Minimal energy for geometrically nonlinear elastic inclusions in two dimensions

We are concerned with a variant of the isoperimetric problem, which in our setting arises in a geometrically nonlinear two-well problem in elasticity. More precisely, we investigate the optimal scaling of the energy of an elastic inclusion of a fixed volume for which the energy is determined by a surface and an (anisotropic) elastic contribution. Following ideas from Conti and Schweizer (Commun. Pure Appl. Math. 59 (2006), 830–868) and Knüpfer and Kohn (Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 467 (2011), 695–717), we derive the lower scaling bound by invoking a two-well rigidity argument and a covering result. The upper bound follows from a well-known construction for a lens-shaped elastic inclusion.


Introduction
This article is concerned with a variant of the isoperimetric problem, for which we investigate the optimal energy of an elastic inclusion of a fixed volume. Here the energy consists of an interfacial and a geometrically nonlinear elastic contribution. The latter is defined by an integral of the stored-energy density function over a domain. As usual, the stored-energy density depends on the strain and describes properties of the material. Physically, the problem is motivated by nucleation phenomena which arise, for instance, in shape-memory materials [8].
The set-up considered in this work is the geometrically nonlinear analogue of the article [40] where the isoperimetric problem for a geometrically linear elastic two-phase inclusion problem had been investigated. Our main aim is to deduce quantitative information on the nucleation problem by studying its scaling properties. The problem of determining the sharp form of the inclusion seems to be more complicated [57]. On top of the presence of non-quasiconvexity as in [40], in the geometrically nonlinear setting under investigation, an additional difficulty is present in the form of the nonlinear gauge group SO (2). In order to deal with this, we hence rely on the geometrically nonlinear rigidity result from [26] in combination with the ideas from [40].
1.1. Model and statement of results. We consider the interior nucleation of a new phase in an elastic material in two space dimensions. More specifically, we consider a material for which two different phases (lattice structures) are energetically preferred. These are represented by the SO(2) orbit of the identity matrix Id ∈ R 2×2 and the SO(2) orbit of another matrix F ∈ R 2×2 \ SO (2). The deformation of the material is described by a function v : R 2 → R 2 . By the Cauchy-Born rule the energy of an elastic material can be represented Date: July 29, 2022. 1 in terms of the gradient of the deformation function v. Following the phenomenological theory of martensite and assuming Hooke's law, we study (volume-constrained) minimizers of the energy (1 − χ)dist 2 (∇v, SO(2)) + χdist 2 (∇v, SO(2)F ). (1) Here χ : R 2 → {0, 1} encodes the location of the new, minority phase. Its variation i.e. the first integral in (1) is the interfacial energy, while the second integral is the elastic energy. Hence, our model includes penalizations of transitions between the phases and deviations from the corresponding material phase. We introduce µ > 0 to denote the volume of the inclusion for the region M := {x ∈ R 2 : χ(x) = 1} associated with the minority phase. In what follows, we will consider minimizers of the energy (1) for a prescribed volume of the minority phase. In order to rule out self-intersections, as the set A m of admissible functions we consider Here bi-Lipschitz with constant m ≥ 1 means that v is a homeomorphism and v and v −1 are Lipschitz continuous functions with Lipschitz constants m ≥ 1. Seeking to model nucleation phenomena, we assume that the strain F is compatible with the identity matrix. In two dimensions this is equivalent to the condition det F = 1. Our main result is the scaling of the minimal energy for prescribed inclusion volume: Theorem 1.1 (Scaling of ground state energy). Suppose that F ∈ R 2×2 \ SO(2) satisfies det F = 1. Let µ be as in (2). Let m ≥ max { F , F −1 } + C for some sufficiently large constant C > 0. Then for any µ > 0 we have Here, we write A ∼ B means that cA ≤ B ≤ CB for two constants c, C > 0 which are independent of µ but may depend on F . The first bound corresponds to the usual isoperimetric regime in which the surface energy dominates while the second estimate for µ ≥ 1 captures the effect of the interaction of the surface and elastic energies. In particular, the role of anisotropy in the elastic contribution in the form of the two physical phenomena of compatibility and self-accommodation are captured in it. We do not track the dependence on F in the energy scaling behaviour.
The result of Theorem 1.1 confirms the similar scalings which had been obtained in the framework of piecewise linear elasticity in [40]. In particular, the result shows that in the framework of geometrically nonlinear elasticity, the model imposes enough rigidity to ensure the same lower bound on the energy as in the geometrically linear model. This is in line with the fact that the only solution for the two gradient problem for two compatible strains are twins in nonlinear elasticity theory [3,Prop. 2] as well as in linear elasticity theory. If we allow for more variants of martensite, the situation is expected to become more intricate since in this case the corresponding many gradient problems possibly allow for a large number of non-trivial solutions and complicated microstructure [8,57].

1.2.
Ideas of the proof. The proof of our main result can be split into two parts: an ansatz-free lower bound estimate and an upper bound construction. On the one hand, in order to verify the lower bound, we observe that without loss of generality, we may assume the deformation F to be symmetric and positive-definite after using the polar decomposition theorem. By a suitable choice of coordinates, F hence takes the form With these normalization results in hand, in the small volume regime, the lower bound follows by the standard isoperimetric inequality. In the large volume setting, we deduce the lower bound by a combination of a segment rigidity argument from [26] (Section 2) and the localization argument from [40] (Subsection 4.1). Working with phase indicator energies as in [40] or [12], see (1), contrary to the energies in [26], we do not directly control the full second derivatives of the deformation v. This additional degeneracy results in a number of small adaptations becoming necessary. For settings with full second derivative control the key localized energy estimate in Proposition 3.1 would directly follow from Corollary 2.5 in [26]. Moreover, in this case also the higher dimensional problem could be treated directly in parallel by invoking the results from [18] or [34]. With our energies this would require adaptations of these strategies which we do not pursue in the present article. The slightly stronger degeneracy of our energy which does not immediately yield the full second derivative control, also accounts for one of the technical reasons for our bi-Lipschitz assumptions in the minimization problem; another reason being the use of approximation theory for bi-Lipschitz functions in Section 3. On the other hand, the upper bound is derived by constructing a deformation v corresponding to a well-known construction for a lens-shaped elastic inclusion (see e.g. [40]) which in our geometrically nonlinear setting leads to an orientation-preserving deformation.
1.3. Relation to the literature. Due to their physical significance and the intrinsic mathematical interest in "non-isotropic" isoperimetric inequalities, nucleation problems for shape-memory materials have been studied in various settings: In a geometrically linearized framework the compatible and incompatible two-well problems (one variant of martensite and one variant of austenite) have been considered in [40], where a localization strategy was introduced. This also forms one of the two core ingredients of our result. Moreover, the nucleation behaviour for the geometrically linearized cubic-to-tetragonal phase transformation was studied in [41] in which Fourier theoretic arguments in the spirit of [11,12] were exploited. Fourier theoretic arguments also underlie the study of the nucleation of multiple phases without gauge invariance in [62]. Using related ideas, the nucleation behaviour at corners of martensite in an austenite matrix was investigated in [6]. We also refer to [4,5,50] for the study of quasiconvexity at the boundary. Further, highly symmetric, low energy nucleation mechanisms have been explored in [25] and [15] both in the geometrically linear and nonlinear theories in two dimensions. In the geometrically nonlinear settings substantially less is known in terms of nucleation properties due to the presence of the nonlinear gauge group. In this context, the incompatible two-well problem was studied in [17] in which an incompatible two-well analogue of the Friesecke-James-Müller rigidity result [33] was used. Moreover, the study of model singular perturbation problems for the analysis of austenite-martensite interfaces in terms of a surface energy parameter [47,46] laid the basis for an intensive, closely related research on singular perturbation problems for shape-memory alloys [16,65,26,53,27,22,29,19,20,60,63,61]. Contrary to the full nucleation problems, in these settings the phenomenon of compatibility plays the main role, while nucleation phenomena in addition require the analysis of the phenomenon of selfaccommodation. Moreover, dynamic nucleation results have been considered in [51,31,30]. We refer to [57] and [52] for further references on these and related results.
Nonlocal isoperimetric inequalities have also been investigated for the Ohta-Kawasaki energy and related models with Riesz interaction. We refer e.g. to [42,39,54,35,9,10,32,36,1]. In these models, above a critical volume minimizers do not exist anymore and the scaling of the energy in terms of the mass is linear. Other related vectorial models where the energy includes both interface type energies as well as a (dipolar) nonlocal interaction are ferromagnetic systems. The nucleation of magnetic domains during magnetization reversal and corresponding optimal magnetization patterns have been investigated in [43,44,45], see also [58]. The competition between a nonlocal repulsive potential and an attractive confining term is found also in other problems, for example in models studying the interaction of dislocations [64,38] or [56,13,14]. Another anisotropic and nonlocal repulsive energy that has been treated variationally using ansatz-free analysis is [14] (based on [56,13]). We finally briefly mention investigations of other physical settings where related nonlocal isoperimetric inequalities have been studied. This includes the works [48,49,59] on compliance minimization, on epitaxial growth (e.g. [7]), on dislocations (e.g. [23]) and superconductors (e.g. [21,24]). We emphasize that the above list of references is far from exhaustive.
By B R (x) we denote the ball of radius R > 0 centered at x ∈ R 2 and we write B R := B R (0). We write M := spt χ ⊂ R 2 to denote the support of the minority phase. For E ⊂ R 2 and v ∈ BV (E), the total variation of v is denoted by ∇v E .

Rigidity
The aim of this section is to find a "good" set in the shape of a rhombus which fulfills a variant of the rigidity estimate from [26]. We first introduce some notation for the elastic energies for the deformation v. We set Then the elastic energy for a 1D or 2D subset E ⊂ R 2 is defined as and the total elastic energy is which we will use in order to deal with estimates for the inverse of v. If the subset is one dimensional we integrate over the 1D Hausdorff measure instead of the Lebesgue measure.
Before stating the central rigidity estimate, we formulate two auxiliary lemmas. First, we note that there is a large set of non-singular points: Proof. This follows by an application of Fubini's theorem and since dist −1 (·, x 0 ) ∈ L 1 loc .
By our bi-Lipschitz assumption, bounds on v can be translated into analogous bounds for its inverse: Assume that Then for Proof. By the transformation formula and since v ∈ A m , (i) follows from By the chain rule for BV functions (cf. Theorem 3.16 in [2]) this implies The claim of (iii) follows by an application of the linear algebra fact from Lemma A.2 (ii). Indeed, using the pointwise identity together with the inverse function theorem, the transformation theorem and with the notationṽ := v −1 , we arrive at for some constant C = C(m, F ) > 0. This completes the proof.
We are now ready to give the key rigidity estimate. It is a variant of the two-well rigidity estimate from [26] and shows that we can find a sufficiently large rhombus such that we control the energy and the change of length on all six connecting lines between the corner points of this rhombus both for the transformation and its inverse: Then there exist four points C := {a, b, c, d} ⊂ B R m ⊂ R 2 with |a − b| ∼ R/m and |c − d| ∼ δR/m, which form the end-points of a symmetric rhombus T such that for all x, y ∈ C and with the notation M = spt χ we have the following properties Furthermore, for (vi) there exist Q ∈ SO(2) and p ∈ R 2 such that Proof. Without loss of generality, by scaling, we may assume that R = m and v(0) = 0. We further choose θ ∈ (0, 1) sufficiently small to be determined below. We argue in several steps based on averaging-type arguments.
Next, we repeat this argument along the vertical lines of the form . Also for this set, we analogously find a volume fractioñ E ⊂ [− 1 2 , 1 2 ] of size 1 − θ such that these vertical line segments satisfy (i)-(ii). Consider now the sets {L hor (r)} r∈E and {L ver (s)} s∈Ẽ of all horizontal and vertical segments with the properties (i)-(ii), respectively (see Fig. 1). Let o(s, r) = L hor (r) ∩ L ver (s) be the intersection point of the corresponding horizontal and vertical line. The point o(s, r) divides both L hor (r) and L ver (s) into two segments denoted by L + hor (r) and L − hor (r) (also L + ver (s) and L − ver (s)). Since E andẼ are sets of positive (close to one) volume fractions, T (ρ) Figure 2. Sketch of a set of rhombiT (ρ), ρ ∈ ( 1 4 , 3 4 ).
there exist r 0 ∈ E and s 0 ∈Ẽ such that |L + hor (r 0 )| ∼ |L − hor (r 0 )| and |L + ver (s 0 )| ∼ |L − ver (s 0 )|. Consequently, we choose L hor and L ver such that o = L hor (r 0 ) ∩ L ver (s 0 ) is the midpoint of L hor as well as the midpoint of L ver . This can be done by (if necessary) cutting exceeding parts of L hor (r 0 ) and L ver (s 0 ); we note that such a modification preserves the conditions |L hor | ∼ 1 and |L ver | ∼ δ.
Step 2: Identification of a "good" rhombus. Let L hor and L ver be the segments forming a symmetric cross and satisfying (i)-(ii) as in the previous step. LetT be the symmetric rhombus given by the convex hull of this cross. We denote byT ρ the homothetically shrunken rhombus with the self-similarity factor ρ ∈ (0, 1] and the same center point. For ρ ∈ ( 1 4 , 3 4 ) =: I, the diagonals (given by the corresponding shortened line segments of the originally constructed cross) of the resulting symmetric rhombiT ρ also satisfy (i)-(ii) by construction, see Fig. 2. After using a Fubini argument as in Step 1, we obtain a subset I 1 of I on which all sides of the rhombus fulfill the properties (i)-(ii).
Next, we seek to ensure that the properties (iii)-(v) are also satisfied on the edges of some of these rhombi. Invoking Lemma 2.1 together with another averaging argument, we obtain another set I 2 ⊂ I of positive volume fraction satisfying (iii). On top of this, since v is bi-Lipschitz and by Lemma 2.2 (i)-(ii), we can repeat Step 1 with the functions v −1 and χ 1 , the energy E elast [χ 1 , v −1 , B m ] and for the line segments [v(x), v(y)], where x, y form the endpoints of the rhombiT ρ for ρ ∈ I. Thus, noting that by the bi-Lipschitz property of v, the length of the lines [v(x), v(y)] is (up to a factor m, m −1 ) comparable to that of [x, y] and after possibly enlarging the constant C > 0, we obtain a subset I 3 of I with the properties (iv)-(v). By choosing the intersection of these subsets of I, we arrive at a subset of I with positive volume fraction such that all sides ofT ρ fulfill (i)-(v) for ρ in this subset, provided η > 0 is sufficiently small.
By the Friesecke-James-Müller rigidity theorem [33] and Poincaré's inequality, there exist Q ∈ SO(2) and p ∈ R 2 such that for constants C δ , Again, the use of a Fubini argument implies that there are many values of ρ such that the resulting rhombiT ρ are "good", in the sense that all lines connecting the corner points of We choose one such "good" rhombus and denote it by T and define its endpoints as the points C := {a, b, c, d} (see Fig. 3). Since v is a continuous function, we obtain from inequality (6) |v As a consequence, by construction the properties (i)-(vi) are satisfied for these endpoints.
Step 3: Proof of (vii). By the fundamental theorem of calculus, for any x, y ∈ C we have Now, we apply the same argument to v −1 (v(x)) − v −1 (v(y)) with x, y ∈ C. Thus, in view of Lemma 2.2, for a constant C = C(δ, m, F ) > 0 we obtain Combining inequality (7) and inequality (8), we obtain the desired estimate (vii). This completes the proof of the lemma.

A Lower Bound for the Elastic Energy
In this section we prove a local lower bound by exploiting the rigidity argument from Lemma 2.3 and the ideas from the proof of Lemma 2.3 in [26]. This local lower bound provides a geometrically nonlinear variant of the central lower bound from Proposition 3.1 in [40]. In Section 4.1 we will combine it with a covering argument as in [40] which will imply the lower bound of Theorem 1.1.

Proposition 3.1 (Lower bound on elastic energy).
There is η > 0 such that for any R > 0 the following holds: Suppose that (χ, v) ∈ A m satisfies Then there are constants α = α(m) ∈ (0, 1) and C = C(F, α, m) > 0 such that Proof. This result essentially follows from an application of a variant of the two-well rigidity result from [26]. Here there are slight adaptations in Steps 1 and 2 in the proof due to the choice of our energies (full gradient control in [26] vs. our phase-indicator energies), while Steps 3 and 4 then follow essentially without changes as in [26]. For self-containedness, we repeat the argument for Proposition 3.1. By scaling we can assume R = m and by the approximation results in [28] for bi-Lipschitz functions we can further assume that v ∈ C 1 (B m ).
Following the argument in [26] in our proof we will construct a rhombus T with B α ⊂ T ⊂ B 1 and show that the corresponding estimate (9) holds for T replaced by B α for some α > 0. We write µ := χ L 1 (Bα) and := E elast [χ, v, B m ]. Moreover, we note that, without loss of generality, we can assume Indeed, if is large e.g. 1 2 ≥ η, then by assumption we have χ L 1 (B 1 ) ≤ η ≤ 1 2 . Then inequality (9) follows immediately.
Without loss of generality we can assume that δ ∈ (0, 1) so that the conditions of Lemma 2.3 with R = m are satisfied. We then consider a rhombus T with corner points C := {a, b, c, d} as obtained in Lemma 2.3, see Fig. 3. Since |c − d| ∼ δ and |a − b| ∼ 1, in particular, By Lemma 2.3 we further have the properties (i)-(vii) for this rhombus.
Step 2: We claim that there exists Q ∈ SO(2) and p ∈ R 2 such that v(x) is close to Qx + p for any point x ∈ C up to an error of order . This implies that there are two isometries x → Q j x + p j with Q j ∈ O(2), p j ∈ R 2 and j ∈ {1, 2} such that for the constant C = C(δ, m, F ) > 0 from Lemma 2.3 we have It remains to argue that Q 1 , Q 2 ∈ SO(2) and p 1 , p 2 ∈ R 2 can be chosen to be equal, respectively.
We first argue that Q j ∈ SO(2). In [26] this follows from the second gradient control and the pointwise estimates in the endpoints of the rhombus. Lacking the control of the full gradient, we here vary the argument slightly. The use of Lemma 2.3 (vi) and the triangle inequality implies that for some constant C = C(F, δ, m) > 0 and for Q ∈ SO(2), p ∈ R 2 we have For η ∈ (0, 1) (depending on δ > 0) and > 0 sufficiently small, this yields a contradiction, if Q 1 ∈ O(2) \ SO (2). Similarly, we also obtain that Q 2 ∈ SO(2). Moreover, since the triangles ∆ cbd with vertices c, b, d and ∆ acd with vertices a, c, d share a common line, we have that Q 1 can be chosen equal to Q 2 and that p 1 = p 2 . A normalization further allows us to suppose that p 1 = p 2 = 0 and Q 1 = Q 2 = Id. As a consequence, we may assume that |v(x) − x| ≤ C 1 2 for x ∈ {a, b, c, d} and for some constant C = C(F, δ, m). (12) Step 3: Smallness estimate for N : As in [26], we claim that where the set N denotes the region where the gradient is closer to the well SO(2)F than to the parent gradient, i.e.
To this end, we use the upper length bounds on v(t), i.e. the fact that v is essentially not length increasing. Let t be any point of [c, d]. By the fundamental theorem of calculus and Lemma 2.3 (ii) we then get for some constant C = C(δ) > 0 Combining this with the triangle inequality and the bound (12) applied to x = c, we obtain |c − v(t)| (12) ≤ |c − t| + C 1 2 for all t ∈ [c, d] and some constant C = C(F, δ, m) > 0. (15) We note that in view of (10) and for η = η(δ) sufficiently small we can assume that Next, we seek to use this to deduce lower bounds on |a − v(t)| + |b − v(t)| for t ∈ [c, d] as above. To this end, we observe that in view of (16), the minimization problem is attained on the line [c, d] and is solved by t * := t − r c−d |c−d| for some r with 0 < r < C 1 2 . Here, the error bound for r is a consequence of (15). Using v(t) as a competitor and inserting the bound for r c,t implies Using again (12) now for x = a and x = b, we infer the following lower bound on the length deformation for points t ∈ [c, d]: We complement this with an upper bound on the length deformation along the segments [a, t] and [t, b], obtained by means of the fundamental theorem. In view of (11) and using where K := SO(2) ∪ SO(2)F and χ N denotes the characteristic function of the set N (cf. (14)) we get Subtracting these estimates from (17) We integrate over all t ∈ [c, d] and change variables from (x 1 , t 2 ) to (x 1 , x 2 ) by the transformation Ψ(x 1 , t 2 ) = (x 1 , t 2 (1 − x 1 a 1 )) (where t = (0, t 2 ), a = (a 1 , 0)) and Φ = Ψ −1 to obtain an integration over the rhombus T . More precisely, denoting by J Φ (x) as the Jacobian determinant of the transformation Φ, we infer Since |J Φ | ∼ dist(x, {a, b}) −1 , and thus, in particular, J Φ ≥ 1, in the left-hand side we can simply drop J Φ . For the right hand side we invoke Lemma 2.3 (iii) which concludes the argument for (13).

Proof of Theorem 1.1
We are ready to give the proof of Theorem 1.1. We split it into two parts and first discuss the lower bound and then provide a matching upper bound construction.

4.1.
Proof of the lower bound in Theorem 1.1. In this section, we provide the proof of the lower bound. To this end, we first observe that in the small volume regime this directly follows from the isoperimetric inequality. It thus suffices to consider the large volume regime µ ≥ 1. Although the proof follows the localization argument as in [40], for the convenience of the reader, we briefly recall its proof.
Proof of Theorem 1.1, lower bound.
Step 1: Strategy. We argue by a localization and covering argument, seeking to invoke Proposition 3.1. We consider a suitably chosen countable family of balls {B R i (x i )} ∞ i=1 covering M := spt χ (see Step 2 below). By a Vitali covering argument, we may assume that {B R i /5 (x i )} ∞ i=1 are pairwise disjoint. Then, we can localize the energy as follows:

It thus remains to argue that
We split this into two steps: Following [40], we prove that where α > 0 is the constant from Proposition 3.1 and for suitably chosen balls B R i (x i ). We note that all the estimates in this proof may depend on the constant α.
Step 2: Choice of radii and center points x i . To this end, without loss of generality, we may assume that all x ∈ M are points of density one of M . Now for any x ∈ M we set where η 0 is sufficiently small constant, which will be fixed later on. By continuity in r and by considering the limit r → ∞, we infer that R(x) ≤ µ 2 3 √ η 0 . Therefore, R(x) is uniformly bounded in terms of µ and the defining infimum actually is a minimum. Similarly as in [40], we note that R = R(x) satisfies one of the following conditions: Either or Obviously, M is covered by ∪ x∈M B R(x) . Since the radii R(x) are uniformly bounded, by Vitali's covering lemma, there is an at most countable subset of points x i ∈ R 2 such that the balls . This yields the balls and radii from Step 1.
Step 3: Proof of the estimate (20). By the definition of R, we obtain the following statements: if |M ∩ B R i (x i )| ≤ 1 and |M ∩ B αR i /5 (x i )| ≤ 1, then The last three obtained estimates together yield bound (20).
Step 4: Proof of the estimate (21). Here, we distinguish three cases: Firstly, we assume that case (23) holds. Since the density of the minority phase is much smaller than one in B R i /5 (x i ), the use of the isoperimetric inequality implies Secondly, we suppose that case (24) and Since Lastly, we assume that case (24) and where means that this estimate requires a small universal constant. Here, choosing η 0 small enough, the assumptions of Proposition 3.1 are fulfilled on B R i /5 (x i ). The use of this proposition results in . Then, inequality (21) follows from the above estimates, which concludes a proof of the lower bound in Theorem 1.1.

4.2.
Proof of the upper bound of Theorem 1.1. We next give the proof of the upper bound in Theorem 1.1. For this, we give an explicit construction for an optimal configuration. It suffices to consider the case µ ≥ 1, since the case µ ≤ 1 follows by simply considering v(x) = x and χ = χ B where B is a ball with |B| = µ. The estimate then follows by using the isoperimetric inequality and noting that 0 < µ ≤ µ 1 2 if µ ≤ 1. We note that similar constructions are well established (e.g. [37]). An upper bound in the setting of geometrically linear elasticity has also been given in [40] for the geometrically linearized theory. We provide an analogous construction for the geometrically nonlinear case and check that the solutions are within our class of admissible functions. We first note that, by a rotation (see Lemma A.1 for more details), we can assume that In particular e 2 is one of the twinning directions for stress-free laminates between F x and x. e 2 (twinning direction) R T ρ u 0 is const. in norm. dir. Figure 4. Sketch of the construction of u 0 and v = ω R u 0 + x.
As in [40] we consider an inclusion which approximately has the shape of a thin disc Q T,R with diameter R and thickness T where T R. The disc is oriented such that the two large surfaces are aligned with the e 2 twinning direction. To be more precise, let x (1) , x (2) ∈ R 2 such that x (1) = −x (2) on the axis x 1 = 0 with distance d := |x (1) − x (2) |. We define χ by where Q T,R is the lens with thickness of order T and diameter of order R given by the intersection B ρ (x (1) ) ∩ B ρ (x (2) ) for some suitable ρ = ρ(R, T ) > 0. We choose T such that it fulfills the volume constraint (2), i.e. |Q T,R | = µ and in particular, RT ∼ µ.
We next define u 0 : R 2 → R 2 such that u 0 (x) = (F − Id)x in Q T,R . Furthermore, outside Q T,R , u 0 is constant on all lines which are normal to the surface ∂Q T,R . Finally, u 0 = 0 in the remaining area which is neither in Q T,R nor reached by any of these lines. The function is sketched in Fig. 4. Furthermore, let ω R ∈ C ∞ (R, [0, ∞]) be a cut-off function with ω R (ξ) = 1 for |ξ| ≤ R and ω R (ξ) = 0 for |ξ| ≥ 2R with |∇ω R | ≤ C R for fixed C > 0. We then define v : Estimates: By construction we have and u 0 L ∞ (R 2 ) T . Since ∇v = F in Q T,R and ∇v = Id in B c 2R we get By (26), since RT ∼ µ and also including the interfacial part of the energy we obtain The asserted upper bound then follows with the choice R ∼ µ 2/3 .

Admissibility:
We need to check that our construction satisfies (χ, v) ∈ A m . In fact, it is enough to check this condition for µ = µ(F ) and correspondingly R ∼ µ 2/3 sufficiently large. We first note that χ ∈ BV (R 2 , {0, 1}) and ∇v L ∞ ≤ C F . We next consider v locally in the different regions defining it and show that in each of these v −1 exists and (∇v) −1 L ∞ ≤ m. To this end, we recall that ) for x ∈ B 2R \Q T,R be the mathematical positive oriented basis where b 2 (x) is the direction of the lines in B 2R \Q T,R where u 0 is constant and with sign convention b 2 (x) · e 2 > 0. By construction we then have |b i (x) − e i | ≤ O( T R ) for i = 1, 2. Since ∇u 0 (x)b 2 (x) = 0 we hence get |∇u 0 e 2 | ≤ C F T R . Since (F − Id)e 1 = 0 we also have |∇u 0 (x)b 1 (x)| ≤ C F T R . Together, this yields ∇u 0 ≤ C F T R . Since |ω R | ≤ 1 and |∇ω R | 1 R , this yields ∇ω R ⊗ u 0 + ω R ∇u 0 ≤ C F (1 + T ) R .
In particular, as R ∼ µ 2 3 , T ∼ µ 1 3 and µ ≥ 1. As a consequence, a Neumann series argument implies that (∇v) −1 (x) ≤ C(1 + ∇v(x) − Id ) ≤ C F and for R = R( F ) sufficiently large. Last but not least, we argue that with the observations for ∇v from above, we obtain that v is globally invertible. To this end, it suffices to prove that v is injective. Assuming that for some x, y ∈ R 2 we have that v(x) = v(y), the fundamental theorem yields that 0 =   1 0 ∇v(tx + (1 − t)y)dt   (x − y).
Since the arguments from above show that ∇v always is a perturbation of an upper triangular matrix, this can only be the case if x = y which hence implies the desired injectivity.

A. Some Auxiliary Linear Algebra Facts
We collect some linear algebra facts which are used in the proofs of the main part of our text.
(ii): By using (i), we have F = R + a ⊗ b for some a, b ∈ R 2 and R ∈ SO(2).
It completes the proof of (ii).
For any F ∈ GL + (2) by polar decomposition there is R ∈ SO(2) and U = U t ∈ R 2×2 positive-definite with F = RU . We give two formulas related to the distance to SO(2): Lemma A.2 (Identities for distance to SO (2)).
Using the above last expressions and SA = A , we hence obtain