Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems

In this paper, we consider the minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. We prove that if the Hamiltonian function $H\in C^2(\Bbb R^{2n}, \Bbb R)$ is super-quadratic and convex, for every number $\tau>0$, there exists at least one $\tau$-periodic brake orbit $(\tau,x)$ with minimal period $\tau$ or $\tau/2$ provided $H(Nx)=H(x)$.


(1.2)
A solution (τ, x) of (1.2) is a special periodic solution of the Hamiltonian system in (1.2), we call it a brake orbit and τ the brake period of x.
The existence and multiplicity of brake orbits on a given energy hypersurface was studied by many Mathematicians. In 1987, P. Rabinowitz in [26] proved that if H satisfies (1.1), Σ = H −1 (1) is star-shaped, and x · H ′ (x) = 0 for all x ∈ Σ, then there exists a brake orbit on Σ. In 1987, V. Benci and F. Giannoni gave a different proof of the existence of one brake orbit on Σ in [1]. In 1989, A. Szulkin in [27] proved that there exist at least n brake orbits on Σ, if H satisfies conditions in [26] of Rabinowitz and the energy hypersurface Σ is √ 2-pinched. Long, Zhang and Zhu in [23] proved that there exist at least 2 geometrically distinct brake orbits on any central symmetric strictly convex hypersuface Σ. Recently, Z.Zhang and the author of this paper in [16] proved that there exist at least [n/2] + 1 geometrically distinct brake orbits on any central symmetric strictly convex hypersurface Σ, furthermore, there exist at least n geometrically distinct brake orbits on Σ if all brake orbits on Σ are non-degenerate.
In his pioneering work [24], P. Rabinowitz proposed a conjecture on whether a superquadratic Hamiltonian system possesses a periodic solution with a prescribed minimal period. This conjecture has been deeply studied by many mathematicians. For the strictly convex case, i.e., H ′′ (x) > 0, Ekeland and Hofer in [6] proved that Rabinowtz's conjecture is true. We refer to [3]- [6], [8], [10], [17]- [19], and reference therein for further survey of the study on this problem. For Rabinowitz' conjecture on the second order Hamiltonian systems, similar results under various convexity conditions have been proved (cf. [5] and reference therein). In [17] and [19], under precisely the conditions of Rabinowitz, Y. Long proved that for any τ > 0 the second order system x + V ′ (x) = 0 possesses a τ -periodic solution x whose minimal period is at least τ /(n + 1). Similar result for the first order system (1.1) is still unknown so far.
It is natural to ask the Rabinowitz's question for the brake orbit problem: for a superquadratic Hamiltonian function H satisfying condition (1.1), whether the problem (1.2) possesses a solution (τ, x) with prescribed minimal period τ for any τ > 0 (brake orbit minimal periodic problem in short).
In this paper we first consider the brake orbit minimal periodic problem for the nonlinear Hamiltonian systems. From Section 3, we have the following result.
In fact, in Section 3 a more general theorem is proved (see Theorem 3.1) where the superquadratic condition (H2) is relaxed to withH satisfying condition (H2), and the convexity condition (H5) is relaxed to H ′′ (x(t)) ≥ 0, ∀t ∈ R and τ /2 0 H ′′ (x(t)) dt > 0 for all brake orbit (τ, x). We also prove some results about the brake orbit minimal periodic problems for the second order Hamiltonian systems in Section 3.
2 §2 Iteration inequalities of the L 0 -index theory We observe that the problem (1.2) can be transformed to the following Lagrangian boundary value problem An index theory suitable for the study of problem (2.1) was established in [13] for any Lagrangian subspace L. As usual, we denote For a symplectic path γ ∈ P(2n), its Maslov-type index associated with a Lagrangian subspace L is assigned to a pair of integers (i L (γ), ν L (γ)) ∈ Z×{0, 1, · · · , n}. We call it the L-index of γ in short. In [23], the index µ j (γ), j = 1, 2 was defined for γ ∈ P(2n), the µ j -indices are essentially the special L-indices for L = L 0 and L = L 1 = R n × {0} ⊂ R 2n up to a constant n, respectively. In order to estimate the period of a brake orbit, we need to estimate the L 0 -index of the iteration path γ k associated to the iterated brake orbit x k . For reader's convenience, we recall the definition of the L 0 -index which was first established in [13]. Some properties for this index theory are listed in the appendix below. For L 0 = {0} ⊕ R n , we define the following two subspaces of Sp(2n) by Since the space Sp(2n) is path connected, and the n × n non-degenerated matrix space has two path connected components, one with det V > 0, and another with det V < 0, the space Sp(2n) * L 0 has two path connected components as well. We denote by We denote the corresponding symplectic path space by Definition 2.1. We define the L 0 -nullity of any symplectic path γ ∈ P(2n) by (1) with the n × n matrix function V (t) defined in (2.1).
We note that rank invertible. We define a complex matrix function by It is easy to see that the matrix Q(t) is a unitary matrix for any t ∈ [0, 1]. We denote by It is clear that M ± ∈ Sp(2n) ± L 0 . For a path γ ∈ P(2n) * L 0 , we first adjoin it with a simple symplectic path starting from J = −M + , i.e., we define a symplectic path bỹ then we choose a symplectic path β(t) in Sp(2n) * L 0 starting from γ(1) and ending at M + or M − according to γ(1) ∈ Sp(2n) + L 0 or γ(1) ∈ Sp(2n) − L 0 , respectively. We now define a joint path bȳ By the definition, we see that the symplectic pathγ starting from −M + and ending at either M + or M − . As above, we definē . We can choose a continuous function∆(t) in [0, 1] such that detQ(t) = e 2 √ −1∆(t) .
By the above arguments, we see that the number 1 π (∆(1) −∆(0)) ∈ Z and it does not depend on the choice of the function∆(t). We note that there is a positive continuous function ρ : [0, 1] → (0, +∞) such that Definition 2.2. For a symplectic path γ ∈ P(2n) * L 0 , we define the L 0 -index of γ by For a L 0 -degenerate symplectic path γ ∈ P(2n) 0 L 0 , its L 0 -index is defined by the infimum of the indices of the nearby nondegenerate symplectic paths.
For any two 2k i × 2k i matrices of square block form, We remind that the unit circle in the complex plane is defined by U = {z ∈ C : |z| = 1}, and the upper(lower) semi In [22], for any M ∈ Sp(2n), Long defined the homotopy set of M in Sp(2n) by The path connected component of Ω(M ) which contains M is denoted by Ω 0 (M ), and is called the homotopy component of M in Sp(2n).
In [15], the following result was proved (cf.
satisfying that all eigenvalues of K located within the arc between 1 and ω including satisfying the condition that all eigenvalues of K located within the closed arc between 1 and ω in for some non-negative integers s and t satisfying 0 ≤ s + t ≤ n − p − r, and some Combining Propositions 2.4 and 2.5, we have the following result.
Proof. By Proposition 2.4, summing the inequalities of (2.6) with ω = ω 2i k , 1 ≤ i < k/2, i ∈ N, we obtain the inequalities (2.7) for odd k and (2.8) for even k. We remind that here we have used the Bott-type formula The equality conditions follow from 2 o and 4 o of Proposition 2.5 together with Corollary 9.2.8 and List 12 in P198 of [21]. We note that from List 12 in P198 of [21], no eigenvalue on U + is Krein positive definite.
Since we should consider the Bott-type iteration formulas in Proposition 2.4 in odd and even cases, the inequalities in Theorem 2.6 is naturally considered in two cases correspondingly. We will see that the inequalities in Theorem 2.6 for even times iteration path are our main difficult to prove that the brake orbit found in Section 3 has minimal period, though we believe this kind brake orbit has minimal period, we can only prove that it has minimal period or it is 2-times iteration of a brake orbit with minimal period. §3 Applications to nonlinear Hamiltonian systems We now apply Theorem 2.6 to the brake orbit problem of autonomous Hamiltonian system where H(N x) = H(x) and B = B 1 0 0 B 2 is a 2n × 2n symmetric semi-positive definite matrix whose operator norm is denoted by B , B 1 and B 2 are n × n symmetric matrices. A solution (τ, x) of the problem (3.1) is a brake orbit of the Hamiltonian system, and τ is the brake period of x. To find a brake orbit of the Hamiltonian system in (3.1), it is sufficient to solve the following problem Any solution x of problem (3.2) can be extended to a brake orbit (τ, x) with the mirror symmetry of L 0 by x(τ /2 + t) = N x(τ /2 − t), t ∈ [0, τ /2] and x(τ + t) = x(t), t ∈ R.
(H2) there are constants µ > 2 and r 0 > 0 such that . Then for every 0 < τ < 2π B , the system (3.1) possesses a non-constant brake orbit (τ, x) satisfying Moreover, if x further satisfies the following condition: We remind that if B = 0, then 2π B = +∞. Proof. We divide the proof into two steps.
Step 1. Show that there exists a brake orbit (τ, x) satisfying (3.3) for 0 < τ < 2π B . Fix τ ∈ (0, 2π B ). Without loss generality, we suppose τ = 2, then τ < 2π B implies B < π. By conditions (H1)-(H4), we can find a non-constant τ -periodic solution x of (3.2) via the saddle point theorem such that (3.3) holds. For reader's convenience, we sketch the proof here and refer the reader to Theorem 3.5 of [15] for the case of periodic solution. We note that the main ideas here are the same as that in the periodic case. We refer the paper [11] for some details. In fact, following P. Rabinowitz' pioneering work [24], let K > 0 and χ ∈ C ∞ (R, R) such that χ(t) = 1 if t ≤ K, χ(t) = 0 if t ≥ K + 1, and χ ′ (t) < 0 if y ∈ (K, K + 1). The number K will be determined later. Set where the constant R K satisfies We set L 2 = L 2 ([0, 1], R 2n ) and define a Hilbert space E := W L 0 = W 1/2,2 L 0 ([0, 1], R 2n ) with L 0 boundary conditions by We denote its inner product by ·, · . By the well-known Sobolev embedding theorem, for any s ∈ [1, +∞), there is a constant C s > 0 such that Define a functional f K on E by Then for large r 1 > 0 and small ρ > 0, ∂Q m and B ρ (0) ∩ E + m form a topological (in fact homologically) link (cf. P84 of [2]). By the condition B < π, we obtain a constant β = β(K) > 0 such that ∀z ∈ ∂Q m . In fact, by (H3), for any ε > 0, there is a δ > 0 such that H K (z) ≤ ε|z| 2 if |z| ≤ δ. SinceĤ K (z)|z| −4 is uniformly bounded as |z| → +∞, there is an M 1 = M 1 (K) such thatĤ K (z) ≤ M 1 |z| 4 for |z| ≥ δ. Hencê So we have Since B < π, we can choose constants ρ = ρ(K) > 0 and β = β(K) > 0, which are sufficiently small and independent of m, such that for z ∈ ∂B ρ (0) ∩ E + m , If r = 0, from condition (H4), there holds If r = r 1 or z = r 1 , then from (H2), We have where b 1 > 0, b 2 are two constants independent of K and m. Then there holds where b 3 , b 4 are constants and b 5 > 0 independent of K and m. Thus there holds So we can choose large enough r 1 independent of K and m such that (see [11] for a proof). Thus by the saddle point theorem (cf. [25]), we see that c K,m ≥ β > 0 is a critical value of f K,m , we denote the corresponding critical point by x K,m . The Morse index of x K,m satisfies m − (x K,m ) ≤ dim Q m = mn + n + 1.
By taking m → +∞, we obtain a critical point is defined to the total number of the eigenvalues of f ′′ K belonging to (−∞, d] for d > 0 small enough, and M 1 is a constant independent of K. Moreover, by the Galerkin approximation method, Theorem 2.1 of [14], we have the d-Morse index satisfying (3.3). By extending the domain with mirror symmetry of L 0 , we obtain a 2-periodic brake orbit (2, x) of problem (3.1).
Remark 3.2. If B = 0, the results of Theorem 3.1 hold for every τ > 0. The following condition is more accessible than (HX) but it implies the condition (HX).
(H6) H ′′ (x) ≥ 0 for all x ∈ R 2n , the set D = {x ∈ R 2n |H ′ (x) = 0, 0 ∈ σ(H ′′ (x))} is hereditarily disconnected, i.e. every connected component of D contains only one point. Similarly, we consider the brake orbit minimal periodic problem for the following autonomous second order Hamiltonian system

(3.7)
A solution (τ, x) of (3.7) is a kind of brake orbit for the second order Hamiltonian system.
In this paper, we consider the following conditions on V : (V1) V ∈ C 2 (R n , R).
Then we take x 0 = λe for large λ such that the above inequalities holds. We define 1] ψ(h(s)).
By using the Mountain pass theorem (cf. Theorem 2.2 of [25]), from the conditions (V2)-(V4) it is well known that there exists a critical point x ∈ W of ψ with critical value c > 0 which is a Mountain pass point such that its Morse index satisfying m − (x, 1) ≤ 1. If we set y =ẋ and z = (x, y) ∈ R 2n , the problem (3.7) can be transformed into the following problem ż = −JH ′ (z), . We note that (V5) implies H(N z) = H(z), so (2, z) is a brake orbit with brake period 2. We remind that in this case the complex structure is −J, but it does not cause any difficult to apply the index theory. By Theorem 5.1 of [13], the Morse index m − (x, 1) of x is just the L 0 -index i L 0 (z, 1) of (1, z). i.e., there holds(see also Lemma 4.6 in the appendix below) We can suppose the minimal period of x is 2/k for k ∈ N. But i L 0 (z, 1/k) = m − (x, 1/k) ≥ 0, and from the convexity condition (V6), we have i 1 (z, 2/k) ≥ n. With the same arguments as in the proof of Theorem 3.1, we get k ∈ {1, 2}.
We note that the functional ψ is even, there may be infinite many solutions (τ, x) satisfying Theorem 3.3. We also note that Theorem 3.3 is not a special case of Theorem 3.1, since the Hamiltonian function H(x, y) = 1 2 |y| 2 + V (x) is quadratic in the variables y, in this case B = 0 0 0 I n with B = 1. Thus when applying Theorem 3.1 to this case, we can only get the result of Theorem 3.3 for 0 < τ < 2π. We now consider the following problem  We note that if we directly solve the problem (3.10) by the same way as in the proof of Theorem 3.3, the formation of the functional is still ψ as defined in (3.8), but the domain should be cos kπt · a k , a k ∈ R n }.
In this time, it is not able to apply the Mountain pass theorem to get a critical point directly due to the fact R n ⊂ W 1 , so the inequality (3.9) is not true. §4 Appendix. Some properties for the indeices 4.1. Some properties of Maslov-type index. For a symplectic path γ ∈ P(2n), its Maslov-type index is a pair of integers (i 1 (γ), ν 1 (γ)) ∈ Z × {0, 1, · · · , 2n} (cf. [20], [21]). If γ ∈ P(2n) is the fundamental solution of a linear Hamiltonian systeṁ with continuous symmetric matrix function B(t), its Maslov-type index usually denoted by (i 1 (B), ν 1 (B)). The following result was proved in [12].
with Lagrangian boundary condition by setting z(t) = (x(t), y(t)) T ∈ R 2n with y = P (t)x ′ (t) − Q(t)x(t): ż = JB(t)z z(0), z(1) ∈ L 0 , (4.5) where B = B(t) is defined by We take the space W = W 1,2 0 ([0, 1], R n ), the subspace of W 1,2 ([0, 1], R n ) with the elements x satisfying x(0) = x(1) = 0. Define the following functional on W The critical point of ϕ is a solution of the problem (4.4), and so we get a solution of the problem (4.5). Denote the Morse index of the functional ϕ at x = 0 by m L 0 (B), which is the total multiplicity of the negative eigenvalues of the Hessian of ϕ at x = 0, and the nullity by n L 0 (B). The following result was proved in [13].
Lemma 4.6. There holds Let E be a separable Hilbert space, and Q = A − B : E → E be a bounded salf-adjoint linear operators with B : E → E a compact self-adjoint operator. N = ker Q and dim N < +∞. Q| N ⊥ is invertible. P : E → N the orthogonal projection. Set d = 1 4 (Q| N ⊥ ) −1 −1 . Γ = {P k |k = 1, 2, · · · } be the Galerkin approximation sequence of A: (1) E k := P k E is finite dimensional for all k ∈ N, (2) P k → I strongly as k → +∞  where m * > 0 is a constant large enough such that the difference in (4.6) becomes a constant independent of m ≥ m * .