# Strong existence for free discontinuity problems in linear elasticity

Manuel Friedrich, Camille Labourie, Kerrek Stinson

► To cite this version:

Manuel Friedrich, Camille Labourie, Kerrek Stinson. Strong existence for free discontinuity problems in linear elasticity. SIAM Journal on Mathematical Analysis, 2025, 57 (2), pp.1652-1679. 10.1137/24M1640847. hal-05044528

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# STRONG EXISTENCE FOR FREE DISCONTINUITY PROBLEMS IN LINEAR ELASTICITY

MANUEL FRIEDRICH, CAMILLE LABOURIE, AND KERREK STINSON

ABSTRACT. In this note we show Ahlfors-regularity for a large class of quasiminimizers of the Griffith functional. This allows us to prove that, for a range of free discontinuity problems in linear elasticity with anisotropic, cohesive, or heterogeneous behavior, minimizers have an essentially closed jump set and are thus strong minimizers. Our notion of quasiminimality is inspired by and generalizes previous notions in the literature for the Mumford-Shah functional, and comprises functions which locally close to the crack have at most a fixed percentage of excess crack relative to minimizers. As for minimizers of the Griffith functional treated by Chambolle, Conti, and Iurlano, our proof of Ahlfors-regularity relies on contradiction-compactness and an approximation result for GSBD functions, showing the robustness of this approach with respect to generalization of bulk and surface densities.

## 1. INTRODUCTION

The equilibrium state of a brittle elastic material can be found by minimizing the Griffith fracture energy, as introduced in the context of linearized elasticity by FRANCFORT AND MARIGO [17]. Letting $\Omega \subset \mathbb{R}^N$ be a bounded open set with Lipschitz boundary, representing the reference configuration of a solid, and $\mathbb{C}$ be a fourth-order elastic tensor, the Griffith energy is defined by

$$ \int_{\Omega} \mathbb{C}e(u) : e(u) \, dx + \mathcal{H}^{N-1}(J_u), \qquad (1.1) $$

for a displacement $u \in \text{GSBD}^2(\Omega)$, where $e(u) := (\nabla u + \nabla u^T)/2$ denotes the symmetrized gradient of $u$, $J_u$ is the associated crack set (i.e., the jump set of $u$), and $\mathcal{H}^{N-1}$ is the $(N-1)$-dimensional surface measure. The energy (1.1) is a weak formulation of the Griffith energy as it does not impose that the crack $J_u$ is closed. The strong formulation, instead, assumes that the cracks are closed in $\Omega$, and this ‘closedness’ acts as a basal regularity property preventing the crack from being diffusely distributed throughout the material domain.

The goal of this paper is to show that a wide variety of free discontinuity problems in linear elasticity with anisotropic, (non-degenerate) cohesive, or heterogeneous behavior have strong minimizers. Precisely, in Theorem 2.7 we show that a GSBD minimizer of the energy

$$ \int_{\Omega} f(x, e(u)) \, dx + \int_{J_u} \psi(x, u^{+}, u^{-}, \nu_u) \, d\mathcal{H}^{N-1} \qquad (1.2) $$

has an essentially closed jump set, i.e., $\mathcal{H}^{N-1}(\overline{J_u} \setminus J_u) = 0$, and thereby is a minimizer of the strong formulation. We frame our result as a ‘weak-to-strong’ improvement in regularity for minimizers, given basic assumptions on the densities, i.e., $\psi: \Omega \times \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{S}^{N-1} \to \mathbb{R}_+$ and $f: \Omega \times \mathbb{R}_{\text{sym}}^{N \times N} \to \mathbb{R}_+$ are Borel functions satisfying $M^{-1} \le \psi \le M$, and

$$ |f(x, \xi) - \mathbb{C}\xi : \xi| \le M(1 + |\xi|^q) \quad \text{for all } x \in \Omega \text{ and } \xi \in \mathbb{R}_{\text{sym}}^{N \times N}, $$

<footnote n="2010">2010 Mathematics Subject Classification. 49J45, 70G75, 74B10, 74G65, 74R10.</footnote>

<footnote>Key words and phrases. Griffith functional, free discontinuity problem, linear elasticity, Ahlfors-regularity.</footnote>
for a fixed constant $M \ge 1$, an elastic tensor $\mathbb{C}$, and an exponent $q \in (0, 2)$. (Here, $\mathbb{R}_{\text{sym}}^{N \times N} \subset \mathbb{R}^{N \times N}$ denotes the subset of symmetric matrices, $S^{N-1} := \{x \in \mathbb{R}^N: |x| = 1\}$, and $\mathbb{R}_+ := [0, +\infty)$. Actually, we can also treat a slightly more general setting, cf. (2.11)-(2.12) below.) Our results also hold for local minimizers, i.e., minimizers with respect to compact perturbations on balls. Note that in this paper we do not address existence of (weak) minimizers for (1.2) as this has been treated thoroughly in the literature by studying compactness [12, 9] and separately the lower semicontinuity of the bulk [9] and the surface term [19, 20], see Remark 2.8.

Let us mention that for functionals with control on the full gradient and not only on its symmetrized part, some strong existence results generalizing the ones for the Mumford-Shah functional [15] are available, namely for energies with specific anisotropic bulk densities [1, 16, 21, 27] covering both the scalar and the vectorial case, and for surface energies with dependence on the traces [6], in connection to free discontinuity problems with Robin conditions. In this work, we extend these results to a large class of admissible bulk and surface densities, particularly requiring only a control on the symmetrized gradient.

In dimension $N = 2$ [11] and $N \ge 2$ [7, 8], it has been shown that minimizers of the Griffith energy (1.1) have an essentially closed jump set and are minimizers of the strong formulation. Practically, this is achieved by proving that the crack is locally Ahlfors-regular, meaning that it satisfies the uniform density estimate

$$C_{\text{Ahlf}}^{-1} r^{N-1} \le \mathcal{H}^{N-1}(J_u \cap B(x_0, r)) \le C_{\text{Ahlf}} r^{N-1} \quad (1.3)$$

in any ball $B(x_0, r) \subset \Omega$ centered at $x_0 \in \bar{J}_u$ with radius $0 < r \le r_0$, for some uniform constants $r_0 > 0$ and $C_{\text{Ahlf}} \ge 1$.

To prove that minimizers of (1.2) have closed crack, we will show that property (1.3) is inherited by more than just minimizers of the energy (1.1), and extends to a flexible class of functions called quasiminimizers. Then, we simply show that (local) minimizers of (1.2) are specific examples of quasiminimizers. This larger class of functions throws away the unnecessary information carried by the densities $f$ and $\psi$, and states that (up to a deviation/gauge) $u$ is an M-quasiminimizer if for all (local) competitors $v$ it holds that

$$\int_{\Omega} \mathbb{C}e(u) : e(u) dx + M^{-1}\mathcal{H}^{N-1}(J_u) \le \int_{\Omega} \mathbb{C}e(v) : e(v) dx + M\mathcal{H}^{N-1}(J_v),$$

see Definition 2.2. Formally, a function is a quasiminimizer if, away from the crack, it has near minimal elastic energy and, close to the crack, it (locally) has at most a fixed percentage of excess crack relative to a local minimizer of (1.1). We emphasize that, in contrast to minimizers, the crack set of a quasiminimizer is in general not smooth for reasons of bi-Lipschitz invariance, see Remark 2.4. Still, as we show in this paper, it satisfies some quantitative properties with uniform bounds, namely Ahlfors-regularity.

Our analysis will follow two parallel notions of quasiminimizer: one in the $GSBD$ setting and one in the classical setting of pairs $(u, K)$ where $K$ is a closed set and $u$ is locally Sobolev outside of $K$. For quasiminimizers in both settings, we will show that the crack is Ahlfors-regular and satisfies a density estimate of the form (1.3), see Theorem 2.3. Note that, while in the case of pairs the crack $K$ is already closed, the Ahlfors-regularity (1.3) is important and provides additional information. In particular, it is the starting point for studying finer regularity properties of crack sets of minimizers, see [18].

To obtain Ahlfors-regularity of the crack set, our approach is motivated by the seminal paper of De Giorgi et al. [15], wherein a contradiction-compactness argument is used to show that in regions with small crack, the energy in a ball $B(x_0, r)$ decays on the order of $r^N$ (like a bulk energy), much faster than the surface energy scaling $r^{N-1}$. Accounting for serious technical difficulties arising from the vectorial nature of the Griffith energy, the contradiction-compactness was also used by CHAMBOLLE et al. [7] for the Griffith energy (see also [11] for the two-dimensional case). Therein, an approximation result for $GSBD$ functions with small crack was introduced, which will also be an essential tool in our argument,
see Lemma 3.3. Going beyond these results, our argument will show that the contradiction-compactness argument is robust and able to absorb large perturbations in the surface energy.

Informed by the work of BONNET on the Mumford–Shah functional [5], where it was shown that blow-up limits are only minimal with respect to competitors satisfying topological constraints, we also cover a weaker minimality condition called topological quasiminimality, see Definition 2.2. In this setting, Theorem 2.3 provides the Griffith counterpart of SIAUDEAU’s result [26], which considers topological quasiminimizers in the scalar setting (see also [13, Chapter 72]). As an application, our result proves Ahlfors-regularity of the crack set for Griffith global minimizers in $\mathbb{R}^N$, see [23, Definition 7.1]. A technical hurdle will be to adapt the aforementioned approximation result introduced in [7] to the case of admissible pairs, which always require closed cracks. Furthermore, we will have to investigate limits of sequences with vanishing jump sets under the weaker assumption of topological quasiminimality.

Finally, we remark that our notion of quasiminimality is inspired by and generalizes [13, Definition 7.21] and [6, Definition 2.1] used in the Mumford–Shah setting, see Remark 2.5. In particular, BUCUR AND LUCKHAUS [6] obtain Ahlfors-regularity of scalar almost quasiminimizers by deriving a monotonicity formula, which is interesting as an alternative approach to the DE GIORGI ET AL. [15] framework, but it is not clear how such a monotonicity formula could be found in the vectorial setting of linearized elasticity.

The paper is organized as follows. In Section 2, we introduce the definition of quasiminimizer and state the main results: Ahlfors-regularity for quasiminimizers and existence of strong solutions for free discontinuity problems. Here we show that the latter is a consequence of the former. In Section 3, we prove that the Griffith energy decays like a bulk energy in regions with small crack and use this to show that local minimizers of (1.2) are specific examples of quasiminimizers of (1.1), hence guaranteeing an Ahlfors-regular crack as in (1.3). The decay of the energy relies on an approximation result whose proof is discussed in Section 4.

## 2. MAIN RESULTS

2.1. **Notation.** For $N \ge 2$ let $\Omega \subset \mathbb{R}^N$ be an open set. We say that a constant is universal if it only depends on $N$. We use $|\cdot|$ for the Euclidean norm of a vector, and given two matrices $\xi, \eta \in \mathbb{R}^{N \times N}$, the notation $\xi : \eta$ denotes the Frobenius inner product of $\xi$ and $\eta$, i.e.,

$$ \xi : \eta := \sum_{i,j=1}^{N} \xi_{ij} \eta_{ij}. $$

The Frobenius norm is given by $|\xi| := \sqrt{\xi : \xi}$. Given $a, b \in \mathbb{R}^N$, we let $a \odot b := ((a_i b_j + b_i a_j)/2)_{ij} \in \mathbb{R}^{N \times N}$. For the entire paper, we fix a coercive fourth-order tensor $\mathcal{C}: \mathbb{R}^{N \times N} \to \mathbb{R}^{N \times N}$ such that

$$ \mathcal{C}(\xi - \xi^T) = 0 \quad \text{and} \quad \mathcal{C}\xi : \xi \ge c_0 |\xi + \xi^T|^2 \quad \text{for all } \xi \in \mathbb{R}^{N \times N} $$

for some constant $c_0 > 0$. Note that $\mathcal{C}$ defines a scalar product on the space $\mathbb{R}_{\text{sym}}^{N \times N}$ of symmetric matrices. We say that a linear map $\mathbb{R}^N \ni x \mapsto a(x) := Ax + b$, where $A \in \mathbb{R}^{N \times N}$ and $b \in \mathbb{R}^N$, is a rigid motion if $A$ belongs to $\mathbb{R}_{\text{skew}}^{N \times N}$, the set of skew symmetric matrices. We use $B(x, r)$ for the open ball of radius $r > 0$ centered at $x \in \mathbb{R}^N$. We denote by $\omega_N$ and $\sigma_{N-1}$ the volume of the $N$-dimensional unit ball and the surface area of the $(N-1)$-dimensional sphere, respectively. By an abuse of notation, we also denote by $|\cdot|$ the Lebesgue measure, and we write $\mathcal{H}^{N-1}$ for the $(N-1)$-dimensional Hausdorff measure. The symmetric difference of two sets $V_1$ and $V_2$ is denoted by $V_1 \Delta V_2$. For a set $V$, we denote the associated characteristic function by $\chi_V$. For a set of finite perimeter $V$, $\partial^* V$ denotes the reduced boundary, see [3, Definition 2.60].
In the following, we consider functions in the space of generalized special functions of bounded deformation, denoted by $GSBD(\Omega)$, which consists of measurable maps $u: \Omega \to \mathbb{R}^N$ having integrable symmetric derivative $e(u)$ and a rectifiable jump set $J_u$, where both of these quantities are interpreted via slicing. We refer to [12] for the precise definition. We denote by $GSBD^2(\Omega)$ those functions $u$ belonging to $GSBD(\Omega)$ such that $e(u) \in L^2(\Omega; \mathbb{R}^{N \times N})$ and $\mathcal{H}^{N-1}(J_u) < \infty$. Naturally, $GSBD^2_{\text{loc}}(\Omega)$ consists of functions belonging to $GSBD^2(\Omega')$ for all $\Omega' \subset \subset \Omega$.

## 2.2. Quasiminimizers and Ahlfors-regularity.
We introduce our notion of quasiminimality, which requires a function to satisfy a minimality condition with respect to an appropriate class of competitors.

We introduce different notions of competitors for functions $u \in GSBD_{\text{loc}}(\Omega)$ and for pairs $(u, K)$, consisting of a relatively closed subset $K \subset \Omega$ and a function $u \in W_{\text{loc}}^{1,2}(\Omega \setminus K; \mathbb{R}^N)$.

**Definition 2.1 (Competitors).** Let $x_0 \in \Omega$ and $r > 0$ be such that $B(x_0, r) \subset \subset \Omega$.

1) A function $v \in GSBD^2_{\text{loc}}(\Omega)$ is a GSBD competitor of $u$ in $B(x_0, r)$ if $\{v \neq u\} \subset \subset B(x_0, r)$.

2) A pair $(v, L)$ in $\Omega$ is a competitor of $(u, K)$ in $B(x_0, r)$ if

$$L \setminus B(x_0, r) = K \setminus B(x_0, r) \quad \text{and} \quad v = u \quad \text{a.e. in} \quad \Omega \setminus (K \cup B(x_0, r)). \qquad (2.1)$$

3) A pair $(v, L)$ in $\Omega$ is a topological competitor of $(u, K)$ in $B(x_0, r)$ if (2.1) holds and in addition the following condition is satisfied:

$$\text{all } x, y \in \Omega \setminus (K \cup B(x_0, r)) \text{ that are separated by } K \text{ are also separated by } L. \qquad (2.2)$$

This means that, if $x, y \in \Omega \setminus (K \cup B(x_0, r))$ belong to different connected components of $\Omega \setminus K$, they also belong to different connected components of $\Omega \setminus L$.

We say that a pair $(u, K)$ is *coral* provided that for all $x_0 \in K$ and for all $r > 0$ we have

$$\mathcal{H}^{N-1}(K \cap B(x_0, r)) > 0. \qquad (2.3)$$

For a given pair $(u, K)$, there is always a coral representative. In particular, such a representative gets rid of any isolated points of $K$ which do not affect the energy but would prevent an estimate of the form (1.3) from holding at all points in the crack. We say a pair $(u, K)$ has *locally finite energy* if $\|e(u)\|_{L^2(B(x_0, r))} + \mathcal{H}^{N-1}(K \cap B(x_0, r)) < \infty$ for all $B(x_0, r) \subset \subset \Omega$. We define a *gauge* as a nondecreasing function $h: (0, +\infty) \to [0, +\infty]$ with $\limsup_{r \to 0} h(r) < \infty$.

**Definition 2.2 (Quasiminimizers).** Let $M \ge 1$ be a constant and let $h$ be a gauge.

1) A function $u \in GSBD^2_{\text{loc}}(\Omega; \mathbb{R}^N)$ is a GSBD local $M$-quasiminizer with gauge $h$ in $\Omega$ if for any open ball $B(x_0, r) \subset \subset \Omega$ and all GSBD competitors $v$ of $u$ in $B(x_0, r)$ we have

$$\int_{B(x_0, r)} \mathbb{C}e(u) : e(u) \,dx + M^{-1}\mathcal{H}^{N-1}(J_u \cap B(x_0, r)) \\ \le \int_{B(x_0, r)} \mathbb{C}e(v) : e(v) \,dx + M\mathcal{H}^{N-1}(J_v \cap B(x_0, r)) + h(r)r^{N-1}. \qquad (2.4)$$

2) A coral pair $(u, K)$ with locally finite energy is a Griffith local $M$-quasiminizer with gauge $h$ in $\Omega$ if for any open ball $B(x_0, r) \subset \subset \Omega$ and all competitors $(v, L)$ of $(u, K)$ in $B(x_0, r)$ we have

$$\int_{B(x_0, r) \setminus K} \mathbb{C}e(u) : e(u) \,dx + M^{-1}\mathcal{H}^{N-1}(K \cap B(x_0, r)) \\ \le \int_{B(x_0, r) \setminus L} \mathbb{C}e(v) : e(v) \,dx + M\mathcal{H}^{N-1}(L \cap B(x_0, r)) + h(r)r^{N-1}. \qquad (2.5)$$
3) A coral pair $(u, K)$ with locally finite energy is a Griffith local topological $M$-quasiminimizer with gauge $h$ in $\Omega$ if for any open ball $B(x_0, r) \subset \subset \Omega$ and all topological competitors $(v, L)$ of $(u, K)$ in $B(x_0, r)$ inequality (2.5) holds.

For convenience, we will often just say ‘quasiminimizer’ instead of ‘Griffith local $M$-quasiminimizer’. As quasiminimal pairs are also topological quasiminimizers, our study of regularity will focus on topological quasiminimizers. However, for our discussion in Remark 2.4 below, it is convenient to distinguish this stronger notion of quasiminimal pair. In the above definition, the gauge plays the role of a normalized deviation from minimality. Whereas in many works a certain decay is needed, i.e., gauges of the form $h(r) = r^\alpha$ for $\alpha > 0$ are considered, we allow for a wide class of gauges. In particular, we do not require $\lim_{r \to 0} h(r) = 0$. The minimality property above is local since conditions (2.4) and (2.5) only hold in balls away from the boundary and it does not refer to an underlying Dirichlet condition on $\partial\Omega$.

Quasiminimizers satisfy an energetic upper bound. For example, given a GSBD $M$-quasiminimizer $u$ with gauge $h$, for all open balls $B(x_0, r) \subset \Omega$, we have

$$ \int_{B(x_0, r)} \mathbb{C}e(u) : e(u) \, dx + M^{-1}\mathcal{H}^{N-1}(J_u \cap B(x_0, r)) \leq (M\sigma_{N-1} + h(r))r^{N-1}. \quad (2.6) $$

This is a straightforward application of (2.4) with $v := u_{\chi_{\Omega \setminus B(x_0, \rho)}}$ for $\rho < r$. Precisely, $v$ is a competitor of $u$ in $B(x_0, t)$ for all $t \in (\rho, r)$. One applies the quasiminimality condition in $B(x_0, t)$ and then let $t \to \rho$, and next $\rho \to r$ to deduce the above bound. For topological quasiminimizers, one has to check additionally that the topological condition (2.2) is satisfied by $L := (K \setminus B(x_0, \rho)) \cup \partial B(x_0, \rho)$, but this is clear.

A primary result of this paper is the following theorem, which in conjunction with (2.6) shows that the crack set of a quasiminimizer is (locally) Ahlfors-regular, and in particular the surface energy is proportional to an $(N-1)$-dimensional surface at a positive and uniform scale.

**Theorem 2.3.** For each $M \ge 1$, there exist constants $\varepsilon_{\text{Ahlf}} \in (0, 1)$ and $C_{\text{Ahlf}} \ge 1$ (depending on $N$, $\mathbb{C}$, $M$) such that the following holds.

1) Let $u$ be a GSBD $M$-quasiminimizer in $\Omega$ with any gauge $h$ in the sense of Definition 2.2. Then, for all $x_0 \in \Omega \cap \overline{J_u}$ and $r > 0$ such that $B(x_0, r) \subset \Omega$ and $h(r) \le \varepsilon_{\text{Ahlf}}$, we have

$$ \mathcal{H}^{N-1}(J_u \cap B(x_0, r)) \ge C_{\text{Ahlf}}^{-1}r^{N-1}. $$

In particular, if $\limsup_{r \to 0} h(r) < \varepsilon_{\text{Ahlf}}$, the jump set is essentially closed, i.e.,

$$ \mathcal{H}^{N-1}(\overline{J_u} \setminus J_u) \cap \Omega = 0. \quad (2.7) $$

2) Let $(u, K)$ be a topological $M$-quasiminimizer in $\Omega$ with any gauge $h$ in the sense of Definition 2.2. Then, for all $x_0 \in K$ and $r > 0$ such that $B(x_0, r) \subset \Omega$ and $h(r) \le \varepsilon_{\text{Ahlf}}$, we have

$$ \mathcal{H}^{N-1}(K \cap B(x_0, r)) \ge C_{\text{Ahlf}}^{-1}r^{N-1}. $$

The proof will be given in Section 3. We now make a variety of remarks on the scope of quasiminimality.

**Remark 2.4** (Bi-Lipschitz invariance). Topological quasiminimizers are related to quasiminimal sets, introduced by DAVID AND SEMMES [14, Definition 1.9], which is a generalization of ALMGREN’s ‘restricted sets’ [2, Definition II.1]. In essence, a set $K$ is quasiminimal if it satisfies

$$ M^{-1}\mathcal{H}^{N-1}(K \cap B(x_0, r)) \le M\mathcal{H}^{N-1}(L \cap B(x_0, r)) \quad (2.8) $$

for all ball $B(x_0, r) \subset \subset \Omega$ and all topological competitors $(0, L)$ of $(0, K)$ in $B(x_0, r)$. As the condition (2.8) is stable under a bi-Lipschitz mapping $\phi$, in the sense that $\phi(K)$ is also a quasiminimal set (with
a bigger constant), any regularity property that holds true for a quasiminimal set must be preserved under bi-Lipschitz mappings. Consequently, since the pair $(0, K)$ is a topological quasiminimizer as in Definition 2.2, the regularity theory for Griffith topological quasiminimizers is limited in a similar fashion and in general smooth crack sets cannot be expected. We note that this limitation is not an artifact of the topological constraint and also applies to the stronger notion of quasiminimizer in the sense of Definition 2.2, 2). Precisely, one can show that quasiminimizers of the Mumford–Shah functional remain quasiminimal under small bi-Lipschitz perturbations, and then identify a quasiminimizer of the Mumford–Shah functional with a Griffith quasiminimizer having only antiplanar shear.

In the case that $M = 1$ and $\lim_{r \to 0} h(r) = 0$, a quasiminimizer is called an almost-minimizer of the Griffith energy. If in addition $h \equiv 0$, we use the term local minimizer. In contrast to Remark 2.4, we expect that the crack of an almost-minimizer is much more than just Ahlfors-regular, as one inherits properties associated with minimality at sufficiently small scales. We refer to [4, 18, 24] for finer regularity properties of (almost-)minimizers of the Griffith energy. We emphasize that we follow the terminology of DAVID [13], and our almost-minimizers with gauge $h(r) = Cr^\alpha$ correspond to what is called quasiminimizers in [3, Definition 7.17] and [8, Definition 5.1].

**Remark 2.5** (Comparison to other notions of quasiminimality). (a) We remark that Definition 2.2 is more general than the definition of quasiminimizers introduced by DAVID [13, Definition 7.21]. Translated to the Griffith setting, the quasiminimality condition [13, (7.20)] for the pair $(u, K)$ says that, for any competitor $(v, L)$,

$$ \mathcal{H}^{N-1}(K \setminus L) \leq M\mathcal{H}^{N-1}(L \setminus K) + \Delta E + ar^{N-1}, \quad (2.9) $$

where

$$ \Delta E := \max \{E(v) - E(u), M(E(v) - E(u))\} \text{ and} $$

$$ E(v) = \int_{B(x_0, r) \setminus L} \mathbb{C}e(v) : e(v) \, dx, \quad E(u) = \int_{B(x_0, r) \setminus K} \mathbb{C}e(u) : e(u) \, dx. $$

In the definition of $\Delta E$, note that $E(v) - E(u)$ can be positive or negative. A direct computation shows that a quasiminimizer in this sense is a quasiminimizer in the sense of Definition 2.2. DAVID's quasiminimality condition (2.9) is natural in the context of the classical Griffith (or Mumford–Shah) energy, where the surface energy density of a crack is a constant, as it does not take into account the contribution of $K \cap L$ present in both pairs $(u, K)$ and $(v, L)$. On the other hand, our notion of quasiminimality in Definition 2.2 applies to local minimizers for a wide class of functionals, as in Theorem 2.7 below.

(b) In [6, Definition 2.1], BUCUR AND LUCKHAUS introduce the closely related notion of a $(\Lambda, \alpha, c_\alpha)$ almost quasiminimizer for the Mumford–Shah functional. Though our definition is slightly more general concerning the class of admissible gauges, one sees that if $u$ is an $M$-quasiminimizer in our sense (adapted to the Mumford–Shah setting) with gauge $h(r) := h(1)r^\alpha$, $\alpha \in (0, 1)$, then $\sqrt{M}u$ is an $(M^2, \alpha, Mh(1))$ almost quasiminimizer.

We finally discuss the difference between the different notions of quasiminimality given in Theorem 2.3.

**Remark 2.6** (Topological versus GSBD quasiminimality). According to Theorem 2.3, if $u$ is a GSBD quasiminimizer with a small gauge $h$, then the pair $(u, \bar{J}_u)$ is a quasiminimizer in the classical sense. Reciprocally, for a quasiminimizer $(u, K)$, one can check that $u \in \text{GSBD}_{\text{loc}}^2(\Omega)$ and that $K = \bar{J}_u$ (with strict inclusion $\bar{J}_u \subset K$, a small ball could be removed from $K$ contradicting minimality). However when translating the minimality condition (2.5) to the GSBD setting, it compares the closure of the jump sets. Although this looks probable, it is not clear how to prove that $u$ is a GSBD quasiminimizer as this would require to compare the jump sets and not their closure. In the case where $(u, K)$ is only a topological quasiminimizer, $\bar{J}_u$ may even be a strict subset of $K$, see the example $(0, K)$ in Remark 2.4.
2.3. Application to models in linear elasticity with surface discontinuities. One of the strengths of Definition 2.2 is that it includes local minimizers of a large class of functionals. In particular, we show that GSBD minimizers of generic fracture energies are in fact strong minimizers. We phrase our result in terms of the near minimal assumptions on the bulk and surface energy densities, and briefly discuss in Remark 2.8 specific examples where minimizers of the weak formulation have been shown to exist.

**Theorem 2.7** ('Weak-to-strong' regularity of local minimizers). Let $u \in \text{GSBD}^2(\Omega)$ be a (local) minimizer of the functional

$$ \int_{\Omega} f(x, e(u)) dx + \int_{J_u} \psi(x, u^{+}, u^{-}, \nu_u) d\mathcal{H}^{N-1}, \quad (2.10) $$

where $\psi: \Omega \times \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{S}^{N-1} \to \mathbb{R}_+$ and $f: \Omega \times \mathbb{R}_{\text{sym}}^{N \times N} \to \mathbb{R}_+$ are Borel functions satisfying $M^{-1} \le \psi \le M$, $\psi(x, a, b, \nu) = \psi(x, b, a, -\nu)$ for all $(x, a, b, \nu) \in \Omega \times \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{S}^{N-1}$, and

$$ |f(x, \xi) - \mathbb{C}\xi: \xi| \le g(|\xi|^2) \quad \text{for all } x \in \Omega \text{ and } \xi \in \mathbb{R}_{\text{sym}}^{N \times N}, \quad (2.11) $$

where $g: [0, +\infty) \to [0, +\infty)$ is a nondecreasing function such that $\int_{1}^{+\infty} s^{-2}g(s) ds < +\infty$. Then $J_u$ is Ahlfors-regular (1.3) and essentially closed (2.7). In particular, $(u, J_u)$ is a (local) minimizer of the strong formulation associated with (2.10).

A particular case of interest for (2.11) are perturbations of type

$$ g(|\xi|^2) = M(1 + |\xi|^q), \quad (2.12) $$

for some constant $M \ge 1$ and exponent $q \in (0, 2)$, cf. also [21]. This is the case for example for densities of the form $f(\xi) = |\xi - S|^2$, where $S: \Omega \to \mathbb{R}_{\text{sym}}^{N \times N}$ is a field of bounded symmetric matrices, arising in the dual formulation of the biharmonic optimal compliance problem [25]. The general surface term can take into account material anisotropy, (non-degenerate) cohesive material properties, or more complex behaviors across cracks, such as in the thermal insulation problem studied in [6, Section 4] (in the scalar case). Note that the second assumption on $\psi$ is needed because the tuple $(x, u^{+}, u^{-}, \nu_u)$ is only determined up to a change of sign of $\nu_u$ and an interchange of $u^{+}$ and $u^{-}$.

**Remark 2.8** (Existence). Suppose that $\Omega$ is a bounded Lipschitz domain and let $\Omega' \supset \Omega$ be a bounded domain such that $\partial_D \Omega = \partial \Omega \cap \Omega'$ is relatively open. We consider $u_0 \in W^{1,2}(\Omega'; \mathbb{R}^N)$ representing Dirichlet boundary values on $\partial_D \Omega$. We suppose that, besides the assumptions in Theorem 2.7, we have that $\psi = \psi(x, \nu)$ is continuous in $x$ and independent of $u^{+}$ and $u^{-}$, and that $f: \Omega \times \mathbb{R}_{\text{sym}}^{N \times N} \to \mathbb{R}_+$ is a Carathéodory function which is symmetric quasiconvex for a.e. $x$, i.e.,

$$ f(x, \xi) \le \frac{1}{\mathcal{L}^N(A)} \int_A f(x, \xi + e(\varphi)(y)) dy \quad \text{for all } x \in \mathbb{R}_{\text{sym}}^{N \times N} \text{ and } \varphi \in W_0^{1,\infty}(A; \mathbb{R}^N) $$

for every bounded open set $A \subset \mathbb{R}^N$. In this setting, existence of (weak) minimizers $u \in \text{GSBD}^2(\Omega)$ of (2.10) (on $\Omega'$ in place of $\Omega$) under the Dirichlet condition $u = u_0$ on $\Omega' \setminus \bar{\Omega}$ is guaranteed, see [9, Proposition 4.4].

The case where $\psi$ additionally depends on the traces $u^{+}$ and $u^{-}$ is more delicate. Here, existence can be shown for a relaxed problem [9, Proposition 4.5], under additional properties for $\psi$ given in [9, Theorem 1.2]. In particular, an assumption on lower semicontinuity of surface integrals is needed [9, (g$_4$)] which is for instance satisfied if $\psi$ is symmetric jointly convex, see [19, Theorem 5.1]. In specific situations, the approach of Theorem 2.7 can also be applied to functionals where $\psi$ is not bounded from below. For instance, in the thermal insulation problem [6], $\psi$ is given by $(u^{+})^2 + (u^{-})^2$ but the authors establish that, if $u$ is a minimizer, this density is actually bounded from below along $J_u$ by a universal constant. Therefore, the minimizers of this problem also fall within the class of quasiminimizers.
**Remark 2.9 (Bulk energy).** The main limitation to the choice of $f$ is that we rely on regularity estimates for local minimizers of

$$v \mapsto \int_{B(0,1)} \mathbb{C}e(v) : e(v) dx.$$

In particular, Theorem 2.7 holds provided that (2.11) is replaced by

$$|f(x, \xi) - f_0(\xi)| \leq g(|\xi|^p) \quad \text{for all } x \in \Omega \text{ and } \xi \in \mathbb{R}_{\text{sym}}^{N \times N},$$

where $f_0$ is a convex $p$-homogeneous function, with

$$\frac{1}{C_0} |\xi|^p \leq f_0(\xi) \leq C_0 |\xi|^p \quad \text{for all } \xi \in \mathbb{R}_{\text{sym}}^{N \times N}$$

for some $C_0 > 0$, such that any local minimizer $u$ of the energy $\int_{B(0,1)} f_0(e(v)) dx$ satisfies the decay estimate

$$\int_{B(0,r)} f_0(e(u)) dx \leq C r^{N-1+\alpha} \int_{B(0,1)} f_0(e(u)) dx \quad \text{for all } r \in (0,1)$$

for some constants $C > 0$ and $\alpha > 0$ depending only on $f_0$ and the dimension. In particular, by [10, Proposition 3.4 and Corollary 4.3], in dimensions $N = 2$ and $3$, our results apply when $f_0(\xi) = (\mathbb{C}\xi : \xi)^{p/2}$ for $p \in (1, +\infty)$. (In fact, as in [7], the approximation result needed in the proof of Theorem 2.3, see Section 4, carries over to general $p$, which we do not perform in this paper for notational convenience. Accordingly, in the decay estimate Proposition 3.1 one needs to replace $\tau^N$ by $\tau^{N-1+\alpha}$ which is enough to prove Theorem 2.3.) Similarly, our results adapt to the fully-elliptic setting when $f(x, e(u))$ is replaced by $f(x, \nabla u)$. In this case, Theorem 2.7 gives existence of strong minimizers to the relaxation of the (nonlinearly) frame indifferent energy

$$\int_{\Omega} \mathrm{dist}^2(\nabla u, \mathrm{SO}(N)) dx + \mathcal{H}^{N-1}(J_u),$$

defined for $u$ in $\mathrm{GSBV}^2(\Omega; \mathbb{R}^N)$, or for energies with more general surfaces densities as considered in (2.10).

**Proof of Theorem 2.7.** We prove that a local minimizer $u \in \mathrm{GSBD}^2(\Omega)$ of the functional (2.10) is a $\mathrm{GSBD}$ $M$-quasiminimizer for a suitable gauge $h$ and $M \geq 1$ in the sense of Definition 2.2. With this, Theorem 2.3 concludes the result. First, let us notice that, since $g$ is nondecreasing, for all $t > 0$ we have

$$\frac{g(t)}{t} \leq \int_{t}^{+\infty} \frac{g(s)}{s^2} ds,$$

which converges to zero as $t \to +\infty$. It follows that there exists $M_1 \geq 1$ (depending on $\mathbb{C}$ and $g$) such that for all $\xi \in \mathbb{R}_{\text{sym}}^{N \times N}$ we have $g(|\xi|^2) \leq (\mathbb{C}\xi : \xi)/2 + M_1$, and thus by (2.11)

$$\mathbb{C}\xi : \xi \leq 2f(x, \xi) + 2M_1 \quad \text{for all } x \in \Omega \text{ and } \xi \in \mathbb{R}_{\text{sym}}^{N \times N}. \qquad (2.13)$$

We fix an open ball $B(x_0, r) \subset \Omega$. Setting $h(r) = +\infty$ if $r > 1$, we can assume without restriction that $r \leq 1$. By comparing $u$ with the competitor $u\chi_{\Omega \setminus B(x_0, r)}$ for $0 < \rho < r$, using $f(x, 0) \leq g(0)$ and $M^{-1} \leq \psi \leq M$, and sending $\rho \to r$, the local minimality of $u$ yields

$$\int_{B(x_0,r)} f(x, e(u)) dx + M^{-1} \mathcal{H}^{N-1}(J_u \cap B(x_0, r)) \leq g(0) \omega_N r^N + M \sigma_{N-1} r^{N-1},$$

where $\omega_N$ is the volume of the unit ball and $\sigma_{N-1}$ is the surface area of the unit sphere. Then, as $r \leq 1$, from (2.13) it follows that

$$\int_{B(x_0,r)} \mathbb{C}e(u) : e(u) dx + M^{-1} \mathcal{H}^{N-1}(J_u \cap B(x_0, r)) \leq C_0 r^{N-1} \qquad (2.14)$$
for some constant $C_0 \ge 1$ which depends on $N$, $M$, $M_1$, and $g(0)$. Now, we let $v$ be a $GSBD$ competitor of $u$ in $B(x_0, r)$ in the sense of Definition 2.1,1). By minimality of $u$, we have

$$ \begin{aligned} & \int_{B(x_0,r)} f(x, e(u)) dx + \int_{J_u \cap B(x_0,r)} \psi(x, u^+, u^-, \nu_u) d\mathcal{H}^{N-1} \\ & \le \int_{B(x_0,r)} f(x, e(v)) dx + \int_{J_v \cap B(x_0,r)} \psi(x, v^+, v^-, \nu_v) d\mathcal{H}^{N-1}. \end{aligned} \quad (2.15) $$

We now show that this implies the quasiminimality condition (2.4) for some gauge $h$ to be specified. In fact, if

$$ \int_{B(x_0,r)} \mathbb{C}e(v) : e(v) dx > C_0 r^{N-1}, $$

inequality (2.4) holds directly for $h \equiv 0$ by (2.14). Otherwise, by (2.11) we have

$$ \int_{B(x_0,r)} f(x, e(v)) dx \le \int_{B(x_0,r)} \mathbb{C}e(v) : e(v) dx + \int_{B(x_0,r)} g(|e(v)|^2) dx. $$

Our goal is to show that

$$ \int_{B(x_0,r)} g(|e(v)|^2) dx \le h(r)r^{N-1}, \quad \text{where} \quad h(r) := C_1 C_0 \int_{1/r}^{+\infty} \frac{g(s)}{s^2} ds, \quad (2.16) $$

for some $C_1 \ge 1$ which depends only on $N$. For this purpose, we estimate the integral separately on $B(x_0, r) \cap \{|e(v)|^2 \le 1/r\}$ and $B(x_0, r) \cap \{|e(v)|^2 > 1/r\}$. For the first part, using the fact that $g$ is nondecreasing, we estimate

$$ \int_{B(x_0,r) \cap \{|e(v)|^2 \le 1/r\}} g(|e(v)|^2) dx \le \omega_N r^N g(1/r) \le \omega_N r^{N-1} \int_{1/r}^{+\infty} \frac{g(s)}{s^2} ds. $$

In order to deal with the second part, for $t > 0$ we define

$$ g^{-1}(t) := \inf \{ s \in (0, +\infty) \mid g(s) \ge t \}. $$

(Clearly, if $g$ is strictly increasing, $g^{-1}$ is the inverse.) If $g(s) \ge t$, we have $s \ge g^{-1}(t)$, and if $s > g^{-1}(t)$, then $g(s) \ge t$. Now, we use the layer cake representation and the assumption $\int_{B(x_0,r)} \mathbb{C}e(v) : e(v) dx \le C_0 r^{N-1}$ to bound

$$ \begin{aligned} \int_{B(x_0,r) \cap \{|e(v)|^2 > 1/r\}} g(|e(v)|^2) dx &= \int_0^{+\infty} \left| \left\{ x \in B(x_0,r) : |e(v)|^2 \ge \max(1/r, g^{-1}(t)) \right\} \right| dt \\ &\le \int_0^{+\infty} \frac{1}{\max(1/r, g^{-1}(t))} \left( \int_{B(x_0,r)} |e(v)|^2 dx \right) dt \\ &\le C C_0 r^{N-1} \int_0^{+\infty} \min(1, 1/g^{-1}(t)) dt. \end{aligned} $$

Using the layer cake representation again and Fubini's theorem, we find

$$ \begin{aligned} \int_0^\infty \min(r, 1/g^{-1}(t)) dt &= \int_0^\infty \int_0^r \chi_{\{0 < s < 1/g^{-1}(t)\}} ds dt \le \int_0^\infty \int_0^r \chi_{\{s: g(1/s) \ge t\}} ds dt \\ &\le \int_0^r g(1/s) ds = \int_{1/r}^{+\infty} \frac{g(s)}{s^2} ds. \end{aligned} $$
This proves (2.16). Similarly, by (2.14), (2.11), and (2.16) we have

$$ 
\begin{aligned}
\int_{B(x_0,r)} f(x, e(u)) \, dx &\geq \int_{B(x_0,r)} \mathbb{C}e(u) : e(u) \, dx - \int_{B(x_0,r)} g(|e(u)|^2) \, dx \\
&\geq \int_{B(x_0,r)} \mathbb{C}e(u) : e(u) \, dx - h(r)r^{N-1}.
\end{aligned}
$$

We may plug the above estimates into (2.15) and use the bound $M^{-1} \leq \psi \leq M$ to see that $u$ is a $M$-quasiminimizer in the sense of (2.4) with a gauge $h$ defined for all $r \leq 1$ as in (2.16). As $s^{-2}g(s)$ is integrable on $[1, +\infty)$, we have $\lim_{r \to 0} h(r) = 0$ and therefore Theorem 2.3 applies. $\square$

### 3. DENSITY LOWER BOUND FOR QUASIMINIMIZERS

In this section we prove Theorem 2.3. The heart of the proof is a decay estimate for the localized Griffith energy in a region only having a small crack. With this decay estimate at hand, we may directly conclude Theorem 2.3. Our proof of the decay estimate requires a study of limits of quasiminimizers with vanishing crack, analogous to [11, 7]. For the case of GSBD quasiminimizers, we will rely on an approximation result proven in [7]. For topological quasiminimizers instead, we will need to ensure that this approximation, found using minimality, respects both the topological constraints for competitors and that the crack is closed. Given the similarity between the two cases, we will focus our attention on the more involved case of topological quasiminimizers and highlight places in the proof where $K$ cannot be replaced simply by $J_u$. For convenience, in place of integrals of the form $\int_{B(x_0,r)\setminus K}$, we always write $\int_{B(x_0,r)}$ which clearly does not affect the integral as $K$ is a Lebesgue-negligible set.

**3.1. Decay property of the Griffith energy.** We adopt the approach used in [11] and [7], which was motivated by the classical result of DE GIORGI ET AL. [15] (also presented in [3]), but also introduced new compactness results for the vectorial Griffith setting. Given an open ball $B(x_0, r)$ and a pair $(u, K)$, we let $G(x_0, r)$ denote the Griffith energy of $(u, K)$ in $B(x_0, r)$, namely

$$ G(x_0, r) := \int_{B(x_0, r)} \mathbb{C}e(u) : e(u) \, dx + \mathcal{H}^{N-1}(K \cap B(x_0, r)). \quad (3.1) $$

We show the following decay estimate for the Griffith energy of a quasiminimizer.

**Proposition 3.1 (Decay estimate).** For each $M \geq 1$ and $\tau \in (0, 1)$, there exists $\varepsilon \in (0, 1)$ (depending on $N, \mathbb{C}, M, \tau$) such that the following holds. If $u$ is a topological (or GSBD) $M$-quasiminimizer with gauge $h$ in a ball $B(x_0, r) \subset \Omega$ and if

$$ \frac{\mathcal{H}^{N-1}(K \cap B(x_0, r))}{r^{N-1}} \leq \varepsilon \quad \text{and} \quad h(r) \leq \varepsilon \frac{G(x_0, r)}{r^{N-1}}, $$

then we have

$$ G(x_0, \tau r) \leq C \tau^N G(x_0, r), $$

for some constant $C \geq 1$ which depends only on $N$ and $\mathbb{C}$.

The essence of the above proposition is that, if the relative amount of crack in a ball is small, then the localized Griffith energy inherits the decay given by the energy with no crack. In particular, the decay scales like that for minimizers of the energy $\int \mathbb{C}e(u) : e(u) \, dx$, which solve a standard linear elliptic PDE. We first prove Theorem 2.3 in Subsection 3.2 and afterwards we give the proof of the decay estimate in Subsection 3.3.
### 3.2. Proof of Theorem 2.3.
Below we prove (2) of Theorem 2.3, and comment on the slight change for GSBD quasiminimizers in the remark following the proof.

**Proof of Theorem 2.3.** We proceed in three steps. In Step 1, we show that one can reduce the problem to considering points in the crack $K$ with their $(N-1)$-dimensional density uniformly bounded away from 0. In Step 2, we find an energy lower bound for the localized Griffith energy. For this, we argue by contradiction and use the decay estimate in Proposition 3.1 to show that, if the Griffith energy is small to start with, the Griffith energy stays small at all smaller scales. This in turn yields small $(N-1)$-dimensional density of the surface energy, contradicting the density bound found in Step 1. Finally, in Step 3, we conclude using a similar argument by contradiction.

**Step 1 (Uniform lower bound on $(N-1)$-dimensional density).** We introduce the set

$$K^* := \left\{ x \in K : \limsup_{r \to 0} \frac{\mathcal{H}^{N-1}(K \cap B(x, r))}{\omega_{N-1} r^{N-1}} \ge 2^{-(N-1)} \right\},$$

where $\omega_{N-1}$ is the measure of the $(N-1)$-dimensional unit disk. As $K$ is $\mathcal{H}^{N-1}$-locally finite (but not necessarily rectifiable) by [3, (2.43)] for $\mathcal{H}^{N-1}$-a.e. $x \in K$ it holds that

$$\limsup_{r \to 0} \frac{\mathcal{H}^{N-1}(K \cap B(x, r))}{\omega_{N-1} r^{N-1}} \ge 2^{-(N-1)},$$

and in particular, $\mathcal{H}^{N-1}(K \setminus K^*) = 0$. We further claim that

$$\overline{K^*} \cap \Omega = K. \qquad (3.2)$$

The inclusion $\overline{K^*} \cap \Omega \subset K$ is clear as $K^* \subset K$ and $K$ is closed. To see that $K \subset \overline{K^*}$, suppose by contradiction that there exists $x \in K \setminus \overline{K^*}$.

Then there is a radius $r > 0$ such that $K^* \cap B(x, r) = \emptyset$ and, since $K^*$ contains $\mathcal{H}^{N-1}$-a.e. point in $K$, we must have

$$\mathcal{H}^{N-1}(K \cap B(x, r)) = 0. \qquad (3.3)$$

But as $K$ is coral, this contradicts the assumption $x \in K$, see (2.3). This concludes the proof of (3.2). Due to (3.2) and a density argument, in the next steps we can focus on points in $K^*$.

**Step 2 (Lower bound on Griffith energy at uniformly positive scale).** We show that we can choose constants $\varepsilon_0 \in (0, 1)$ and $C_0 \ge 1$ (depending only on $N, \mathbb{C}, M$) such that for all $x \in K$ and $r > 0$ such that $B(x, r) \subset \Omega$ and $h(r) \le \varepsilon_0^3$, we have

$$G(x, r) \ge C_0^{-1} r^{N-1}. \qquad (3.4)$$

We fix a parameter $\tau_0 \in (0, 1)$ to be chosen later (depending only on $N$ and $\mathbb{C}$, see right after (3.7)). With this, we fix $\varepsilon_0$ with $0 < \varepsilon_0 < \min\{\tau_0^{N-1}, \omega_{N-1}(\tau_0/2)^{N-1}\}$ such that statement of Proposition 3.1 holds with respect to $\tau_0$. Finally, we note that by (3.2) it is sufficient to prove (3.4) for points $x \in K^*$.

**Substep 2.1 (One step).** Let $x \in K^*$. We claim that, whenever $B(x, r) \subset \Omega$ with

$$G(x, r) \le \varepsilon_0 r^{N-1} \quad \text{and} \quad h(r) \le \varepsilon_0^2 \tau_0^{N-1}, \qquad (3.5)$$

then

$$G(x, \tau_0 r) \le \varepsilon_0 (\tau_0 r)^{N-1}. \qquad (3.6)$$

Indeed, condition (3.5) implies $\mathcal{H}^{N-1}(K \cap B(x, r)) \le \varepsilon_0 r^{N-1}$, and then two possibilities can occur. If we have $h(r) \le \varepsilon_0 r^{-(N-1)} G(x, r)$, Proposition 3.1 and (3.5) show that

$$G(x, \tau_0 r) \le C \tau_0^N G(x, r) \le C \varepsilon_0 \tau_0^N r^{N-1} \qquad (3.7)$$
for some constant $C \ge 1$ that depends on $N$ and $\mathbb{C}$. Taking $\tau_0$ small enough such that $C\tau_0 \le 1$, inequality (3.7) gives the desired estimate (3.6). If on the other hand $h(r) > \varepsilon_0 r^{-(N-1)}G(x, r)$, then we trivially have

$$G(x, \tau_0 r) \le G(x, r) \le h(r)\varepsilon_0^{-1}r^{N-1},$$

and the assumption $h(r) \le \varepsilon_0^2 \tau_0^{N-1}$ yields the estimate (3.6).

**Substep 2.2 (Iteration).** We deduce by iteration that, if there exists some $r > 0$ such that $B(x, r) \subset \Omega$, $h(r) \le \varepsilon_0^2 \tau_0^{N-1}$, and $G(x, r) \le \varepsilon_0 r^{N-1}$, then for all integers $k \ge 0$,

$$G(x, \tau_0^k r) \le \varepsilon_0 (\tau_0^k r)^{N-1}.$$

This follows from Substep 2.1 and the fact that $h$ is nondecreasing. For all $0 < \rho \le r$, there exists an integer $k \ge 0$ such that $\tau_0^{k+1}r \le \rho \le \tau_0^k r$, and thus

$$G(x, \rho) \le G(x, \tau_0^k r) \le \varepsilon_0 (\rho/\tau_0)^{N-1}.$$

In turn, this implies that

$$\limsup_{\rho \to 0} \frac{\mathcal{H}^{N-1}(K \cap B(x, \rho))}{\rho^{N-1}} \le \varepsilon_0 \tau_0^{-(N-1)} < \omega_{N-1} 2^{-(N-1)},$$

where we used $\varepsilon_0 < \omega_{N-1}(\tau_0/2)^{N-1}$. This contradicts the assumption $x \in K^*$. We conclude that for all $x \in K^*$ and all $r > 0$ such that $B(x, r) \subset \Omega$ and $h(r) \le \varepsilon_0^3$ (note $\varepsilon_0^3 \le \varepsilon_0^2 \tau_0^{N-1}$), (3.4) holds with $C_0 = 1/\varepsilon_0$.

**Step 3 (Lower bound on surface energy at uniformly positive scale).** We take $\tau \in (0, 1)$ as a small constant to be chosen below (depending on $N$, $\mathbb{C}$ and $M$) and let $\varepsilon > 0$ be the constant associated with $\tau$ coming from Proposition 3.1 (which depends on $N$, $\mathbb{C}$, $M$).

If there exists $x \in K$ and $r > 0$ with $B(x, r) \subset \Omega$ such that $h(r) \le \min\{C_0^{-1}\varepsilon, \varepsilon_0^3\}$ and

$$\mathcal{H}^{N-1}(K \cap B(x, r)) \le \varepsilon r^{N-1}, \qquad (3.8)$$

then by (3.4) we have $h(r) \le \varepsilon r^{-(N-1)}G(x, r)$, and thus Proposition 3.1 shows that

$$G(x, \tau r) \le C \tau^N G(x, r).$$

By (2.6), we have $G(x, r) \le Cr^{N-1}$ (for a constant $C > 0$ depending on $M$), so

$$G(x, \tau r) \le C \tau^N r^{N-1}$$

for $C > 0$ depending on $M$, $N$, and $\mathbb{C}$. If $\tau$ is taken small enough (depending on $N$, $\mathbb{C}$ and $M$) this contradicts (3.4). Consequently, (3.8) cannot hold for any $r > 0$ with $B(x, r) \subset \Omega$ and $h(r) \le \min\{C_0^{-1}\varepsilon, \varepsilon_0^3\}$. Thus, the statement holds for $C_{\text{Ahlf}} := \varepsilon^{-1}$ and $\varepsilon_{\text{Ahlf}} := \min\{C_0^{-1}\varepsilon, \varepsilon_0^3\}$. $\square$

**Remark 3.2 (The GSBD case).** Proceeding as in Step 1 of the above proof for GSBD quasiminimizers, with $K$ replaced by $J_u$ (or occasionally $\overline{J}_u$), one must show $\overline{J}_u^* \cap \Omega = \overline{J}_u \cap \Omega$, the analogue of (3.2). Herein, (3.3) becomes

$$\mathcal{H}^{N-1}(J_u \cap B(x, r)) = 0 \qquad (3.9)$$

for some $x \in J_u \setminus \overline{J}_u^*$ and $r > 0$ small enough such that $B(x, r) \subset \Omega$ and $h(r) < \infty$. To see that $x$ cannot be long to $J_u$, note that by the definition of GSBD via slices (see [12]) and (3.9) the distributional symmetric derivative is absolutely continuous, and by Korn's inequality, one finds that $u \in W^{1,2}(B(x, r); \mathbb{R}^N)$. But this contradicts the assumption $x \in J_u$ thanks to Proposition A.1 in Appendix A (note that (A.1) holds by (2.6)). Steps 2 and 3 remain unchanged, replacing $K$ by $J_u$. Provided that $\limsup_{r \to 0} h(r) < \varepsilon_{\text{Ahlf}}$, we eventually get (2.7) as a standard consequence of the density lower bound along $\overline{J}_u$, see [3, (2.42)].
### 3.3. Proof of the decay estimate.
The proof of the decay estimate, i.e., Proposition 3.1, is achieved by a contradiction and compactness argument. For this, we will need to understand the limit behavior of quasiminimizing sequences with asymptotically vanishing crack.

We start with explaining the proof idea. Working on the unit ball and supposing that the conclusion of Proposition 3.1 fails, we have a sequence of pairs $(u_n, K_n)$ with vanishing crack, in the sense that $\lim_{n \to \infty} \mathcal{H}^{N-1}(K_n) = 0$, and such that

$$G_n(0, \tau) > C\tau^N G_n(0, 1), \quad (3.10)$$

where $\tau \in (0, 1)$ is fixed and $G_n$ is the Griffith energy of $(u_n, K_n)$ as in (3.1) and the constant $C > 0$ will be chosen appropriately. To derive a contradiction, we will show that $u_n$ converges to $u \in W^{1,2}(B(0, 1); \mathbb{R}^N)$ minimizing $\int_{B(0,1)} \mathbb{C}e(u) : e(u) dx$ subject to its own boundary conditions. As $u$ solves a linear elliptic PDE, we have in particular that

$$\int_{B(0,\tau)} \mathbb{C}e(u) : e(u) dx \le C_0 \tau^N \int_{B(0,1)} \mathbb{C}e(u) : e(u) dx \quad (3.11)$$

for a universal constant $C_0 > 0$. We will pass to the limit in (3.10) to find a contradiction with (3.11). For a contradiction to arise, we must know that the energies converge, and in particular that

$$\lim_{n \to \infty} G_n(0, \tau) = \int_{B(0,\tau)} \mathbb{C}e(u) : e(u) dx. \quad (3.12)$$

Showing that the left-hand side of (3.12) exceeds the limit's elastic energy follows from lower semicontinuity associated with weak convergence, but showing the reverse inequality will require a careful choice of competitors in the quasiminimality condition for $(u_n, K_n)$, see (2.5). The main technical tool to accomplish this is a more regular approximation of $u_n$ given by $\tilde{u}_n$ such that $\tilde{u}_n - a_n \to u$ in $L^2_{\text{loc}}$ for a sequence of rigid motions $a_n$. We remark that in general it is not possible to show that $u_n - a_n$ converges in $L^2$ to the limit, as small inclusions in the domain may be cut out by the crack, and subsequently travel off to infinity.

We use the approximation technique introduced by CHAMBOLLE ET AL. in [7] for GSBD functions (see also [11]). We state the result for pairs $(u, K)$, and note that the analogous statement for GSBD functions may be found by replacing $K_n$ by $J_{u_n}$. We emphasize that in the GSBD setting, except for the statement being on a ball (and not a cube), it is as in [7, Corollary 4.1]. We defer the discussion of the proof of this result to Section 4.

**Lemma 3.3.** Let $(u_n, K_n)$ be a sequence of pairs on $B(0, 1)$ such that

$$\sup_{n \in \mathbb{N}} \int_{B(0,1)} |e(u_n)|^2 dx < \infty \quad \text{and} \quad \mathcal{H}^{N-1}(K_n) \to 0.$$

Let $t \in (0, 1]$. After extracting a subsequence, there exist a sequence of pairs $(\tilde{u}_n, \tilde{K}_n)$ in $B(0, 1)$, a sequence of rigid motions $a_n$, and a function $u \in W^{1,2}(B(0, t); \mathbb{R}^N)$ such that

(1) $\{\tilde{u}_n \neq u_n\} \subset B(0, t)$ and $K_n \Delta \tilde{K}_n \subset B(0, t)$;

(2) $\tilde{u}_n - a_n \to u$ in $L^2_{\text{loc}}(B(0, t); \mathbb{R}^N)$ and $u_n - a_n \to u$ locally in measure in $B(0, t)$;

(3) for all $r < s < t$, we have

$$\mathcal{H}^{N-1}(\tilde{K}_n \cap A_{s,t}) \le (1+o(1))\mathcal{H}^{N-1}(K_n \cap A_{s,t})$$

and

$$\int_{A_{s,t}} \mathbb{C}e(\tilde{u}_n) : e(\tilde{u}_n) dx \le (1+o(1)) \int_{A_{r,t}} \mathbb{C}e(u_n) : e(u_n) dx,$$

where $A_{s,t} := B(0, t) \setminus B(0, s)$;
(4) for all $s \in (0, t]$ we have

$$ \int_{B(0,s)} \mathbb{C}e(u) : e(u) \, dx \leq \liminf_{n \to \infty} \int_{B(0,s)} \mathbb{C}e(u_n) : e(u_n) \, dx. $$

We will need to apply this result on nested balls, and it will be important to ensure that the limit function $u$ and the sequence of affine maps can be chosen to be same.

**Remark 3.4** (Compactness on nested balls). Applying Lemma 3.3 in $B(0, 1)$ (with $t \equiv 1$) and subsequently in $B(0, t)$ for some $t \in (0, 1)$, we find (corresponding to a subsequence of a subsequence) functions $\tilde{u}_n$, $a_n$, and $u$ along with $\tilde{u}_n^t$, $a_n^t$, and $u^t$, respectively. We show that we may take $a_n^t = a_n$ and $u^t = u$.

We know that $u_n - a_n$ converges to $u$ in measure and $u_n - a_n^t$ converges to $u^t$ in measure in $B(0, t/2)$, so $a_n^t - a_n$ converges to $u - u^t$ in measure in $B(0, t/2)$. As $a_n^t - a_n$ is a sequence of affine maps, convergence in measure actually implies convergence in $L^2(B(0, 1); \mathbb{R}^N)$, see for instance [18, Lemma 3.7]. The limit of $a_n^t - a_n$ is necessarily a rigid motion, denoted by $a$, and we have $u = u^t + a$ in $B(0, t)$. It follows that $\tilde{u}_n^t - a_n \to u$ in $L_{\text{loc}}^2(B(0, t); \mathbb{R}^N)$. We conclude that, even if we apply the lemma in different nested balls, we can always take the same rigid motions $a_n$ and the same limit function $u$.

We can now investigate limits of quasiminimizing sequences with vanishing jump sets. The proof is close to [11, Proposition 3.4] and [7, Theorem 4], but we allow for different weights $M$ and $M^{-1}$ on the surface energy. In addition, we account for the topological constraint within the minimality condition using a trick observed in the scalar setting by SIDEAU [26], see also [13, Chapter 72]. When we later apply the proposition, the constant $c_n$ will arise from a renormalization of the Griffith energy. As before, we state the result for pairs $(u_n, K_n)$, but the same statement holds for GSBD quasiminimizers when one replaces $K_n$ by $J_{u_n}$.

**Proposition 3.5.** Let $(u_n, K_n)$ be a sequence of pairs in $B(0, 1)$ with vanishing crack, i.e., $\mathcal{H}^{N-1}(K_n) \to 0$. We assume that there exist some $M \geq 1$, a sequence $c_n \geq 0$, and a sequence $\varepsilon_n \to 0$ such that

$$ \sup_{n \in \mathbb{N}} \left( \int_{B(0,1)} |e(u_n)|^2 dx + c_n \mathcal{H}^{N-1}(K_n) \right) < +\infty, \quad (3.13) $$

and that for all $n \in \mathbb{N}$, all $t \in (0, 1)$, and all topological competitors $(v, L)$ of $(u_n, K_n)$ in $B(0, t)$, we have

$$ \int_{B(0,t)} \mathbb{C}e(u_n) : e(u_n) \, dx + c_n M^{-1} \mathcal{H}^{N-1}(K_n \cap B(0,t)) \\ \leq \int_{B(0,t)} \mathbb{C}e(v) : e(v) \, dx + c_n M \mathcal{H}^{N-1}(L \cap B(0,t)) + \varepsilon_n. \quad (3.14) $$

Then, up to extracting a subsequence, there exists a function $u \in W^{1,2}(B(0, 1); \mathbb{R}^N)$ such that for all $t \in (0, 1)$,

$$ \lim_{n \to \infty} \int_{B(0,t)} \mathbb{C}e(u_n) : e(u_n) \, dx = \int_{B(0,t)} \mathbb{C}e(u) : e(u) \, dx, \quad (3.15) $$

$$ \lim_{n \to \infty} c_n \mathcal{H}^{N-1}(K_n \cap B(0,t)) = 0, $$

and such that

$$ \int_{B(0,1)} \mathbb{C}e(u) : e(u) \, dx \leq \int_{B(0,1)} \mathbb{C}e(v) : e(v) \, dx \quad (3.16) $$

for all $v \in u + W_0^{1,2}(B(0, 1); \mathbb{R}^N)$.
Proof. Step 1 (Approximating functions and subsequences). Since the functions

$$ 
\begin{aligned}
\alpha_n & : t \mapsto \int_{B(0,t)} \mathbb{C}e(u_n) : e(u_n) \,dx \\
\beta_n & : t \mapsto c_n \mathcal{H}^{N-1}(K_n \cap B(0,t))
\end{aligned}
 $$

are non-decreasing and uniformly bounded on $[0, 1]$ by (3.13), we may find two non-decreasing functions $\alpha, \beta : [0, 1] \to \mathbb{R}_+$ and extract a subsequence (not relabeled) such that for all $t \in [0, 1]$ it holds that

$$ \lim_{n \to \infty} \int_{B(0,t)} \mathbb{C}e(u_n) : e(u_n) \,dx = \alpha(t), \qquad (3.17a) $$

$$ \lim_{n \to \infty} c_n \mathcal{H}^{N-1}(K_n \cap B(0,t)) = \beta(t). \qquad (3.17b) $$

By Lemma 3.3 and Remark 3.4, there exist a sequence of rigid motions $(a_n)_n$ and a Sobolev function $u \in W^{1,2}(B(0,1); \mathbb{R}^N)$ such that

$$ \int_{B(0,t)} \mathbb{C}e(u) : e(u) \,dx \le \alpha(t) \quad \text{for all } t \in (0, 1], \qquad (3.18) $$

and for all $t \in (0, 1]$, we can extract a further subsequence (which depends on $t$ and is not relabeled) to find pairs $(\tilde{u}_n, \tilde{K}_n)$ in $B(0, 1)$ satisfying

$$ (1) \{\tilde{u}_n \neq u_n\} \subset B(0,t) \text{ and } \tilde{K}_n \Delta K_n \subset B(0,t); $$

$$ (2) \text{ the functions } \tilde{u}_n - a_n \text{ converge to } u \text{ in } L_{\text{loc}}^2(B(0,t); \mathbb{R}^N); $$

$$ (3) \text{ for all } r, s \text{ such that } r < s < t, \text{ we have} $$

$$ \limsup_{n \to \infty} \int_{B(0,t) \setminus B(0,s)} \mathbb{C}e(\tilde{u}_n) : e(\tilde{u}_n) \,dx \le \alpha(t) - \alpha(r) \qquad (3.19a) $$

and

$$ \limsup_{n \to \infty} c_n \mathcal{H}^{N-1}(\tilde{K}_n \cap B(0,t) \setminus B(0,s)) \le \beta(t) - \beta(s). \qquad (3.19b) $$

Step 2 (Construction of competitors). Now we fix $t \in (0, 1)$ as a point of continuity of $\alpha$ and $\beta$ (almost every $t$ works because these functions are non-decreasing) and let $(\tilde{u}_n, \tilde{K}_n)$ be as above. We let $v \in W^{1,2}(B(0,1); \mathbb{R}^N)$ be such that $\{v \neq u\} \subset B(0,t)$. In particular, there exists $s < t$ such that $\{v \neq u\} \subset B(0,s)$. We let $\varphi \in C_c^\infty(B(0,t))$ be a cut-off function such that $\varphi = 1$ on $B(0,s)$, and we introduce the $(s$-dependent) competitor $(v_n, L_n)$ given by

$$ v_n := \varphi(v + a_n) + (1 - \varphi)\tilde{u}_n, \quad L_n := \tilde{K}_n \setminus B(0, s). \qquad (3.20) $$

We are going to estimate the energy of the competitors, i.e.,

$$ \limsup_{n \to \infty} \int_{B(0,t)} \mathbb{C}e(v_n) : e(v_n) \,dx + M\mathcal{H}^{N-1}(L_n \cap B(0,t)), $$

and subsequently modify each competitor $(v_n, L_n)$ to ensure that it is a topological competitor, see (2.2).

Step 3 (Elastic energy of the competitors). We have

$$ e(v_n) = \varphi e(v) + (1 - \varphi)e(\tilde{u}_n) + \nabla \varphi \odot (u + a_n - \tilde{u}_n), $$

where we use that $v = u$ on $\{\nabla \varphi \neq 0\}$. The function $\mathbb{C}\xi : \xi$ comes from a scalar product on the space $\mathbb{R}_{\text{sym}}^{N \times N}$ of symmetric matrices and it is temporarily more convenient to work with the underlying norm, so for $\xi \in \mathbb{R}_{\text{sym}}^{N \times N}$ we set

$$ |\xi|_{\mathbb{C}} := \sqrt{\mathbb{C}\xi : \xi}. $$
By the triangle inequality we have

$$|e(v_n)|_{\mathbb{C}} \leq \varphi|e(v)|_{\mathbb{C}} + (1-\varphi)|e(u_n)|_{\mathbb{C}} + C|\nabla\varphi||\tilde{u}_n - a_n - u|$$

for some constant $C \geq 1$ that depends on $\mathbb{C}$. It is elementary to check that

$$(a+b)^2 \leq (1+\varepsilon)a^2 + (1+\varepsilon^{-1})b^2 \quad \text{for all } a, b \geq 0 \text{ and for all } \varepsilon > 0. \qquad (3.21)$$

Using (3.21) and the convexity of $t \mapsto t^2$, we estimate

$$\int_{B(0,t)} |e(v_n)|_{\mathbb{C}}^2 dx \leq \int_{B(0,t)} \left( (1+\varepsilon)(\varphi|e(v)|_{\mathbb{C}} + (1-\varphi)|e(\tilde{u}_n)|_{\mathbb{C}})^2 + C(1+\varepsilon^{-1})|\nabla\varphi|^2 |\tilde{u}_n - a_n - u|^2 \right) dx \\ \leq (1+\varepsilon) \int_{B(0,t)} \varphi|e(v)|_{\mathbb{C}}^2 + (1-\varphi)|e(\tilde{u}_n)|_{\mathbb{C}}^2 dx \\ + C(1+\varepsilon^{-1}) \int_{B(0,t)} |\nabla\varphi|^2 |\tilde{u}_n - a_n - u|^2 dx,$$

for a bigger constant $C$. Then, by construction of $\varphi$, this simplifies to

$$\int_{B(0,t)} |e(v_n)|_{\mathbb{C}}^2 dx \leq (1+\varepsilon) \int_{B(0,t)} |e(v)|_{\mathbb{C}}^2 dx + (1+\varepsilon) \int_{B(0,t) \setminus B(0,s)} |e(\tilde{u}_n)|_{\mathbb{C}}^2 dx \\ + C(1+\varepsilon^{-1}) \int_{B(0,t)} |\nabla\varphi|^2 |\tilde{u}_n - a_n - u|^2 dx.$$

Returning to the notation $\mathbb{C}\xi : \xi$ and using $\tilde{u}_n - a_n \to u$ in $L_{\text{loc}}^2(B(0,t); \mathbb{R}^N)$, $\varphi \in C_c^\infty(B(0,t))$, along with (3.19a), this yields, for all $r, s$ such that $0 < r < s < t$,

$$\limsup_{n \to \infty} \int_{B(0,t)} \mathbb{C}e(v_n) : e(v_n) dx \leq (1 + \varepsilon) \int_{B(0,t)} \mathbb{C}e(v) : e(v) dx + (1 + \varepsilon)(\alpha(t) - \alpha(r)).$$

In the previous inequality, $\varepsilon$ can be sent to 0 and $r \uparrow s$ to find

$$\limsup_{n \to \infty} \int_{B(0,t)} \mathbb{C}e(v_n) : e(v_n) dx \leq \int_{B(0,t)} \mathbb{C}e(v) : e(v) dx + (\alpha(t) - \alpha(s-)), \qquad (3.22)$$

where $\alpha(s-) := \lim_{r \uparrow s} \alpha(r)$.

*Step 4 (Surface energy of the competitors).* By definition of $L_n$ we have

$$L_n \cap B(0, t) = \tilde{K}_n \cap B(0, t) \setminus B(0, s).$$

Thus, (3.19b) gives

$$\limsup_{n \to \infty} c_n \mathcal{H}^{N-1}(L_n \cap B(0, t)) \leq \beta(t) - \beta(s). \qquad (3.23)$$

*Step 5 (Topological constraint).* The pair $(v_n, L_n)$ may not be a topological competitor of $(u_n, K_n)$ in $B(0, t)$ because we have wiped out $K_n$ in the interior, possibly creating holes. To rectify this, we fix a parameter $t' \in (t, 1)$, which will later be sent down to $t$, and modify $L_n$ to construct a topological competitor in $B(0, t')$. We emphasize that there is no need for this step in the case of GSBD quasiminimizers because topological competitors are not considered in that case.

Let $(V_i)_{i \in \mathbb{N}_0}$ denote the open and connected components of $B(0, t') \setminus K_n$, ordered in such a way that $|V_i| \geq |V_{i+1}|$ (we omit the dependence on $n$). We show that the volume of $V_0$ is big in the sense that

$$|B(0, t') \setminus V_0| \leq C \mathcal{H}^{N-1}(K_n)^{N/(N-1)}, \qquad (3.24)$$
where $C > 0$ is a dimensional constant. Indeed, for each $i$, we have $\partial V_i \cap B(0, t') \subset K_n$ and $\mathcal{H}^{N-1}(K_n) < +\infty$, so by [3, Proposition 3.62] $V_i$ has finite perimeter in $B(0, t')$ with $\partial^* V_i \cap B(0, t') \subset K_n$. As $(V_i)_{i \in \mathbb{N}_0}$ forms a Caccioppoli partition of $B(0, t')$, by [3, Theorem 4.17] it follows that

$$ \sum_{i \ge 0} \mathcal{H}^{N-1}(\partial^* V_i \cap B(0, t')) \le 2 \mathcal{H}^{N-1}(K_n \cap B(0, t')). \quad (3.25) $$

From the ordering properties of the sequence, we see that for all $i \ge 1$,

$$ |V_i| \le |V_0| \le |B(0, t') \setminus V_i|, $$

and thus we can estimate by the relative isoperimetric inequality

$$ |V_i| \le C \mathcal{H}^{N-1}(\partial^* V_i \cap B(0, t'))^{N/(N-1)}. $$

We sum the above inequality over $i \ge 1$, use the superadditivity of $t \mapsto t^{N/(N-1)}$, and apply (3.25) to obtain the desired estimate (3.24).

We define the relatively closed set $F_n := B(0, t') \setminus V_0$ and note by the coarea formula that

$$ |F_n| = \int_0^{t'} \mathcal{H}^{N-1}(F_n \cap \partial B(0, \rho)) \, d\rho. $$

Using this equality and (3.24), we find a radius $\rho_n \in (t, t')$ such that

$$ \mathcal{H}^{N-1}(F_n \cap \partial B(0, \rho_n)) \le (t' - t)^{-1} |F_n| \le C(t') \mathcal{H}^{N-1}(K_n)^{N/(N-1)}. \quad (3.26) $$

We then replace $L_n$ by

$$ G_n := L_n \cup (F_n \cap \partial B(0, \rho_n)). $$

Since $c_n \mathcal{H}^{N-1}(K_n)$ is uniformly bounded and $\mathcal{H}^{N-1}(K_n) \to 0$, we use (3.26) to deduce that

$$ \lim_{n \to \infty} c_n \mathcal{H}^{N-1}(F_n \cap \partial B(0, \rho_n)) = 0, \quad (3.27) $$

hence the contribution of $F_n$ to the crack will disappear in the limit.

We now justify that $(v_n, G_n)$ is a topological competitor of $(u_n, K_n)$ in the ball $B(0, t')$ and, in particular, satisfies (2.2). Arguing by contraposition, we consider two points $x$ and $y$ in $B(0, 1) \setminus (B(0, t') \cup K_n)$ that are connected by a continuous path which is disjoint from $G_n$. If the path does not intersect $\partial B(0, \rho_n)$, then it stays in the complement of $\overline{B(0, \rho_n)}$ where $G_n$ coincides with $K_n$. Therefore, this continuous path does not meet $K_n$, and $x$ and $y$ are also connected in $B(0, 1) \setminus K_n$. If the path meets $\partial B(0, \rho_n)$, it can only be at a point of $V_0$. Considering the portion of the path starting from $x$ to the first time it meets $\partial B(0, \rho_n)$, we see that $x$ is connected to $V_0$ in the complement of $K_n$. Likewise, we see that $y$ is also connected to $V_0$ in the complement of $K_n$. Since $V_0$ is an open connected set disjoint from $K_n$, $x$ and $y$ are also connected in $B(0, 1) \setminus K_n$. This shows that (2.2) holds, as desired.

*Step 6 (Limit passage using quasiminimality).* By item (1) in Step 1 and (3.20) we get $e(v_n) = e(u_n)$ a.e. in $B(0, t') \setminus B(0, t)$. This along with (3.17a) and (3.22) gives

$$ \limsup_{n \to \infty} \int_{B(0, t')} \mathbb{C}e(v_n) : e(v_n) \, dx \le \int_{B(0, t)} \mathbb{C}e(v) : e(v) \, dx + (\alpha(t') - \alpha(s-)). $$

Similarly, by (3.17b), (3.23), and (3.27), we have

$$ \limsup_{n \to \infty} c_n \mathcal{H}^{N-1}(G_n \cap B(0, t')) \le \beta(t') - \beta(s). $$
Finally, we apply the quasiminimality property of $(u_n, K_n)$ in $B(0, t')$, see (3.14), to find

$$ \begin{aligned} & \int_{B(0,t')} \mathbb{C}e(u_n) : e(u_n) \,dx + c_n M^{-1} \mathcal{H}^{N-1}(K_n \cap B(0,t')) \\ & \leq \int_{B(0,t')} \mathbb{C}e(v_n) : e(v_n) \,dx + c_n M \mathcal{H}^{N-1}(G_n \cap B(0,t')) + \varepsilon_n. \end{aligned} $$

Taking the lim sup of both sides and using (3.17a)-(3.17b), we find

$$ \alpha(t') + M^{-1}\beta(t') \leq \int_{B(0,t)} \mathbb{C}e(v) : e(v) \,dx + (\alpha(t') - \alpha(s-)) + M(\beta(t') - \beta(s)). $$

As $t$ is a point of continuity for both $\alpha$ and $\beta$, we can let $t' \downarrow t$ and $s \uparrow t$ to recover

$$ \alpha(t) + M^{-1}\beta(t) \leq \int_{B(0,t)} \mathbb{C}e(v) : e(v) \,dx. \quad (3.28) $$

This holds a priori for almost all $t \in (0, 1)$ and all $v \in W^{1,2}(B(0, 1); \mathbb{R}^N)$ such that $\{u \neq v\} \subset B(0, 1)$. Since $\alpha$ and $\beta$ are non-decreasing and the right-hand side of (3.28) is continuous in $t$, we deduce that (3.28) actually holds for all $t \in (0, 1)$. Taking $v = u$ and recalling (3.18), we find that

$$ \alpha(t) = \int_{B(0,t)} \mathbb{C}e(u) : e(u) \,dx \quad \text{and} \quad \beta(t) = 0 $$

for all $t \in (0, 1)$. This along with (3.17a)-(3.17b) shows (3.15). Using these two equalities in (3.28), we have for all $v \in W^{1,2}(B(0, 1); \mathbb{R}^N)$ with $\{v \neq u\} \subset B(0, t)$ that

$$ \int_{B(0,t)} \mathbb{C}e(u) : e(u) \,dx \leq \int_{B(0,t)} \mathbb{C}e(v) : e(v) \,dx. $$

Then, it follows easily that $u$ minimizes the elastic energy $v \mapsto \int_{B(0,1)} \mathbb{C}e(v) : e(v) \,dx$ subject to its own Dirichlet condition on $\partial B(0, 1)$, giving (3.16) and concluding the proof. $\square$

Before turning to the proof of the decay estimate, we observe that quasiminimizers have a natural scaling coming from the energy.

**Remark 3.6 (Rescaling).** If a pair $(u, K)$ is a topological $M$-quasiminimizer in $B(x_0, r_0)$ with gauge $h$, then the pair $(\hat{u}, \hat{K})$ defined by

$$ \hat{u}(x) := r_0^{-1/2} u(x_0 + x r_0) \quad \text{and} \quad \hat{K} := r_0^{-1} (K - x_0) $$

is a topological $M$-quasiminimizer in $B(0, 1)$, with gauge $\hat{h}(t) := h(r_0 t)$. Moreover, one has for all $t \in (0, 1]$,

$$ \begin{aligned} & \int_{B(0,t) \setminus \hat{K}} \mathbb{C}e(\hat{u}) : e(\hat{u}) \,dx + \mathcal{H}^{N-1}(\hat{K} \cap B(0,t)) \\ & = \frac{1}{r_0^{N-1}} \left( \int_{B(x_0, tr_0) \setminus K} \mathbb{C}e(u) : e(u) \,dx + \mathcal{H}^{N-1}(K \cap B(x_0, tr_0)) \right). \end{aligned} $$

The same scaling is at play for $GSBD M$-quasiminimizers, except it only requires defining $\hat{u}$ as above, as this implicitly includes the rescaling for the jump set $J_u$.

We finally pass to the proof of the decay estimate for the energy.
Proof of Proposition 3.1. By scaling (Remark 3.6), it suffices to prove the result in $B(0, 1)$. We proceed by contradiction and assume that for some fixed $M \ge 1$, $\tau \in (0, 1)$, and $C > 0$, there exist a sequence $\varepsilon_n \to 0$ and a sequence of pairs $(u_n, K_n)$ such that each $(u_n, K_n)$ is M-quasiminimal with gauge $h_n$ in $B(0, 1)$ and

$$ \mathcal{H}^{N-1}(K_n \cap B(0, 1)) \le \varepsilon_n, \quad h_n(1) \le \varepsilon_n G_n(0, 1) $$

but

$$ G_n(0, \tau) > C\tau^N G_n(0, 1), \qquad (3.29) $$

where

$$ G_n(0, t) := \int_{B(0,t)} \mathbb{C}e(u_n) : e(u_n) \,dx + \mathcal{H}^{N-1}(K_n \cap B(0,t)) \quad \text{for all } t \in (0, 1). $$

We will derive a contradiction by choosing $C > 0$ large enough (depending only on $N$, $\mathbb{C}$ but not on $\tau$ or $M$). We set $g_n := G_n(0, 1)$ and observe using (3.29) that $g_n \in (0, \infty)$. We then define $w_n := g_n^{-1/2}u_n$. One can check that

$$ \int_{B(0,1)} \mathbb{C}e(w_n) : e(w_n) \,dx + g_n^{-1} \mathcal{H}^{N-1}(K_n \cap B(0,1)) = 1, \qquad (3.30) $$

$$ \int_{B(0,\tau)} \mathbb{C}e(w_n) : e(w_n) \,dx + g_n^{-1} \mathcal{H}^{N-1}(K_n \cap B(0,\tau)) > C\tau^N \qquad (3.31) $$

and that for all $t \in (0, 1)$, for all pairs $(v, L)$ with $\{v \ne w_n\} \subset \subset B(0, t)$ and $L \Delta K_n \subset \subset B(0, t)$, we have

$$ \int_{B(0,t)} \mathbb{C}e(w_n) : e(w_n) \,dx + M^{-1}g_n^{-1} \mathcal{H}^{N-1}(K_n \cap B(0,t)) \\ \le \int_{B(0,t)} \mathbb{C}e(v) : e(v) \,dx + Mg_n^{-1} \mathcal{H}^{N-1}(L \cap B(0,t)) + \varepsilon_n. $$

By an application of Proposition 3.5 (for $c_n = g_n^{-1}$), we find a subsequence (not relabeled) and a function $w \in W^{1,2}(B(0, 1); \mathbb{R}^N)$ such that for all $t \in (0, 1)$,

$$ \lim_{n \to \infty} \int_{B(0,t)} \mathbb{C}e(w_n) : e(w_n) \,dx = \int_{B(0,t)} \mathbb{C}e(w) : e(w) \,dx, \\ \lim_{n \to \infty} g_n^{-1} \mathcal{H}^{N-1}(K_n \cap B(0,t)) = 0, $$

and $w$ is a minimizer of the elastic energy $v \mapsto \int_{B(0,1)} \mathbb{C}e(v) : e(v) \,dx$ subject to its own boundary conditions. Thanks to Korn's inequality, the elasticity tensor $\mathbb{C}$ is coercive in the sense that $\|\varphi\|_{H^1}^2 \le C \int \mathbb{C}e(\varphi) : e(\varphi)$ for all test functions $\varphi$. This allows the application of the standard regularity theory of elliptic systems [22, Section 4 and 5], and in particular there exist a constant $C_0 \ge 1$ (depending on $N$, $\mathbb{C}$) such that

$$ \int_{B(0,\tau)} \mathbb{C}e(w) : e(w) \,dx \le C_0\tau^N \int_{B(0,1)} \mathbb{C}e(w) : e(w) \,dx. \qquad (3.32) $$

But passing to the limit in (3.30) and (3.31) yields

$$ \int_{B(0,1)} \mathbb{C}e(w) : e(w) \,dx \le 1, \quad \int_{B(0,\tau)} \mathbb{C}e(w) : e(w) \,dx \ge C\tau^N. $$

Plugging these into (3.32), we obtain a contradiction by taking $C = 2C_0$. $\square$
## 4. APPROXIMATION OF PAIRS WITH A SMALL CRACK SET

The goal of this section is to establish the compactness Lemma 3.3 for pairs $(u, K)$ and $GSBD$ functions. This was shown by CHAMBOLLE ET AL. [7] for $GSBD$ functions on the unit cube. Here we discuss how the proof can be adapted to the unit ball and, for topological quasiminimizers, how one ensures that the constructed approximations $(\tilde{u}_n, \tilde{K}_n)$ are admissible in the sense that $\tilde{u}_n$ is Sobolev away from the crack and that the constructed crack is closed.

As in [7], an essential step in the proof is a mollification procedure to construct an approximation of a function with small crack. We adapt this approach to our setting and emphasize the necessary changes to account for (1) working on the unit ball and (2) ensuring that the pair $(\tilde{u}_n, \tilde{K}_n)$ is admissible. These are the only changes in the proof, and they do not deal with the topological nature of the competitors in any other way. Given that our result is an adaptation of the original technical proof for $GSBD$ in [7, Theorem 3], we assume that the reader interested in these details is already familiar with the proof in [7], and we discuss how the proof can be adapted to our setting.

**Proposition 4.1.** There exist constants $\eta > 0$, $c > 1$, $s > 0$ (depending on $N$ and $\mathbb{C}$) such that the following holds. Let $(u, K)$ be a pair in $B(0, 1)$ with

$$ \int_{B(0,1)} |e(u)|^2 dx + \mathcal{H}^{N-1}(K) < +\infty. $$

Assume that $\delta := \mathcal{H}^{N-1}(K)^{1/N} < \eta$. Then, there exist a radius $R \in (1 - \sqrt{\delta}, 1)$, a pair $(\tilde{u}, \tilde{K})$ in $B(0, 1)$, and a Borel set $\tilde{\omega} \subset B(0, R)$ such that

$$ (1) \quad \tilde{u} \text{ is } C^\infty \text{ in } B(0, 1 - \sqrt{\delta}) \text{ and } (\tilde{u}, \tilde{K}) = (u, K) \text{ in } B(0, 1) \setminus B(0, R); $$

$$ (2) \quad \textit{we have} $$

$$ \mathcal{H}^{N-1}(\tilde{K} \setminus K) \le c\sqrt{\delta}\mathcal{H}^{N-1}(K \setminus B(0, 1 - \sqrt{\delta})) \quad (4.1) $$

and for all open set $\Omega \subset B(0, 1)$,

$$ \int_{\Omega} \mathbb{C}e(\tilde{u}) : e(\tilde{u}) dx \le (1 + c\delta^s) \int_{\Omega_{\delta}} \mathbb{C}e(u) : e(u) dx + c\sqrt{\delta} \int_{B(0,1) \setminus B(0,1-\sqrt{\delta})} \mathbb{C}e(u) : e(u) dx, \quad (4.2) $$

where $\Omega_{\delta} = \{x \in B(0, 1) : \text{dist}(x, \Omega) < c\delta\}$;

$$ (3) \quad \textit{we have} $$

$$ |\tilde{\omega}| \le c\delta\mathcal{H}^{N-1}(K) $$

and

$$ \int_{B(0,1)\setminus\tilde{\omega}} |\tilde{u} - u|^2 dx \le c\delta^2 \int_{B(0,1)} |e(u)|^2 dx. $$

*Outline of proof.* We outline the proof and take the liberty to omit necessary details spelled out in [7, Theorem 3], and instead emphasize why the setting of pairs only requires a couple of adaptations.

**Step 1 (Whitney covering).** To define the approximation $(\tilde{u}, \tilde{K})$, the idea is to find a good radius $R \in (1 - \sqrt{\delta}, 1)$ via an averaging argument with energetically favorable properties. In particular, $R$ is chosen such that

$$ \mathcal{H}^{N-1}(K \cap \partial B(0, R)) = 0. \quad (4.3) $$

Then, we will make no changes to $(u, K)$ outside of the ball $B(0, R)$. To modify $(u, K)$ inside $B(0, R)$, we construct a Whitney covering of this region made up of cubes $\{q_i\}_{i \in \mathbb{N}}$ with each side of $q_i$ is of length $r_i$. Further, we assume $q'_i \subset B(0, R)$, where $q'_i$ is a cube with the same center as $q_i$ and side length given by $3r_i/2$. Importantly, all cubes intersecting $B(0, 1 - \sqrt{\delta})$ have side length $\delta$ and the cubes refine as
they approach $\partial B(0, R)$, meaning diam$(q_i) \to 0$ as the center of $q_i$ approaches $\partial B(0, R)$. This is used to ensure that the trace of the modified function coincides with the trace of $u$. In [7], this step is done using an explicit covering made of dyadic cubes. Working on the ball, as we do, requires only minor changes, namely using a Whitney covering for the ball.

**Step 2 (Good and bad cubes).** We partition the cubes $q_i$ into good cubes and bad cubes, depending on whether the cube contains a relatively large proportion of the crack. Precisely, fixing $\eta$ as $\eta = \frac{1}{2}8^{-N}c_*^{-1}$ with $c_*$ from [7, (6)], $q_i$ is a good cube if

$$ \mathcal{H}^{N-1}(K \cap q_i') \leq \eta r_i^{N-1}, \quad (4.4) $$

and a bad cube otherwise. We denote the indices of good cubes by $\mathbb{G}$ and, similarly, $\mathbb{B}$ for the bad cubes.

**Step 3 (Modified function).** We consider a partition of unity $\{\phi_i\}_{i \in \mathbb{G}}$ on the good cubes, subordinate to $\bigcup_{i \in \mathbb{G}} q_i' \setminus \bigcup_{i \in \mathbb{B}} q_i$. In each good cube $q_i$, we consider a smoothed out function $\rho_{\delta_i} * (u\chi_{q_i' \setminus \omega_i} + a_i\chi_{\omega_i})$, where $\omega_i \subset q_i'$ and $a_i$ is a rigid motion associated to the cube through the Korn inequality for functions with a small jump set (see [7, (13)-(14)]), and $\rho_{\delta_i}$ is a mollifier supported in $B(0, \delta_i)$ with $\delta_i \approx r_i/10$. The modified function $\tilde{u}$ is defined as

$$ \tilde{u}(x) := \begin{cases} \sum_{i \in \mathbb{G}} \phi_i(x) (\rho_{\delta_i} * (u\chi_{q_i' \setminus \omega_i} + a_i\chi_{\omega_i}))(x) & \text{if } x \in \bigcup_{i \in \mathbb{G}} q_i \setminus \bigcup_{i \in \mathbb{B}} q_i, \\ u(x) & \text{if } x \in (\bigcup_{i \in \mathbb{B}} q_i) \cup (B(0, 1) \setminus B(0, R)). \end{cases} \quad (4.5) $$

As we have blindly stitched together the mollified function in the good cubes with the original function $u$ on the bad cubes, we cannot prevent a jump in the function across these boundaries. So, the new jump set must be defined as

$$ \tilde{K} = K \cup \overline{\bigcup_{i \in \mathbb{B}} \partial q_i}, \quad (4.6) $$

where we must take the closure to ensure that $\tilde{K}$ is closed. The assumption $\delta < \eta$ ensures that all cubes intersecting $B(0, 1 - \sqrt{\delta})$ lie in $\mathbb{G}$, and thus $\tilde{u}$ is $C^\infty$ in $B(0, 1 - \sqrt{\delta})$. Indeed, as all cubes $q_i$ intersecting $B(0, 1 - \sqrt{\delta})$ have side length $\delta$, we get $\mathcal{H}^{N-1}(K \cap q_i') \leq \delta^N < \eta\delta^{N-1} = \eta r_i^{N-1}$, and thus (4.4) holds.

**Step 4 (Size of the closed crack).** Estimating the size of the new crack proceeds as in the original proof of [7, Theorem 3], except one must now additionally make sure that there is no charge arising on the boundary of $\partial B(0, R)$ coming from taking the closure of $\bigcup_{i \in \mathbb{B}} \partial q_i$ in (4.6). We show that

$$ \overline{\bigcup_{i \in \mathbb{B}} \partial q_i} \cap \partial B(0, R) \subset K \cap \partial B(0, R), \quad (4.7) $$

which then by (4.3) gives

$$ \mathcal{H}^{N-1} \left( \overline{\bigcup_{i \in \mathbb{B}} \partial q_i} \cap \partial B(0, R) \right) = 0. $$

To see (4.7), if $x \in \overline{\bigcup_{i \in \mathbb{B}} \partial q_i} \cap \partial B(0, R)$, then there is a sequence of bad cubes $(q_{i_k})_{k \in \mathbb{N}}$ with $i_k \in \mathbb{B}$ and $\text{dist}(x, q_{i_k}) \to 0$ as $k \to \infty$. Since $x \in \partial B(0, R)$, we also have $\text{diam}(q_{i_k}) \to 0$. Finally, as (4.4) fails for each bad cube, $K \cap q_{i_k} \neq \emptyset$, and we may combine the above facts to conclude that $\text{dist}(x, K) \to 0$. This implies $x \in K \cap \partial B(0, R)$ because $K$ is closed, concluding (4.7). This along with [7, (12)] shows (4.1).

**Step 5 (Locally Sobolev function).** We need to ensure that $\tilde{u} \in W_{\text{loc}}^{1,2}(\Omega \setminus \tilde{K}; \mathbb{R}^N)$. If $x \notin \tilde{K} \cup \partial B(0, R)$, as the Whitney covering is locally finite in the interior of $B(0, R)$ and no change is made outside of $B(0, R)$, it is clear that there is a radius $r > 0$ such that $\tilde{u} \in W^{1,2}(B(x, r); \mathbb{R}^N)$. By definition of $\tilde{K}$ in (4.6), if $x \in \partial B(0, R) \setminus \tilde{K}$, then $x \in \partial B(0, R) \setminus K$ by (4.7). Consequently, there is a radius $r > 0$ such that $B(x, r)$ does not intersect $K$ or, thereby, any bad cube. As in the original proof, one can then verify that the trace of $\tilde{u}$ coincides with $u$ on $\partial B(0, R)$ implying $\tilde{u} \in W^{1,2}(B(x, r); \mathbb{R}^N)$. However, there is a simpler option in the setting of pairs $(u, K)$. If $q_i \cap K = \emptyset$, one can simply replace $\rho_{\delta_i} * (u\chi_{q_i' \setminus \omega_i} + a_i\chi_{\omega_i})$
by $u$ in the definition of $\tilde{u}$ in (4.5), so that $\tilde{u} = u$ in $B(x, r)$ for $r > 0$ sufficiently small. This discussion also shows $(\tilde{u}, \tilde{K}) = (u, K)$ in $B(0, 1) \setminus B(0, R)$ concluding the proof of (1).

Step 6 (Remaining arguments). The proof of (4.2) and item (3) remains the same as in [7], see in particular [7, (26)-(27)]. □

Finally, we turn to the proof of the compactness result used to obtain the decay estimate.

*Proof of Lemma 3.3.* By rescaling it suffices to perform the proof for $t = 1$. We let $\delta_n := \mathcal{H}^{N-1}(K_n)^{1/N}$. For all sufficiently large $n$, we may apply Proposition 4.1. We obtain a sequence of pairs $(\tilde{u}_n, \tilde{K}_n)$ and of sets $\tilde{\omega}_n$ satisfying properties (1), (2), and (3) of Proposition 4.1. By (1) we get $\{\tilde{u}_n \neq u_n\} \subset \subset B(0, 1)$ and $K_n \Delta \tilde{K}_n \subset \subset B(0, 1)$. From (2), we observe that for all $s \in (0, 1)$,

$$ \mathcal{H}^{N-1}(\tilde{K}_n \cap A_{s,1}) \leq \mathcal{H}^{N-1}(K_n \cap A_{s,1}) + c\sqrt{\delta_n}\mathcal{H}^{N-1}(K_n \setminus B(0, 1 - \sqrt{\delta_n})) $$

and for $n$ large enough, $K_n \setminus B(0, 1 - \sqrt{\delta_n}) \subset K_n \cap A_{s,1}$, whence

$$ \mathcal{H}^{N-1}(\tilde{K}_n \cap A_{s,1}) \leq (1 + o(1))\mathcal{H}^{N-1}(K_n \cap A_{s,1}). $$

Similarly, for $0 < r < s < 1$, we observe that for all $n$ sufficiently large

$$ (A_{s,1})_{\delta_n} = \{x \in B(0, 1): \operatorname{dist}(x, A_{s,1}) < c\delta_n\} \subset A_{r,1} $$

and $B(0, 1) \setminus B(0, 1 - \sqrt{\delta_n}) \subset A_{r,1}$, so that again by (2) we have

$$ \begin{aligned} \int_{A_{s,1}} \mathbb{C}e(\tilde{u}_n) : e(\tilde{u}_n) \,dx &\leq (1 + c\delta^s) \int_{A_{r,1}} \mathbb{C}e(u_n) : e(u_n) \,dx + c\sqrt{\delta} \int_{A_{r,1}} \mathbb{C}e(u_n) : e(u_n) \,dx \\ &\leq (1 + o(1)) \int_{A_{r,1}} \mathbb{C}e(u_n) : e(u_n) \,dx. \end{aligned} $$

According to (3), the sets $(\tilde{\omega}_n)_n \subset B(0, 1)$ are such that $|\tilde{\omega}_n| \to 0$ and

$$ (\tilde{u}_n - u_n)\chi_{B(0,1)\setminus\tilde{\omega}_n} \to 0 \text{ in } L^2(B(0,1); \mathbb{R}^N). \quad (4.8) $$

Property (1) shows that for any fixed $t \in (0, 1)$, the functions $\tilde{u}_n$ belong to $C^\infty(B(0, t); \mathbb{R}^N)$ for all $n$ sufficiently large. Since $e(\tilde{u}_n)$ is uniformly bounded in $L^2$ by (2), we deduce that, up to extracting a subsequence, there exist a sequence of rigid motions $a_n$ and a function $u \in W^{1,2}(B(0, t); \mathbb{R}^N)$ such that

$$ \tilde{u}_n - a_n \to u \text{ in } L^2(B(0, t); \mathbb{R}^N) \quad (4.9) $$

and

$$ e(\tilde{u}_n) \to e(u) \text{ in } L^2(B(0, t); \mathbb{R}^{N \times N}). $$

Using (4.8), (4.9), and $|\tilde{\omega}_n| \to 0$, we conclude that

$$ (u_n - a_n)\chi_{B(0,1)\setminus\tilde{\omega}_n} \to u \text{ in } L^2(B(0, t); \mathbb{R}^N), $$

and in particular, $u_n - a_n$ converges to $u$ in measure in $B(0, t)$.

As in Remark 3.4, we note that, if we apply the above argument in $B(0, t)$ and subsequently in $B(0, t')$, where $t' \in (t, 1)$, we may take the sequence of rigid motions $(a_n)_n$ (now associated to a subsequence of a subsequence) and the limit function $u$ to be the same in each ball. Diagonalizing on a sequence $t \uparrow 1$, we may find $u \in W^{1,2}(B(0, 1); \mathbb{R}^N)$ and a subsequence (not relabeled) such that

$$ \tilde{u}_n - a_n \to u \text{ in } L^2_{\text{loc}}(B(0, 1); \mathbb{R}^N) \quad \text{and} \quad e(\tilde{u}_n) \to e(u) \text{ in } L^2_{\text{loc}}(B(0, 1); \mathbb{R}^{N \times N}), $$
and $u_n - a_n$ converges locally in measure to $u$ in $B(0, 1)$. Finally, as $e(\tilde{u}_n) \to e(u)$ in $L^2_{\text{loc}}(B(0, 1); \mathbb{R}^{N \times N})$, for all $r \in (0, 1)$ we get

$$ \int_{B(0,r)} \mathbb{C}e(u) : e(u) \, dx \le \liminf_{n \to \infty} \int_{B(0,r)} \mathbb{C}e(\tilde{u}_n) : e(\tilde{u}_n) \, dx, $$

and applying (4.2) for $\Omega = B(0, r)$, we also have for all $s \in (r, 1]$ that

$$ \liminf_{n \to \infty} \int_{B(0,r)} \mathbb{C}e(\tilde{u}_n) : e(\tilde{u}_n) \, dx \le \liminf_{n \to \infty} \int_{B(0,s)} \mathbb{C}e(u_n) : e(u_n) \, dx. $$

We therefore find

$$ \int_{B(0,r)} \mathbb{C}e(u) : e(u) \, dx \le \liminf_{n \to \infty} \int_{B(0,s)} \mathbb{C}e(u_n) : e(u_n) \, dx, $$

and we can let $r \uparrow s$ on the left-hand side to prove the last item of the statement. $\square$

<figure><img src="image_2.png" /></figure>

## ACKNOWLEDGEMENTS

This work was supported by the DFG project FR 4083/5-1 and by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategies EXC 2044 - 390685587, Mathematics Münster: Dynamics-Geometry-Structure and EXC 2047/1 - 390685813, and project 211504053 - SFB 1060. The second author was also funded by the French National Research Agency (ANR) under grant ANR-21-CE40-0013-01 (project GeMfaceT).

## APPENDIX A. POINTWISE DEFINITION OF GSBD QUASIMINIMIZERS

In this section, we justify that a GSBD quasiminizer can be defined pointwise everywhere outside of the crack set. We note that, due to the error terms in the definition of quasiminimizers (coming from the gauge $h$), the displacement field does not solve an elliptic PDE away from the crack set and standard regularity theory does not show that $u$ is smooth. In the scalar case, using the upper bound on the elastic energy, namely

$$ \int_{B(x_0,r)} |\nabla u|^2 \, dx \le Cr^{N-1}, $$

and CAMPANATO's characterization of Hölder spaces, one can deduce that $u$ is Hölder regular in any small ball $B(x, r)$ for which $H^{N-1}(B(x, r) \cap J_u) = 0$, which leads to the stronger conclusion $B(x, r) \cap J_u = \emptyset$. In the vectorial case, similar reasoning shows that every point outside of the crack set is a Lebesgue point, see also [8, Proof of Theorem 5.7, Part 2]. As we are interested in the behavior of the displacement $u$ "away from the crack", we consider a ball $B(x_0, r)$ with $H^{N-1}(B(x_0, r) \cap J_u) = 0$. By slicing, this implies that $u$ is a Sobolev function. By rescaling (see Remark 3.6) we may replace $B(x_0, r)$ by the unit ball $B(0, 1)$.

**Proposition A.1.** Let $u \in W_{\text{loc}}^{1,2}(B(0, 1); \mathbb{R}^N)$ and assume that there exists a constant $C \ge 1$ such that for any ball $B(x_0, r) \subset B(0, 1)$, we have

$$ \int_{B(x_0,r)} |e(u)|^2 \, dx \le Cr^{N-1}. \qquad (A.1) $$

Then, for every $x_0 \in B(0, 1)$, there exists $b_{x_0} \in \mathbb{R}^N$ such that

$$ \lim_{r \to 0} \int_{B(x_0,r)} |u - b_{x_0}|^2 \, dx = 0. $$

In the language of BV, this implies $J_u = S_u = \emptyset$.
Proof. We use C as a generic constant greater than 1 that depends only on N and the constant in (A.1). We recall that for all balls $B(x_0, r) \subset \mathbb{R}^N$ and for all linear maps $a(x) = A(x - x_0) + b$, we have

$$C^{-1} (|b|^2 + r^2 |A|^2) \leq \int_{B(x_0, r)} |a(x)|^2 dx \leq C (|b|^2 + r^2 |A|^2). \quad (A.2)$$

by equivalence of norms in a finite dimensional space. Now, let us fix $x_0 \in B(0, 1)$ and $R > 0$ such that $B(x_0, R) \subset B(0, 1)$. For $0 < r \leq R$, the Korn-Poincaré inequality and (A.1) tell us that

$$\int_{B(x_0, r)} |u - a_{x_0, r}|^2 dx \leq Cr^2 \int_{B(x_0, r)} |e(u)|^2 dx \leq Cr, \quad (A.3)$$

where the rigid motion $a_{x_0,r}(x) = A_{x_0,r}(x - x_0) + b_{x_0,r}$ is defined by

$$A_{x_0,r} := \int_{B(x_0,r)} \frac{\nabla u - \nabla u^T}{2} dx \quad \text{and} \quad b_{x_0,r} := \int_{B(x_0,r)} (u(x) - A(x - x_0)) dx.$$

We consider $0 < s \leq r \leq R$. If $s \in [r/2, r]$, we estimate

$$\int_{B(x_0, s)} |a_{x_0, s} - a_{x_0, r}|^2 dx \leq C \int_{B(x_0, s)} |u - a_{x_0, s}|^2 dx + C \int_{B(x_0, r)} |u - a_{x_0, r}|^2 dx \leq Cr^{N+1}$$

and therefore $|b_{x_0,s} - b_{x_0,r}| \leq Cr^{1/2}$ and $|A_{x_0,s} - A_{x_0,r}| \leq Cr^{-1/2}$ by (A.2). Next, for any $s \in (0, r]$, there exists an integer $k \geq 0$ such that $s \in [2^{-(k+1)}r, 2^{-k}r]$. According to the previous estimate, we have

$$|b_{x_0,s} - b_{x_0,2^{-k}r}| \leq C(2^{-k}r)^{1/2}$$

and, for all $0 \leq i < k$,

$$|b_{x_0,2^{-i}r} - b_{x_0,2^{-i}r}| \leq C(2^{-i}r)^{1/2},$$

whence

$$|b_{x_0,s} - b_{x_0,r}| \leq C \sum_{i \leq k} (2^{-i}r)^{1/2} \leq Cr^{1/2}. \quad (A.4)$$

A similar iteration argument show that for all $s \in (0, r)$ it holds that

$$|A_{x_0,s} - A_{x_0,r}| \leq C \sum_{i \leq k} (2^{-i}r)^{-1/2} \leq Cs^{-1/2}. \quad (A.5)$$

By (A.4), $b_{x_0,r}$ is a Cauchy sequence, and we see that the limit $b_{x_0} := \lim_{r \to 0} b_{x_0,r}$ exists, and in particular,

$$\lim_{r \to 0} \int_{B(x_0, r)} |b_{x_0, r} - b_{x_0}|^2 dx = 0. \quad (A.6)$$

Inequality (A.5) shows that $|A_{x_0,r}| \leq Cr^{-1/2} + |A_{x_0,R}|$ for all $0 < r \leq R$, and in conjunction with (A.6), we have

$$\limsup_{r \to 0} \int_{B(x_0, r)} |a_{x_0, r} - b_{x_0}|^2 dx \leq C \limsup_{r \to 0} r^2 |A_{x_0, r}|^2 = 0. \quad (A.7)$$

Combining (A.3) and (A.7), we see that

$$\lim_{r \to 0} \int_{B(x_0, r)} |u - b_{x_0}|^2 dx = 0.$$

This concludes the proof. □
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(Manuel Friedrich) DEPARTMENT OF MATHEMATICS, FRIEDRICH-ALEXANDER UNIVERSITÄT ERLANGEN-NÜRNBERG. CAUER-STR. 11, D-91058 ERLANGEN, GERMANY, & MATHEMATICS MÜNSTER, UNIVERSITY OF MÜNSTER, EINSTEINSTR. 62, D-48149 MÜNSTER, GERMANY

Email address: manuel.friedrich@fau.de

(Camille Labourie) DEPARTMENT OF MATHEMATICS, FRIEDRICH-ALEXANDER UNIVERSITÄT ERLANGEN-NÜRNBERG. CAUER-STR. 11, D-91058 ERLANGEN, GERMANY

Email address: camille.labourie@fau.de

(Kerrek Stinson) HAUSDORFF CENTER FOR MATHEMATICS, UNIVERSITY OF BONN, ENDENICHER ALLEE 62, 53115 BONN, GERMANY

Email address: kerrek.stinson@hcm.uni-bonn.de