Line-tension limits for line singularities and application to the mixed-growth case

We study variational models for dislocations in three dimensions in the line-tension scaling. We present a unified approach which allows to treat energies with subquadratic growth at infinity and other regularizations of the singularity near the dislocation lines. We show that the asymptotics via Gamma convergence is independent of the specific choice of the energy and of the regularization procedure.


Introduction
Variational models depending on fields that can have topological singularities are of particular interest in materials science. The asymptotic analysis of such models is often based on tools from geometric measure theory and permits to derive effective energies concentrated on sets of lower dimension. Important examples range from the study of vortices in superconductors, to grain boundaries, fractures and other interfaces in solids, as well as line defects and disclinations in liquid crystals. Here we consider the codimension-two case, and in particular integral energies in three dimensions for fields that are curl-free away from a one-dimensional set, having in mind the important application of the study of dislocations in crystals.
Dislocations are the main mechanism for plastic deformation in metals. They are one-dimensional singularities of the strain field, with discrete multiplicity arising from the structure of the underlying crystal lattice. We treat dislocation models of the type Ω W (β)dx, (1.1) where Ω ⊆ R 3 , W : R 3×3 → [0, ∞) is an elastic energy density, β : Ω → R 3×3 characterizes the strain field and obeys curl β = µ. In turn, µ is the density of dislocations; which is a divergence-free finite measure of the form εθ ⊗ τ H 1 γ, with γ ⊂ Ω a one-rectifiable curve, θ ∈ L 1 (γ, H 1 ; B) the discrete, vectorial multiplicity, τ ∈ L ∞ (γ, H 1 ; S 2 ) tangent to γ. Here B is a three dimensional Bravais lattice which describes the underlying crystalline structure and the small parameter ε represents the ratio between the atomic distance and the size of the sample. In a finite setting we understand Ω as the deformed configuration, and β as the inverse of the elastic strain; the energy density W is then invariant under the action of SO(3) on the right. In the entire paper curl f , for f a distribution taking values in R 3×3 , denotes the rowwise distributional curl, which is also a distribution taking values in R 3×3 . If W has quadratic growth, the energy (1.1) is infinite whenever µ = 0. In reality, the discrete nature of crystals shows that the appropriate model would be discrete. The divergence is a nonphysical result due to the fact that one uses a continuum approximation which in the region close to the singularity is not appropriate. Therefore a number of regularizations of the continuum model have been proposed. They include a subquadratic growth of W at infinity, the replacement of µ by a mollified version at scale ε, and the elimination of a core region from the integration domain. For example, the latter corresponds to where (γ) ε denotes an ε-neighbourhood of the curve γ. The above models are all considered semi-discrete models, in the sense that they are continuum models that still contain the discrete parameter ε and some discrete effects, namely, the quantization of the measure and the regularization, both at scale ε. In turn, the energy density W can be treated in a geometrically linear setting, or with finite kinematics. Ultimately, all these variants have the same leading-order behaviour.
In this paper, we provide a unified treatment of the three-dimensional continuum models which covers many different regularizations, as for example the one with mixed growth conditions, and clarifies the equivalence of all these possible approximations and corresponding regularizations. Our general approach produces also a simpler and more direct proof of some results from the literature [CGO15,GMS21]; we expect that this unified approach will prove helpful in further future generalizations. We assume that dislocations are dilute, in the sense that the curve γ on which µ is supported has some regularity, which may however degenerate in the limit. Our mechanical model includes frame indifference and is formulated in the deformed (spatial) configuration Ω. As a measure of elastic strain at a point x ∈ Ω, we use β(x) ∈ R 3×3 , which maps directions in the spatial configuration to directions in the lattice configuration; in the language of continuum mechanics this is the inverse elastic strain seen as a spatial field, β = (F e ) −1 s , as discussed in Section 3.1 below (the subscript s indicates that the field depends on the spatial coordinates).
The topological singularity leads to a logarithmic divergence of the energy in the regularization parameter ε. After rescaling by log 1 ε , we show Γ-convergence to an energy which depends on a constant rotation Q, a measure µ = θ⊗τ H 1 γ representing the dislocation density, and a curl-free field η representing the elastic strain. The limiting energy takes the form The second term represents the line-tension energy, which can be obtained as the relaxation of where ψ C is the line-tension energy per unit length of infinite straight dislocations and can be computed from the matrix of elastic constants C by a onedimensional variational problem (see [CGO15] and Section 2 below for details). The first term involves the elasticity matrix C, rotated by Q.
We remark that our kinematic treatment of dislocations in finite kinematics is different from the one used in recent related mathematical literature, as for example [SZ12,GMS21]. Indeed, we work in the deformed configuration and our limiting energy has the integrand ψ C (θ, Qτ ), whereas the cited literature works in the reference configuration and obtains a functional containing (in our notation) ψ C (Q T θ , τ ). The two can be made to coincide if θ = Qθ, τ = Qτ ; as θ takes value in the lattice B this suggests that the expression ψ C (Q T θ , τ ) needs the multiplicity θ to take values in the rotated lattice QB. We remark that our approach closely matches the one presented in [MSZ15,(7.6)-(7.11)].
In Section 3 we state the main Γ-convergence result for the energies with mixed growth, we discuss the other models for alternative regularizations in the nonlinear and linear setting, which have the same Γ-limit, and we give a sketch of the main ideas of our approach. The proofs of the Γ-convergence results are then collected in Section 7.
In Section 4 we show that for a single straight dislocation the leadingorder term of the energy can be characterized by a cell problem under general assumptions for the energy density W . After rescaling, this term leads to (1.4). Therefore the linearization of a nonlinear elastic energy arises naturally in the limit. In particular this provides (Section 5) the lower bound for the Γ-limit of a nonlinear elastic model with energy densities of mixed growth, i.e., that behave as the minimum between dist 2 (·, SO(3)) and dist p (·, SO(3)) for some p ∈ (1, 2). For this model we also obtain a compactness statement, that asserts that sequences β ε with uniformly bounded energy have a converging subsequence in the relevant topology.
Moreover in Section 6 we show that the upper bound can be obtained by a pointwise limit. This proves that there is indeed a complete separation of scales between the relaxation, which happens at the line-tension level and only involves the line-tension energies ψ C and ψ rel C , and the concentration, which relates the elastic energy W integrated on a three-dimensional volume to the line-tension energy ψ C integrated on one-dimensional sets. This was already apparent in the two-dimensional Nabarro-Peierls setting of [CGM11].
The general approach presented here will also allow, in a forthcoming paper [CGO22], to treat a discrete model, confirming that the semi-discrete models are a good description of discrete ones outside the core region.
In two dimensions dislocations are point singularities, and a similar analysis for dilute configurations was performed for linear models in [CL05,GLP10] and for nonlinear models in [SZ12,MSZ15], both in the energy regimes scaling as log 1 ε and as log 2 1 ε . These results were extended in [DGP12,Gin19a,Gin19b] to general configurations, exploiting the connection between models for dislocations and Ginzburg-Landau models for vortices [ACP11] and refining the ball construction due to Sandier and Jerrard [San98,Jer99,SS07] to the context of elasticity in order to obtain compactness and optimal lower bounds. Similar techniques have been used first in [Pon07] and then in [ADGP14] in order to study a discrete two-dimensional model for screw dislocations, see also [HO15]. Similar problems arise in the context of spin systems, see for instance [BCLP18]. The two scales mentioned above give rise to mesoscopic limits where dislocations are identified with points (with energy scaling as log 1 ε ) or with densities (log 2 1 ε ). Results which obtain a characterization of grain boundaries in polycrystals exhibit a different scaling. Lauteri and Luckhaus [LL17] characterized the optimal scaling for the elastic energies of small-angle grain boundaries in nonlinear kinematics and gave a rigorous derivation of the Read-Shockley formula (see [GS] for the proof of the corresponding Γ-convergence result and [FPP20] for the result in the context of linear elasticity with diluteness assumptions).
The extension from two dimensions to three dimensions in the context of Ginzburg-Landau models is based on a slicing argument [ABO05,BJOS13] which cannot be directly used for the anisotropic vectorial problem of elasticity, also because of the degeneracy arising from frame indifference and the relaxation of the line-tension energy. Therefore the question whether our asymptotic analysis can also be obtained without the diluteness condition in dimension three may be very difficult. An intermediate step in this direction is the case in which the kinematics is restricted to a plane. Here the model reduces to a nonlocal phase-field model and was studied without diluteness assumptions in both regimes in [GM06, CG09, CGM11, CGM16, CGM20].

Line dislocations in linear elasticity
We first collect some results concerning the continuum study of dislocations in linear elasticity. Precisely, we recall the definition of the strain field in the presence of defects and its properties, then we give a characterization of the strain field for a straight infinite dislocation and of the associated line tension per unit length.
As usual, we assume that C ∈ R 3×3×3×3 sym is degenerate on linearized rotations and is coercive on symmetric matrices, in the sense that Working in R 3 , dislocations are measures concentrated along one-dimensional sets. More precisely a distribution of dislocations in Ω ⊂ R 3 is a measure µ in the set M 1 (Ω) defined below.
Definition 2.1. We denote by M(Ω; V ), with V a finite-dimensional vector space, the set of finite V -valued Radon measures on the open set Ω ⊆ R 3 . We write M 1 (Ω) for the set of µ ∈ M(Ω; R 3×3 ) which obey div µ = 0 distributionally and have the form The condition div µ = 0 automatically implies that τ is tangent to γ µalmost everywhere. We refer to the multiplicity θ in (2.2) as the Burgers vector of the dislocation line γ.
distributionally. The solution β satisfies and such that the unique solution β satisfies (for x ∈ supp µ).
(iii) If additionally µ ∈ M 1 (R 3 ) and µ = i b i ⊗ t i H 1 γ i for countably many segments γ i , then for x ∈ supp µ If the number of segments is finite then β ∈ L p (R 3 ; R 3×3 ) for all p ∈ [3/2, 2).
The constant c depends only on C.
We remark that in point (iii) of [CGO15, Theorem 4.1] the vectors b i were required to be in a lattice. This assumption was never used in the proof. Assertion (ii) is proven in (4.7) and the following equations of [CGO15].

Characterization of the strain for straight dislocations
From the representation formula (2.6) one also obtains a representation for the strain field of an infinite straight dislocation of Burgers vector b along the line Rt, for some t ∈ S 2 and b ∈ R 3 , which is then a distributional solution of (2.8) In the entire paper we denote by B r (x) and B r (x) the balls of radius r > 0 centered at x in R 3 and R 2 respectively. For x = 0 we simply write B r and B r .
where γ R is the union of the segment [−Rt, Rt] with a half-circle which has [−Rt, Rt] as diameter and τ : γ R → S 2 is a tangent vector to γ R , oriented so that it coincides with t in [−Rt, Rt]. One easily checks that µ R ∈ M 1 (R 3 ) and µ R * µ as R → ∞.
Equation (2.16) in particular shows that β b,t does not depend on the choice of the matrix Q t .
Proof. We need to prove thatβ as defined in (2.9) coincides with β b,t . This involves two steps: we first show thatβ has the same structure as β b,t , and then that it obeys the Euler-Lagrange equations of the variational problem that defines β b,t , which has a unique solution by [CGO15, Lemma 5.1].
We claim that the definition ofβ implies that there is G ∈ C 1 per ([0, 2π]; R 3×3 ) such that for any r > 0, θ and z, with Φ t as in (2.12), To see this, we observe that by definition Φ t (r, θ, z) = rQ t e r (θ) + zt, so that the definition ofβ yieldŝ where we used the change of variables s := (s − z)/r and (2.5). As the integral depends only on θ, and the dependence is smooth, (2.18) is proven. Next we show that the condition thatβ is locally curl free away from Rt implies that G has a special structure. We compute DβQ t e r = − 1 r 2 G , DβQ t e θ = 1 r 2 G , DβQ t e 3 = 0.
As above, we compute and, computing again the integral and recalling that R ψ 3 dz = 0 and Therefore B = 0 and the proof of the first assertion is concluded.
It remains to prove (2.17). This follows immediately from the fact that N is −2-homogeneous and continuous, . (2.27)

Line tension energy of straight dislocations
The variational problem that characterizes β b,t provides the energy per unit length of the straight infinite dislocation, ψ C (b, t), which we will call the unrelaxed line-tension energy density. Indeed using β b,t we can compute the elastic energy induced by a straight dislocation along Rt in a cylinder with axis Rt. Precisely given r, R, h ∈ (0, ∞) with r < R we define the hollow cylinders (2.28) and the full cylinders We recall that B ρ denotes the 2-dimensional disk of radius ρ > 0, B ρ := {(x 1 , x 2 ) : x 2 1 + x 2 2 < ρ 2 }, and that before (2.12) we introduced Q t as a fixed rotation with Q t e 3 = t. Therefore using the radial structure of β b,t , from (2.15) we have With this energy density one can associate to any µ ∈ M 1 (Ω) the linetension energy γ ψ C (θ, τ )dH 1 . (2.31) As shown in [CGM15], this functional may not be lower semicontinuous and needs to be relaxed (see Section 3.3 formula (3.42) below). A variational characterization of ψ C , in which the elastic energy is minimized over cylinders, has been obtained in [CGO15, Lemma 5.5]. To explain it, for C as in (2.1), b ∈ R 3 , t ∈ S 2 , h ≥ R > r > 0 we define (2.32) Then ψ C arises as limit of Θ for r → 0. We improve over [CGO15,Lemma 5.5] removing one error term, and prove the following.
Lemma 2.5. Assume that C obeys (2.1). (2.33) (ii) There is c > 0, depending only on C, such that (iii) There is c > 0, depending only on C, such that for all b ∈ R 3 , t, t ∈ S 2 , we have (2.36) Proof. (i): Since any function u ∈ W 1,1 (Q t T R,r h ; R 3 ) can be extended to a function in (2.37) (2.38) Moreover for R > R > r it holds Taking the limit as r → 0, since lim r→0 for all R > 0, and therefore for R = 0. At the same time, by (2.30) we obtain Θ(b, t, h, R, r) ≤ ψ C (b, t) for all r ∈ (0, R), and the proof of (2.33) is concluded.
(ii): The bounds in (2.34) follow from [CGO15, Lemma 5.1(iii)] (the upper bound can be also easily proven inserting the bound in (2.17) into (2.30) and integrating). The remaing bound (Eq. (2.35)) is proven using Lemma 2.4 and (2.30). We first observe that by (2.16) we have β b,t = β b ,t + β b−b ,t . Therefore for any δ > 0, fixing some r < R ≤ h from (2.30) we obtain (2.39) In turn, by the upper bound in (2.34) we have 3 Model and main results

Dislocations in finite kinematics
We work within the general framework of continuum mechanics, using finite kinematics. In the presence of dislocations, there is no smooth bijection between the reference configuration and the deformed configuration. Therefore any choice of a reference configuration includes a high degree of arbitrariness. We use spatial variables, with an energy obtained integrating over the deformed configuration Ω, and focus on the strain β : Ω → R 3×3 , with β(x) seen as a linear map from the tangent space to the deformed configuration to the tangent space to the reference configuration, or the so-called (fictitious) "intermediate configuration" or "lattice configuration". Precisely, for x ∈ Ω (a point in the physical space occupied by the material) and t ∈ S 2 a direction in the same space, β(x)t is the corresponding vector in the intermediate configuration, as illustrated in Figure 1. It is then natural to expect that the line integral of β over a closed (spatial) curve results in a lattice vector, the Burgers vector. Therefore we consider the distributional rotation curl β as a measure of the density of dislocations. Indeed, this is how normally the Burgers circuit argument is formulated, see for example [HB11, Fig. 1.19], [AB00, (7)], or [LM10] for a mathematical treatment. We can relate curl β to the dislocation density tensor as used in continuum mechanics. Within the customary multiplicative decomposition of the strain into an elastic and a plastic part F = F e F p , the density of dislocations (in the sense of the Nye tensor) is given by where curl denotes the rotation in the spatial configuration, whereas Curl denotes the rotation in the material configuration. We refer for example to [CG01,Sve02,MM06,RC14] for these formulas and their equivalence (to compare formulas it is useful to keep in mind that in [CG01] curl is defined columnwise instead of rowwise as it is here, and that cof F = F −T det F for any invertible matrix F ). In this language, the variable β we use corresponds to F −1 e , seen as a spatial field. Whereas α measures the density of (geometrically necessary) dislocations seen as the total Burgers vector crossing a certain area in the reference configuration, curl β measures the total Burgers vector crossing a certain area in the deformed configuration. We prefer this version for consistency with the fact that we work in the spatial representation, and because curl β (at variance with curl β cof β −1 ) can be easily understood distributionally.
In order to correctly formulate the variational model it is important to understand how frame indifference acts in this setting. In the usual material formulation of (dislocation-free) nonlinear elasticity, one considers a (bijective) deformation field u : ω → Ω, with ω ⊆ R 3 the reference configuration and Ω ⊆ R 3 the deformed (spatial) configuration. In this simple setting, β : Ω → R 3×3 is defined by β(u(x)) = (Du(x)) −1 , and it is the gradient of the map v : Ω → ω which is the inverse to u. Superimposing a rotation amounts to replacing u by u Q (x) := Qu(x), u Q : ω → QΩ, so that the deformation gradient Du(x) gets replaced by . Differentiating this expression one obtains Dv Q (Qy)Q = Dv(y), which is the same as This expression shows the action of rotations on the field β. The above computation based on the chain rule shows that the multiplication of β by a rotation on the right is naturally coupled to a change of variables by the same rotation. An hyperelastic material can then be modeled by the energy and material frame indifference leads to the requirement With a change of variables, one can relate this expression to the usual integration over the reference configuration: whereŴ (F ) = W (F −1 ) det F . One immediately sees that (3.4) is equivalent to the usual right-invarianceŴ (QF ) =Ŵ (F ), and the requirement that If the bijective map u does not exist globally, the same procedure can be performed locally, away from the dislocation cores (or using the intermediate configuration). In the spatial formulation, we consider β : Ω → R 3×3 . It maps directions t in the spatial configuration (in Ω) onto directions βt in the (fixed) lattice configuration. If we insert a rotation, mapping Ω to QΩ, the direction t becomes Qt, and a point y ∈ Ω becomes Qy ∈ QΩ. However, the vector in the lattice configuration is not modified, therefore necessarily β Q (Qy)Qt = β(y)t for all t ∈ R 3 , which is the same as (3.2).
We next address how the distribution of dislocations, understood as curl β, transforms under rotations, and how the line energy ψ C transforms. The key fact, in terms of the field introduced in (3.2), is (curl β Q )(Qy)Q = curl β(y).
(3.6) Figure 1: A Burgers circuit in finite elasticity. Left: spatial (deformed) configuration. A closed path goes around the dislocation. The path is composed of finitely many segments t i that join an atom to a neighbour. Right: representation in the reference (more precisely, intermediate) configuration. Here the atoms are exactly on the reference Bravais lattice, each of the segments t i is mapped to a corresponding lattice vector which (in the continuum limit) is βt i . The path does not close, the sum of the segments is the Burgers vector, which is naturally an element of the (undeformed) crystal lattice. Therefore the contour integral of β equals the Burgers vector of the dislocation.
If curl β is a (matrix-valued) measure, then so is curlβ, and for any Borel (iii) For any C which obeys (2.1) and any Q ∈ SO(3) the line-tension energy where C Q is defined by (3.10) We remark that (3.7) is the same as Proof. (i): We first observe that (summing over repeated indices) , and e i ∧ e i+1 = e i+2 (with indices taken modulo 3). We now compute (again with implicit sums)β ak (x) = F ab β bp (λQx + v)Q pk and (3.14) the general case follows by density. Further, We recall (2.32), which can be equivalently written as For any β as above we defineβ bŷ , and by (ii) we have With a change of variables (3.19) and taking the infimum over all choices of β gives (3.20) Taking the limit r → 0 with Lemma 2.5 (i) we obtain and analogous computation proves the converse inequality.
We remark that in the geometrically linear setting one identifies the reference with the deformed configuration, hence the discussion above becomes largely irrelevant. The linearization procedure is based on considering deformations u close to the identity, in the sense that u(x) = x + δv(x), with v the scaled displacement and δ the linearization parameter. One then introduces the strain measure ξ(x) := Dv(x), computes F e (x) = Id + δξ(x) and β(x + δv(x)) = (Id + δξ(x)) −1 . A Taylor series for δ → 0, if all fields are regular, gives β(x) = Id − δξ(x) + O(δ 2 ), and in the linear theory the strain measure is identified with ξ(x) = − β(x)−Id δ . In the presence of dislocation, the passage through u and v can only be done locally, but linearization still amounts to ξ = − β−Id δ . Obviously curl ξ = − 1 δ curl β is then the easiest measure of the dislocation density. If one uses a quadratic energy the prefactor δ can be dropped, and the minus sign amounts to a minor change in the definition of the Burgers vector.
In this paper we provide a unified mathematical approach for the linear and the nonlinear setting, using different assumptions on the energy density W presented in the next section.

Main assumptions
We introduce the class of dilute dislocations identified with a class of divergencefree measures concentrated on polyhedral curves.
Definition 3.2. Given two positive parameters α, h > 0 and an open set Ω ⊆ R 3 we say that a polyhedral curve γ ⊂ Ω is (h, α)-dilute if it is the union of finitely many closed segments s j ⊂ Ω such that (i) each s j has length at least h; (ii) if s j and s i are disjoint then their distance is at least αh; (iii) if the segments s j and s i are not disjoint then they share an endpoint, and the angle between them is at least α; (iv) γ does not have endpoints inside Ω.
The diluteness condition given in the Definition 3.2 allows to define the set of compatible configurations. The asymptotic analysis will be performed assuming that the diluteness parameters h and α are much larger then the lattice spacing ε, in a sense made precise in (3.38) below.
Given a Bravais lattice B (i.e., a set of the form We consider an elastic energy density W in one of two natural frameworks. To simplify notation in the presence of a mixed growth condition we define for (3.22) Assumption H W Finite : In a geometrically nonlinear setting, W : R 3×3 → [0, ∞) is minimized at the identity and is invariant under the right action of SO(3), We assume also that W is Borel, twice differentiable in a neighbourhood of Id, and for some p ∈ (1, 2] and c > 0.
Remark. If H W Finite holds, then the tensor satisfies condition (2.1), and there is a modulus of continuity ω : It is easy to see that it also obeys condition (2.1).
Assumption H W Lin : In a geometrically linear setting, W : R 3×3 → [0, ∞) is minimized at zero and only depends on the symmetric part of its argument, We assume also that W is Borel, twice differentiable in a neighbourhood of 0, and 1 for some p ∈ (1, 2] and c > 0.
Remark. If H W Lin holds, then the tensor satisfies condition (2.1). It will be useful to have some properties of Φ p . We denote by Φ * * p its convex envelope, which obeys for some c p > 0.
To see this, it suffices to set a := f χ {|f |≤1} , and b := f − a.

Γ-convergence to a line tension model
We now introduce a small parameter ε > 0, which in this semidiscrete model represents the lattice spacing, and (given an open set Ω ⊆ R 3 and a Bravais lattice B ⊂ R 3 ) the class of admissible configurations (3.37) and write briefly F ε [β] := F ε [β, Ω]. The asymptotic analysis will be performed for any diluteness parameters α ε and h ε that obey For pairs in the set of admissible configurations A * ε we shall use the following notion of convergence.
and there are Q ε ∈ SO(3) such that Q ε → Q and The local weak- * convergence in (3.39) can be equivalently defined testing with elements of C 0 c (Ω). In what follows, for b ∈ B and t ∈ S 2 , the function ψ rel C (b, t) denotes the H 1 -elliptic envelope of ψ C (b, t) and it is given by In [CGM15] it is proven that ψ rel C provides the energy density of the relaxation of the line tension energy given in (2.31). By a change of variables, one easily sees that the same holds for the functionψ(b, t) := ψ(b, Qt), for any Q ∈ SO(3).
The main result of the paper is then the following compactness and Γconvergence statement in the subcritical regime p < 2.
Theorem 3.6. Let Ω ⊂ R 3 be a bounded Lipschitz set and assume that (h ε , α ε ) obeys (3.38). Assume also that W obeys H W Finite for some p ∈ (1, 2). Then the functionals Γ-converge, with respect to the convergence in finite kinematics with p growth in the sense of Definition 3.5, to ⊗ tH 1 γ and curl η = 0, and ∞ otherwise, C Q as in (3.25) and (3.28), ψ rel C as in (3.42). Further, if Ω is connected then any sequence with F subcr ε [µ ε , β ε ] bounded has a subsequence that converges in the same topology.
If instead W obeys H W Lin , then the corresponding assertions hold with respect to convergence in infinitesimal kinematics with p growth, with (3.44) replaced by We remark that there are several equivalent ways of treating the sequence of rotations Q ε → Q. One simple observation is that by Lemma 3.1(iii), , which can be inserted in the second term of (3.44). One can also replace C Q by C in the first term, if the fields are redefined accordingly; in order to keep the differential constraints one should also rotate the domain of integration. Specifically, given a sequence (µ ε , β ε ) ∈ A * ε (Ω) (defined as in (3.36)) that converges as above, for each ε one considers the pair (the fact that the domain changes along the sequence is not a problem for L q loc convergence since Q ε → Q). The limiting functional in (3.44) then takes the form andt the unit tangent toγ := Qγ. For simplicity we stick to the formulation in which the integration domain Ω is fixed. Related results have been proven before in [CGO15,GMS21]. We present here a more general argument that permits to prove Theorem 3.6 in a unified way for finite and infinitesimal kinematics. Our argument also provides the Γconvergence result for different types of core regularizations, which are needed in the case p = 2 and are discussed in Subsection 3.4.
In Section 4 we give a unified treatment for the cell problem formula. Building on this, compactness and lower bound are then proven in Section 5. In particular, we shall introduce in (5.21) an auxiliary functional, for which we will prove the lower bound result. The auxiliary functional is chosen so that, after rescaling, it is below F subcr ε , with corresponding bounds holding for other regularizations as discussed below, and then all results follow at once. Analogously, in Section 6, the upper bound will be proven for a second auxiliary functional defined in (6.32), which after rescaling is larger than F subcr ε , up to a small error term which can be controlled. Moreover we show that the recovery sequence needed for this upper bound has a good decay near the dislocation line, so that different types of core regularization do not change its energy asymptotically.
In Section 7 we will collect the proofs of the Γ-convergence results (Theorem 3.6 and Theorem 3.10 below) that are obtained as straightforward consequences of the results proved in Section 4, Section 5 and Section 6. All proofs are given explicitly in the case H W Finite . The case H W Lin is very similar therefore we only point out a few relevant differences.

Extension to different core regularizations
In the case of quadratic growth (in the sense p = 2) one needs a different regularization and possibly a different set of admissible configurations. This requires different convergence properties and compactness results. The Γ-convergence result instead can be obtained as a consequence of the arguments developed to prove Theorem 3.6.
In the literature one considers configurations (µ, β) ∈ M 1 (Ω) × L 1 (Ω; R 3×3 ) with curl β = µ and the associated energy where (supp µ) ε := {x ∈ Ω : dist(x, supp µ) < ε}. This is known as the coreregion approach. Alternatively, one can replace the condition curl β = µ with a different condition that only involves the behavior of β away from the singularity, that we will call ρ-compatibility of the pair (µ, β), see Definition 3.7 below. Since the energy (3.49) does not depend on the value of β inside the core region, one can show that the asymptotics of the rescaled energy does not depend on the set of admissible configurations chosen. Nevertheless for the compactness result an extension argument is needed, which goes beyond the scope of the present work. A third option is a regularization via mollification of the measure, which smears out the singularity on a scale ε.
where βμ is the solution obtained fromμ via Theorem 2.2.
In the set of admissible configurations A core ρε,ε (with ρ ε → 0) we introduce the following convergence, which is a variant of the one in Definition 3.5. The key difference is that now the value of β ε in the core region (supp µ ε ) ρε is ignored.
and there are Q ε ∈ SO(3) such that Q ε → Q and loc (Ω; R 3×3 ) in infinitesimal kinematics with p growth and radius ρ ε if ρ ε → 0, (3.56) holds, and We then have the following Γ-convergence result which includes the critical case p = 2.
(ii) The same holds also for the functionals otherwise.
(3.61) (iii) The same holds also for the functionals with respect to the convergence of Definition 3.5.
If instead W obeys H W Lin , then the corresponding assertions hold with respect to convergence in infinitesimal kinematics, with F Finite replaced by F Lin defined in (3.45).
We remark that this result does not contain a compactness statement. This requires different arguments and will be addressed elsewhere. Only in the case of assertion (iii) compactness can be obtained from the rigidity estimate as in the case of Theorem 3.6.

Straight dislocations in a cylinder
This section deals with the cell problem, which describes a single straight dislocation in a cylinder. We first present in Lemma 4.1 a rigidity estimate with mixed growth for an hollow cylinder, then in Lemma 4.2 the standard coercivity statement which shows the origin of the logarithmic divergence. Afterwards, in Proposition 4.3, we prove a lower bound, relating the energy in a cylinder around a straight dislocation to the line-tension energy ψ C defined in (2.15).

Rigidity and coercivity
The first result shows that the constant in the rigidity estimate with mixed growth from [CDM14, Theorem 1.1] for domains T R,r h with r ≤ 1 2 R and R ≤ h can be estimated with h 2 /R 2 , and in particular does not depend on r. We recall the definition of the tubes in (2.28) and (2.29), and the definition of the function Φ p in (3.22) and its properties (see Remark 3.4).
Lemma 4.1. For any p ∈ (1, 2] there is c = c(p) > 0 such that for any r, R, h > 0 with 2r ≤ R ≤ h, any Q ∈ SO(3), and any β ∈ Proof. We first show that it suffices to prove both assertions for Q = Id.
. Therefore the assertions forβ translate in the desired estimates for β. In the rest of the proof we deal with Q = Id. By scaling we can assume R = 1. We next show that it suffices to prove both assertions for h = 1. Indeed, given h > 1 we fix 0 = z 0 < z 1 < · · · < z N = h − 1 such that z i ≤ z i−1 + 1 2 for 1 ≤ i ≤ N , z i ≥ z i−2 + 1 2 for 2 ≤ i ≤ N , and N = 2h − 2 (for example, z i = i/2 for i < N , and z N = h − 1). We then apply (4.1) to each cylinder T 1,r 1 + z i e 3 and obtain matrices Q i ∈ SO(3) such that (4.6) Using z i ≥ z i−2 + 1 2 we see that the overlap is finite, hence (4.7) By (3.33) with δ = 1/N , for 1 ≤ i ≤ N we have and iterating Therefore, using i ≤ N and (4.7), (3)))dx (4.10) for each i ∈ {0, . . . , N } and, since N ≤ 2h, we conclude, applying (3.34), (4.10) and (4.4), (4.11) It remains to prove (4.1) in the case R = h = 1, Q = Id. Assume first that r = 2 −N for some integer N ≥ 1. If N = 1 then, since curl β = 0, using [CDM14, Theorem 1.1] on two overlapping simply connected subsets of T 1,1/2 1 and then (3.34) one obtains that there is Q ∈ SO(3) such that (4.12) The argument is the same in the linear case, using [CDM14, Theorem 2.1]. For the same reason, for any ρ > 0 and any β ∈ L 1 (T (4.13) by scaling the constant does not depend on ρ.
For the same reason, (0, 1) = (0, 2 N r) ∪ (1 − 2 N r, 1). Therefore We apply the rigidity estimate to each of these three cylinders separately and obtain rotations Q 1 , Q 2 , Q 3 . As the overlap between them is not uniformly controlled we introduce a fourth cylinder. We let ρ := 2 −N +r 2 be the average between the two values of the inner radius and set We then apply the rigidity estimate also to C 4 , and obtain another rotation (4.20) The constant depends only on W , p, and B.
Proof. We prove the statement for the case h = R. If h > R, it suffices to apply this bound on each of the h/R disjoint subsets of T R,r h which have the same shape as T R,r R , and then to use that h/R ≤ 2 h/R . For b = 0 there is nothing to prove. Recalling that b ∈ B, we see that we can assume |b| ≥ c > 0.
Since curl β = 0 in T R,r R , by H W Finite and Lemma 4.1 there is a rotation Q ∈ SO(3) such that (4.21) In the linear case the same holds with S ∈ R 3×3 skew . Moreover by condition (4.19), for every circle C with radius ρ ∈ (r, R), centered in a point of the segment (0, Rt), and contained in the plane orthogonal to t, ε|b| ≤ C |β − Q|dH 1 . (4.22) By monotonicity and Jensen's inequality, we then have where Φ * * p is the convex envelope of Φ p . Integrating over all circles C by Fubini's theorem we easily get (4.24) If p = 2 then Φ * * 2 (t) = Φ 2 (t) = |t| 2 and integrating the left-hand side concludes the proof.
It remains to estimate the left-hand side of (4.24) for p < 2. By (3.32) there are η, η > 0 such that (4.26) We observe that s 2−p ≥ (2 − p) log s for all s > 0. Using that ε ≤ r < R, Combining these two concludes the proof in this case. Finally we consider the case 2r < ε|b| < R. We have Estimating the two integrals as in (4.25) and (4.26) and observing that for some c p > 0 it holds, since |b| ≥ log |b|, with ε ≤ r we conclude.

Lower bound for a cylinder
We present in Proposition 4.3 the key estimate for the lower bound. We work in a unified framework able to treat jointly the case of finite and infinitesimal kinematics and therefore frame the problem via a generic energy density V . In the rest of the paper we use the results of this section only via Corollary 4.4.
(in the sense of Definition 3.7, for the domain Q t T R h ) and Q ∈ SO(3) then (4.32) Assume that W satisfies H W Lin for some p ∈ (1, 2] and let C := D 2 W (0). Then the same holds by replacing (4.32) with (4.33) Proof. Assume H W Finite holds. Without loss of generality, R ≤ h ≤ 2R. Otherwise we decompose the cylinder in h/R pieces with height h ∈ [R, 2R), apply the result to each of them and sum. By r-compatibility, there are an extensionμ ∈ M 1 (R 3 ) of µ and β 0 ∈ L 1 (Q t T R h ; R 3×3 ) with curl β 0 = 0 in Q t T R h and β = βμ + β 0 in Q t T R,r h . We use βμ + β 0 to define an extension (4.34) By Lemma 2.5 there is c u > 0 such that ψ C (b, t) ≤ c u |b| 2 for all b and t. If QtT R,r h W (β)dx ≥ c u ε 2 h|b| 2 log R r , we are done. Therefore we can assume the converse inequality holds. By the rigidity estimate in Lemma 4.1 and (3.24), there is a rotation Q * ∈ SO(3) such that We define so that V (ξ) = W (βQ T * ) = W (β) and D 2 V (0) = C Q * , which was defined from C = D 2 W (Id) in (3.28). We observe that (4.34) gives which is (4.29) with Q T * b in place of b. Equation (4.35) implies (4.28) with c K := cc u , which depends only on C. By Proposition 4.3 with 0 := min{|b| : b ∈ B \ {0}} and the above estimates we obtain (4.38) . By Lemma 2.5(iii) there is c > 0, depending only on C, such that (4.41) From (4.35), treating separately the part with |β − Q * | < 1 and the one with |β − Q * | ≥ 1, we have where p is the conjugate of p (i.e., p = p p−1 ). Therefore, recalling that h ∈ [R, 2R] and that log t ≤ t 2 for all t > 0, (4.43) Plugging (4.43) into (4.41) we easily obtain (4.32). The proof of (4.33) is similar but simpler. Indeed, one obtains from Lemma 4.1 a matrix S * ∈ R 3×3 skew with the property corresponding to (4.35). One then sets ξ := β − S * , which obeys curl ξ = curl β. Therefore Proposition 4.3 gives, using , and (4.33) follows.
Proof of Proposition 4.3. We first show that there is c ≥ 1 such that (4.44) To prove this, assume first H W Finite , and to shorten notation let a :=QaQ T , b :=QbQ T . From (3.24) we obtain (4.45) Using dist(Id + a + b , SO(3)) ≤ dist(Id + a , SO(3)) + |b |, monotonicity of Φ p , (3.34) and then (3.24) again we have ≤ 2cΦ p (|a|) + 2cΦ p (|b|) ≤ 2c 2 V (a) + 2cΦ p (|b|). (4.47) This concludes the proof of (4.44). We reason by contradiction and assume that there exist ρ > 0 and sequences r j , R j , h j , ε j , b j , t j , ξ j as in the statement such that, writing Q j : and Since ψ C V is positively two-homogeneous in the first argument (see (2.15) and (2.16), or [CGO15, Lemma 5.1(iii)]), (4.51) After passing to a subsequence, we can assume that b j |b j | → b * and Q j → Q * , for some b * ∈ S 2 , Q * ∈ SO(3); this implies t j = Q j e 3 → t * := Q * e 3 . By continuity of ψ C V (which follows from (2.15) and (2.16)), (4.52) Choose now λ > 0 such that λc K ≤ ρψ C V (b * , t * ). By (4.52) and (4.49), (4.53) Let δ ∈ (0, 1 2 ] be fixed. We observe that the sets In particular for any such k we have (4.55) We choose k j (depending on j) such that and then i j (depending on j) with (4.57) Inserting (4.57) in (4.53) and taking the limit, using R j /r j → ∞ and that h j ≥ R j and k j ≥ 1 imply (4.58) We now rescale to a fixed tube T 1,δ 1 . As in the entire proof, the maps are defined and integrable over the entire cylinder T 1 1 , but the estimates are only on the restriction to the tube T 1,δ 1 . Precisely, we defineξ j ∈ L 1 (T 1 1 ; R 3×3 ) bȳ and observe that it obeys (see Lemma 3.1(ii), and recall that Q T j t j = e 3 ) (4.60) Using (4.58), with a change of variables we obtain lim sup and, for sufficiently large j, In the rest of the argument we shall use (4.60), (4.61), (4.62) and the definition of ψ C V to reach a contradiction. By Remark 3.4(ii) and (4.62) there are A j , B j : T 1,δ 1 → R 3×3 such that |A j | ≤ 1 everywhere, |B j | ≥ 1 on the set {B j = 0}, and for sufficiently large j Recalling that ε j |b j |/r j → 0 and that r j /R j → 0, we compute with (4.55) From (4.64) we obtain A j → 0 in L 2 (T 1,δ 1 ; R 3×3 ) and B j → 0 in L p (T 1,δ 1 ; R 3×3 ). We defineÂ j ,B j ∈ L 1 (T 1,δ 1 ; R 3×3 ) bŷ (4.66) With (4.64) and (4.65) we obtain that (4.67) Passing to a further subsequence,Â j Â * weakly in L 2 (T 1,δ 1 ; R 3×3 ). Let η ∈ (0, 1) be fixed and set We recall that A j → 0 in L 2 and B j → 0 in L p . In particular, they converge to zero in measure, and therefore L 3 (T 1,δ 1 \ E η j ) → 0. On E η j we have B j = 0 and in particularB j =B j χ T 1,δ 1 \E η j 0 in L p (T 1,δ 1 ; R 3×3 ). By the differentiability of V in a neighbourhood of the origin, if η is sufficiently small we have where C V := D 2 V (0) and ω : [0, ∞) → [0, ∞) is monotone, continuous, with ω(0) = 0. We shall use this estimate with F = Q j A j Q T j on E η j . As F = 0 is a minimizer of V , V (0) = 0 and DV (0) = 0, so that which is the same as (4.73) for every η ∈ (0, 1) sufficiently small and therefore for η = 0. Hence recalling (4.61), and with C V Q * defined from C V as in (3.28), for any δ ∈ (0, 1 2 ] we have To conclude it remains to relate the left-hand side to ψ C . We shall show below that there is u * ∈ W 1,1 (T 1,δ 1 ; R 3 ) such that with β b,t defined as in Lemma 2.4. We recall the definition of Θ in (2.32), and see that Du * + β Q T * b * ,e 3 is an admissible test field, so that (4.74) implies (4.76) Taking the limit δ → 0 with Lemma 2.5, we obtain Recalling that by Lemma 3.1(iii) one has ψ C V Q * and that Q * e 3 = t * gives the desired contradiction.
The cell problems from [CGO15,GMS21] are all immediate consequences of Proposition 4.3, via Corollary 4.4. As an easy consequence one can also prove a lower bound when the dislocation is straight.

Compactness and lower bound
In this section we provide the proofs of the compactness and the lower bound needed to prove Theorem 3.6. The proof is similar to the one of [CGO15, Prop. 6.6] using Proposition 4.3 (via Corollary 4.4) instead of [CGO15, Lemma 5.8].
We discuss the various arguments separately in order to be able to reuse them.

Rigidity
We first recall a rigidity result from [CG21], which is based on the Friesecke-James-Müller geometric rigidity [FJM02] and on the Bourgain-Brezis critical integrability bound [BB07], and extend it to smaller exponents.

(5.2)
Proof. The important case is the critical exponent q = 3 2 , which was proven in [CG21]. The case q ∈ [1, 3 2 ) follows from the same argument, using s = q in the proof of Theorem 1 in [CG21], and using Hölder's inequality in (8). Then after scaling Eq. (11) becomes q −2 is uniformly bounded since Q r ⊆ Ω. We remark that the second term in [CG21, Eq. (11)] incorrectly contains Du instead of β. The rest of the argument does not make use of the specific value of s and is unchanged. The modification in the linear case is the same.

Compactness
We recall the definition of the class A * ε of admissible pairs in (3.36) and that of the elastic energy E elast [β, A] in (3.37) for fields β ∈ L 1 (Ω; R 3×3 ) and Borel sets A ⊆ Ω.

(5.4)
Then there is c > 0 such that for any Ω ⊂⊂ Ω lim sup Moreover there are µ ∈ M 1 B (Ω) and a subsequence ε k such that Remark 5.3. Proposition 5.2 still holds true if we replace A * ε with A core ρε,ε or with A moll ε defined in (3.50) and (3.51) respectively. Indeed, in both cases is suffices to extend (separately for each i) the restriction β ε | T i ε \T i ε to a functioñ β ε in L 1 (T i ε ; R 3×3 ) with curlβ ε = µ T i ε as was done at the beginning of the proof of Corollary 4.4.

7)
A i ε an affine isometry that maps S i ε e 3 into γ i ε and the midpoint of S i ε e 3 to the midpoint of γ i ε . We define the cylinders Disjoint segments in the family {γ i ε } i are separated by α ε h ε R ε . Since the angle between two non-disjoint segments is at least α ε , and for small ε we have δ ε tan 1 2 α ε > R ε we obtain that the sets {T i ε } i are pairwise disjoint and T i ε ∩γ j ε = ∅ for all j = i. Further, for ε sufficiently small we have 2h (5.9) By Lemma 4.2, Since Therefore there is a subsequence such that ε −1 µ ε converges to a limiting measure µ ∈ M(Ω ; R 3×3 ), with |µ|(Ω ) ≤ cE.
Assume W obeys H W Lin for some p ∈ (1, 2]. Then the same holds with respect to the convergence in infinitesimal kinematics with p growth and radius ρ ε k in the sense of Definition 3.9 with ψ rel C (b, Qt) replaced by ψ rel C (b, t) and C Q by C = D 2 W (0).

Upper bound
A key technical result of this section, which permits a refinement of the argument used for the upper bound in [CGO15,GMS21] is the following statement. It shows that the upper bound for the unrelaxed problem can be taken at constant β.
Proposition 6.1. Let C obey (2.1). Let µ = b ⊗ tH 1 γ ∈ M 1 B (R 3 ) be polygonal, and let β ∈ L 3/2 (R 3 ; R 3×3 ) be the solution to curl β = µ, div Cβ = 0. (6.1) Let ν ε be the positive measure defined by Then ν ε converges weakly- * to In particular, if Ω ⊂ R 3 is an open set with |µ|(∂Ω) = 0, then Further, the positive measureŝ are uniformly locally bounded, in the sense that for all R > 0 one has and for any s ∈ (0, 1) obey Proof. Since µ is polygonal, we have that γ = ∪ i γ i , with the γ i being finitely many pairwise disjoint segments in R 3 . For the same reason, b ∈ L ∞ (γ, H 1 ; B). By Theorem 2.2(iii), for any x ∈ γ we have . (6.7) In particular, this shows that We fix an index i and estimate ν i ε separately near γ i and far away. Let δ, R > 0.
We recall that we denote by B r (x) and B r (x) the balls of radius r > 0 centered at x in R 3 and R 2 respectively; for x = 0 we simply write B r and B r . Let now x ∈ γ i , r > 0. Then ≤ 4πr + 4πr log (r/ε) log(1/ε) . (6.10) Note that the first integral in the first inequality of (6.10) is needed to estimate ν i ε (B r (x)) in the case that x is near the endpoints of γ i . For any δ ∈ (0, H 1 (γ i )) we can cover (γ i ) δ with at most 3H 1 (γ i )/δ balls B 2δ (x k ), with x k ∈ γ i . Therefore, using (6.10) with r = 2δ and (6.9), for all R > 0. Recalling (6.8), this proves (6.5).
For any s ∈ (0, 1) the same computation, using δ = ε s in (6.10), shows that lim sup (6.12) and similarly lim sup ε→0 ν i ε ((γ j ) ε s ) = 0 for j = i. This proves (6.6). By (6.11), after taking a subsequence we can assume that for each i ν i ε * ν i 0 for some Radon measure ν i 0 . By (6.9) we obtain ν i 0 (B R \ (γ i ) δ ) = 0 for any δ and R; taking δ → 0 and R → ∞ this gives ν i 0 (R 3 \ γ i ) = 0. By (6.10) we obtain Therefore ν i 0 H 1 γ i . Summing over i (and possibly taking a further subsequence), ν ε ν 0 with ν 0 H 1 γ. In order to conclude the proof it suffices to show that for any i we have for H 1 -almost every x ∈ γ i , (6.13) where we recall that b i ∈ B and t i ∈ S 2 are defined by µ = i b i ⊗ t i H 1 γ i . For x ∈ γ i we denote by T x r a cylinder with axis parallel to t i , radius r > 0 and height 2r centered at x, where as usual Q t i is an element of SO(3) with Q t i e 3 = t i . By the Radon-Nikodým theorem, for H 1 -almost every x ∈ γ i there is a sequence r k → 0 such that ν 0 (∂T x r k ) = 0 and .
We can assume that x is not an endpoint of γ i . For sufficiently large k, , the definition of ν ε , and dist(x, γ j ) > 0 for j = i, (6.14) By Theorem 2.2(ii) Assume first j = i, so that dist(x, γ j ) > 0. Since |N (z)| ≤ c/|z| 2 , we have |β j | ≤ c in T x r k for sufficiently large k. The term β i instead is compared with β b i ,t i . By Lemma 2.4 we have that which implies, using again |N (z)| ≤ c/|z| 2 , for sufficiently large k (with c depending on x and i).
We next deal with a technical issue related to the intersection of γ and the boundary of Ω. Indeed, in the above proposition one has integrated over the complement of an ε-neighbourhood of the entire curve γ ⊂ R 3 , whereas in E ub ε one integrates over the complement of an ε-neighbourhood of the part of γ which is contained within Ω. It is apparent that this may generate difficulties if γ is tangential to ∂Ω, including the extreme case in which one of the segments composing γ is contained in ∂Ω. This situation is obviously not generic, and indeed we show in the next result that in the generic situation this boundary effect vanishes in the limit.
Proof of Lemma 6.3. (i): There are finitely many points in γ which belong to more than one segment, and for almost all choices of a none of them belongs to the 2-dimensional set ∂Ω. For the rest of the argument we can consider each of the segments composing γ individually. Analogously, since ∂Ω is covered by finitely many Lipschitz graphs, it suffices to prove the assertion for one of them. Therefore it suffices to show the following: if v ∈ S 2 , F : R 2 → R 3 is Lipschitz, and r > 0, then for L 3 -almost every a ∈ R 3 the following holds: the equation F (y ) ∈ a + vR : y ∈ B r (6.26) has finitely many solutions y 1 , . . . , y K , at each y k the function F is differentiable, with v not contained in the space spanned by ∂ 1 F (y k ) and ∂ 2 F (y k ). We remark that one can replace L 3 -almost every a ∈ R 3 with H 2 -almost every a ∈ v ⊥ . To see this, one considers the Lipschitz function G : B r → v ⊥ ⊆ R 3 defined by G(y ) := P v F (y ), where P v := Id−v ⊗v is the projection onto the space v ⊥ . For a ∈ v ⊥ , equation (6.26) can be rewritten as G(y ) = a. As F is Lipschitz, it is almost everywhere differentiable, and G({y ∈ B r : DF (y ) does not exist}) is an H 2 -null set. By Sard's Lemma, G({y ∈ B r : rank DG(y ) ≤ 1}) is also an H 2 -null set. We observe that, since DG = (Id − v ⊗ v)DF , DG has full rank exactly when v is not contained in the space spanned by ∂ 1 F and ∂ 2 F . By the area formula [AFP00, Th. 2.71] applied to the function G we obtain Pick c * ∈ (0, 1) with c * c < 1. We claim that for ε sufficiently small one has Ω ∩ (γ) c * ε \ (Ω ∩ γ) ε = ∅. (6.28) We prove (6.28) by contradiction. If not, there are ε k → 0, y k ∈ Ω, z k ∈ γ with |y k − z k | < c * ε k and dist(y k , Ω ∩ γ) ≥ ε k . As c * < 1 the last two conditions imply z k ∈ γ \Ω. By compactness ofΩ, passing to a subsequence we can assume that y k and z k converge to the same limit, which belongs to Ω ∩ (γ \ Ω) = ∂Ω ∩ γ. Therefore there is i ∈ {1, . . . , M } with z k → x i , which means that for sufficiently large k we have z k ∈ γ ∩ B ρ (x i ) and therefore z k = x i + t k v i for some t k → 0. By (6.19) we have This contradicts ε k ≤ dist(y k , Ω ∩ γ) ≤ |y k − x i | and concludes the proof of (6.28). We now observe that (6.28) implies Ω \ (Ω ∩ γ) ε ⊆ Ω \ (γ) c * ε and therefore For any s ∈ (0, 1), if ε is sufficiently small then (c * ε) s ≥ ε. Using first (6.4) and then (6.6), (6.31) As s ∈ (0, 1) was arbitrary, this concludes the proof of (6.17).
Proof. We prove the thesis in the finite case as the linear case can be treated similarly. As in [CGO15, Proposition 6.8], the proof is divided in two steps. In the first one we provide an explicit construction attaining the unrelaxed limiting energy, and under the additional assumption that η is essentially bounded. In the second step we conclude by using a diagonal argument.
By frame indifference and Taylor expansion near the identity there is ω with ω(δ) → 0 as δ → 0 such that where in the last step we used Lemma 3.1(iii). Since η ∈ L ∞ (Ω; R 3×3 ) and ξ ∈ L 1 (Ω; R 3×3 ) we derive that  As ω(δ k ) → 0, also the last error term vanishes in the limit.

Proofs of the main results
We start with the proof of our main result, Theorem 3.6.