Energy Scaling Law for a Singularly Perturbed Four-Gradient Problem in Helimagnetism

We study pattern formation in magnetic compounds near the helimagnetic/ferromagnetic transition point in case of Dirichlet boundary conditions on the spin field. The energy functional is a continuum approximation of a J1-J3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_1-J_3$$\end{document} model and was recently derived in Cicalese et al. (SIAM J Math Anal 51: 4848–4893, 2019). It contains two parameters, one measuring the incompatibility of the boundary conditions and the other measuring the cost of changes between different chiralities. We prove the scaling law of the minimal energy in terms of these two parameters. The constructions from the upper bound indicate that in some regimes branching-type patterns form close to the boundary of the sample.


Introduction
We study a continuum variational problem that arises from a statistical mechanics description of magnetic compounds and describes pattern formation in case of incompatible boundary conditions. We prove a scaling law for the minimal energy in terms of the problem parameters. Such scaling results have proven useful in a huge variety of singularly perturbed non-(quasi-)convex models for pattern forming systems Communicated by Paul Newton. where explicit minimizers cannot be easily determined analytically or numerically, see, e.g., (Kohn 2007) for some examples. Pattern formation is then often related to competing terms in the energy functional, favoring rather uniform or highly oscillatory structures, respectively. The proofs of the scaling laws often involve branching-type constructions where structures oscillate on refining scales near the boundary, among many others see for instance the scaling laws in , Conti (2000Conti ( , 2006, Otto (2009, 2012), Chan and Conti (2015), Knüpfer et al. (2013), Bella and Goldman (2015), Conti and Zwicknagl (2016), Conti et al. (2020), Rüland and Tribuzio (2022)) for martensitic microstructure, (Kohn andWirth 2014, 2016) for compliance minimization, (Choksi et al. 2004(Choksi et al. , 2008) for type-I-superconductors, (Ben Belgacem et al. 2002;Bella and Kohn 2014;Bourne et al. 2017;Conti et al. 2005) for compressed thin elastic films, (Conti and Ortiz 2016;Conti and Zwicknagl 2016) for dislocation patterns, and (Brancolini and Wirth 2017;Brancolini et al. 2018) for transport networks.
We point out that in particular a variety of magnetization patterns (including branching structures) have been successfully explained via scaling laws of continuum micromagnetic energies, see, e.g., , , Dabade et al. (2019), DeSimone et al. (2006a), DeSimone et al. (2006b), Knüpfer and Muratov (2011), Otto and Steiner (2010), Otto and Viehmann (2010), Venkatraman et al. (2020)). While these models typically contain local and non-local terms, we will focus on a purely local model that arises -at least heuristically -from a frustrated spin system, see Diep (2013), Diep (2015) for the general context and Cicalese and Solombrino (2015), Cicalese et al. (2019) for the specific setting considered here. Precisely, starting from a 2-dimensional square lattice εZ 2 with lattice width ε > 0, we consider spin fields v : εZ 2 → S 1 and a configurational energy of the form (also called J 1 − J 3 -model) with some positive parameter α > 0, where the summation is taken over indices i, j ∈ Z 2 ∩ 1 ε . While the first term favors nearest neighbors to have aligned spins, the second term favors next-to-nearest neighbors (horizontally and vertically) to have opposite spins. Note that the model considered here does not take into account diagonal interactions, for a recent analysis in that case see (Cicalese et al. 2021) and the references therein. Depending on the size of the parameter α, different minimizers arise. Precisely, the analysis in Cicalese and Solombrino (2015), Cicalese et al. (2019) shows that (at least locally) the energy is minimized by ferromagnetic configurations, i.e., constant spin fields, if α ≥ 4, while for small α < 4, the energy is minimized by helimagnetic configurations, i.e., spin fields in which spins rotate at a fixed angle φ = ± arccos(α/4) between horizontal and vertical nearest neighbors, respectively. Such helical structures have recently been observed experimentally, see, e.g., Schoenherr et al. 2018;Uchida et al. 2006. Of particular interest is the transition point α 4 where the ground state changes from a helimagnetic to a ferromagnetic structure. Mathematically, an asymptotic analysis in the sense of -convergence in this transition regime in the limit of vanishing lattice spacing has been performed in Cicalese et al. (2019). In the sequel, we briefly sketch the idea as outlined there, for details and references see Cicalese et al. (2019). Heuristically, it can be shown that the appropriately rescaled normalized energy E ε − min E ε can be rewritten in terms of an appropriately rescaled version u of the angular liftingũ, given by v = (cosũ, sinũ), as an energy of the form with τ = √ 2ε √ 4−α . Here, the preferred derivatives ∂ 1 u = ±1 and ∂ 2 u = ±1 are the order parameters describing the chiralities, i.e., they correspond to helical structures rotating clockwise (−1) and counterclockwise (+1) between horizontally and vertically adjacent spins, respectively. We note that this is a heuristic simplification where in particular discrete derivatives are approximated by continuous ones and we assume that there are no vortices in the spin field. However, the rigorous analysis of Cicalese et al. (2019), Cicalese and Solombrino (2015) supports such a perspective, at least in certain parameter regimes. Roughly speaking, by the classical Modica-Mortola result, one expects that the functional I τ converges in the sense of convergence as τ → 0 to a functional that is finite only on fields ∇u ∈ BV satisfying the differential inclusion ∇u ∈ {(±1, ±1)}, see Cicalese et al. (2019) for the rigorous derivation. A respective rigorous result in terms of -convergence in the regime τ → τ 0 ∈ (0, ∞), relating the discrete J 1 − J 3 -spin model with boundary conditions to a continuum functional of the form (1) holds also true (see Cicalese and Solombrino 2015 for a one-dimensional local result and Ginster et al. (in preparation) for the two-dimensional setting considered here). While the analysis in Cicalese et al. (2019) focuses on the local behavior, we study the system under Dirichlet boundary conditions on the spin field. More precisely, we start from the continuum model (1) on a square domain = (0, 1) 2 , and derive the scaling law of the minimal energy among configurations satisfying affine boundary conditions u(0, y) = (1−2θ)y at the left boundary. Here, the parameter θ ∈ (0, 1/2) is a compatibility parameter, where for θ = 0, the boundary condition is compatible with the helical structures (±1, 1), θ = 1/2 corresponds to a ferromagnetic configuration (at least in vertical direction), and θ ∈ (0, 1/2) indicates that the spin field on the boundary rotates in vertical direction with an angle that is smaller than the optimal angle φ = arccos(α/4). For the ease of notation, we present the proof of the scaling law for a slightly simplified functional, namely The four preferred gradients can be combined so that the corresponding function is zero outside the rotated square. Right: Rescaled versions of the rotated square can be used to cover (0, 1) 2 so that the resulting function u satisfies u = 0 on ∂(0, 1) 2 and ∇u ∈ {(±1, ±1)} a.e with affine boundary conditions on one boundary. Precisely, we show that there are two scaling regimes for the minimal energy, The second scaling is attained, for example, by the affine functions u(x, y) = (1 − 2θ)y ± x, while the first one, which is relevant for small σ , is attained by a branching-type construction. It turns out that in contrast to many other branching-type constructions, the length scale on which patterns form, depends only on the compatibility parameter θ but not on σ . Also, our upper bound construction does not show equi-partition of energy but indicates that the surface term plays a major role. We note that the second scaling implies that minimizers in this regime just fail to have gradients in BV, see Rüland et al. (2019). Let us briefly comment on the differences of J σ compared to I τ . First, the doublewell potential penalizing deviations from the preferred gradients (±1, ±1) is different. However, the main difference lies in the growth for large arguments which play no role in our estimates, and we can easily transfer our results to the original doublewell potential, see Sect. 4.2. Next, the higher-order term in I τ does not control the full Hessian but only the two diagonal components. This also does not influence the scaling properties, see Remark 3. Finally, we work in J σ with a BV-type regularization while I τ contains a quadratic regularization term. As is well known for related problems (see, e.g., Schreiber 1994; Zwicknagl 2014) this usually does not qualitatively change the scaling regimes of the minimal energy, see also Remark 2. The functional J σ is a four-gradient functional, and hence formally lies "in between" very well-studied problems, namely scalar models for martensitic microstructures (preferred gradients (1, ±1)), see, e.g., Kohn and Müller (1992), , and the Aviles-Giga functional (preferred gradients in S 1 ), see, e.g., Aviles et al. (1987). While for the two-gradient problem, the minimal energy scales as min{σ 2/3 θ 2/3 , θ 2 } (this follows by a change of variables 1 directly e.g. from Zwicknagl 2014), the minimal energy for the Aviles-Giga functional in our setting is 0. 2 Thus the scaling we prove here for the four-gradient setting indeed lies "in between." However, while any test function for the two-gradient problem yields a test function for J σ , this functional has much more flexibility which comes from the high compatibility of the four gradients, see Fig. 1. Roughly speaking, this allows to construct test functions with low energy by covering the domain with building blocks using only the preferred gradients in the spirit of a simplified convex integration for differential inclusions, see, e.g., Conti (2008), Müller and Šverák (1999), Pompe (2010), Rüland et al. (2018), Rüland et al. (2020), Rüland and Tribuzio (2022). This relation will be explored in Sect. 5. We remark that a similar functional with corresponding four preferred magnetizations, in which patterns form due to non-local terms, has been studied in Dabade et al. (2019), Venkatraman et al. (2020). The rest of the article is structured as follows: After briefly collecting the notation in Sect. 2, the main result will be proven in Sect. 3. We state the energy scaling law, discuss the regimes and prove the upper bound in Sect. 3.1 and the lower bound in Sect. 3.2. In Sect. 4, several generalizations are considered, including p-growth, different doublewell potentials, boundary conditions on the full boundary, and rectangles. Finally, in Sect. 5, consequences for solutions of the related differential inclusion as derived in Cicalese et al. (2019) are collected.

Notation and Preliminaries
We will write C or c for generic constants that may change from line to line but do not depend on the problem parameters. The notation c i with an index i indicates that these are fixed constants which do not change within a proof. We write log to denote the natural logarithm. For the ease of notation, we always identify vectors with their transposes. For a measurable set B ⊂ R n with n = 1, 2, we use the notation | B | or L n (B) to denote its n-dimensional Lebesgue measure. For σ > 0 and θ ∈ (0, 1/2], we set The expression | D 2 u | ( ) in the second term of the functional E σ,θ denotes the total variation of the vector measure D 2 u. Note that u ∈ A θ in particular implies that u ∈ W 1,1 ((0, 1) 2 ) and ∇u ∈ BV . Hence, u has a continuous representative on the closed square [0, 1] 2 , see, e.g., (Conti and Ortiz 2016, Lemma 9). We will always identify such functions with their continuous representatives. For a Borel set B ⊂ R 2 and u ∈ W 1,2 (B) with ∇u ∈ BV , we use the notation E σ,θ (u; B) for the energy on B, i.e., 3 Energy Scaling on the Square (0, 1) 2 Our main result is the following scaling law for the minimal energy.
Theorem 1 There exists a constant C T > 0 such that for all σ > 0 and all θ ∈ (0, 1 2 ], We will prove the upper bound in Proposition 2 and the lower bound in Proposition 3.

Remark 1
We note some properties of the scaling regimes in Theorem 1.

The Upper Bound
In this section, we provide test functions to prove the upper bound in Theorem 1. Before we start the proof, let us briefly explain the heuristics of the construction of the test function in the regime σ < θ 2 in which the affine function does not yield the optimal scaling. Instead, we provide a branching construction which (up to a small interpolation layer) only uses the four preferred gradients. In particular, in the yvariable the function is a saw-tooth function with slope ±1, where the volume fraction of slope +1 is 1−θ and of slope −1 is θ to match the slope 1−2θ = (1−θ)·1+θ ·(−1) on the boundary, see Fig. 2. If we assume that ∂ 1 u = ±1, we observe that from the boundary condition, we have | u(x, y) − (1 − 2θ)y |≤ x. On the other hand, assuming that ∂ 2 u(x, y) = ±1, one obtains that the number of jumps of the y-derivative on the slice {x} × (0, 1) is of order θ x , see Fig. 4. Following these estimates, we present a self-similar construction that refines in the k-th step from x ≈ θ k approximately θ −k+1 jumps of the y-derivative into approximately θ −k jumps at x ≈ θ k+1 . If θ = 1 m for some m ∈ N (i.e., θ −k ∈ N) this can be done in an exact manner, see Fig. 2, for other θ a modification is needed leading to slightly more complicated branching patterns, see Fig. 3. Moreover, we note that although other branching constructions are in principle possible, the construction presented below yields in every construction step an approximate balance between the occurring horizontal and vertical interfaces. The proof of the lower bound (see Sect. 3.2) indicates that this is essential for a function providing the optimal scaling.

Proof
Step 1: Preparation. We first note that the affine function In view of Remark 1(i), it hence suffices to consider the case σ < θ 2 and to construct a function u such that E σ,θ (u) ≤ C U σ |log σ | |log θ| + 1 with a constant C U chosen below. Let k ∈ N be such that θ k+1 ≤ σ < θ k . To simplify notation, set We note that we always have 1/δ ∈ N and δ = 1/m > θ/(θ + 1), and hence We point out that many of the expressions below simplify if θ −1 ∈ N since then δ = θ . For an illustration of the construction described in the next steps for θ = 1 3 see Fig. 2 (the case θ = 1/2 is sketched in Fig. 6). Finally, we fix some N ∈ N to be chosen later (see (14)).
Step 2: Construction of the building block. As in many branching constructions (see, e.g., , we first construct an auxiliary function that acts as a building block for the construction of u. For the ease of notation, we describe an admissible function via its gradient field. By the boundary condition u(0, x 2 ) = (1 − 2θ)x 2 , this uniquely determines the function u.
Precisely, we define V : (δθ, θ ] × R → R 2 as the function which is 1-periodic in y-direction and satisfies the following (see Fig. 3): Consequently, V is a gradient field on (δθ, θ ) × R, and additionally, V (x, y) ∈ K for almost all (x, y), and We will use in the next step that for the second component V (2) of V , we have Step 3: Branching construction.
where ν is the measuretheoretic normal to J V N , see also (8) and Fig. 3. Moreover, V N (x, y) ∈ K for almost all (x, y)∈ (δ N θ, 1), and Finally, we interpolate linearly in x to satisfy the boundary condition, and eventually define u N : Step 4: Estimate for the energy. By (11) and (10) In particular, we have Next, by (9), and since L 1 y ∈ (0, 1) : Now fix Since 0 < σ < θ 2 < 1 we have log σ θ log δ > 0 and consequently N ≥ 1. Combining (12) and (13) and using that | log δ |≥| log θ |, we obtain This concludes the proof of the upper bound with C U :=35.
Remark 2 If we replace the BV-type regularization σ | D 2 u | by the smoother one σ 2 (D 2 u) 2 dL 2 , we can use slight modifications of the above-constructed test functions to obtain the same upper bound on the energy scaling. Clearly, the function u = (1 − 2θ)y ± x produces again an energy of order θ 2 . Hence, it remains to consider the branching regime σ < θ 2 . Starting with the branching construction u N : (0, 1) 2 → R (N chosen as in (14)) which we can extend in y-direction so that ∇u N is 1-periodic, we set and smooth this function with a symmetric mollifier of support B σ (0). For the smoothed function a straightforward computation shows that one can estimate the term σ 2 (D 2 u) 2 dL 2 , up to a constant, by σ N . On the other hand, in addition to the region (0, 4σ ) × (0, 1) (outside of (0, 3σ ) × (0, 1) we have ∇ũ ∈ K) the gradient of the mollified function agrees with one of the preferred gradients except for a tube Step 2 for θ = 2/5 and m = 3. Note that the left region of size θ − δ is not needed in the construction but is rather added for the sake of an easier notation in the proof. On the other hand, due to Proposition 3 deleting this region from the construction cannot lead to an improved energy scaling. Right: The corresponding branching construction for N = 3 with width 2σ around the jump set of J ∇ũ . Thus, we can estimate the second term in the energy, up to a constant, by σ N + 4σ . Recalling the computation at the end of the proof of Proposition 2 leads to the claimed energy scaling.

The Lower Bound
In this section, we prove the ansatz-free lower bound in Theorem 1. Precisely, we show the following statement.
Remark 3 A careful inspection of the upcoming proof shows that the same lower bound holds true even if the term σ | D 2 u | ( ) is replaced by the term Proof The proof is split in several steps. By Lemma 4, for fixed θ 0 On the other hand, Corollary 7 shows that there exist θ 0 ∈ (0, 1/2], k 0 ∈ N with k 0 ≥ 2 and c B > 0 (depending only on k 0 ) such that for all σ > 0 and all θ ∈ (0, θ 0 ], we have Note that by Remark 1, this indeed implies the assertion: and consequently, by (15), (iii) If σ ∈ (θ k+1 , θ k ) for some 2 ≤ k 0 ≤ k then again by Remark 1(ii), we have and consequently, by (15), Hence, choosing c L = min{c A , c B k 0 +2 } concludes the proof.
As outlined above, we will prove the lower bound (15) separately for large and small θ , respectively. We build on some techniques that have been used for example in the derivation of scaling laws for martensitic microstructures, see, e.g., Conti (2006), Conti et al. (2017), Conti et al. (2020), Zwicknagl (2014).
Proof Let u be an admissible function such that (Otherwise we are done.) Note that there exist measurable functions ρ x , ρ y : Step 1: Comparison of u on vertical slices to the boundary data For almost every x ∈ (0, 1) and almost every y ∈ (0, 1) we have by the fundamental theorem of calculus that Consequently, it holds for almost every x ∈ (0, 1) that Journal of Nonlinear Science (2023) 33:20 Step 2: A lower bound on the energy on many vertical slices. We fix a constant Letx ∈ (0, 1) be such that We claim that this implies thatx ≤ c 1 θ 0 min {σ (| log σ | +1) , 1} orx ≥ 2c 1 θ 0 , so that in particular for almost allx ∈ ( c 1 θ 0 min {σ (| log σ | +1) , 1} , 2c 1 θ 0 ) at least one of the two properties fails. Note that by (17), this is an interval of length at least c 1 θ 0 that is completely contained in (0, 1). To see the claim, we proceed similarly to Conti et al. (2017) and subdivide the interval (0, 1) into the three subsets Since the three sets form a partition of the interval (0, 1), one of them has measure at least 1 3 . From property 2. it follows immediately that | M 3 |≤ 1 12 < 1 3 . We consider the remaining two cases separately.

Consequently, there exists a family of disjoint open intervals
In this case, we obtain that in each interval I k , and the rest follows as in case (a).
The proof of the lower bound in the case θ 1 is split into two lemmata which combined lead to the estimate in Corollary 7. We start with a general estimate which will be relevant for the lower bound only if σ ≥ θ 32 .
Step 3: A lower bound for the energy.
We consider the two cases separately.
Since u(x, y 2 ) − u(x, y 1 ) the claimed lower bound on E σ,θ (u) follows as in (31) (with an even better constant). This concludes the proof of the lemma.
Step 1: Choice of representative vertical slices and reduction to auxiliary statement. By Fubini's theorem and standard slicing arguments we can find points For an illustration, see Fig. 5. Note that by the assumption θ ≤ 1/2, the intervals ( 1 2 θ 2i , 3 2 θ 2i ) for 1 = 1, . . . , K are pairwise disjoint and contained in (0, 1). In the subsequent steps, we will prove the following result: There exists a constant c 3 > 0 (not depending on σ and θ ) such that for each 1 ≤ i ≤ K − 1 we have (recall that we chose c 2 above) Note that this indeed implies the assertion since (using the choice of x i ) Therefore, from now on, we fix some 1 ≤ i ≤ K − 1 and assume that (32) does not hold, i.e., we have In the rest of the proof, we will show that (33) holds for a constant c 3 > 0 chosen below.
Combining the estimates of Lemma 5 and 6, we obtain the following lower bound.

General p
We consider for 1 ≤ p < ∞ the energy E σ,θ : Journal of Nonlinear Science (2023) 33:20 For an upper bound one can use the constructions for p = 2 from Proposition 2. Clearly, the function u(x, y) = (1 − 2θ)y ± x produces an energy of order θ p . On the other hand, it can be seen from the proof of Proposition 2 that the function constructed via branching u N satisfies ∇u N ∈ K except for an interpolation region of size θ N on which it holds | ∇u N |≤ 5. Hence, one obtains again and | D 2 u N | ((0, 1) 2 ) ≤ C N . Setting N = |log σ | |log θ| yields as in the proof of Proposition 2, E p σ,θ (u N ) ≤ Cσ |log σ | |log θ| + 1 . The lower bounds can be shown analogously to the case p = 2.

Classical Double-Well Potential
We define the function W : The following corollary shows that the scaling law for min E σ,θ and min F σ,θ are the same.
Corollary 9 There exists a constant C > 0 such that for all σ > 0 and all θ ∈ (0, 1 2 ] it holds Proof First note that Consequently, inf E σ,θ ≤ inf F σ,θ and the lower bound follows from Theorem 1. Again, it can be easily checked that the competitors from the proof of Proposition 2 produce a corresponding upper bound for min F σ,θ .

Boundary Conditions on Whole Boundary
In this section we show that in the symmetric case θ = 1 2 , we can replace the boundary condition u(0, ·) = 0 by the more restrictive boundary condition u = 0 on ∂(0, 1) 2 . Fig. 7 Sketch of a competitor such that u = 0 on ∂(0, 1) 2 . The construction is obtained by subdividing the square (0, 1) 2 through diagonals into four triangles. On each of those triangles, a version of the construction sketched in Fig. 6 is used Corollary 10 Let σ > 0. Then there exists u ∈ W 1,∞ ((0, 1) 2 ) such that u = 0 on ∂(0, 1) 2 such that Proof Clearly u = 0 meets the more restrictive boundary conditions and satisfies E σ,1/2 (u) = 2.
For the other two regimes, we use the test function u N : (0, 1) 2 → R constructed in the proof of Proposition 2 and note that for some value α ∈ R. We now define the auxiliary function u : Note that this construction resembles the truncated branching construction used for martensitic microstructures (Conti and Zwicknagl 2016;Conti et al. 2020;Zwicknagl 2014). We claim that restrictions of the so-defined function u yield the respective energy scalings. We consider the cases from the definition of s(L, θ, σ ) separately.
Step 1: L ≥ 1. In this case, the proof follows from the lower bound for E σ,θ on (0, 1) 2 . We will from now on focus on the case L < 1.
This in particular concludes the proof of the lower bound in the case L < min{θ, σ }.
Step 3: L ∈ [θ, 1). In this case, the proof of the lower bound follows as for L = 1. More precisely, for large θ ≥ θ 0 both, θ and L are of order one, and we can proceed similarly to the proof of Lemma 4 to obtain a lower bound E σ,θ (u; L ) ≥ c min{σ (| log σ | +1), 1}. For σ ≥ θ k for some k ≥ 33, we can directly use Lemma 6 since this proof uses only the energy on a domain that is contained in (0, θ) × (0, 1) ⊆ L .
With this notation a lower bound can then be proven with minor modifications along the lines of part (B) in the proof of Proposition 3.
While our scaling shows that the resulting gradient is not in BV (cf. also Rüland et al. 2019), we can in the spirit of Rüland et al. (2018), Rüland et al. (2019), Rüland et al. (2020) exploit regularity properties of it.