The Tapering Length of Needles in Martensite/Martensite Macrotwins

We study needle formation at martensite/martensite macro interfaces in shape-memory alloys. We characterize the scaling of the energy in terms of the needle tapering length and the transformation strain, both in geometrically linear and in finite elasticity. We find that linearized elasticity is unable to predict the value of the tapering length, as the energy tends to zero with needle length tending to infinity. Finite elasticity shows that the optimal tapering length is inversely proportional to the order parameter, in agreement with previous numerical simulations. The upper bound in the scaling law is obtained by explicit constructions. The lower bound is obtained using rigidity arguments, and as an important intermediate step we show that the Friesecke–James–Müller geometric rigidity estimate holds with a uniform constant for uniformly Lipschitz domains.


Introduction
Shape-memory materials are special alloys that undergo a diffusionless solidsolid phase transformation upon a change of temperature or stress. During nucleation, complex microstructures emerge, and these microscopic patterns seem to be closely linked to macroscopic properties of the materials [3,9,32]. The patterns are usually modeled in the framework of the phenomenological theory of martensite [6], based on finite or linearized elasticity. The linearized theory is widely used and has been proven to arise naturally as -limit of the nonlinear theory for small displacements [26]. While the linearized theory often provides a good approximation to the physical (nonlinear) setting, several mathematical results indicate that in various situations the linear theory can lead to qualitatively different predictions, see e.g. [3,4,10,16,20,24]. During nucleation of martensite in an austenitic matrix, various interfaces between austenite and martensite or between different martensitic variants are formed. We  35 shape-memory alloy, from Boullay, Schryvers and Kohn [12,Fig. 5]. Right: Experimental (optical microscopy with polarized light) image of needles in the Cu 14 Al 3.9 Ni shape-memory alloy, courtesy of Chu and James [18,19]. The two pictures show a similar phenomenon at very different length scales, of the order of a few nanometers on the left, and of a fraction of a millimeter on the right focus on interfaces between different regions of laminated martensites, so called martensite/martensite macrotwins. At such interfaces, one often observes needletype microstructures, i.e., laminates where the minority variant drops out at the interface (see Fig. 1 and [12]). We point out that in many experiments, the needles show a more complex topological structure, and in particular branch close to the interface. This leads to different mathematical challenges [13,35,36], but we will focus here on simple needle structures as sketched in Fig. 2. The mathematical treatment of such structures within the framework of the phenomenological theory started with the work in [11,12,34,47,48]. In [12], the authors used a static variational model based on linearized elasticity and were able to predict the bending angle at which the needle meets the interface, in terms of the measured tapering length of the needle. We point out that the authors here did not intend to make predictions on the tapering length from the theory. Related problems of optimal needle shapes near martensite/martensite or martensite/austenite interfaces have since then been studied extensively, both in the analytical and the numerical literature, see e.g. [11,15,21,22,29,[39][40][41]43,[46][47][48][49]. In many numerical simulations, it has been observed that numerical schemes with geometrically linearized elastic energies are unstable or do not reproduce the experimental results while geometrically nonlinear models appear more appropriate. We follow here the recent approach from the numerical study in [21]. We aim at a better understanding of the length scale of such needle structures, in particular the tapering length. To determine this length in terms of the material parameters, a shape optimization problem for the parametrized needle shape is considered.
Let us finally comment on some simplifications that have been proposed in the literature. The most popular simplification is the linearization of the elastic energy (see, e.g., [12]). In this note, we will underline the numerical findings from [21] that the linearized theory is not appropriate to determine the tapering length. More precisely, we show (see Theorem 3.1) that for any energy-minimizing sequence in the geometrically linearized setting, the tapering lengths tend to infinity. This is in Sketch of the geometry. The right picture is based on data from the numerical simulation in [21] Fig. 3. Sketch of the geometry around a martensite/twinned martensite interface. This geometry is often called a macrotwin accordance with several other numerical findings in which it was observed that the geometrically linear model does not yield the expected results (see [21,29,43]). In [51,52], a related problem of needle-type microstructures near austenite/martensite interfaces has been investigated. There, the situation is simplified by assuming constant gradient in the very thin needles. Although one does not expect a large contribution to the elastic energy from these small domains, our analysis here indicates that the optimal energy scaling is not preserved under this simplification. Indeed, a significant effect seems to come from rotations at rather large angles (comparable to the shear in the variants). Let us start with a brief qualitative explanation of the relevant effects. We work in two dimensions, and denote the eigenstrains of the two martensitic variants by We start from the large-scale picture, as summarized in Fig. 2 and 3. On the far right there is a laminate between A δ and B δ , with volume fraction θ ∈ (0, 1); its average deformation is F := θ A δ + (1 − θ)B δ . On the left of the macro interface we have only variant A δ , but with a different rotation Q * ∈ SO (2). Compatibility of the macro interface implies that Q * A δ − F is rank-one, and the orientation e θ,δ of the macro interface is characterized by the condition (Q * A δ − F)e θ,δ = 0. This fixes both Q * ∈ SO(2) and e θ,δ as functions of θ and δ, details are given in Lemma 2.1 below (see also Fig. 2). As Q * = Id, the two regions in the A δ variant are not rank-one compatible, rank(A δ − Q * A δ ) = 2.
In the geometry of Fig. 2, needles are structures by which the volume fraction of the A δ phase varies from 0 close to the macro interface to the asymptotic value θ at large x 1 . For this qualitative discussion we use non-orthogonal coordinates and assume that x 1 = 0 corresponds to the macro interface. The entire construction is affine-periodic in the direction of the macro interface, with a periodicity condition given by the macroscopic deformation on the left, u(x +e θ,δ ) = Q * A δ e θ,δ = Fe θ,δ . For definiteness, let us assume that the period is 1, and denote by > 0 the characteristic length scale in the x 1 direction. Given these boundary conditions, one then determines the optimal deformation by minimizing the total elastic energy jointly over the elastic deformation u and the shape of the interfaces, see Fig. 2 for the result in a specific situation. The precise functionals we minimize are introduced below, and given in (3.9) in the geometrically linear case, and in (4.12) for the geometrically nonlinear one. For simplicity, in this introduction we do not include a description of the detailed shape of the interfaces, but only a simplified characterization, based on volume fractions at fixed x 1 . For any x 1 , let a(x 1 ) be the volume fraction (averaged over one period) of the A δ phase at given x 1 ; correspondingly b(x 1 ) for the B δ phase. Obviously, a(x 1 )+b(x 1 ) = 1 for all x 1 , and by the boundary conditions a(0) = 0, whereas a(x 1 ) ∼ θ and b(x 1 ) ∼ (1 − θ) for x 1 . If no rotation is present, and the energy is exactly zero, we necessarily have that ∂ 2 u 1 = δ in phase A δ , and −δ in phase B δ . The vertical average of ∂ 2 u 1 over one period is then (a − b)δ, which matches the periodicity requirement only if a = θ and b = 1 − θ . We next include infinitesimal rotations in the picture, assuming that the rotation angle depends on x 1 but not on x 2 . In particular, if β(x 1 ) is the average lattice rotation angle at given x 1 , then one has ∂ 2 u 1 = δ + β in the A δ and ∂ 2 u 1 = −δ + β in the B δ phase, with ∂ 1 u 2 = −β everywhere. Equating the vertical average to the one required by the periodicity leads to This relation between tapering profile and rotation was first studied in [12], it permits in particular to express β in terms of a and b. Treating the individual layers as elastic plates, we see that the change in lattice rotation β (x 1 ) generates a bending energy depending on the curvature, and proportional to Inserting the previous expression for β leads to minimizing . Therefore the total energy is proportional to δ 2 θ 2 / , and the optimal value for is ∞ (see Theorem 3.1 below for a precise statement). The geometrically linear model is unable to predict a finite length scale.
In finite elasticity, one considers an energy density which behaves as min R∈SO(2) in the A δ phase, and similarly in the B δ phase (see below, and (4.12) in particular, for a precise definition) [5,8,44]. A new term enters the picture in the product R A δ . Indeed, in (and the same with B δ ) there are more nontrivial entries than in the previous description based on linear elasticity. The ∂ 2 u 1 term, similarly to (1.2) above, prescribes β(x 1 ) in terms of a(x 1 ) − b(x 1 ); the ∂ 1 u 1 and ∂ 1 u 2 terms are not important for the same reason as above. However, periodicity requires ∂ 2 u 2 to have average 1, The total energy density is then of order β 2 + β 2 δ 2 , and balancing terms one obtains that the characteristic length scale is ∼ 1/δ. We refer to Proposition 4.3 below for a precise construction. As in (1.2) we have β ∼ δθ, and therefore this results in a total energy scaling as θ 2 δ 3 , as specified in Theorem 4.2 below.
These results are made precise below. We first formulate a precise mathematical model for the needle geometry following [21,34,51], with boundary conditions that fix the long-range structure and the topology but leave the shape of the domain boundaries free. The key assumptions are affine-periodic boundary conditions along the macro interface, a boundary data corresponding to a laminate at large x 1 , and the fact that the phase boundaries are Lipschitz functions. We then provide, separately in the linear and in the nonlinear case, explicit upper-bound constructions which make the above arguments rigorous. We finally show that these constructions are, up to universal factors, optimal, by providing corresponding lower bounds on the energy. This involves minimizing not only over the elastic deformation, but also over the phase interfaces, and hence over the shape of the domains of the different phases. We present in Sect. 2 the model, in Sect. 3 the linear results, and in Sect. 4 the nonlinear results.
One important difficulty in proving the lower bound is that one has to deal with Sobolev functions on varying domains. Our argument in particular uses a trace theorem, a Poincaré inequality, and a geometric rigidity inequality with constants which are uniform for uniformly Lipschitz domains. Whereas the first two are already present in the literature, the geometric rigidity estimate has, to the best of our knowledge, up to now only been proven with domain-dependent constants, even if uniformity of the constant for certain classes of domains has been formulated in the literature without explicit proof (for example, [2, Prop. 1]), and used in the study of microstructures near austenite/martensite interfaces (see e.g. [51,52]). We provide in Sect. 5 a full proof of the uniform geometric rigidity inequality for a general class of domains, which we believe to be of independent interest. The key ingredient is a uniform weighted Poincaré inequality, which also permits to easily obtain as byproducts uniform trace and Poincaré estimates.

Kinematics and Reduction to a 2D Problem
We describe the crystallographic situation under consideration (see also [12]) building on the crystallographic theory of martensite (see [6]). We consider a macrotwin using two martensitic variants, given by wells SO(3)U 1 and SO(3)U 2 . We make the standard assumption that the transformation stretch matrices are positive definite with det U 1 = det U 2 and nontrivial in the sense that U 1 ∈ SO(3)U 2 . If laminates with gradients in the two wells are possible, then the wells are rank-one connected, i.e., there existR ∈ SO(3), and non-zero vectors a, n ∈ R 3 such that It is shown in [8] that under these assumptions, we may without loss of generality (up to a linear change of variables) restrict to matrices of the form and for some orthogonal unit vectors e, e ⊥ ∈ R 3 with δ > 0. Here, δ > 0 measures the shear, and we will focus on the case of small and moderate strains, |δ| 1, and in particular on the limit δ → 0. While we consider the three-dimensional setting, it turns out that in our analysis, we may restrict to a two-dimensional simplification since, as shown in [7], the optimal strains are plane strains, cf. Remark 2.2(iii). Specifically, we shall work in the plane spanned by e and e ⊥ , and denote by f ∈ R 3 a unit vector perpendicular to that plane. Following [6], the normals to the laminate and macrotwin interfaces can be determined from the crystallographic theory of martensite. For that, we collect the necessary linear algebra results in this section.

Nonlinear elasticity
In the setting of nonlinear elasticity, the direction of the macro interface is not orthogonal to the one of the laminate, as illustrated in Fig. 2. We denote the orientation of the macro interface (in the plane orthogonal to f ) by e δ,θ := 1 This expression arises as the unique (up to a sign) nontrivial solution of the rankone compatibility condition between SO(3)A δ and the weighted average θ A δ + (1 − θ)B δ . We summarize the relevant algebraic conditions in the next Lemma. All assertions can be checked by direct computation; see, for example, [5].
Lemma 2.1. Let (e, e ⊥ , f ) be an orthonormal basis of R 3 , δ ∈ R\{0}, θ ∈ (0, 1), and let A δ and B δ be the matrices given in (2.1). Then it holds that and In particular, The rotation R 0 is given by (2.8) (iv) All matrices act trivially on f , i.e., In Sect. 4 we shall consider a macrotwin with normal e ⊥ δ,θ , and twin plane normal e ⊥ deep in the laminate on the right hand side of the macrotwin, see Fig. 3.

Remark 2.2.
(i) It follows from (2.2) that e δ,θ → e ⊥ as δ → 0, but e δ,θ = e ⊥ for finite δ. (ii) If |δ| and θ are small, then the rotation Q * as given in (2.5) is a rotation by roughly 2δ(1 − θ), while R 0 is a rotation by roughly −2δθ. In particular, both rotations are of order δ. (iii) Motivated by assertion (iv) of Lemma 2.1, we make the following simplification: We consider only deformations u satisfying ∇u f = f , and, slightly abusing notation, we identify the matrices introduced above with their restrictions to the plane spanned by e and e ⊥ .

Linearization
We now turn to the geometrically linearized setting. Precisely, for the small strain case |δ| 1, we linearize around the identity and define the strain matrices (2.10) We denote by ξ sym := 1 2 (ξ + ξ T ) the symmetric part of a matrix, so that We note the geometrically linearized compatibility properties.

Lemma 2.3.
Suppose that θ ∈ (0, 1), and let A lin sym and B lin sym be given by (2.11). Then it holds that skw if and only if v is parallel to either e or e ⊥ .
Proof. The proof follows from a direct calculation.
In particular, in the geometrically linearized setting, both, the macrotwin and the laminates can have normals e or e ⊥ . The main difference to the geometrically nonlinear setting is that the compatibility plane between the two variants is aligned with the compatibility plane of the macrotwin.

Energy Scaling in the Geometrically Linearized Setting
Following [12], we first consider the geometrically linearized setting. With the strain matrices A lin sym and B lin sym as defined in (2.11), we are led to study the geometrically linearized shape optimization problem involving the displacement. We consider a periodic cell of a macrotwin using linearized kinematics (see, e.g., [3]) and briefly recall the setting. By Lemma 2.3(i), there are two possible directions for A lin sym /B lin sym laminates, given by the normals e and e ⊥ . Deep in the twinned region on the right-hand side of the macrotwin, we choose the twin planes to be parallel to e, and the macrotwin plane parallel to e ⊥ . Since we work in an orthogonal coordinate system, for the ease of notation, we set e 1 := e and e 2 := e ⊥ , and for x ∈ span{e, e ⊥ } = R 2 , we use the notation x 2 := x · e 2 = x · e ⊥ , and We describe the needle by the two confining curves f ± : [0, ∞) → R which we assume to be measurable and satisfy see Fig. 4. Note that we do not impose any regularity assumptions on f ± but to get closer to the experimental results, we could also impose a length or Lipschitz condition without changing the results, see Remark 3.2(i).
We assume that the displacement v ∈ W 1,2 loc (R 2 ; R 2 ) obeys the periodicity condition and we consider the set We further assume that there is > 0 such that for x 1 the deformation coincides with a simple laminate, in the sense that there is η ∈ R such that These conditions characterize the class of admissible configurations As the experimental results indicate that is large (see Fig. 1), we restrict ourselves to the case 1. By periodicity, for the computation of the energy it suffices to integrate over one period, and therefore to consider (alternatively, one could take for x 1 > 0 only the k = 0 contribution in (3.3) and (3.4), by periodicity the two choices are equivalent). The elastic shape optimization problem is then to minimize lin . Note that we minimize with respect to both the configuration given by f ± and the displacement v. Here we denote the symmetric part of the gradient by e(v) := 1 2 (∇v + ∇ T v), and C represents the elastic modulus which satisfies the standard boundedness and coercivity properties, i.e. C(ξ − ξ T ) = 0 for all ξ and there exists α > 0 such that The energy functional (3.9) does not contain any interfacial energy term penalizing the lengths of the interfaces parametrized by f ± . Typically such terms are necessary to identify the appropriate length scale on which the twin structures form. In our setting of a periodic cell, however, it is of higher order as long as the curves are sufficiently regular. We find the following scaling law for the minimal energy.
Remark 3.2. (i) We derive the scaling law for the minimal energy without regularity assumptions on f ± but the upper bound uses only a Lipschitz profile with Lipschitz constant bounded by one. (ii) If on the right boundary we impose boundary conditions only on f ± (see (3.1)) but not on the deformation (see (3.6)) then there is no minimizing configuration and the infimum of the energy is zero. The reason for that is that in the linearized setting we can have a strain-free A lin sym /B lin sym interface along the macrotwin plane {x 1 = 0} (a fact that will also be used in the proof of the upper bound below). Precisely, consider for n ∈ N the configuration and extended to R 2 via (3.2). Then f ± n satisfy (3.1) and (3.5) with η = 0, and v satisfies the periodicity condition (3.2) by construction.
We note that the sequence of profiles for these competitors does not have uniformly bounded Lipschitz constants. If one assumes a bound on the Lipschitz constants L of f ± , then a respective construction has energy ∼ θ 2 /L. Hence, for large , the construction of the proof of Theorem 3.1 has lower energy. (iii) The scaling law in Theorem 3.1 implies that the tapering length of needles is not determined by linearized elasticity. Precisely, setting = ∞, the infimum of the energy equals zero, which implies that the optimal tapering length of the needle in this linearized setting is infinite, contradicting the experimental findings. (iv) If one interprets the linearized energy as an approximation to the nonlinear energy (cf. (2.10)), the resulting energy scaling is δ 2 θ 2 / .

Proof. Upper bound.
To prove the upper bound, i.e., the second inequality in the assertion, we use a special case of the construction from [12, Figure 4], which makes precise the sketch discussed in the introduction (see (1.2)-(1.4)). We set Then f ± satisfy (3.1) and (3.5). By periodicity, it suffices to describe the associated and We then extend the displacement constantly in As (3.6). For the gradients, we have, inside the needle, i.e., for 0 x 1 and 0 and thus, (3.14) Outside the needle for 0 x 1 and θ x 1 / x 2 1, we have and thus, , v is continuous in R 2 and is admissible. By (3.14) and (3.15), we find that there is a constant (not depending on or θ ) such that by (3.10) This concludes the proof of the upper bound. For later reference we remark that lin be an arbitrary admissible configuration. If the elastic energy on the left hand side of the interface is large, i.e., then the assertion follows. Hence, from now on, we assume that Thus, by Korn's inequality, there exists a constant c K > 0 and an infinitesimal rotation W := w(e 2 ⊗ e 1 − e 1 ⊗ e 2 ) ∈ R 2×2 skew with some w ∈ R such that and hence, in particular, By Fubini's theorem and Hölder's inequality, there exists By (3.2), this implies that We finally note that using Poincaré's inequality and (3.17) With (3.18) we can eliminate w and obtain We now consider the slice at x 1 = . By (3.6) and the condition θ 1/2, there exists an interval (t, t + 1/4) ⊂ (0, 1) (depending on η) of length 1/4 such that ∂ 2 v 1 ( , ·) = (B lin ) 12 = −1 and therefore v 1 ( , is close in L 1 to a different affine function than v 1 ( , x 2 ). We thus estimate the energy from below with this difference using Hölder's and triangle inequality: Ifc > 0 is chosen small enough such thatc 1/2 1 128c and hence, This concludes the proof of the lower bound.
The fact that the minimal energy tends to zero as → ∞ indicates that we cannot expect existence of minimizers for the problem on the infinite domain. We show that this is indeed the case, at least if we prescribe that the phase boundaries are uniformly Lipschitz.
and there exists no minimizer.
be a minimizing sequence. Then by the Lipschitz condition and (3.1), we have a uniform bound sup n f ± n C 0,1 ([0, ]) < ∞. By Arzelà-Ascoli, there exists a subsequence (not relabeled, the same subsequence for f + and L. From boundedness of the energy, we get that sup n e(v n ) L 2 ((−∞, )×(0,1)) < ∞. Since the periodicity condition (3.2) fixes that the average of (∇v n ) 12 is −1 + 2θ , this implies a bound on the full gradients, sup n ∇v n L 2 ((a, )×(0,1)) < ∞ for all a < 0. By adding a constant, we can assume without restriction that all v n have mean zero over (0, 1) 2 , and thus by Poincaré's inequality, we obtain a subsequence that converges weakly in W 1,2 loc (R 2 ; R 2 ) to an admissible function v ∈ W 1,2 loc (R 2 ; R 2 ). The boundary condition and the periodicity condition immediately pass to the limit. It remains to estimate the energy of the limit. Let ε > 0. Then, by uniform convergence, there exists N ∈ N such that for all n N , we have graph and analogously in ω Taking ε → 0, the assertion follows. (ii) By the upper bound of Theorem 3.1, we obtain for → ∞ a sequence lin , and by the lower bound of Theorem 3.1,

Energy Scaling in the Geometrically Nonlinear Setting
It appears that in the geometrically nonlinear setting, the qualitative behavior of the minimal energy is rather different. On a technical level, the main difference seems to be that the macrotwin habit plane e δ,θ is not parallel to a plane of compatibility of the two wells e ⊥ . Recall that this property was in particular used to extend the test function in the upper bound of Theorem 3.1 with vanishing energy to the left-hand side of the interface.
We first introduce the setting, using the notation from Lemma 2.1. As in Sect. 3, we assume without loss of generality e := e 1 and e ⊥ = e 2 . We recall the definitions e δ,θ := 1 We shall impose the following periodicity condition on admissible deformations: To parametrize the needle shapes, let f ± : [0, ∞) → R be measurable and such that (Later on, we will assume that they are L-Lipschitz.) The periodicity condition (4.2) suggests that we use the non-orthogonal coordinates introduced by the macrotwin, see Fig. 2. We define the linear map T δ,θ : We also define d, g : (4.5) which is equivalent to Note that d(e 1 ) = 0 and d(e δ,θ ) = 1, so that we can use d(x) ∈ Z to label the cell of periodicity that contains x. In turn, g(x) denotes the coordinate along the needle, which corresponds to x 1 in the geometrically linear setting. We set =T δ,θ y ∈ R 2 : y 1 0 or y 1 > 0 and (4.8) By periodicity, for the computation of the energy it suffices to integrate over one period, and therefore to consider the sets The class of admissible configurations is given by Note that it depends implicitly on δ and θ via (4.2). For L > 0 we further set The resulting variational problem then is to minimize over this set the functional Here, W : R 3×3 → [0, ∞) is a typical nonlinear elastic energy density satisfying with some constant c W > 0.
There is c f > 0 such that, if δ = 0, the same holds if one imposes that, for Proof. The upper bound follows from Proposition 4.3, the lower bound from Proposition 4.9.

Proof. For
To make an ansatz for the deformation on the right-hand side of the interface, we consider a rotation and a shift as independent parameters. Let The reason for this choice will become clear in (4.23) below. With these quantities, we define the deformation u : where We note that this yields a continuous function in ω * To obtain an admissible configuration, the functions R, f + and w have to satisfy the following properties: first, f + should be 1-Lipschitz with 0}. This requires that 20) so that the periodicity condition is equivalent to From now on, we restrict to {g(x) > 0}. Before we give the explicit constructions, we provide an estimate for the energy within this ansatz. We observe that ∇d = 1 e δ,θ ·e 2 e 2 , ∇g = −(e δ,θ · e 2 ) −2 e ⊥ δ,θ , and −e 1 ⊗ e ⊥ δ,θ + e δ,θ ⊗ e 2 = (e 2 · e δ,θ ) Id . The definition of φ (see (4.16)) was chosen so that φ = (e 2 · e δ,θ )Re 1 = (e 2 · e δ,θ )R A δ e 1 , which -using (4.22) -implies φ (g(x)) ⊗ ∇g + R(g(x))A δ e δ,θ ⊗ ∇d = 1 e 2 · e δ,θ R(g(x))A δ (−e 1 ⊗ e ⊥ δ,θ + e δ,θ ⊗ e 2 ) =R(g(x))A δ .
(4.23) Therefore, and similarly, Hence, the elastic energy of such an admissible test function is estimated by Finally, we specify how to choose R, w and f + . We consider the periodicity condition (4.21) and divide it into two equations, testing with e 1 and e 2 . First, we set w · e 1 = 0, (4.25) and take the scalar product of (4.21) with e 1 . Using that (A δ − B δ )e δ,θ = 2δ √ 1+(δθ) 2 e 1 and B δ e δ,θ = e 2 −δ(1+θ)e 1 √ 1+(δθ) 2 , we obtain, multiplying by 1 + (δθ ) 2 and skipping the arguments g(x) everywhere, We let α := e 1 · Re 1 , β := e 1 · Re 2 and solve this equation for f + , which leads to the definition Since R is a rotation, we have |α| = 1 − β 2 , and we choose α = 1 − β 2 . Roughly speaking, we expect that for large arguments approximately β = 0 and α = 1, which correspond to f + = θ . On the other hand, in view of Remark 2.2(ii) since R 0 is a rotation by roughly −2δθ, we expect that for small arguments, β ≈ 2δθ which is positive for δ > 0 and negative for δ < 0. Hence, we assume that β δ is monotonically decreasing and α is monotonically increasing, so that by (4.26) also f + is monotonically increasing. We shall fix the value β(0) from the condition f + (0) = 0, so that monotonicity of f + implies 0 f + θ everywhere and in particular (4.19).

Lower Bound
For the lower bound, we need some auxiliary statements.

Technical Preliminaries
Proof. This follows immediately by the fact that all matrices considered are conformal. For clarity we present a short explicit computation. By scaling we can assume |v| = 1. Let φ ∈ (−π, π] be such that Q = cos φ Id + sin φ J . Then Then concludes the proof.
The next lemma concerns stability of the rank-one directions.
The next two statements are uniform geometric rigidity and trace statements on domains which are appropriate sections of the setsω A ,ω B defined in (4.7)-(4.8).
For clarity we present here the specific assertion used in the lower bound, postponing to Sect. 5 the proof in a more general context and the specific definition of (L , R)-Lipschitz sets. and define ω f,g := {x : fix any θ ∈ (0, 1 2 ] and any δ ∈ [−1, 1]. Then the sets ω f,g and T δ,θ (ω f,g ) are (L,R)-Lipschitz. Further, for any u ∈ W 1,2 (ω f,g ; R 2 ), and any F ∈ {A δ , B δ , Id}, there is Q F u ∈ SO(2) such that and, for some d u ∈ R 2 , By Lemma 5.3 we obtain that the sets T δ,θ (ω f,g ) are (6(2L + 1), 6R)-Lipschitz. The result for F = Id follows then immediately from Theorem 5.10. Consider now F = A δ . For notational simplicity we prove the statement for ω f,g , the argument for By Lemma 5.3, using that |A δ |, |A −1 δ | 3, we obtain that the sets A −1 δ ω f,g are (c(2L + 1), cR)-Lipschitz. Therefore Theorem 5.10 implies that there is Q A δ u ∈ SO(2) such that Using (4.43) and a change of variables, this implies (4.45) and concludes the proof. The case F = B δ is identical. The second bound follows immediately from Theorem 5.8.
For completeness, we finally note a rescaling property of the trace norm.
Proof. For = 1, this follows from Lemma 5.4, Theorems 5.8 and 5.9. The general case follows from rescaling f :

Proof of the Lower Bound
We start introducing some notation. Recall that the periodicity condition (4.2) is the same as u(T δ,θ (y + e 2 )) = u(T δ,θ (y)) The constant may depend on L.
Proof. For brevity in this proof we write E(I ) for E[I ; ( f ± , u)]. We can assume L 1 in the proof (otherwise we cover I * with c L subintervals of length 1/4 and use the result for L = 1 in each of them).
Step 1: Estimate on ω I * A . Let t * be the midpoint of I * and * its length. We assume that ( f + − f − )(t * ) We write ω A := ω I * A , and c L for a generic constant that may change from line to line but depends only on L. Proposition 4.6 can be applied (with M = 4) to the set ω A , and there is Q ∈ SO(2) such that . (4.50) Note that this concludes the proof in the degenerate case ω I * B = ∅, i.e., if I * ∩{ f − + 1 > f + } = ∅. In the other case, we note that there is d A ∈ R 2 such that (4.51) Step 2: Estimate on ω I * B . For any Borel set I ⊆ I * we write, recalling (4.47) and the short-hand notation E(I ) = E[I ; ( f ± , u)], It is clear that E andÊ are measures on I * , and thatÊ(I * ) c L E(I * ). We shall first obtain estimates on suitable subintervals of I * , and then cover I * by countably many such subintervals. Let M > 0 be a fixed number, we shall choose M = 16 below.
Assume that I ⊆ I * is an interval of length ∈ (0, * ] such that , there is d I B ∈ R 2 such that, setting γ I c L E(I ). (4.55) Analogously, the trace theorem onω I A := T δ,θ ({y : y 1 ∈ I, f + (y 1 ) − 4 < y 2 < f + (y 1 )}), which by (4.49) and * 1/4 is contained in ω I A , gives so that with a triangular inequality and E Ê , we obtain Therefore there is v ∈ R 2 with v 1 1 3 and |v 2 | Lv 1 such that Combining (4.54) and (4.59) we conclude that (4.60) be covered (up to null sets) by countably many intervals I with the property (4.53) and finite overlap.
contains t and obeys the property (4.53) with M = 3. By the Besicovitch covering theorem this family contains a countable set of intervals (I k ) k which covers I * ∩ { f − + 1 > f + } and has finite overlap. The intervals (I k ∩ I * ) k obey property (4.53) . Therefore, by (4.60), and with (4.51) andÊ(I * ) c L E(I * ) the proof is concluded.
Proof. For brevity, in this proof we write E( Step 1. Piecewise affine approximation. Consider the intervals By Proposition 4.8 there are rotations Q j ∈ SO(2) such that, for any j ∈ N, one has with the constant c L (here and in all following estimates) depending only on L.
We shall use this estimate and the periodicity condition to obtain four different bounds, which are stated in (4.64), (4.65), (4.67) and (4.68).

Existence of Minimizers
In this section, we show existence of minimizers of the shape optimization problem (4.12) on the set defined in (4.11) under the additional assumption that the energy density W is quasiconvex. The notion of quasiconvexity was introduced by Morrey [42] and is a central concept in the calculus of variations since it guarantees existence of minimizers for variational problems of the form min u W (∇u) dx under rather weak additional assumptions on W , see for example [23,44] for general presentations. Proof. We proceed along the lines of the proof of Proposition 3.
nl be a minimizing sequence. By subtracting constants, we can assume without loss of generality that (4.70) After passing to a subsequence, the functions f ± j converge locally uniformly to L-Lipschitz functions f ± * which by uniform convergence satisfy (4.3). For every m ∈ N, by the lower bound in (4.13), there is a uniform bound on the L 2 -norms and hence, by (4.70), there is a subsequence that converges locally weakly in W 1,2 to u * ∈ W 1,2 loc (R 2 ; R 2 ). By Rellich's theorem, the limiting function u * satsifies the periodicity condition (4.2). Let ω * A and ω * B denote the respective domains induced by f ± * , which are defined as in (4.7)-(4.9). By quasiconvexity of W and the growth condition (4.13), we get lower semi-continuity of the energy restricted to compact sets in (ω * A ∪ ω * B )\graph( f ± * )\(Re δ,θ ), and hence, choosing a diagonal sequence as in the proof of Proposition 3.3, we find, for every m > 0, , m), ( f ± * , u * )) : m > 0}, this concludes the proof.
It would be natural to ask if for a minimizer ( f ± * , u * ) the functions f ± * (x 1 ) have a finite limit as x 1 → ∞. This does not follow from the above proof. Another open question is whether the condition that f ± are Lipschitz is needed, or if it can be replaced by a term of the form which represents the additional length of the interfaces with respect to the "flat" situation.

Uniformly Lipschitz Domains
We show that certain estimates on Sobolev functions hold uniformly for a class of bounded open sets which are uniformly Lipschitz. We focus on bounded sets and use the following definition, which is a variant of the one in [38,Def. 13.11]. For x ∈ R n , we use the notation x = (x , x n ) with x ∈ R n−1 and x n ∈ R. Similar statements holds for other sets whose size is uniformly controlled by ε.
(v) One can reduce to the case that only finitely many functions f x appear, after changing R to 2R. Indeed, if ⊂ R n is (L , R)-Lipschitz then (using the notation of Definition 5.1) the balls B ε/2 (x), x ∈ ∂ , cover ∂ . By Vitali's covering theorem there is a subset x 1 , . . . , x M such that ∂ ⊆ ∪ M i=1 B ε/2 (x i ) and B ε/10 (x i ) ∩ B ε/10 (x j ) = ∅ for i = j. Since ∂ is contained in a ball of radius Rε, we have M (10 · R + 1) n . As for every Proof. Let ε > 0 as in Definition 5.1 for and set ε := ε/|F −1 |. Pick y ∈ ∂(F ) and let x := F −1 y ∈ ∂ . We first show that To see this, we pick z ∈ B ε (y), consider z := F −1 z, and compute |z − x| = Fix again y ∈ ∂(F ). We need to show that for some isometry I y and some g ∈ Lip(R n−1 ; R) with Lip(g) L ; the precise value of L is given below.
We let x := F −1 y as above. By property (ii) in Definition 5.1 there are an isometry A x and an L-Lipschitz function f : R n−1 → R such that The isometry A x can be written as A x z = b + Rz for some R ∈ O(n). We set η := |F Re n | ∈ (0, |F|], pick a rotation Q ∈ SO(n) such that F Re n = ηQe n , and let T := Q T F R ∈ R n×n . Then F R = QT and QT e n = ηQe n , which implies T e n = ηe n . We shall show below that with g as stated after (5.2). This implies that where I y w := Fb + Qw, which together with (5.4) concludes the proof of (5.2) and therefore of the Lemma. It remains to construct g ∈ Lip(R n−1 ; R) such that (5.5) holds. We observe that T is invertible, with |T −1 | = |F −1 | and T −1 e n = 1 η e n , and write for some s ∈ R n−1 and S ∈ R (n−1)×(n−1) , invertible. These obey max{|S|, |s|} |T −1 | = |F −1 |.
is the same as Assume that α g − f β for some α, β > 0, and that Lip( f ), Lip(g) L. Then ω f,g is (2L + 1, R)-Lipschitz for some R which depends only on α, β and L.
We consider points close to the lower-left corner, in the sense that we show property (ii) of Definition 5.1 for x in (see Fig. 6) (5.11) the other three corners can be treated analogously. We extend f to R, setting f (t) = f (0) for t < 0 and f (t) = f ( ) for t > . By the choice of ε we see that for all x ∈ A L L the ball B ε (x) does not intersect {z : z 2 = g(z 1 )} and {z : z 1 = }, in the sense that After a translation, we can assume f (0) = 0. In order to make the mentioned boundary the graph of a Lipschitz function we need a nontrivial rotation, as illustrated in Fig. 6. Let Q ∈ SO(2) be such that Qe 2 bisects the angle between e 2 and (1, L), which means that Q is a clockwise rotation by α := 1 2 ( π 2 − arctan L). Then there is a unique function F : R → R such that obviously F(0) = 0. One can check that F is L := tan( π 2 − α)-Lipschitz, and that L = L + √ 1 + L 2 1 + 2L. At the same time, by the definition of Q and L {z ∈ R 2 : z 1 > 0} = Q{y ∈ R 2 : y 2 > −L y 1 }. (5.14) Recalling (5.12), we see that it suffices to intersect these two sets. We defineF (t) := max{−L y 1 , F(y 1 )}, which is also L -Lipschitz. Then for every x ∈ A L L we have This concludes the proof.

Weighted Poincaré Inequality
The next result is a quantitative version of the estimate in [ We first prove the result in one dimension, by an explicit computation.
Proof. If the right-hand side is finite, then the function ϕ can be extended continuously to (a, b]. Let β := ϕ(b). For any x ∈ I by the fundamental theorem of calculus applied to |ϕ − β| p we have We integrate over all x ∈ I and use Fubini's theorem to get that so that, as |ϕ − β| p−1 By a triangular inequality, so that, with a further triangular inequality, which concludes the proof.
Inserting (5.29) and B r ∩ V ⊆ U in (5.28) concludes the proof.
Proof of Theorem 5.5. It suffices to consider the scalar case; by density it suffices to prove the inequality if u is C 1 ( ). For brevity, let A := dist(·, ∂ )∇u L p ( ) . Let ε be as in Definition 5.1 and fix r B := ε/(12(L + 1)) (the reason will become clear below). We first show that for every x ∈ there is α(x) ∈ R such that with c depending only on n, p, L and R. To see this, we distinguish two cases. If dist(x, ∂ ) 2r B this follows from the usual Poincaré inequality applied to the ball B r B (x) ⊂ , with dist(·, ∂ ) r B in B r B (x). If not, we fix x * ∈ ∂ with |x * − x| < 2r B and use Lemma 5.7 to B 3r B (x * ) (this is the point where the size of r B is fixed). As B r B (x) ⊆ B 3r B (x * ), this concludes the proof of (5.31).
By Vitali's covering theorem, there is a finite set x 0 , . . . , x K ∈ such that ⊂ ∪ K k=0 B k , B k := B r B /2 (x k ), with the smaller balls B r B /10 (x k ) pairwise disjoint. In particular, since they are all centered in and the diameter of is bounded by Rε, we obtain K (1 + 10Rε/r B ) n cR n L n . Let α k := α(x k ). We claim that, for every k = 1, . . . , K , one has r n/ p B |α 0 − α k | cK A. (5.32) To see this, fix k, and let j 0 := 0, j 1 , . . . , j H := k be finitely many indices in {0, K } such that B j h ∩ B j h+1 ∩ = ∅ for all h. They exist since is connected, which means that there is a continuous curve in which joins a point of B 0 ∩ with a point of B k ∩ ; as the curve is compact it is covered by finitely many of the balls. We can further assume the indices j h to be distinct. Indeed, if j h = j h for some h < h , we can remove h, h + 1, . . . , h − 1 from the set. As there are at most K balls, we obtain H K . In turn, B j h ∩ B j h+1 ∩ = ∅ implies that the larger balls have significant overlap. Indeed, for each x ∈ one has L n ( ∩ B r B /2 (x)) c L r n B , and recalling that the radius of the balls B k is r B /2, we obtain (5.33) Using (5.31) on these two balls and then a triangular inequality, which, as H K , implies (5.32). Finally, using that the balls B 0 , . . . , B K cover , Many well-known results from the literature follow easily from the weighted Poincaré inequality and its proof above. We start with a Poincaré inequality (see for example [45,Theorem 1.2]). Proof. This follows from Theorem 5.5, using that dist(x, ∂ ) diam( ) for all x ∈ .
With the same strategy as above we can obtain a uniform estimate on the trace operator, as a map from W 1, p to L p of the boundary. Also this result is well-known in the literature; see, e.g., [38,Theorem 18.40]. Proof. This can be proven along the same lines as Theorem 5.5. One starts showing that, in the setting of Lemma 5.6, one has |ϕ(a) − α| p c p 1 L 1 (E) E |ϕ − α| p dt + c p (L 1 (I )) p−1 I |ϕ | p (t)dt. (5.37) To see this, it suffices to observe that for any t ∈ E the fundamental theorem of calculus in (a, t ) ⊆ I gives |ϕ − α|(a) |ϕ − α|(t ) + I |ϕ |(t)dt. (5.38) We take the p-th power, average over all t ∈ E, and use Hölder's inequality in the last term to obtain (5.37).
In the next step we observe that, with r as in Lemma 5.7, for any x ∈ ∂ , . Finally, we cover ∂ with no more than cR n (ε/r ) n such balls, as in the proof of Theorem 5.5, and conclude the proof.

Rigidity
We prove a version of geometric rigidity from [28] and of Korn's inequality with uniform constant on all (L , R)-Lipschitz sets. Instead of repeating the entire proof, and checking that the variuos constants depend on the domain only through the parameters L and R, we show that the estimate for a general domain can be reduced to the one for a cube. We refer to [14] for the proof for general p and a review of the literature on Korn's inequality and rigidity. The explicit dependence of the constant on the shape of the domain was analyzed, in the specific case of long and thin domains, in [30,31]. Korn's inequality was derived for John domains and related classes using different techniques, see [1,25,33] and references therein. The constant c Rig depends only on n, p, L and R.
We recall that both estimates do not hold for p = 1 and p = ∞.
Proof. For the case that is a cube both results are well known. We prove the first inequality, the proof of the second one is almost identical.
The key observation is that the constant depends on the domain only via the weighted Poincaré estimate. The general structure of the argument, using a Whitney decomposition of the domain, is similar to the one used in [28] for the harmonic part. Indeed, there rigidity was first proven in small cubes, and then the second derivative was estimated using harmonic estimates. Here we instead construct a new function (called β below) which interpolates between the values of the rotation in each cube, using a partition of unity. This permits to avoid discussing the dependence on the domain of the constants involved in the initial truncation in the proof of [28]. This argument is almost identical to the one used in [17], we repeat it here for completeness.
We intend to construct a partition of unity subordinated to a Whitney covering of , done as in [50,Section VI.1] or [27,Sect. 6.5]. Precisely, we select countably many cubes Q j := x j + (−r j , r j ) n such that, lettingQ j := x j + (− 1 2 r j , 1 2 r j ) n denote cubes with the same center and half the size, The constant c in (5.42) and (5.43) depends only on the dimension n. These conditions imply that the cubes have finite overlap, and that if Q j ∩ Q k = ∅ then 1/c r j /r k c. We fix ϕ * ∈ C ∞ c ((−1, 1) n ; [0, 1]) with ϕ * = 1 on (− 1 2 , 1 2 ) n , letφ j (x) := ϕ * ((x − x j )/r j ) and ϕ j :=φ j / kφ k . Using (5.42) one obtains ϕ j ∈ C ∞ c (Q j ), j ϕ j = 1 in , and |∇ϕ j | c/r j . By the estimate for the cube, for each j there is R j ∈ SO(n) such that ∇u − R j L p (Q j ) c n, p dist(∇u, SO(n)) L p (Q j ) . (5.44) We define β : → R n×n as a smooth interpolation between the R j , β := j ϕ j R j . Using j ϕ j = 1 in , ϕ j 1 and the finite overlap, At the same time, the distance between ∇u and the R j controls the derivative of β. Indeed, from j ϕ j = 1 we obtain j ∇ϕ j = 0 on , so that ∇β = j ∇ϕ j R j = j ∇ϕ j (R j − ∇u). (5.47)