Uniqueness result for a weighted pendulum equation modeling domain walls in notched ferromagnetic nanowires

We prove an existence and uniqueness result for solutions $\varphi$ to a weighted pendulum equation in $\mathbb{R}$ where the weight is non-smooth and coercive. We also establish (in)stability results for $\varphi$ according to the monotonicity of the weight. These results are applied in a reduced model for thin ferromagnetic nanowires with notches to obtain existence, uniqueness and stability of domain walls connecting two opposite directions of the magnetization.


Introduction
We consider a weight a : R → R that is a bounded positive measurable function (not necessarily continuous) satisfying Motivated by a reduced model for notched ferromagnetic thin nanowires (see [2]) where a represents the area of transversal sections in the nanowire (the area function a may have jumps in that model), we associate to the magnetization m = (m 1 , m 2 , m 3 ) : R → S 2 the following energy functional We are interested in the analysis of domain walls that are transition layers connecting the opposite directions ±e 1 , where e 1 = (1, 0, 0). Up to a rotation and a translation (eventually yielding a translated weight), we fix the center of the domain wall at the origin by imposing m(0) = e 2 = (0, 1, 0). Our first theorem is the following uniqueness result for optimal domain walls: This minimizer has the form m = (sin ϕ, cos ϕ, 0) where ϕ : R → R is an increasing Lipschitz function with ϕ(0) = 0 and ϕ(±∞) = ± π 2 (2) and ϕ solves the weighted pendulum equation If in addition, a is even in R and non-decreasing in R + = (0, +∞), then ϕ is odd in R, (3) holds in the entire R and m is a stable critical point of F , i.e., for every v ∈ H 1 (R, R 3 ) with v · m = 0 in R, where λ(x) = a(x)|∂ x m| 2 + a(x)m 2 2 is the Lagrange multiplier for the constraint |m| = 1.
Theorem 1 is based on the following uniqueness result for solutions of the weighted pendulum equation (3).

Remark 3.
In Theorems 1 and 2, the even symmetry of the weight a is imposed to have the odd symmetry of the solution ϕ yielding the equation (3) to hold in the entire R. Moreover, the monotonicity of a is imposed to have the stability of the solutions ϕ and m.
Without the assumption that a is non-decreasing in R + , the solution ϕ in Theorem 2 can be unstable, i.e., Q(η) < 0 in some direction η ∈ H 1 (R) (yielding also the instability of the constraint minimizer m in Theorem 1). We give the following example for a non-increasing weight a in R + : Proposition 4. Let a : R → R be the even function given by a = 2 in (−1, 1) and a = 1 in R \ [−1, 1]. Then the solution ϕ in Theorem 2 is unstable, i.e., Q(η) < 0 in some direction η ∈ H 1 (R). Consequently, the constraint minimizer m in Theorem 1 is unstable, i.e., T (v) < 0 for some In the case of an even weight a that is C 1 smooth in R and non-decreasing in R + , existence and stability results for domain walls are proved by Carbou and Sanchez in [2]. They address the uniqueness of domain walls as an open question. Theorems 1 and 2 give positive results for the question of uniqueness. The proof is based on a variational method for non-smooth and nonmonotonous weight a (instead of the shooting method used in [2] where the regularity of a is essential). The difficulty here consists in the heterogeneity of the non-smooth weight a for which the equipartition of the two terms in the energy is in general lost for optimal domain walls (in contrast to the case of homogeneous weight yielding an autonomous ODE in (3)).

Claim 1.
We prove − π 2 ≤ ϕ ≤ π 2 in R. For that, assume by contradiction that there is a point in R where ϕ is larger than π 2 . By continuity of ϕ and (2), it means that there is a non-empty interval J = (x 0 , y 0 ) such that ϕ > π 2 in J and ϕ(x 0 ) = ϕ(y 0 ) = π 2 . If we cut-off at π 2 and set ϕ := ϕ in R \ J and ϕ = π 2 in J , then ϕ satisfies the constraints (2) and G(ϕ) > G( ϕ) (as the energy of ϕ in J close to x 0 and y 0 is positive while the energy of ϕ vanishes in those regions) which contradicts the minimality of ϕ. Thus, ϕ ≤ π 2 in R. A similar argument shows that ϕ ≥ − π 2 in R. Claim 2. We prove that 0 is the only vanishing point of ϕ in R.
We prove now the uniqueness result for the weighted pendulum equation: Step 1: We show that ∂ x ϕ is a non-negative bounded function in R. We prove it in R + (a similar argument yields also the conclusion in Then for every x > y > 0, we have for every large n (so that , ∂ x ϕ ∈ L ∞ (R + ) and non-negative.
Step 2: We show ∂ x ϕ > 0 a.e. in R. First we prove it in R + . Assume by contradiction that there exists a Lebesgue point (2). Then the unique continuation principle 1 for the A similar argument yields also the conclusion in R − which finishes Step 2.

Uniqueness of domain walls
We use Theorem 2 to prove the uniqueness of domain walls in Theorem 1. Step 2: If a is even in R and non-decreasing in R + = (0, +∞), then m is a stable critical point of F . By Theorem 2, we know that ϕ is odd in R, (3) holds in the entire R implying that (6) holds in the entire R (so m is a critical point of F in R) and Q(η) ≥ 0 for every η ∈ H 1 (R).
. As m(0) = e 2 , we have v 2 (0) = 0. The second variation T of F at m is given by where ( · , · ) is the duality (H −1 , H 1 ) in R and L 2 is given in (8). By Step 4 in the proof of Theorem 2, we know that ( Case 1: v 1 is Lipschitz with compact support in R and v 2 is Lipschitz with compact support in R \ {0}. In this case, the tangential constraint v 1 sin ϕ + v 2 cos ϕ = 0 yields a Lipschitz function η with compact support in R \ {0} (in particular, η ∈ H 1 (R)) such that η = v 1 cos ϕ = − v 2 sin ϕ in R. Then one checks

Case 2. The general case. We can approximate
As v n is not necessarily orthogonal to m in every point, we consider the projection v n = v n −(m ·v n )m that also converges to v in H 1 (R) (as m, ∂ x ϕ ∈ L ∞ (R)) and satisfies the tangential constraint v n · m = 0 in R. As m is Lipschitz, Case 1 applies to v n and T ( v n ) ≥ 0. By the continuity of T in H 1 (R) (as a and λ are bounded in R), we conclude that T (v ) ≥ 0.

Example of an unstable solution
We choose the even weight a = 2 in (−1, 1) and a = 1 in R \ [−1, 1] that clearly is non-increasing in R + . The aim is to prove that the solution ϕ in Theorem 2 (which is odd and satisfies (3) in the entire R) is unstable. An important feature for this weight is the non-existence of minimizers in (4) if the constraint ϕ(0) = 0 is dropped; this yields the non-existence of optimal domain walls connecting ±e 1 if the center of the domain wall is not fixed.

Proof of Proposition 4.
We divide the proof in several steps.

Some open questions
In Theorem 2, we proved existence and uniqueness of the minimizer ϕ in (4), in particular, under the constraint of a fixed center at the origin. A natural question is whether ϕ is a minimizer of G under the only two constraints ϕ(±∞) = ± π 2 . The answer is negative for some weights a as shown in Proposition 4 where ϕ is unstable and moreover, no minimizers of G exist under the constraints ϕ(±∞) = ± π 2 . However, the answer is positive for the homogeneous weight a where ϕ is the unique minimizer and also, the unique critical point (up to a translation of the center) of G under the constraints ϕ(±∞) = ± π 2 (see e.g. [3]). Open Question 1. Under which additional condition on the weight a satisfying (1), is it true that the solution ϕ in Theorem 2 is a minimizer of G under the constraints ϕ(±∞) = ± π 2 ? In that case, under which further conditions on a, ϕ is the unique minimizer (or more, the unique critical point) of G under the only two constraints ϕ(±∞) = ± π 2 ? This addresses in particular the question of existence of a minimizer ϕ of G under the two constraints ϕ(±∞) = ± π 2 . Such problem is solved in general by using the concentration-compactness lemma à la Lions. For the homogeneous weight a, we recall the following compactness result that handles the constraints ϕ(±∞) = ± π 2 , i.e., transitions between two different states: Lemma 6 (Doering-Ignat-Otto [4]). Let ϕ n ∈Ḣ 1 (R) be such that lim inf x +∞ ϕ n (x) > 0 and lim sup x −∞ ϕ n (x) < 0 for every n ∈ N. If lim sup n→∞ ∂ x ϕ n L 2 (R) < ∞, then for a subsequence, there exists a zero z n of ϕ n and a limit ϕ ∈Ḣ 1 (R) such that ϕ(0) = 0 and ϕ n (· + z n ) → ϕ locally uniformly in R and weakly inḢ 1 (R) and lim inf x +∞ ϕ(x) ≥ 0 and lim sup x −∞ ϕ(x) ≤ 0.

Open Question 3. If a satisfies
Proof. Note first that η is Lipschitz with compact support in R. Integrating by parts, we have: which is the desired identity.