ON THE FORMATION OF MICROSTRUCTURE FOR SINGULARLY PERTURBED PROBLEMS WITH 2 , 3 OR 4 PREFERRED GRADIENTS

. In this manuscript, singularly perturbed energies with 2, 3 or 4 preferred gradients subject to incompatible Dirichlet boundary conditions are studied. This extends results on models for martensitic microstructures in shape-memory alloys ( N = 2), a continuum approximation for the J 1 − J 3 -model for discrete spin systems ( N = 4), and models for crystalline surfaces with N different facets (general N ). On a unit square, scaling laws are proven with respect to two parameters, one measuring the transition cost between different preferred gradients, the other measuring the incompatibility of the set of preferred gradients and the boundary conditions. By a change of coordinates, the latter can also be understood as as an incompatibility of a variable domain with a fixed set of preferred gradients. Moreover, it is shown how simple building blocks and covering arguments lead to upper bounds on the energy and solutions to the differential inclusion problem on general Lipschitz-domains.

Proving scaling laws for the minimal energy has been proven useful to explain the formation of patterns in a variety of problems in which the energy is non-quasiconvex and identifying the minimizers by analytical or numerical methods is not possible, c.f. [34].Often the formation of patterns is related to the competition of a part of the energy that favors rather uniform structures and a non-quasiconvex part of the energy that favors oscillations on fine scales.Constructions of good competitors for the upper bound often involve branching construction that refine in a self-similar manner.A non-exhaustive list of references where this technique has been successfully applied includes [2,6,7,10,16,17,20,25,32,37,44] for martensitic microstructure, [36,38] for compliance minimization, [12,13,26,27,28,33,40,41,46] for micromagnetism, [11,14,24] for type-Isuperconductors, [3,4,5,19] for compressed thin elastic films, [23,25] for dislocation patterns and [30,31] for helimagnets.
Here, the term of order 1 corresponds to very uniform configurations such as u = 0, whereas the bound of order σ 2/3 can be shown by a self-similarly refining branching construction.
Again, the term of order 1 is associated to rather uniform structures such as u = 0, whereas the bound σ(| log σ| + 1) corresponds again to a self-similarly refining branching construction.However, the scales of the construction are very different from the one for N = 2.In particular, in contrast to N = 2, for N = 4 it is possible to construct branching configurations such that (up to an interpolation region of order σ) ∇u ∈ K π/4,N .
For general N the energy functional (1.1) is very closely related to the (small slope approximation of the) free energy of a faceted crystalline surface in R 3 with N different facets parameterized by z = u(x, y) where W has N distinct minima in S 1 , see, for example, [48] and the references therein.Typically, replacing the term σ|D 2 u|((0, 1) 2 ) by σ 2 ´(0,1) 2 |D 2 u| 2 dxdy does no qualitatively change corresponding scaling law results, see [49] and [45].Hence, up to replacing the Laplacian by a full Hessian it is to be expected that the scaling law results presented in this paper are also valid for F σ above (c.f. also the discussion in [30]).A study of the dynamics of F σ , coarsening rates and simulations can, for example, be found in [47,48].
For general γ ∈ (−π, π], the energy (1.1) can by a simple change of variables alternatively be written as ˆQγ 0 −γ dist(∇u, K γ0,N ) 2 + σ|D 2 u|(Q γ ), where Q γ0−γ evolves from (0, 1) 2 by a counterclockwise rotation with angle γ 0 − γ.Hence, our studies can be understood as a generalization of the scaling laws (1.2) and (1.3) (cf.[35,49] and [30]) either to different domains or to a different set of preferred gradients.Our main result is the following scaling law min where the range of γ is by symmetry considerations restricted to the interval around 0 such that The first term | sin(γ)| 2 corresponds for all N to a uniform structure, i.e., u(x, y) = x.The second term for N = 2 still corresponds to a rather uniform structure, where the function u behaves as u(x, y) = x close to x = 0 and has one transition to u(x, y) = − cos(γ)x − sin(γ)y shortly after, see Figure 1.For the last term, branching structures play a role.The corresponding upper bounds can be proven using variants of the self-similar constructions for γ = π/2 (N = 2) and γ = π/4 (N = 4), for a sketch see Figure 3 for N = 2 and Figure 6, 8 and 7 for N = 3, 4. Similarly to before, there are significant differences between the constructions for N = 2 and N = 3, 4, mainly stemming from the fact that for N > 2 it is possible to construct self-similarly refining competitors such that (up to a small interpolation region) ∇u ∈ K γ,N .For N = 2, this is not possible as ∇u ∈ K γ,N implies that u is constant in direction e iγ+iπ/2 .This is also reflected in the different scaling laws for N = 2 and N = 3, 4. It is to be expected that a similar behavior a for N = 3, 4 is also true for N > 4. It would then be interesting to understand the scaling behavior with respect to the number of preferred gradients N .
Additionally, for N = 3, 4 we discuss how upper bounds can be constructed using simple building blocks and covering arguments.More precisely, the structures of K γ,N allow us to construct functions u : T → R (where T is a (rotated) square if N = 4 or a particular triangle for N = 3) such that u = 0 on ∂T , ∇u ∈ K γ,N a.e. and |D 2 u|(T ) ≤ C (see Figure 9), c.f. also [8,9,18,22,39,42,43] and the references therein for constructions in the significantly more complicated vectorial setting.Then simple covering arguments allow us to construct upper bounds on more general domains and give rise to solutions of the differential inclusion u = 0 on ∂Ω and ∇u ∈ K γ,N .Similarly to [43] (c.f. also [30]), by interpolation inequalities regularity of these solutions can be established in certain fractional Sobolev spaces.
In the following, we will fix notation and state the precise model under consideration.In Section 3 we will present our main results and discuss the organization of the proofs.

Notation and setting of the problem
We will write C or c for generic constants that may change from line to line but do not depend on the problem parameters.We write log to denote the natural logarithm.We write For the ease of notation, we always identify vectors with their transposes.Moreover, we will identify C with R 2 and denote by e 1 and e 2 the two canonical basis vectors for R 2 .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.In addition, for B ⊆ R 2 we write conv(B) ⊆ R 2 for its convex hull and int(B) for its interior.For γ ∈ [−π, π] and N ∈ {2, 3, 4} we set The set of admissible functions is defined as The expression |D 2 u|(Ω) in the second term of the functional E σ,γ,N denotes the total variation of the vector measure D 2 u.By symmetry considerations it will be enough to consider the angles γ Note that u ∈ A 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.[23,Lemma 9].We will always identify such functions with their continuous representatives.For an open set B ⊆ R 2 and u ∈ W 1,2 (B) with ∇u ∈ BV (B), we use the notation E σ,γ,N (u; B) for the energy on B, i.e., In addition for x ∈ (0, 1), I ⊆ (0, 1) and u ∈ A Note that since ∇u ∈ BV ((0, 1) 2 ) this formula makes sense for almost every x ∈ (0, 1) in the sense of slicing of BV -functions, see [1].Similarly, we write for y ∈ (0, 1) and u ∈ A Eventually, we define for B ⊆ (0, 1) 2 open with Lipschitz boundary the set A 0 (B) := u ∈ W 1,2 (B) : ∇u ∈ BV (B), u = 0 on ∂B .

Main result
Our main result is the following scaling law for the minimal energy.
Theorem 3.1.There exists constants C, c > 0 such that for all σ > 0 and γ ∈ Γ N it holds: (1 ( The proof of the theorem is split into different sections.Bounds that are valid for all N ∈ {2, 3, 4} are collected in Secion 4. Specific upper and lower bounds for N = 2 are proven in Section 5.The lower bound for N = 3, 4 can be found in Proposition 6.3 in Section 6.The corresponding upper bound is shown in Proposition 6.4 in Section 6.
Below, we identify the different regimes that appear in the bounds in Theorem 3.1.

Remark 1.
(1) We note that it holds for γ ∈ Γ 2 Hence, we find that (2) For N = 3, 4 and γ ∈ Γ N it holds since the mapping t → t | log(t)| is increasing for t ∈ (0, 1).Hence, it follows and consequently, In particular, we have Additionally, we also consider for N = 3, 4 the minimization problem subject to u = 0 on ∂(0, 1) 2 .In this situation the scaling laws for N = 3 or N = 4 differ due to different incompatibilities of K γ,N with respect to the full boundary of (0, 1) 2 .Theorem 3.2.There exists constants C, c > 0 such that it holds for all γ ∈ Γ 4 Moreover, it holds for all γ ∈ Γ 3 Upper bounds can be shown by means of optimal coverings with suitable building blocks.The same arguments lead to upper bounds and solutions to the differential inclusion problem on general Lipschitz domains.(2) There exists u ∈ W 1,∞ 0 (Ω) such that ∇u ∈ BV loc (Ω; K γ,N ) and ∇u ∈ W s,q for all 0 < s < 1, q ∈ (0, ∞) satisfying 1 q > s.
The proofs are discussed in Section 7.
On the other hand, where we used the definition of Γ N for the second inequality.Consequently, we find In this section we prove the scaling law for N = 2 claimed in Theorem 3.1.The additional upper in the regime σ ≤ | sin(γ)| 3 is proven using a variant of the celebrated Kohn-Müller branching construction developed in [35], c.f. also [21,49].The lower bound uses a variant of the argument from [17], c.f. also [21,49].
We start with proving the lower bound for all relatively large (in absolute value) angles γ ∈ Γ N .Here, the term | log | sin(γ)|| in the denominator of the lower bound will not play a role as it is uniformly bounded.Proof.Let u ∈ A. We fix α 0 = min{sin(γ 0 ), cos(π/4), 1/16} and c 1 = 16 min{σ(| log σ| + 1), 1}.Otherwise there is nothing to show.Let 0 < s ≤ c 1 .Then find an interval I ⊆ (0, 1) of length s c1 such that Then one of the following three conditions has to hold on I: Let us first assume that α0 8 |I| 2 ≤ ∥u(s, •)∥ L 1 (I) .We estimate By the definition of c 1 it follows s ≤ α0 16 |I|.Consequently, we obtain .
Then one of the following holds in (y, y + t) (recall that the minimal distance between two points in Suppose first that (1) holds.Then and thus the claim follows in this case.
For the rest of the proof we will now assume that (2) holds.First, recall that by the definition of Γ N we have Hence, it holds one of the following: Let us first assume that (a) holds.We estimate For the third estimate we used that by the choice of x i and γ 0 it holds E σ,γ,N (u; Next, observe that it holds by ( 2) and Poincaré's inequality for Consequently, since ∥u( 2) and ( 6.3) that which yields the claim if ( 2) and (a) hold.Hence, from now on we will assume that (b) holds i.e., ∥u(x i , s) − u(x i , s + t/2)∥ L 1 (y,y+t/2) ≥ 1 8 |ξ 2 |t 2 .
Step 3: An estimate for horizontal difference quotients.First, observe that it holds for all s ∈ (y, y + t/2) where we used that | sin(γ) For the fourth inequality we used that we have kσ Next, find for s ∈ (y, y + t) a value s ∈ A i such that |s − s| ≤ 1 80 t.Then we obtain where we used similarly to above that it holds E σ,γ,N (u; In particular, we obtain for almost all s ∈ (y, y + t/2) that On the other hand, it holds by (b) and (6.4) that We define Combining (6.5) and (6.6) yields Hence, for a subset (y, y + t/2) of size at least t 64 it holds It follows for a subset of (y, y + t) whose measure is at least t 64 that Step 4: Lower bound for E σ,γ,N (u; (x i+1 , x i ) × (0, 1).Let us now fix such an s ∈ (y, y + t) for which (6.7) holds.Moreover, let us assume that |∂ 1 ∇u(•, s)|(x i+1 , x i ) ≤ 1/10.By (2) it follows that |∇u(x, s) − ξ| ≤ 6 dist(∇u(x, s), K γ,N ) for a.e.x ∈ (x i+1 , x i ).Recalling that Consequently, we obtain Next, assume that σ ≤ | sin(γ)| 2 and let γ 0 > 0 and K ≥ 2 be as in Proposition 6.2.Then by Proposition 6.1 there exists c B > 0 such that for all γ ∈ Γ N with |γ| ≥ γ 0 , For γ ∈ Γ N with |γ| ≤ γ 0 and σ ≤ | sin(γ)| K it holds by Proposition 6.2 Eventually, we notice that for

This concludes by
Hence, it remains to consider the case | sin(γ)| 2 ≥ σ.Additionally, we will only treat the case γ ≥ 0. As for N = 2, we define the set Again, we will first use a self-similar branching construction to define a function on S γ .However, in this setting the constructed competitor will have gradients in K π/2,N in a large part of the domain (c.f. the branching constructions in [30]).Then -up to a rotation -we will use this function on the set {(x, y) ∈ (0, 1) 2 : (x, y − 1) • e iγ ≤ 0} and glue it to a function with constant gradient e iγ on the rest of (0, 1) 2 .

K
, where ν is the measure-theoretic normal to J V (4)
Step 2a : Building block for S γ .Let m = ⌈| sin(γ)| −1 ⌉ and define the set Then there exists a function V (3) : B γ → R 2 such that (see Figure 8 (middle) and T Step 3: Branching construction on (0, 1) 2 for N = 3, 4. We define u : (0, 1) 2 → R as (c.f. also Figure 4) In this section, we discuss how upper bounds for the energy E σ,γ,N (•, Ω) for N = 3, 4 and a bounded Lipschitz-domain Ω ⊆ R 2 can be shown using a good covering of Ω with specific building blocks that allow for simple functions which only use the preferred gradients and satisfy zero boundary conditions.Iterating this construction leads to solutions of the differential inclusion ∇u ∈ K γ,N and u = 0 on ∂Ω whose regularity can be controlled through interpolation.We start by constructing the building blocks.Lemma 7.0.1.
(1) Let Q γ = e iγ (−1/2, 1/2) 2 be the unit cube centered at 0 and with two sides parallel to e iγ .Then there exists a function (2) Let T Proof.For a visualization of the constructions for γ = 0, see Figure 9.
where u (i) : T where u : Q γ → R is the function from lemma 7.0.1 (1).Then u
Proof.Let us assume that γ > 0. For a visualization of the construction, see Figure 11.For ℓ ∈ N, let . Then define the families This shows all the needed properties of F 1 .To construct F K , we notice that (0, 1) 2 \ Q∈F1 Q can be written as the disjoint union k∈N T k where T k is a rotated (by multiples of π 2 ), dilated and translated version of the triangle T = conv 0 0 , 0 1 , sin(γ) cos(γ) sin(γ) such that k∈N H 1 (∂T k ) ≤ C, see Figure 11.In order to construct the families F K for K ≥ 2 inductively, it is then enough to show that there exists a disjoint family F of dilated and translated versions of Q γ such that T \ a+λQγ ∈F a + λQ γ can be written as the disjoint union of translated and dilated versions of T such that (see Figure 11) λ ≤ 1 and L 2 (T \ a+λQγ ∈F a + λQ γ ) ≤ sin(γ)L 2 (T ).
Then one can inductively define families FK for T such that (cf.

Figure 2 .
Figure2.Left: Sketch of the building block for the branching construction on S γ for N = 2. Right: Sketch of the gradient of the function u ∂ used to moderate between the sawtooth functions (in vertical direction) achieved by the branching construction and the boundary condition u = 0 on the line conv{(− cos(γ), sin(γ)), (0, 0)}.This construction is used if 0 ≤ γ ≤ π/4 is relatively small.The red and green regions indicate that the y-derivative is ±1, respectively.

Figure 3 .
Figure 3. Sketch of the Kohn-Müller type branching constructions.Top: Sketch of the regions appearing in the definition of u K in (1) (brown), (2) (blue), (3) (beige), (4) (pink), and (5) (orange) on the domain S γ .In the regions (1) and (3) the function is constant in x-direction.Middle: Sketch for small γ.The interpolation between the boundary values and the branching construction uses an extension to region (4) which is constant in x and the function u ∂ as a building block in region(5).The regions with ∂ 2 u K = 1 is colored in red, the region with ∂ 2 u K = −1 is colored in green.Bottom: Sketch for large γ.In region (4) a linear interpolation in x to u K = 0 is used.Then u K is the extended by 0 into region(5).The regions with ∂ 2 u K = 1 is colored in red, the region with ∂ 2 u K = −1 is colored in green.

Figure 4 .
Figure 4. Sketch of the construction of the test function u in Step 1c.The purple area extends T γ and is a rotated version of S γ .In this region, we use a rotated version of the branching constructions from Step 1a and 1b in the proof of the upper bound of Theorem 3.1 for N = 2, Step 1 in the proof of Proposition 6.4 (N = 4) and Step 2 in the proof of Proposition 6.4 (N = 3).

Figure 5 .
Figure 5. Sketch of the choices made in the proof of the lower bound in Theorem 3.1 for N = 2.

| 2 | 2 cosFigure 6 .Figure 7 .
Figure 6.Sketch for the construction of the building block for N = 4 and m = 8.The different appearing gradients are color-coded.Left: A version of the construction of the building block for a rectangle of height sin(γ).It is immediate that |D 2 V (3) | ≤ C(sin(γ)m + 1).Right: Construction of the building block for S γ , N = 4 and m = 8.The essential difference to the construction on the left is that after each branching of the construction an extra horizontal gap of length cos(γ)/m has to be bridged by horizontal interfaces creating an extra interface of length cos(γ)/m in horizontal direction and at most √ 2 sin(γ) in diagonal direction.Hence, the total surfaces created in the construction can be estimated up to a constant by | sin(γ)|m + 1.