PERIODIC SOLUTIONS OF INHOMOGENEOUS SCHR¨ODINGER FLOWS INTO 2-SPHERE

. In this paper,we consider the so called generalized inhomogeneous Schr¨odinger ﬂows from a closed Riemann surface M into the standard 2-sphere S 2 associated with the energy functional given by

In local charts, we have ). The tension field of u can be written as where ∆ g M is the Laplace-Beltrami operator on (M, g M ), Γ α βγ are the Christoffel symbols of the Riemannian connection on (N, h). The following flow is called the Schrödinger flow on maps from M into N (see [5]). In some cases, the flow is equivalent to a nonlinear Schrödinger equation(see [3]). One has gotten some results on the existence of solutions, precisely we refer to [2,5,6,16,17,24,25].
It is easy to see that the above flow preserves that following energy where f is a smooth real function on M and P is a real function on N . Let S 2 ⊂ R 3 be the unit 2-sphere. Since S 2 is a Kähler manifold and is the standard complex structure on S 2 , where × denotes the cross product in R 3 , then, the generalized Schrödinger flow from M into S 2 can be written as the following equation u t = u × τ f,P (u).
These flows are also closely related to mechanics and physics ( [9,13,19]). For f ≡ 1 and P ≡ 0, the Schrödinger flow into S 2 is just the well-known Heisenberg spin chain system (also called ferromagnetic spin chain system or Landau-Lifshitz equations [19]) which arises in the description of the magnetization of a ferromagnet. In this case, the tension field can be expressed by τ (u) = ∆ g M u + |∇u| 2 u, hence, the Landau-Lifshitz system can be written simply as In the case the coupling function f ≡ 1 and P ≡ 0, the flow (2) is just so-called inhomogeneous Heisenberg spin chain system, which also attracted one's attention for long time. We refer to [17], [23] and references therein.
In the case f ≡ 1 and P (u) = 1 2 (1 − u 2 3 ), the flow (2) still stems from physics and is called Landau-Lifshitz system with an external magnetic field( [19]). Gustafson and Shatah discussed the existence and stability problem for the special periodic solutions to the Landau-Lifshitz system from R 2 into S 2 in [9].
Recently, in [7,24] Ding and Yin studied the existence of some special periodic solutions to the flow on maps from an oriented closed Riemann surface M , which has an isometric convolution T : M → M (i.e. T is an isometry and T 2 = I) such that the fixed point set of T consists of a finite number of embedded closed geodesics, into S 2 . When M = S 2 , they employed the Sacks-Uhlenbeck's perturbation method, Palais's symmetric criticality principle and so called odd extension to approach this problem and showed the following theorem.
Theorem A ( [Ding-Yin]) The Schödinger flow on maps from S 2 into S 2 admits for any λ > 0 an infinite number of inequivalent periodic solutions with period 2π λ . Inspired by [7] we will consider the existence of the special periodic solutions to the generalized Schrödinger flows from an oriented closed Riemann surface, which has a certain symmetry of above isometric convolution, into S 2 . We also need to assume that the P (y) is a function of the third variable y 3 of y, i.e., P (y) ≡ P (y 3 ). Then, in this case the energy functional is of the form In this paper, we restrict us to the case the domain manifold M is a an oriented closed Riemann surface. Our task is to show the existence of some special periodic solutions to the generalized Schrödinger flow from M with convolution symmetry (or M = S 2 ) into S 2 when f and P satisfy some reasonable conditions respectively. In the present situation, we will encounter some new difficulties which come from the coupling function f and the polynomial P . Especially, f makes it harder than [7] to construct the supersolution and the subsolution of the related elliptic problem from S 2 into S 2 . Fortunately, we can find considerably general sufficient conditions on the positive smooth function f such that we may distinguish the different solutions corresponding different topological degree (see the subsection 3.2).
In order to state our main results, we choose the spherical polar coordinates (r, θ) centered at the north pole with metric g = dr 2 + sin 2 rdθ 2 on M = S 2 ⊂ R 3 . Then f can be expressed as f (r, θ). We say that f is radial symmetric if f = f (r). We always assume that P (u 3 ) = Σ p i=0 a i u i 3 is a polynomial with respect to u 3 . Theorem 1.1. Assume f is a positive radial symmetric function on S 2 and f (r) = f (π − r). For any constant λ > 0 and any positive integer d, suppsoe that the coefficient a i s of the polynomial P (u 3 ) satisfy: 1]; and that the coupling function f satisfies: For the well-known inhomogeneous Heisenberg spin chain system, i.e., f ≡ 1 and P (u 3 ) ≡ 0, as a corollary we obtain the following which generalized Theorem A due to Ding and Yin ([7]): Theorem 1.2. Suppose f is a positive radial symmetric function which satisfies f (r) = f (π − r) and P (u 3 ) ≡ 0. For any constant λ > 0 and any positive integer d, if then the inhomogeneous Schrödinger flow (2) from S 2 into S 2 admits d inequivalent periodic solutions with period 2π λ .
We also have the following: The rest of the paper is organized as follows. In the section 2 we discuss the variational reductions of the generalized Schrödinger flows. We prove our main theorem and corollaries in the section 3. We will show that when M has a convolution symmetry which splits M into two parts M + and M − , the reduced variational problem can be further reduced to one on M + or M − with Dirichlet condictions. Employing the Sacks-Uhlenbeck's perturbation method [22] we can show the direct minimization may lead to a nontrivial solution provided the infimum of the functional I satisfies an inequality (Lemma 3.1). By what we call the odd extension the original variational problem also be solved (The approach is similar with [7]). Then, in Theorem 3.1 we prove the key inequality holds true under some assumptions on f , the parameter λ and the polynomial P , hence the solution existence follows. In the last section we remark something on the stationary solution related to the flows.

2.
The reduced variational problems for N = S 2 . In this section, following the reduction method in [7] we will reduce equation (2) to an elliptic one by making use of the Killing vector field of the target manifold S 2 . The solutions of the elliptic equation correspond to global solutions of (2). Usually we consider S 2 as a unit sphere in R 3 centered at the origin. Then the rotation (anti-clockwise) around x 3axis produces a family of holomorphic isometries S t , which is periodic with period 2π, i.e. S t ≡ S t+2π . Precisely, we have Lemma 2.1. Let S t be the one-parameter group of rotations (holomorphic isometries) around x 3 -axis with S 0 = I (the identity map) and the corresponding holomorphic Killing vector field on S 2 denoted by V . Then u(t) = S t •g with g : M → S 2 is a solution to (2) if and only if g is a solution to the equation Proof. We know the Euler-Lagrange operator of E f,P can be written by , here e 3 = (0, 0, 1) is the north pole of S 2 . Since S t is an one-parameter subgroup of the rotation group SO(2) on S 2 which leaves the direction e 3 fixed, then, for any t ∈ R we can see easily that there holds On the other hand, Furthermore, we have The last equality is implied by the group property S s • S t = S s+t . In fact, differentiating the identity with respect to s at s = 0 leads to Hence, we obtain the following J(u)u t = J(S t • g)dS t (V (g)) = dS t (J(g)V (g)) .
Here we used the fact that S t is holomorphic, hence J • dS t = dS t • J. Combining (6)- (7), we get Since dS t is an isomorphism on S 2 , then we must have we have proved the lemma.
Generally, it is difficult to solve the above equation (5) if it is not of a variational structure. In fact, in [1] Chen and Jost have ever considered the existence of the following where W is a smooth vector field defined on the target manifold N . They established some existence results on the Dirichlet problems of the equation in the case the domain manifold M is a compact manifold with boundary and N is a compact manifold with nonpositive curvature.

Remark 1.
It is well-known that S 2 is a Kähler-Einstein manifold with constant curvature 1 and that JV is the gradient of the first eigenfunction F (u) = u 3 (see [12]). We denote JV by ∇F with the form As S λt+2π = S λt , then S λt is of period 2π λ , so is the solutions of Schrödinger flow the Euler-Lagrange equation of the Energy Functional . By Reduction Lemma 2.1, the reduced equation can be given by which is the Euler-Lagrange equation of the energy functional We are interested in the nontrivial solutions. Notice that if P (v 3 ) = λv 3 , the solutions are just the inhomogeneous harmonic maps which correspond to stationary (t-independent) solutions of inhomogeneous Schrödinger flows. Even if P (v 3 ) = λv 3 , (11) may have trivial solutions which maps M into a single point so that both sides of (11) vanish. There are at least two such points on S 2 , the north pole N and the south pole S (v 3 ≡ ±1).
In the following we always assume that M is an oriented closed Riemann surface which has an isometric convolution T , i.e. T 2 = I such that the fixed point set of T consists of an embedded closed geodesic. We say the Riemann surfaces of such property is of convolution symmetry. We also always suppose that f satisfies f • T = f . We will denote the fixed closed geodesic of T by C.
In the case M is an unit sphere in R 3 , i.e. M = S 2 . Obviously, S 2 is of an isometric convolution T : 1 + x 2 2 = 1} fixed and exchanges the upper hemisphere S 2 + and the lower hemisphere S 2 − . Then, under the spherical polar coordinates the condition f • T = f is just the assumption f (r) = f (π − r) in Theorem 1.1.
We will employ the so called odd extension method in [7] to approach our problem (11). Our main idea is to minimize the functional I among all mappings g : M + → S 2 which satisfy the boundary value condition We only need to consider the case λ > 0 since it is the same in the case λ < 0 except for the south pole in (13) is replaced by the north pole N .
And then if we have a smooth solution g + : M + → S 2 of (11) which satisfy the boundary value condition (13), we can define the odd extension as Lemma 2.
2. Assume f • T = f . Then, g is a smooth solution of (11) on M .
Proof. Since T : M → M and S π : S 2 → S 2 are isometries, it is easy to see that ) is a solution to (11) on S 2 − satisfying the same boundary condition (13). On the boundary C, condition (13) implies that tangent maps for both g + and g − vanish along the tangent directions of C, so g is C 1 along tangent directions. Then we need to check on the normal directions. Let n be the normal unit vector at a point y ∈ C. We have Since T is an isometric convolution and y is a fixed point of T , we have dT (y)n = −n.
On the other hand, since g + (y) = S ∈ S 2 is fixed under the rotation S π and dS π = −I on T S S 2 , we get that dg − (y)n = dg + (y)n i.e. g is C 1 smooth along the normal directions too. On the other hand, f = f • T implies that df (y)n = 0. Hence, it is easy to see that g is a smooth solution to (11) on M + ∪ M − . Indeed, to verify that g is a weak solution to (11) in the sense of distribution it suffices to verify g is a distributional solution on any neighbourhood Ω ⊂ M which satisfies that Ω ∩ C = ∅. Since we have known that g is a smooth solution on M + and M − respectively, for any φ ∈ C ∞ 0 (Ω, g −1 T S 2 )), we have Therefore, in view of the fact on C by integrating by parts we infer from the above integral equalities So, g is a weak solutions on M . As g is C 1 smooth on M , by the standard elliptic theory we know that g is a smooth solution of (11) on M .
3. The existence of special periodic solutions into S 2 . 3.
1. An existence result. For any positive integer d, we define the function space The degree of v is considered like this: by the boundary condition (13), we treat M 2 + as a closed manifoldM 2 + , and v can be viewed as a continuous map fromM 2 + into S 2 . Then the degree is the standard degree for mappings between closed manifolds. It is known that (cf. [20]) Then we have If then equation (11) has a nontrivial solution on M .
Proof. Using the odd extension, we need only consider the case on M + with the boundary condition as (13). Following the perturbation method of Sacks-Uhlenbeck [22]. The perturbation functional is chosen as For α > 1, we define : v(C) = S, degree(v) = d}. As in [22], I α has a smooth minimizer g α in X α d . The problem is to study what may happen when α goes to 1. Since 2α > 2, the Sobolev embedding theory ensures that the integral will not make any trouble to the convergence. So, the blow-up analysis of Sacks and Uhlenbeck (see [11,22]) can be used here because f is smooth and positive. A similar argument as in [4,11] says that, as α → 1, there may exist a finite number of points (p 1 , · · · , p k ) in M + or on the boundary C satisfying g α → g 1 in C ∞ (M + \{p 1 , · · · , p k }).
We have Since g α is weakly convergent to g 1 in H 1,2 , we have And since the degree of g α is d, we get On the other hand, using the fact that ∪ α>1 X α d is dense in X d , so It contradicts the assumption (16). So (16) gives us a nontrivial solution.
By the above discussion we know that f 1 is a smooth solution of (11) with finite many isolated singularities. If p i is an interior singularity of g 1 , the following Lemma 3.2 claims that p i is removable as f is smooth and positive. If p i is on the boundary of M + , due to the special boundary condition, we can employ the odd extension method to extend the domain to a punctured disk around p i and use Lemma 3.2 to remove the singularity. Hence g 1 is a smooth and the odd extension of g 1 is a smooth solution on entire M .
where β ∈ C 0 (D) and g ∈ L p (D, T N ) for some p > 2, then u can be extended to the disk D as a W 2,p map. Moreover, if β and g is smooth, u can be extended smoothly.
The proof of the lemma was given in [11]. In the present case, we may apply it since f is smooth and g is smooth and bounded.
Next, we need to prove the following Proof. Lemma 3.1 tells us that it is sufficient to verify (16). Following [7] we will construct a test function in X d to verify (16). Pick a point p away from the boundary C, and choose an isothermal coordinate system (x, y) centered at p on a neighborhood D away from C. For simplicity we assume D = {(x, y) : |x 2 + y 2 | < 1}. The metric g has the following form in D We will define the test map u : M + → S 2 which maps M + \D into S and satisfies i)u(∂D) = S; ii)u(D) covers S 2 d times (d ≥ 1); We choose the map u : D → S 2 so that it is rotationally symmetric and has the form u(r, θ) = (sin h(r) cos dθ, sin h(r) sin dθ, cos h(r)).

PING-LIANG HUANG AND YOUDE WANG
We have Next we choose a family of test function h c (r) with c > 0 as h c (r) = 2 arctan r d c + 2 arctan(cr 2d ).
It is easy to see that h c (1) = π, so the corresponding u c satisfies i) and ii).
We claim that if f satisfies such that I(h c ) < 4dπ min f + (P (−1) + λ)V ol(D). In order to prove the above assertion we need to estimate the following integrals with respect to small parameter c. By a direct computation we have For the first term of the above equality, we have and it is easy to see that As for C 1 , because Hence, we have To estimate other integral terms in E(h c ), we need to make the following calculations.
We need only to calculate By substitute r d = ct, we get Then we have Combining ( One can check that  1]. The mean value theorem says that where the last inequality follows from | 2c 2 c 2 + r 2d | ≤ 2. By the assumption a i satisfies the condition (1) in Theorem 1.1, i.e., b 1 + Σ p i=2 2 i−1 |b i | < 0. Combining the last inequality with the estimates on the integral terms of E f,P (h c ) we obtain the following From the above inequality we can easily see that, to make the following inequality hold true it is sufficient that ω(f ) and c satisfy simultaneously and From the last equality we obtain i.e.
From the above inequality (21), it follows Substituting (22) and (23) into the last inequality, we derive then condition (3) is satisfied and we have finished the proof.

Remark 2.
From the proof of Theorem 3.1, it is easy to see that δ depends only on min f , the metrics on D, d, P (·) and λ. For the case M = S 2 , the parameter c depends only on min f , d, a i and λ.

3.2.
Existence of several special periodic solutions on S 2 . Although we have had an existence result, we did not know whether there are blow-ups or not, the weak limit may be in a different homotopy class. In this subsection, we use the rotational symmetry of the domain manifold S 2 to ensure that we obtain different solutions for different d's.
In the spherical polar coordinates centered at the north pole, the S 1 -action as a rotation around x 3 -axis is represented as S t (r, θ) = (r, θ + t).
A map u : S 2 + → S 2 is called S 1 -invariant iff it satisfies for some d = 0 Since S is fixed under the S 1 -action, it must be mapped to either S or N of the target S 2 . we still use the Banach manifold It is not hard to see that (25) implies that u is uniquely determined by real functions h(r) and l(r) as u(r, θ) = (h(r), dθ + l(r)).
(26) We can assume h(π/2) = π so that the boundary condition u(C) = {S} is met. We will choose the connected component in Z α d with h(0) = 0, denote it by Σ α d , i.e. Σ α d = {u ∈ X α d : u(r, θ) = (h(r), dθ + l(r)), h(0) = 0, h(π/2) = π}. Then for x ∈ Σ α d , we have then we can express the integral terms in E α f by h and l. We still use the form When we minimize I α in Σ α d , we assume l(r) ≡ 0 since it will decrease E α f by (27). Moreover, we have Proof. Let u n be a minimizing sequence for I α in Σ α d with the form (26) where l n = 0. We show that we can modify h n into the form (28). In fact, for any u and corresponding h, we can definē Check that the correspondingū satisfies E α f (ū) = E α f (u) by (27). We also have P (ū 3 ) = P (u 3 ) and . similarly, changing h n intoh n will not change the value of I α . So we can assume h n ≤ π. Next we may denote h = h n and definē Since λ > 0, replacing h n byh n will decrease I α , so we can assume that h n satisfies (28).
It is standard that we may assume that u n converge weakly in W 1,2α and uniformly in C 0 to some u α . It follows that the corresponding h α satisfies 28. Moreover, since E α f (u) is weakly lower-semi-continuous and P (u 3 ) is weakly continuous, u α is a critical point of I α in Σ α d . Finally, we note that I α is invariant under the S 1 -action on X α d , and Σ α d is just the submanifold of fixed points of this action, by Palais' Principle of symmetric criticality [14], u α is also a critical point of I α on X α d .
The arguments in the proofs of Lemma 3.2 and Theorem 3.1 can be used to show that as α → 1, u α converges weakly in H 1 to some u 1 which is a critical point of I with the form (26) (l ≡ 0), except h 1 (0) = 0. If h 1 (0) = 0, we must have h 1 (0) = π. Lemma 3.1 ensures us that u 1 is nontrivial, so h 1 is not identically π. In this case, we will construct a "super solution" by u 1 , then we can get a critical point u of I with the form (26) that u ∈ Σ 1 d ⊂ X 1 d ≡ X d . Notice that if u is a critical point of I with the form (26), then the function h satisfies the following Euler-Lagrange equation Assume h 1 is a solution of (29) and satisfies 0 ≤ h 1 ≤ π, h 1 (0) = π, h 1 (π/2) = π, we will construct a weak super-solution [20] with h(0) = 0 as follow.
Definition 3.5. w(r) ∈ C 1 (0, π/2) is called a weak super-solution of equation (29), ) dr ≤ 0. We must have h 1 (r) > 0 for r ∈ [0, π/2], or otherwise h 1 (r 0 ) = 0 at some point r 0 ∈ (0, π/2). h 1 achieves its minimum at r 0 , so that (h 1 ) r (r 0 ) = 0, by the uniqueness of the solution of regular ODEs, h 1 ≡ 0, it is a contradiction. Now we can construct the weak super-solution. Set This function is strictly increasing on r ∈ [0, π/2] with φ c (0) = 0 and φ c (π/2) < π. And it satisfies In fact, the map corresponding φ c is just the S 1 -invariant harmonic map of degree d from S 2 to S 2 . We will use this later. By continuity of h 1 , there exist a constant δ 1 so that h 1 > δ 1 . For c small enough, h 1 > δ 1 > φ c . Since u 1 is nontrivial, h 1 is not constant and must be smaller than π at some point. Then for c large enough, there exists r ∈ (0, π/2) that h 1 (r) < φ c (r). So there must be a c 0 in between such that h 1 (s) = φ c0 (s) at some s ∈ (0, π/2). Let s be the first point of intersection between h 1 and φ c0 , thus we have Then we denote Notice that w is continuous at s and smooth elsewhere. If f satisfies on [0, π/2]\{s}. Next, for any φ ≥ 0 belonging to C ∞ 0 ([0, π/2]), integrating by part in [0, s), we have On (s, π/2], Since w is a weak super solution and v = 0 is a trivial sub-solution, and w > 0 = v on (0, π/2), using Perron's method directly ( [20,10]), one can show the existence of h which is a solution of (29) with h(0) = 0.
The remaining is that under the assumptions in Theorem 1.1 we need to verify whether f r satisfies the inequality appeared in the above discussion or not, and for different d whether the corresponding u d differs from each other or not.
Fortunately, by a simple computation we have By the assumption of Theorem 1.1, we have So, it ensures that the following inequality holds true Notice that S 2 + is isometric to (D 2 , ρ(|x|)ds 2 ), where ds 2 denotes the Euclidean metric of R 2 and ρ(|x|) = 4 (1 + |x| 2 ) 2 . Thus, in Theorem 3.1 we have V ol(D) = 2π and a = 1, and then δ can be written explicitly by ∀λ > 0, a i satisfy conditions (1) and (2), and f satisfies ω(f ) < δ and f r ≤ g(r) on [0, s], we find a solution on S 2 + of (29), by using the Reduction Lemma and the method of odd extension we get the special periodic solution from S 2 to S 2 . Moreover, notice that is decreasing with respect to d, and + 29 min f 4π + 29 is also decreasing with respect to the mapping degree d, thus we know that δ is decreasing with respect to d. It is obvious that g(r) = K sin r d is decreasing with respect to d too. Therefore, for each positive integerd < d, we can find a solution ud. As the solutions are of S 1 -invariant form (26), different d corresponds to different u. Thus we have found d different periodic solutions and finish the proof of Theorem 1.1.

Remark 3.
It is easy to see that for any fixed λ there exist many polynomials P satisfy the conditions (1) and (2) in Theorem 1.1. It is clear that if (a 1 − λ) is small enough then the conditions is met. For example, for P (x) = 1 + a 1 x + x 2 + x 3 , to make (1) and (2) hold true, we only need a 1 < λ − 9.
Remark 4. In Theorem 1.1, while P satisfies some suitable conditions, the restriction on f is quite loose. It is easy to check that if a 1 → −∞ or λ → ∞, we have δ → ∞ and g(r)| r =0,π → ∞. So, for sufficiently large −a 1 or sufficiently large λ, we only need f to satisfy f r | r=0 = f r | r=π = 0 besides smoothness, positivity and symmetry. For radial function f , it is obvious that its gradient vanishes at the poles.
Next, we give the proofs of Theorem 1.2, Theorem 1.3 and Corollary 1: The Proof of Theorem 1.2. Since P ≡ 0, we have a i = 0. Then b 1 = −λ and b i = 0 with i ≥ 2. So, from the proof of Theorem 1.1 we can see easily 16πλ + 29d min f (4π + 29)d , and g(r) = K sin r d = λ sin r d .
Thus we complete the proof of Theorem 1.2.
The Proof of Theorem 1.3. For the case P (u 3 ) = 1 2 (1−u 2 3 ), it is easy to verify λ > 1 implies that (1) and (2) hold true. Since f ≡ 1, according to Theorem 1.1 it follows that (2) admits solutions from S 2 into S 2 and for different d's we obtain different solutions. Then we get an infinite number of inequivalent periodic solutions, and finish the proof.
The Proof of Corollary 1. As f ≡ 1 and P satisfies the conditions (1) and (2), for every integer d > 0 Theorem 1.1 says (2) admits a corresponding S 1 -invariant periodic solution u d and u d is different from each other. Thus we get an infinite number of different solutions, and finish the proof. 4. Some remarks on stationary solutions. We know that the periodic solutions to the Schrödinger flows contain a class of special solutions, i.e. the stationary solutions (harmonic maps). Obviously, the stationary solutions of the flow (2) is just a harmonic map u from M into S 2 with a prescribed potential P (u), which satisfies the following τ (u) − ∇P (u) = 0. Especially, for the two dimensional Landau-Lifshitz system in [19] the above elliptic system can be written by ∆u + |∇u| 2 u + u 3 (e 3 − u 3 u) = 0, since the energy functional for this case is of the form Let's recall that in [18] Peng and Wang have discussed the existence for such harmonic maps from S 2 into S 2 with a symmetric potential. Concretely, they obtained that if the potential P : S 2 → R is symmetric with respect to a plane containing the origin point of R 3 and the restriction of P on the fixed point set with respect to the symmetry (i.e. the intersecting circle of S 2 and the plane) is constant, then the existence problem of the harmonic maps of degree 1 with the symmetric potential P is solvable. It follows that there exist solution maps from S 2 into S 2 of degree 1 to (32) as P (u) = 1 2 M (1 − u 2 3 ) is symmetric with respect to the plane (x 1 , x 2 , 0)