More precisely, he considered the stochastic Mather minimizationproblemwith respect to the controlled Markov diffusiondx = v(t)dt +σdw,whereμis a Borel probability measure on T~n×R~n, and satisfiesfor allφ∈C~2(T~n).
We attempt to generalize Gomes' results (uniform diffusion case) to non-uniform diffusion case (σis a function of x). It means that we consider the stochastic Mather minimization problem with respect to the controlled Markov diffusiondx = v(t)dt +σ(x)dw.Here w(t) is an n-dimensional Brownian motion, v(t) is a bounded progressively measurable control, and the diffusion rateσ(x)≥0 for all x∈T~n. We also assumeσis weakly differentiate andσ(x)≠0 almost everywhere (Lebesgue measure).
We prove that the stochastic Mather minimization problem above admits minimizers, i.e., stochastic action minimizing measures, and each minimizer has the following property:
Proposition 1 Each stochastic action minimizing measure is supported in the graph {(x, -D_pH(x, D_xu))}, where u is the solution of(?), (3)and (?) is the stochastic Ma(?)'s critical value.
By the above proposition we can obtain another property of stochastic action minimizing measures.
Theorem 4 Letμbe a stochastic action minimizing measure, and v = proj_(T~n)μ. Then we haveσ~2(x)dv =θ{x)dxfor someθ∈W~(1,2)(T~n). Furthermore,θis a weak solution ofwhere w = -D_pH(x, D_xu) and u is the solution of (3).
Furthermore, we construct a stochastic action minimizing measure by using a variational approach from the stochastic minimax formula. The method is valid for both uniform and non-uniform diffusion. At last, we define the stochastic versions of Mather's functions and discuss the differentiability of them. The stochastic Mather theory is well developed now.
在第二章,我们首先将法国数学家A.Fathi建立的弱KAM理论(与Mather理论密切相关)参数化,对于一般形式的Hamilton函数得到低维作用极小测度的存在性定理.然后,将该定理局部地应用于近可积Hamilton系统,从而对动力系统理论的创始人,著名的法国博学家H.Poincar(?)提出的可积系统的稳定性机制有多少在小扰动下保持下来这一动力学基本问题作出进一步的回答.在第三章,我们研究了一类特殊的平面Lagrange系统极小解集的构造.Lagrange系统的极小解对应着相应Hamilton系统Poincar(?)映射的Mather集.我们给出了连接相邻的以0为旋转数的周期极小解的极小异宿解的一种新的变分构造.在第四章,我们研究了相应于具有变扩散系数的受控Markov扩散的Mather理论,讨论了随机作用极小测度的性质,给出了随机作用极小测度的变分构造,定义了相应的Mather函数,并且讨论了它们的可微性.
The study on the dynamical behaviour of area-preserving mappings is always on the cutting edge of the study on Hamiltonian systems and symplectic geometry. The class of area-preserving mappings with monotone twist property has been paid more attention by mathematicians. So-called Aubry-Mather theory of the mappings mentioned above is one of the most fascinating developments in this field.
Area-preserving mappings were first considered by H. Poincare in his work on the restricted three-body problem. The celebrated "last geometric theorem of Poincare" states that an area-preserving homeomorphism of the annulus, which rotates the two boundary circles in opposite directions, must have at least two fixed points. It was conjectured by H. Poincare (1912) from a consideration of the three-body problem in celestial mechanics and proved by G. Birkhoff (1913), who generalized the result in 1925. Poincar(?)-Birkhoff geometric theorem implies the existence of periodic orbits of the monotone twist mappings. Afterwards, in the field of area-preserving mappings we must mention the famous work on stability theory of A. Kolmogorov [90]-V. Arnold [12]-J. Moser [124]-KAM theory. Moser's twist mapping theorem [124] guarantees the existence of closed invariant curves of nearly integrable monotone twist homeomorphisms. Obviously, there should be a new kind of solutions between periodic orbits and closed invariant curves. Aubry-Mather theory confirmed the assertion.
Physicists S. Aubry and P. Le Daeron published their theory of minimum energy states for Frenkel-Kontorova models on《Physica D: Nonlinear Phenomena》in 1983. The theory is a beautiful body of work that has greatly enhanced understanding of how structure is determined in the solid state and played a key role in the interplay between condensed matter physics and dynamical systems theory and in our careers. They proved that the set of recurrent minimum energy states of mean spacing is either a curve or a Cantor set. Some of these results were also proved by J. Mather (starting independently of Aubry). Mathematician J. Mather obtained the existence of quasi-periodic orbits of monotone twist mappings of the annulus in his work, which was published on《Topology》in 1982. He proved that for each given numberαin the twist interval, there exists an invariant Mather set of rotation numberα. Mather sets are closed and invariant. For the smallest Mather set there are only three cases: when the rotation number is rational, it is an order-preserving periodic orbit, which is also called Birkhoff periodic orbit; when the rotation number is irrational, it is either a closed invariant curve or a Denjoy Cantor set. The works by S. Aubry and J. Mather were begun independently and with different motivations but led to similar results by different methods. So the subject is often called Aubry-Mather theory. Afterwards, many people generalized the theory in various directions. In particular, J. Mather [113] proposed a generalization of Aubry-Mather theory to periodic Hamiltonian systems in more degrees of freedom.
In this thesis, we make attempt to study several problems in Aubry-Mather theory. In Chapter 2, we parametrize weak KAM theory (closely related to Mather theory [113]), developed by A. Fathi [60, 61, 62, 63], using a variational approach, and then by applying the parametrized weak KAM theory to nearly integrable Hamiltonian systems locally, we obtain the existence of lower dimensional action minimizing measures, which gives a definite answer to the basic problem of dynamics raised by H. Poincar(?). In Chapter 3, we study the structure of the set of minimal solutions to a plane Lagrangian system. Minimal solutions of Lagrangian systems correspond to Mather sets of Poincare map of Hamiltonian systems. We give a constructive proof of the existence of minimal heteroclinic solutions connecting two adjacent periodic minimal solutions of rotation number 0. In Chapter 4, we study the stochastic Mather theory with respect to the non-uniform controlled Markov diffusion. More precisely, we discuss the properties of stochastic action minimizing measures and construct a stochastic action minimizing measure by using a variational approach. At last, we define the stochastic versions ofMather's functions and discuss the differentiability of them. Now let us introduce the main results of the thesis.
Lower dimensional action minimizing measures
LetΩ= T~n×R~n×X×X, where X is a real Hilbert space. Assume that Hamiltonian H :Ω→R~1, H＝H(x,p,α,β), is smooth, and satisfies the following two conditions:
(a) uniformly strict convexity in p: There exists a constantθ> 0, such thatfor each x∈T~n, p,ξ∈R~n,α,β∈X;
(b) growth bounds: There exists a constant C > 0, such thatfor each x∈T~n, p∈R~n,α,β∈X.
For given P∈R~n, (?), we obtain a Borel probability measureμonΩby using a variational approach. Setν= proj_(TT~n)μ, where TT~n= T~n×R~n. We have the following result:
Theorem 1 Probability measure v is Euler-Lagrange flow invariant (lower dimension), i.e.,for allφ∈C~1(T~n). Furthermore, v is an action minimizing measure on TT~n of rotation vector∫_(TT~n) qdv.
Consider the nearly integrable Hamiltonian systemH(x,y) = H_0(y) +εP(x,y), (1)where y∈G (?) R~n, x∈T~n, H_0 and P are C~3 functions defined on the closed bounded region G and T~n×G, respectively. Assume that H_0 is strictly convex. P is a perturbation andε> 0 is a small parameter.
For a given rank m_0 subgroup g of Z~n, set m = n -m_0 andis called a g-resonant surface.
Applying Theorem 1 to the nearly integrable Hamiltonian system (1), we have:
Theorem 2 For each y∈O(g,G), there exist at least m_0 + 1 action minimizing measures {μ_y~i}_(m=1)~(m_0+1) on T~m×K~m for the nearly integrable Hamiltonian system (1).
It is well known that most of the nonresonant tori of the integrable system persist under a small perturbation, and the resonant tori are destroyed. In the late 60's of the last century, V. Melnikov [123] used the then relatively new ideas of KAM theory to find lower dimensional invariant tori residing in the resonant surfaces of analytic, nearly integrable Hamiltonian systems. Since then, such tori have become a crucial element in attempts to establish a rigorous foundation for "Arnold diffusion" (an instability mechanism whose explication is considered by many to be the outstanding problem in the theory of nearly integrable Hamiltonian systems). Partly motivated by applications to the problem of Arnold diffusion, a number of authors (D. Bernstein and A. Katok [22], C. Cheng [37], L. Chierchia and G. Gallavotti [39], R. de la Llave and E. Wayne [46], L. Eliasson [51], S. Graff [78], J. P(?)schel [134], M. Rudnev and S. Wiggins [139], D. Treschev [146], and others) have given more refined results on the persistence of lower dimensional invariant tori. For the most part, these results are restricted to the multiplicity one resonant case and require hyperbolicity conditions in addition to the usual nondegeneracy conditions. For the multiplicity m_0 > 1 resonant case, in [40] the authors prove the persistence of lower dimensional tori in such systems under standard nondegeneracy conditions and without the requirement of hyperbolicity. More precisely, they proved that for most frequencies on the resonant surface (a Cantor subset of positive measure), the resonanttorus foliated by nonresonant lower dimensional tori is not destroyed completely and that there are some lower dimensional tori which survive the perturbation. Then what happens to the points outside the Cantor set in O(g,G)? Theorem 2 tells us that for each y∈O(g,G), there exist at least m_0 +1 action minimizing measures on T~m×R~m, which support certain either quasi-periodic motions or weak quasi-periodic motions. Therefore, Theorem 2 gives a definite answer to the basic problem of dynamics put forward by H. Poincar(?).
Minimal heteroclinic solutons
Next let us discuss the structure of the set of minimal solutions to plane Lagrangian systems. First we give the concept of minimal solutions. Given a Lagrangian L : R~3→R~1, the corresponding Euler-Lagrange equation reads
Definition 1 A C~1 function q : R~1→R~1 is called a minimal solution of (LS), if for each [a, b] andξ∈C_0~1([a, b]).The concept was given by M. Giaquinta and E. Guisti in [70].
Consider the standard Lagrangian L(t, q, (?)) = (?) + U(t, q) in classical mechanics. Assume that U satisfies the following three conditions: (U1) U∈C~2(R~1×R~1); (U2) U is 1-periodic t, q; (U3) U is even in t
The Euler-Lagrange equation is Newton equation(?). (2)LetΞ_0 = {q∈W_(loc)~(1,2)(R~1) : q is 1-periodic in t}.Consider a minimization problemFrom a well known result of Moser [125], we know that there exist minimizers of the problem above, and each minimizer q∈Ξ_0 is a C~2 solution to (2).
Set c_0 = inf{I_0(g) : q∈Ξ_0 }. It is easy to see that c_0∈R~1. Then the set of periodic minimal solutions of rotation number 0 of (2) is M(0) = {q∈Ξ_0 : I_0(q) = c_0}. Furthermore, M(0) is a nonempty ordered set. Consider the following condition:
(*) gap condition: There exist adjacent p_0, r_0∈M(0) with p_0 < r_0.
For i∈Z, set
Theorem 3 Let U satisfy (U1)-(U3) and gap condition (*) hold. Then there exists a classical solution Q of (2), such thatτ_iQ (?) p_0 and (?)r_0 in W~(1,2)([0,1]) as i→∞. Furthermore, Q is a minimal solution of (2).
Since minimal solutions of Lagrangian systems correspond to Mather sets of Poincare map of Hamiltonian systems, studying the structure of the set of minimal solutions is important for understanding of Aubry-Mather theory. Theorem 3 tells us that there exists a minimal heteroclinic solution connecting two adjacent periodic minimal solutions of rotation number 0. And the proof of Theorem 3 is constructive.
Stochastic Mather theory: non-uniform diffusion case
D. Gomes [74] generlized the work of J. Mather [113] on the action minimizing measures of positive definite Lagrangian systems to the stochastic case. More precisely, he considered the stochastic Mather minimizationproblemwith respect to the controlled Markov diffusiondx = v(t)dt +σdw,whereμis a Borel probability measure on T~n×R~n, and satisfiesfor allφ∈C~2(T~n).
We attempt to generalize Gomes' results (uniform diffusion case) to non-uniform diffusion case (σis a function of x). It means that we consider the stochastic Mather minimization problem with respect to the controlled Markov diffusiondx = v(t)dt +σ(x)dw.Here w(t) is an n-dimensional Brownian motion, v(t) is a bounded progressively measurable control, and the diffusion rateσ(x)≥0 for all x∈T~n. We also assumeσis weakly differentiate andσ(x)≠0 almost everywhere (Lebesgue measure).
We prove that the stochastic Mather minimization problem above admits minimizers, i.e., stochastic action minimizing measures, and each minimizer has the following property:
Proposition 1 Each stochastic action minimizing measure is supported in the graph {(x, -D_pH(x, D_xu))}, where u is the solution of(?), (3)and (?) is the stochastic Ma(?)'s critical value.
By the above proposition we can obtain another property of stochastic action minimizing measures.
Theorem 4 Letμbe a stochastic action minimizing measure, and v = proj_(T~n)μ. Then we haveσ~2(x)dv =θ{x)dxfor someθ∈W~(1,2)(T~n). Furthermore,θis a weak solution ofwhere w = -D_pH(x, D_xu) and u is the solution of (3).
Furthermore, we construct a stochastic action minimizing measure by using a variational approach from the stochastic minimax formula. The method is valid for both uniform and non-uniform diffusion. At last, we define the stochastic versions of Mather's functions and discuss the differentiability of them. The stochastic Mather theory is well developed now.
引文
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