Effective Theories for Brittle Materials : A Derivation of Cleavage Laws and Linearized Griffith Energies from Atomistic and Continuum Nonlinear Models [1 ed.] 9783832591625, 9783832540289

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28

Augsburger Schriften zur Mathematik, Physik und Informatik

Effective Theories for Brittle Materials: A Derivation of Cleavage Laws and Linearized Griffith Energies from Atomistic and Continuum Nonlinear Models

Manuel Friedrich

λογος

Eective Theories for Brittle Materials

A Derivation of Cleavage Laws and Linearized Grith Energies from Atomistic and Continuum Nonlinear Models Dissertation zur Erlangung des akademischen Grades

Dr. rer. nat.

eingereicht an der

Mathematisch-Naturwissenschaftlich-Technischen Fakultät

der Universität Augsburg

von

Manuel Friedrich Augsburg, März 2015

Augsburger Schriften zur Mathematik, Physik und Informatik Band 28 herausgegeben von: Professor Dr. F. Pukelsheim Professor Dr. B. Aulbach Professor Dr. W. Reif Professor Dr. B. Schmidt Professor Dr. D. Vollhardt

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ISBN 978-3-8325-4028-9 ISSN 1611-4256

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Contents Introduction

1

I An analysis of crystal cleavage in the passage from atomistic models to continuum theory 15 1 The model and main results 1.1 1.2 1.3 1.4 1.5 1.6

The discrete model . . . . . . . . . . . . . . . . . . . . . . . . Boundary values and scaling . . . . . . . . . . . . . . . . . . . Limiting minimal energy and cleavage laws . . . . . . . . . . . A specic model: The triangular lattice in two dimensions . . Limiting minimal congurations . . . . . . . . . . . . . . . . . Limiting variational problem . . . . . . . . . . . . . . . . . . . 1.6.1 Convergence of the variational problems . . . . . . . . 1.6.2 Analysis of a limiting variational problem . . . . . . . . 1.6.3 An application: Fractured magnets in an external eld

2 Preliminaries 2.1 2.2 2.3 2.4

Elementary properties of the cell energy Interpolation . . . . . . . . . . . . . . . An estimate on geodesic distances . . . . Cell energy of the triangular lattice . . .

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3 Limiting minimal energy and cleavage laws 3.1 3.2

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Warm up: Proof for the triangular lattice . . . . . Estimates on a mesoscopic cell . . . . . . . . . . 3.2.1 Mesoscopic localization . . . . . . . . . . . 3.2.2 Estimates in the elastic regime . . . . . . . 3.2.3 Estimates in the intermediate regime . . . 3.2.4 Estimates in the fracture regime . . . . . . 3.2.5 Estimates in a second intermediate regime Proof of the cleavage law . . . . . . . . . . . . . . Examples: mass-spring models . . . . . . . . . . . i

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3.4.1 3.4.2 3.4.3

Triangular lattices with NN interaction . . . . . . . . . . . Square lattices with NN and NNN interaction . . . . . . . Cubic lattices with NN and NNN interaction . . . . . . . .

4 Limiting minimal energy congurations 4.1 4.2 4.3

Fine estimates on the limiting minimal energy . . . . . Sharp estimates on the number of the broken triangles Convergence of almost minimizers . . . . . . . . . . . . 4.3.1 The supercritical case . . . . . . . . . . . . . . . 4.3.2 The subcritical case . . . . . . . . . . . . . . . . 4.3.3 Proof of the main limiting result . . . . . . . . .

5 The limiting variational problem 5.1 5.2

Convergence of the variational problems . . 5.1.1 The Γ-lim inf -inequality . . . . . . . 5.1.2 Recovery sequences . . . . . . . . . . Analysis of the limiting variational problem

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II A quantitative geometric rigidity result in SBD and the derivation of linearized models from nonlinear Grith energies 115 6 The model and main results 6.1 6.2 6.3 6.4

Rigidity estimates . . . . . . . . . . . . . . . . Compactness . . . . . . . . . . . . . . . . . . Γ-convergence and application to cleavage laws Overview of the proof . . . . . . . . . . . . . . 6.4.1 Korn-Poincaré-type inequality . . . . . 6.4.2 SBD-rigidity . . . . . . . . . . . . . . . 6.4.3 Compactness and Γ-convergence . . . .

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7 Preliminaries 7.1 7.2

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118 121 124 126 126 129 131

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Geometric rigidity and Korn: Dependence on the set shape . . . . 133 A trace theorem in SBV2 . . . . . . . . . . . . . . . . . . . . . . . 140

8 A Korn-Poincaré-type inequality 8.1 8.2 8.3

117

Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modication of sets . . . . . . . . . . . . . . . . . . . . . . . . . Neighborhoods of boundary components . . . . . . . . . . . . . 8.3.1 Rectangular neighborhood . . . . . . . . . . . . . . . . . 8.3.2 Dodecagonal neighborhood . . . . . . . . . . . . . . . . . Proof of the Korn-Poincaré-inequality . . . . . . . . . . . . . . . 8.4.1 Conditions for boundary components and trace estimate ii

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144 146 150 151 161 164 165

8.5

8.4.2 8.4.3 Trace 8.5.1 8.5.2 8.5.3 8.5.4 8.5.5

Modication algorithm . . . . . . . . . . . . . . . . . . . . Proof of the main theorem . . . . . . . . . . . . . . . . . . estimates for boundary components . . . . . . . . . . . . . Preliminary estimates . . . . . . . . . . . . . . . . . . . . . Step 1: Small boundary components . . . . . . . . . . . . Step 2: Subset with small projection of components . . . . Step 3: Neighborhood with small projection of components Step 4: General case . . . . . . . . . . . . . . . . . . . . .

9 Quantitative SBD-rigidity 9.1 9.2

9.3 9.4

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Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . A local rigidity estimate . . . . . . . . . . . . . . . . . . . 9.2.1 Estimates for the derivatives . . . . . . . . . . . . . 9.2.2 Estimates in terms of the H1 -norm . . . . . . . . . 9.2.3 Local rigidity for an extended function . . . . . . . Modication of the deformation . . . . . . . . . . . . . . . SBD-rigidity up to small sets . . . . . . . . . . . . . . . . 9.4.1 Step 1: Deformations with least crack length . . . . 9.4.2 Step 2: Deformations with a nite number of cracks 9.4.3 Step 3: General case . . . . . . . . . . . . . . . . . Proof of the main SBD-rigidity result . . . . . . . . . . . .

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Compactness of rescaled congurations . . . . . . . . . . . . Admissible & coarsest partitions and limiting congurations Derivation of linearized models via Γ-convergence . . . . . . Application: Cleavage laws . . . . . . . . . . . . . . . . . . .

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10 Compactness and Γ-convergence 10.1 10.2 10.3 10.4

166 176 178 178 181 183 191 194

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A Functions of bounded variation and Caccioppoli partitions

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B Rigidity and Korn-Poincaré's inequality

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A.1 (G)SBV and (G)SBD functions . . . . . . . . . . . . . . . . . . . 271 A.2 Caccioppoli partitions . . . . . . . . . . . . . . . . . . . . . . . . 276

iii

Introduction The main focus of this thesis lies on the derivation of eective models for brittle materials in the simultaneous passage from discrete-to-continuum and nonlinear to linearized systems. Such materials show an elastic response to very small displacements and develop cracks already at moderately large strains. Typically there is no plastic regime in between the restorable elastic deformations and complete failure due to fracture. In spite of its importance in applications, a thorough understanding of the cleavage behavior of brittle crystals remains a challenging problem in theoretical mechanics. In particular, it is fundamental to identify critical loads for failure and to analyze the geometry of crack paths that occur in the fractured regime. In variational fracture mechanics displacements and crack paths are determined from an energy minimization principle. Following the pioneering work of Grith [52], Francfort and Marigo [44] have introduced an energy functional comprising elastic bulk contributions for the intact regions of the body and surface terms that assign energy contributions on the crack paths comparable to the size of the crack of codimension one. Subsequently, these models have been investigated and extended in various directions. Among the vast body of literature we only mention the work of Dal Maso and Toader [37]; Francfort and Larsen [43]; Dal Maso, Francfort and Toader [35] and refer to [11] for further references. Determining energy minimizers of such functionals leads to solving a free discontinuity problem in the language of Ambrosio and De Giorgi [38] as the crack path, i.e., the set of discontinuity of the diplacement eld is not pre-assigned but has to be found as a solution to the variational problem. In particular, these models also lead to ecient numerical approximation schemes, cf., e.g., [7, 10, 58, 59, 66]. In the continuum setting many Grith energies contain anisotropic surface terms (see e.g. [2, 22, 42, 58]) modeling the fact that due to the crystalline structure of the materials certain directions for the formation of cracks are energetically favored. Indeed, under tensile boundary loads fracture typically occurs in the form of cleavage along crystallographic planes of the atomic lattice. Ultimately, such a continuum model should be identied as an eective theory derived from atomistic interactions. One aim of this thesis is to provide a rigorous study of discrete systems for the validity and failure of crystal cleavage in the multidimensional framework with vector valued deformations. 1

In the engineering literature such discrete systems had been analyzed computationally in [53, 55] and formally by renormalization group techniques in [61]. Braides, Lew and Ortiz [18] then showed analytically that in the continuum limit the energy satises a certain cleavage law with a universal form independent of the specic choice of the interatomic potential. In all these models the crack geometry is pre-assigned and fracture may only occur along planes leading eectively to a one-dimensional problem which is much easier to analyze. However, in order to understand the physical and geometrical cause for cleavage in the fracture regime it is indispensable to examine vectorial problems in more than one space dimension. In our model we assume that the macroscopic region occupied by the specimen is a cuboid subject to uniaxial tensile boundary conditions. Our focus on such boundary values corresponds to one of the basic experiments in determining e.g. the Poisson ratio and is naturally motivated by our main goal of analyzing cleavage behavior. We suppose that in our discrete model the atoms in the reference conguration are given by the portion of a Bravais lattice lying in that region. The interaction of the material points will be described by a general class of `cell energies' including well known mass-spring models where the pair interaction of neighboring atoms is modeled by potentials of Lennard-Jones type. We prove that under uniaxial tension in the continuum limit the energy satises a cleavage law of a universal form essentially only depending on the stiness and toughness of the material which may be deduced from the cell energy. The limiting energy exhibits quadratic response to small boundary displacements followed by a sharp constant cut-o beyond some critical value. Similarly as in the one-dimesional seminal paper √ [18] we nd that the most interesting regime for the elastic strains is given by ε (ε denotes the typical interatomic distance) as in this particular regime the elastic and the crack energy are of the same order. This is in accordance to the observation that brittle materials develop cracks already at moderately large strains. Suitable test congurations then show that asymptotically optimal congurations are given by homogeneous elastic deformations for subcritical boundary values and by congurations cleaved along specic crystallographic hyperplanes beyond critical loading. It seems that our analysis provides a rst multidimensional result for the validity of crystal cleavage. However, it turns out that there are non-generic models where also other, much more complex crack geometries are energetically optimal whence in the general case a full characterization of all asymptotically minimizing sequences seems currently out of reach. Consequently, for a deeper investigation we restrict our analysis to a specic two-dimensional model problem where the discrete system is given by the portion of a triangular lattice in a rectangular strip interacting via next neighbor Lennard-Jones type potentials. Indeed, in this case we can show that any sequence of minimizers converges (up to subsequences) to a homogeneous continuum deformation for subcritical boundary values, while it converges to a contin2

uum deformation which is cracked completely and does not store elastic energy in the supercritical case. We prove that in the generic case cleavage occurs along a unique crystallygraphic line, whereas for specic symmetric orientations of the crystal cleavage might fail. Nevertheless, also in these special cases we obtain a complete characterization of all possible limiting crack geometries. The model under investigation leads, in particular, to congurations respecting the Poisson eect, which would not be possible in scalar models. These results justify rigorously the aforementioned assumptions in the derivation of cleavage laws as, e.g., in [18]. Even though the uniaxial tension test is a natural set-up for investigating cleavage phenomena, it is desirable to also incorporate more general boundary conditions and to identify limiting continuum congurations and energies in the same energy regime which are not necessarily asymptotically energy minimizing. While the passage from discrete systems to eective continuum models via Γconvergence (see [12, 33]) is by now well understood for one-dimensional brittle chains, see e.g. [14, 15, 16], not much is known on discrete-to-continuum limits for models allowing for fracture in more than one dimension. The farthest reaching developments in that direction seem to be results for scalar valued models (see [17]) and approximations of vector valued free discontinuity problems where the elastic bulk part of the energy is characterized by linearized terms (see [2]) or by a quasiconvex stored energy density (see [42]). However, in more than one dimension the energy density of discrete systems such as well-known mass spring models is in general not given in terms of a discretized continuum quasiconvex function. For large strains these lattices typically become even unstable, see e.g. the basic model discussed in [50]. Consequently, in the regime of nite elasticity it is a subtle question if minimizers for given boundary data exist at all. On the other hand, for suciently small strains one may expect the Cauchy-Born rule to apply so that individual atoms do in fact follow a macroscopic deformation gradient, see [50, 27]. In particular this applies to the regime of innitesimal elastic strains. For purely elastic interactions this relation has also been obtained in the sense of Γ-convergence for a simultaneous passage from discrete to continuum and linearization process in [19, 65]. In the present context the investigation of cleavage laws already showed that √ the most interesting regime for the elastic strains is given by ε. Consequently, a passage from discrete-to-continuum systems naturally involves a linearization process. Indeed, we will prove that in the small displacement regime the energies associated to the discrete energies under consideration can be related to a continuum Grith energy functional with anisotropic surface contributions. In our analysis √ we rst make the simplifying assumption that we consider deformations lying ε-close to the identity mapping. Indeed, we discuss physically interesting applications where such a smallness assumption can be justied rigorously, e.g. (1) a boundary value problem describing uniaxial extension and (2) fractured magnets in an external eld. However, in general a small distance 3

form the identity mapping can not be inferred from energy bounds and identifying all possible limiting continuum congurations and energies is a subtle task. The main challenge is to establish suitable rigidity estimates being essential in the passage from nonlinear to linearized theory (see [19, 65]). One aim of this thesis is the derivation of such quantitative geometric rigidity estimates for brittle materials which allows to establish a general Γ-limit result in the passage from nonlinear to linearized energies in fracture mechanics without any a priori assumption on the deformation and the crack geometry. To avoid further complicacies of technical nature concerning the topological structure of cracks in higher dimensions and to concentrate on the essential diculties arising from the frame indierence of the energy density, we will tackle this problem in a continuum setting in two dimensions with isotropic crack energies. However, we believe that our results can be extended to discrete systems with anisotropic surface terms whereby we can justify a posteriori the aforementioned smallness assumptions. Clearly, such a derivation is not only important in the context of discrete systems but interesting on its own. Indeed, for many realistic models in fracture mechanics being genuinly nonlinear it is desirable to identify an eective linear theory and in this way to rigorously show that in the small displacement regime the neglection of eects arising from the non-linearities is a good approximation of the problem. In fact, the propagation of crack was studied in the framework of linearized elasticity since the seminal work of Grith and led to a lot of realistic applications in engineering. Also from a mathematical point of view the theory is well developed (see [5, 8]) and adopted in many recent works in applied analysis (see e.g. [8, 26, 54, 66]) since, as discussed above, such models are often signicantly easier to treat as their nonlinear counterparts due to the convexity of the bulk energy density. In general, the analysis of fracture models is often very involved due to the two coexistent, competing energy forms, the elastic and the crack energy, showing dierent scaling properties. Therefore, as a rst approach to the nonlinear-tolinear limit we consider each of the regimes separately and discuss the results which are available in the literature. For elastic bodies not exhibiting cracks the passage from nonlinear to linearized models is by now well understood and was rst rigorously derived by Dal Maso, Negri and Percivale in [36] in a continuum setting using Γ-convergence (cf. also [19, 65]). The main ingredient in their analysis is a quantitative geometric rigidity estimate by Friesecke, James and Müller [49] which allows to establish a compactness result for rescaled displacement elds. For congurations with small elastic energy the frame indierence of the energy density induces that the deformation gradient is pointwise near a rotation. The result states, loosely speaking, that then the deformation is globally near one single rigid motion. In the framework of fracture mechanics one diculty arises form the fact that global rigidity may fail if the body is disconnected by the jump set. Under the 4

constraint that the material does not store elastic energy Chambolle, Giacomini and Ponsiglione [25] could show that the body behaves piecewise rigidly, i.e. the only possibility that global rigidity can fail is that the body is divided into various parts each of which subject to a dierent rigid motion. The goal of our analysis is to combine the aforementioned results and to tackle the problem for general Grith models where both energy forms are coexistent. As a rst observation we see that, without passing to rescaled congurations, in the small strain limit the energies converge to a limiting functional which is nite for piecewise rigid motions and measures the segmentation energy which is necessary to disconnect the body. Consequently, in order to arrive at a limiting model showing coexistence of elastic in surface energy the strategy is to pass to appropriate rescaled displacement elds similarly as in [36]. The farthest reaching result in this direction seems to be a recent work by Negri and Toader [60] where a nonlinear-to-linear analysis is performed in the context of quasistatic evolution for a restricted class of admissible cracks. In particular, in their model the dierent components of the jump set are supposed to have a least positive distance rendering the problem considerably easier. In fact, one can essentially still employ the rigidity estimate [49] and the specimen cannot be separated into dierent parts eectively leading to a simple relation between the deformation and the rescaled displacement eld. In the present work we do not presume any a priori assumptions on the jump set and thus treat the full free discontinuity problem. The major challenge is the derivation of a  to the best of our knowledge  new kind of geometric rigidity result in the framework of geometric measure theory. We call this estimate an SBD-rigidity result as it is formulated in terms of special functions of bounded deformation (see [5, 8]). The derivation is very involved as among other things one has to face the problems that (1) the body might be disconnected by the jump set, (2) the body might be still connected but only in a small region where the elastic energy is possibly large, (3) the crack geometry might become extremely complex due to relaxation of the elastic energy by oscillating crack paths and innite crack patterns occurring on dierent scales. The common diculty of all these phenomena is the possible high irregularity of the jump set. Even if one can assume that the domain can be decomposed into dierent sets with Lipschitz boundary (e.g. by a density argument), there are no uniform bounds on the constants of several necessary inequalities such as the Poincaré and Korn inequality and the rigidity estimate [49]. The rigidity result provides the relation between the deformation of a material and the rescaled displacements, which measure the distance from piecewise rigid motions being constant on each connected component of the cracked body. It proves to be the fundamental ingredient for the derivation of linearized Grith models via Γ-convergence in a small strain limit. As before the limiting congurations consist of a piecewise rigid motion with corresponding segmentation energy. Additionally, on each component of the partition there is an associated 5

displacement eld whose energy is of Grith-type in the realm of linearized elasticity. Finally, we discuss that the general Γ-limit result can be applied to solve boundary value problems of uniaxial compression which is as the uniaxial tension test a natural problem. Hereby we complete the picture about our derivation of cleavage laws in the passage from discrete-to-continuum systems.

Outline We now give a more specic outline of the content of this thesis. In Chapter 1 we introduce the discrete models and present our main results about the derivation of cleavage laws and the establishing of a discrete-tocontinuum Γ-limit. Although the investigation of the general nonlinear-to-linear limit is a related problem, we prefer to postpone the presentation of the results to Chapter 6 as we base our analysis on a slightly dierent setting and wish to avoid confusion between the dierent models. In Section 1.1 we introduce our multidimensional discrete model. We suppose that the atoms in the reference conguration are given by the portion εL ∩ Ω, where the macroscopic region Ω ⊂ Rd occupied by the body is a cuboid and εL is some Bravais lattice scaled by the typical interatomic distance ε  1. The main structural assumption is that the energy of a deformation y : εL ∩ Ω → Rd may be decomposed as a sum over cell energies, i.e. X ¯ Q ). Eε (y) = Wcell (∇y| Q

Here the sum runs over the scaled cells Q ⊂ Ω induced by εL and the cell ¯ encoding all the relative energy on its part depends on the discrete gradient ∇y displacements of atoms in a cell and satises some reasonable assumptions in the elastic regime (see e.g. [27]), particularly the frame indierence. As forces between well separated atoms are governed by dipole interactions we assume that for large deformation gradients the cell energy reduces to a pair interaction energy neglecting multiple point interactions. In Section 1.2 we show by a heuristic √ argument that the most interesting regime of boundary values is given by ε as in this case the energies of typical elastic deformations and congurations with cleavage are of the same order. This scaling was rst proposed by Nguyen and Ortiz [61], who investigated the problem with renormalization group techniques. In Section 1.3 we present our main cleavage law. In Section 1.4 we introduce a specic two-dimensional model where the atoms in the reference conguration form a triangular lattice. Although being a model problem we emphasize that it contains the main features essential for our analysis. Indeed, it is (1) is frame indierent in its vector-valued arguments in more 6

than one dimension, (2) gives rise to non-degenerate elastic bulk terms and (3) leads to surface contributions sensitive to the crack geometry with competing crystallographic lines. Moreover, we will discuss an application to the stability of brittle nanotubes under interior expansive pressure and observe that such twodimensional lattice surfaces naturally appear in the analysis of thin structures. In Section 1.5 and Section 1.6 we state the main results about the characterization of minimizing sequences and the convergence of the variational problems, respectively. Here we study an application to fractured magnets in an external eld and briey discuss that the ndings on crystal cleavage can be re-derived by investigating a limiting variational problem. Chapter 2 is devoted to preliminaries. In Section 2.1 we rst derive formulae for the essential constants appearing in the cleavage law characterizing the stiness and the toughness of the material. Here we already see that it is optimal to cleave along a crystallographic hyperplane. In Section 2.2 we introduce interpolations both for the elastic regime following the methods in [65] and for the fracture regime being adapted for the application of slicing techniques. In Section 2.3 we state a short lemma about the length of Lipschitz curves in sets of bounded variation being substantially important in dimensions d ≥ 4. Finally, in Section 2.4 we derive some elementary properties for the cell energy of the triangular lattice. In particular, we introduce a lower-bound comparison energy providing ne estimates on the discrete minimal energies. Chapter 3 is entirely devoted to the derivation of cleavage laws for the limiting minimal energy. As a warm-up we rst give the proof for the triangular lattice in Section 3.1 by reducing the problem to one-dimensional segments using projection and slicing arguments. Due to the isotropy of the linearized elastic energy and the planar geometry whereby cracks may not concentrate on lower dimensional structures the result can be established in a comparatively elementary way. By way of contrast, the analogous result in arbitrary dimensions with general lattices and interaction potentials requires (1) new projection estimates for the size of cracks in the specimen, (2) a thorough analysis of all possible crack modes of a lattice unit cell and in particular (3) a full dimensional analysis of an auxiliary problem on a `mesoscopic cell' whose size is carefully chosen between the microscopic scale ε and the macroscopic magnitude of the specimen. On the one hand, by choosing this size of the mesoscopic cell small enough it is possible to separate the eects arising from the bulk elastic and the surface crack energy and to apply elaborated methods in the various regimes, including rigidity estimates [49] and slicing techniques for special functions of bounded variation (SBV) (see e.g. [6]). On the other hand, given that the size is large with respect to ε we can exploit the validity of the Cauchy-Born-rule for suciently small strains which means, loosely speaking, that every single atom follows the mesoscopic deformation gradient and atomistic oscillations are eectively excluded (see [27, 7

50]). More precisely and in mathematical terms, passing simultaneously from discrete to continuum theory and from nite to innitesimal elasticity the discrete gradient of the atomic displacements reduces to a classical gradient leading to a simpler description of the stored elastic energy (cf. [65]). With the help of tailor-made interpolations depending on individual crack modes, it can be shown that the fracture energy consisting of all contributions from pair interactions of neighboring atoms reduces to a surface energy in the continuum limit which only depends on the crack geometry (cf. also [17]) and is minimized for a specic crystallographic hyperplane. Section 3.2 contains the essential technical estimates providing a lower comparison potential for the energy of a `cell of mesoscopic size' under given averaged boundary conditions. The proof is mainly divided into three parts each of which dealing with one particular regime: The elastic regime where we show that linear elasticity theory applies, the fracture regime where we use a slicing argument in the framework of SBV functions and an intermediate regime. Beyond that, in the case d ≥ 4 an additional intermediate regime has to be introduced due to the fact that in higher dimensions it becomes more dicult to derive uniform bounds on the dierence of boundary values. Section 3.3 is devoted to the proof of the main theorem which relies on the application of the comparison energy derived in Section 3.2 and a slicing argument in the space direction were the tensile boundary conditions were imposed. In Section 3.4 we give further examples of mass-spring models to which the aforementioned results apply and provide the limiting minimal energy as well as asymptotically optimal congurations. We rst analyze the nearest and next-tonearest neighbor interaction in a square lattice and see that in addition to the Poisson-eect elastic minimizers generically also show a shear eect due to the anisotropy of the linearized elastic energy. Whereas the energetically favorable crack line in the triangular lattice was exclusively determined by the geometry of the problem, we nd that for the square lattice two competing crystallographic lines occur due to possible dierent microscopic structures of fracture. Afterwards, we apply our results to a general nearest and next-to-nearest neighbor model in 3d considered e.g. in [63, 65]. In Chapter 4 we reduce our analysis to the triangular lattice and provide a characterization of all minimizing sequences for the boundary value problem of uniaxial extension. As a preparation we rst establish ner estimates on the discrete minima by deriving higher order terms (see Section 4.1). In particular, our proof illustrates the typical behavior of brittle materials already seen in the continuum cleavage law also in a discrete framework: There is essentially no plastic regime besides the elastic and the crack regime.√ More precisely, we see that for almost minimizers the deformation is either ε-close to the identity mapping (representing elastic response) or springs between adjacent atoms are elongated by a factor scaling like √1ε (leading to fracture in the limit description). 8

Here we can already see that homogeneous deformations or cleavage along specic lines are asymptotically optimal. We then proceed to show that, under appropriate assumptions, in terms of suitably rescaled displacement elds indeed all discrete energy minimizers converge strongly to such continuum deformations. As alluded to above, the main challenge is to establish a suitable compactness result which is typically based on geometric rigidity estimates. In this specic setting we do not employ our general SBD rigidity result, but pursue a straighter way which eectively leads to stronger results. The strategy is to provide a ne characterization of the crack, i.e. of the number and position of largely elongated springs (see Section 4.2). In the subcritical case the contribution of such springs is abitrarily small such that the purely elastic theory applies. In the generic case, for supercritical boundary values largely deformed springs lie in a small stripe in direction of the optimal cristallographic line in such a way that the two components on the right and on the left of the stripe essentially behave elastically whence the rigidity result by Friesecke, James and Müller [49] is applicable. This characterization is the key ingredient to prove strong discrete-to-continuum convergence results in the various regimes (see Section 4.3). Chapter 5 is devoted to the derivation and investigation of a limiting variational problem. In Section 5.1 we establish a corresponding continuum energy functional via Γ-convergence. It turns out that this problem is an issue similar to those considered in [2, 17, 42]. Nevertheless, we believe that the present Γ-convergence result is interesting as (1) it gives rise to a limiting Grith functional in the realm of linearized elasticity which can be explicitly investigated for cleavage, (2) there are applications to systems with small displacements for small energies and (3) to the best of our knowledge our approach to the problem diers from techniques which are predominantly used when treating discrete systems in the framework of fracture mechanics. The reduction to one-dimensional sections using slicing properties for SBV functions turned out to be a useful tool not only to derive general properties of these function spaces but also to study discrete systems and variational approximation of free discontinuity problems. E.g., the original proofs of the main compactness and closure theorems in SBV (see [3]) as well as the Γ-convergence results in [17, 42] make use of this integral-geometric approach. Similar to the fact that there are simplied proofs of these compactness theorems being derived without the slicing technique (see [1]), we show that in our framework the lower bound of the Γ-limit can be achieved in a dierent way. In fact, we carefully construct the crack shapes of discrete congurations in an explicit way which allows us to directly appeal to lower semicontinuity results for SBV functions. The elastic part can then be treated similarly as in [49, 65]. In Section 5.2 we analyze the continuum problem under tensile boundary values. A similar problem has been studied recently by Mora-Corral in [57], 9

where he investigates a rectangular bar of brittle, incompressible, homogeneous and isotropic material subject to uniaxial extension and shows that, depending on the loading, the minimizers are either given by purely elastic congurations or deformations with horizontal fracture. We extend these results to anisotropic and compressible materials and moreover re-derive in part the aforementioned convergence results of Chapter 4. A careful analysis of the anisotropic surface contribution shows that in the generic case there is a unique optimal direction for the formation of fracture, while in a symmetrically degenerate case cleavage fails and all energetically optimal crack geometries can be characterized by specic Lipschitz curves. As in [57] the proof makes use of a qualitative rigidity result for SBV functions (see [25]) and of the structure theorem on the boundary of sets of nite perimeter by Federer [41]. In Chapter 6 we present the main results about geometric rigidity in SBD and the derivation of linearized Grith energies from nonlinear counterparts. We adopt a slightly dierent point of view and consider continuum fracture models of the form ˆ 1 W (∇y(x)) dx + H1 (Jy ), Eε (y) = ε Ω where W is a frame-indierent energy density and H1 (Jy ) denotes the size of the jump set of the deformation y . We briey note that the small parameter ε denotes not only the order of the elastic energy, but in models arising from discrete systems as considered in Chapter 1 it again represents the typical interatomic distance. In fact, the length scale ε plays an important role in our analysis as the system shows remarkably dierent behavior on scales smaller and larger than the atomistic unit. We also discuss that besides the main nonlinear rigidity and compactness theorems there can be established associated results in a linearized regime which are much easier to prove and interesting on its own (see Section 6.1 and Section 6.2). As particularly the proof of the rigidity result is very long and technical, we present a thorough overview and highlight the principal proof strategies for the convenience of the reader (see Section 6.4). Chapter 7 is devoted to some preliminaries. We have already discussed in the rst part of the introduction that the constants in certain inequalities crucially depend on the shape of the domain. We carry out a careful analysis for the constants of the geometric rigidity estimate and a Korn-Poincaré inequality in BD (see [62]). At this point we notice that easy counterexamples to rigidity estimates in SBD can be constructed if one does not admit a small modication of the deformation. Moreover, we establish a trace theorem in SBV which allows to control the L2 -norm of the functions on the boundary. As a rst approach to the proof of the SBD-rigidity result it is convenient to replace the nonlinear problem by a linearized version. In Chapter 8 we establish a Korn-Poincaré-type inequality in SBD which measures the distance of the 10

displacement from an innitesimal rigid motion in terms of ε. This problem is signicantly easier as (1) the estimate only involves the function itself and not its derivative and (2) the set of innitesimal rigid motions is a linear space in contrast to SO(d). It turns out that this estimate will be one of the key ingredients to derive our main result which can be compared with the fact that in elasticity theory the linearized rigidity estimate, called Korn's inequality (see [29]), is one of the fundamental steps to establish the geometrically nonlinear result in [49]. To the best of our knowledge our Korn-Poincaré inequality diers from other inequalities of this type available in the SBV-setting as it is not based on a truncation argument for the congurations (see [39, 23]), but on a modication of the jump set (see Sections 8.1 - 8.4) and a subsequent determination of the jump heights (see Section 8.5). In particular, the set where the function and the modication dier has a rather simple geometry being the union of a nite number of rectangles. Consequently, in contrast to the recently established theorem in [24], the estimate is suitable for the application of compactness result for (G)SBD functions (see e.g. [34]). Although tailor-made for the applications in Chapter 9 and Chapter 10, we believe that this result is of independent interest and may contribute to solve related problems in the future, especially concerning fracture models in the realm of linearized elasticity which are related to problems in SBV where Poincaré inequalities (see [39]) have proved to be useful. Chapter 9 is devoted to the proof of the SBD-rigidity estimate. It turns out that the result can only be established under the additional condition that we admit an arbitrarily small modication of the deformation. One essential point is to derive an inequality for the symmetric part of the gradient. We also see that in general it is not possible to gain control over the full gradient which is not surprising as there is no analogue of Korn's inequality for SBV functions. In addition, we provide an L2 -bound for the congurations. In contrast to the setting in elasticity theory this is highly nontrivial as Poincaré's inequality cannot be applied due to the possibly present complicated crack geometry. The main strategy of the proof is to establish local rigidity results on cells of mesoscopic size (Section 9.2) which together with the Korn-Poincaré inequality allows to replace the deformation by a modication where the least length of the crack components has increased (Section 9.3). Repeating the arguments on various mesoscopic scales becoming gradually larger it is possible to show that the modied deformation behaves rigidly on each connected component of the domain (Section 9.4). The fact that we analyze the problem on dierent length scales is indispensable to understand specic size eects correctly such as the accumulation of crack patterns on certain scales. Moreover, we briey note that similarly as in Section 3.2 a mesoscopic localization technique proves to be useful to tackle problems in the framework of brittle materials as hereby eects arising from the bulk and the surface contributions can be separated. Basically, this is enough the establish the requirements for compactness results 11

in the space of SBD functions. However, as we are interested in the derivation of eective linearized models we have to assure that we do not change the total energy of the deformation during the modication procedure. In particular, for the surface energy this is a subtle problem and in Section 9.5 a lot of eort is needed to show that the modied congurations can be constructed in a way such that the crack length does not increase substantially. We observe that many arguments in the proof are valid also in dimension d ≥ 3. The essential reason why we restrict ourselves to the two-dimensional framework is the derivation of the Korn-Poincaré-type inequality in Chapter 8 where a lot of technical diculties concerning the topological structure of the crack geometry occur. Nevertheless, we believe that we provide the principal techniques which are necessary to prove the result in arbitrary space dimension. Moreover, we are condent that our methods, in particular the small modication of the deformation and the jump set, may contribute to the solution of related problems. Finally, Chapter 10 is devoted to the identication of limiting congurations and to the derivation of linearized Grith energies via Γ-convergence. In Section 10.1 we present the main compactness result showing that the congurations consist of piecewise rigid motions and corresponding displacement elds. As there is no uniform bound on the functions it turns out that the limiting displacements are generically not summable and we naturally end up in the space of GSBD functions (for the denition and basic properties we refer to [34]). We believe that our results are interesting also outside of this specic context as they allow to solve more general variational problems in fracture mechanics. Typically, for compactness results in function spaces as SBV and SBD one needs L∞ or L1 bounds on the functions (see [4, 8, 34]). However, in many applications, in particular for atomistic systems and for models dealing with rescaled deformations, such estimates cannot be inferred from energy bounds. Nevertheless, we are able to treat problems without any a priori bound by passing from the deformations to displacement elds whose distance from rigid motions can be controlled. In Section 10.2 we discuss the properties of the limiting partition which is related to the piecewise rigid motion. We observe that an even ner segmentation may occur if on a connected component of the partition the jump set of the corresponding displacement further disconnects the domain. Here it becomes apparent that we treat a real multiscale model as the jump√heights at the boundaries associated to the coarse partition are √ of order  ε, whereas the jump heights of the ner partition are of order ε. In particular, it is evident that the choice of the limiting partition is not unique. However, we propose a selection principle and show that the partition can always be chosen in a way such that a further coarsening is not possible. In Section 10.3 we derive the main Γ-limit which is almost immediate due to the preparations in Section 10.1. Finally, in Section 10.4 we return to our 12

discussion about cleavage laws and investigate a specic boundary value problem, where we essentially follow the proof presented in Section 5.2 (cf. also [57]). Whereas the strategy followed in Chapter 3, which basically relied on a slicing technique, was not appropriate to treat the case of compression, at least in a continuum setting for isotropic surface energies the full Γ-limit allows to extend the results obtained in Section 1.6 to the case of uniaxial compression. It turns out that in the linearized limit the behavior for compression and extension is virtually identical. We briey note that to avoid unphysical eects such as selfpenetrability further modeling assumptions would be necessary.

Acknowledgement It is a great pleasure to thank my supervisor Prof. Bernd Schmidt for having proposed interesting topics for my thesis, and for his guidance, continuous advice and encouragement. I would also like to thank the whole group "Nichtlineare Analysis" in Augsburg for many interesting discussions and joint activities. The nancial support of this thesis by a scholarship from the Universität Bayern e.V. is gratefully acknowledged. Moreover, I would like to thank the Stiftung Maximilianeum in Munich for oering me much more than an accommodation during the years of my studies. Finally, I am grateful to my family and friends, in particular to Michaela Hofmann, for their support, encouragement and patience in the last years.

13

Part I An analysis of crystal cleavage in the passage from atomistic models to continuum theory

15

Chapter 1 The model and main results 1.1 The discrete model Let Ω ⊂ Rd be the macroscopic region occupied by the body under consideration. To simplify the exposition we assume that Ω = (0, l1 ) × . . . × (0, ld ) is rectangular, but remark that all our results extend without diculty to more general geometries as Ω = (0, l1 ) × ω , ω ⊂ Rd−1 open, for which cleavage boundary values as discussed in Section 1.2 below may be imposed. Let L be some Bravais lattice in Rd , i.e. there are linearly independent vectors v1 , . . . , vd ∈ Rd such that

L = {λ1 v1 + . . . λd vd : λ1 , . . . , λd ∈ Z} = AZd , where A is the matrix (v1 , . . . , vd ). Without restriction we may assume that the vectors vi are labeled such that det A > 0. The portion of the scaled lattice Lε = εL lying in Ω represents the positions of the specimen's atoms in the reference position. Here ε is a small parameter measuring the typical interatomic distance eventually tending to zero. Note that Lε partitions Rd into cells of the form εA(λ + [0, 1)d ) for λ ∈ Zd . The shifted lattice εA(( 21 , . . . , 12 )T + [0, 1)d ) consisting of the midpoints of the cells is denoted by L0ε . For x ∈ Rd we denote by x ¯ = x¯(x, ε) the center of the ε-cell containing the point x and set Qε (x) = x ¯(x, ε)+εA[− 12 , 12 ).  1 1 d We choose a numbering z1 , . . . , z2d of the corners A − 2 , 2 of the reference cell A[− 21 , 12 )d and set

Z = (z1 , . . . , z2d ),

Z = {z1 , . . . , z2d } .

(1.1)

For subsets U ⊂ Ω we dene the following lattice subsets with respect to the midpoints L0ε and the corners Lε :

L0ε (U ) = {¯ x ∈ L0ε : Qε (¯ x) ∩ U 6= ∅} , Lε (U ) = L0ε (U ) + ε {z1 , . . . , z2d } ,  (L0ε (U ))◦ = x¯ ∈ L0ε : Qε (¯ x) ⊂ U , (Lε (U ))◦ = (L0ε (U ))◦ + ε {z1 , . . . , z2d } . S We call Qε (¯ x) for x¯ ∈ (L0ε (Ω))◦ an inner cell and set Ωε = x¯∈(L0ε (Ω))◦ Qε (¯ x). 17

The deformations of our system are mappings y : Lε ∩ Ω → Rd . Given x ∈ Ωε and the corresponding midpoint x ¯ ∈ (L0ε (Ω))◦ we denote the images of the atoms in Qε (x) by yi = y(¯ x + εzi ) for i = 1, . . . , 2d and view (1.2)

Y (x) = (y1 , . . . , y2d ). d d ¯ as elements of Rd×2 . We dene the discrete gradient ∇y(x) ∈ Rd×2 by d

¯ := ε−1 (y1 − y¯, . . . , y2d − y¯), ∇y

2 1 X yi . y¯ := d 2 i=1

(1.3)

¯ is a function on Ωε , which is constant on each cube Qε (¯ In particular, ∇y x) , 0 ◦ x¯ ∈ (Lε (Ω)) . We also need to keep track of the atomic positions within subsets of Z . Thered fore, for a given matrix G = (g1 , . . . , g2d ) ∈ Rd×2 and Z˜ ⊂ Z we dene ˜

˜ = (gj ) ˜ ∈ Rd×#Z . G[Z] zj ∈ Z

(1.4)

In cells with large deformation it will be convenient to measure the distance of d dierent subsets of the atoms forming the cell. For G ∈ Rd×2 and Z1 , Z2 ⊂ Z we set

d(G; Z1 , Z2 ) := min {|gi − gj | : zi ∈ Z1 , zj ∈ Z2 } .

(1.5)

We now dene the set of interaction directions

V = A{−1, 0, 1}d \ {0} and characterize the crystallographic hyperplanes spanned by the corners of a unit cell by their normal vectors. Let S d−1 = {ξ ∈ Rd : |ξ| = 1} and set

P := {ξ ∈ S d−1 : ∃u1 , . . . , ud−1 ∈ V, span{u1 , . . . ud−1 } = ξ ⊥ }. Note that every hyperplane is represented twice in P , by ξ and −ξ . Our basic assumption is that the energy associated to deformations y : Lε ∩ d Ω → Rd can be written as a sum over cell energies Wcell : Rd×2 → [0, ∞] in the form X ¯ x)). Eε (y) = Wcell (∇y(¯ (1.6) x ¯∈(L0ε (Ω))◦

For convenience the energy is dened as a sum over the inner cells only as the energy contribution of cells with midpoints lying in L0ε (Ω)\(L0ε (Ω))◦ are negligible in our model for uniaxial extension. We briey note that Wcell is of order one in atomic units and therefore we will have to consider a suitably scaled quantity of Eε to arrive at macroscopic energy expressions for small ε. This will be discussed in the next section. 18

Remark 1.1.1. A decomposition as in (1.6) is, in particular, possible for many mass spring models, as will be exemplied in Section 1.4 and Section 3.4: The energy stored in an atomic bonds which lies on a face of more than one unit cell will then be equidistributed to the energy contribution of all adjacent cells. Moreover, energy functionals of the form (1.6) can also incorporate bond angle dependent energy terms. We let

 ¯ ¯ = RZ : R ∈ SO(d) ⊂ Rd×2d , SO(d) := R

where Z is as dened in (1.1) and now describe the general assumptions on the cell energy Wcell in detail.

Assumption 1.1.2. (i) Wcell : Rd×2d d → [0, ∞] is invariant under translations and rotations, i.e. for G ∈ Rd×2 we have Wcell (G) = Wcell (RG + (c, . . . , c))

for all R ∈ SO(d) and c ∈ Rd . (ii) Wcell (G) = 0 if and only if there exists R ∈ SO(d) and c ∈ Rd such that G = RZ + (c, . . . , c). ¯ (iii) Wcell is continuous and C 2 in a neighborhood of SO(d) . The Hessian Qcell = 2 D Wcell (Z) at the identity is positive denite on the complement of the subspace spanned by translations (c, . . . , c) and innitesimal rotations HZ , with H + H T = 0. ˙ n such that min1≤i 0. We will see that

αA =

det(Q) , ˆ det(Q)

(1.16)

where, roughly speaking, Q is the projection of Qcell onto the linear subspace ˆ arises from Q by cancellation of the orthogonal to innitesimal rotations and Q rst row and column. This will be stated more precisely in Lemma 2.1.2 below. √ √ Theorem 1.3.1. Let l1 ≥ L( AT A, Wcell , l2 , . . . , ld ) and suppose aε / ε → a ∈ [0, ∞]. The limiting minimal energy is given by Qd n1 o j=2 lj Elim (a) := lim inf{Eε (y) : y ∈ A(aε )} = min l1 αA a2 , βA . ε→0 det A 2 22

A detailed proof of this result will be given in Section 3.3 and is content of the paper [47]. As discussed above, for a ∈ {0, ∞} either the elastic or the fracture regime is energetically favorable. The more interesting case is a ∈ (0, ∞) where both energies are of the same order. In particular, we are interested in the behavior of the specimen when a is near the critical value of boundary displacements r 2βA acrit = . (1.17) l1 αA We briey indicate asymptotically optimal congurations. In the subcritical case a ≤ acrit we consider the sequence of congurations

yεel (x) = x + F¯ (aε ) x,

x ∈ Lε ∩ Ω,

where F¯ (aε ) is the solution of the minimization problem (1.15) with r = aε (see Lemma 2.1.2 below). The deformations behave purely elastically and as we will see in Corollary 1.5.2 and in the examples in Section 3.4 show elongation in e1 -direction and contraction in the other space directions, a manifestation of the Poisson-eect. Moreover, the congurations illustrate the validity of the Cauchy-Born-rule in this regime as each individual atom follows the macroscopic deformation gradient. In the supercritical case a ≥ acrit there is some ξ ∈ P and c ∈ R such that the hyperplane Π = {x ∈ Rd : x · ξ = c} satises Π ∩ Ω ⊂ Ω \ (B1ε ∪ B2ε ) and the congurations ( x, x · ξ < c, yεcr (x) = x ∈ Lε ∩ Ω, (1.18) x + l1 aε e1 , x · ξ > c, are asymptotically optimal. As ξ ∈ P , we conclude that Π is a crystallographic hyperplane, as desired. Let us also remark that our class of atomistic interactions is rich enough to model any non-degenerate linearly elastic energy density, respectively, any preferred cleavage normal in P (not perpendicular to e1 ) in the continuum limit: In the elastic regime, this has in fact been observed in [20, Prop. 1.10]. Now suppose that ξ ∈ P with ξ · e1 6= 0 is orthogonal to span{u1 , . . . , ud−1 } with u1 , . . . , ud−1 ∈ V . Then, if β(u1 ), . . . , β(ud−1 ) are much larger than β(ν) for all ν ∈ V \ {u1 , . . . , ud−1 }, it is elementary to see that the minimum of P β(ν)|ν · ς| min ν∈V , ς∈P |e1 · ς| is attained at ς = ξ , so that indeed ξ denes a crack normal for an asymptotically optimal conguration as in (1.18). The main idea in the proof of Theorem 1.3.1 is based on the derivation of a lower comparison potential for a certain cell energy depending on the expansion 23

in e1 -direction and on the application of a slicing argument. Testing either with elastic deformations or congurations forming jumps along specic hyperplanes as given above, we will then see that this lower bound is sharp. Actually, it will ˜ in the regime turn out that the lower bound coincides with the reduced energy Q ˜ of innitesimal elasticity. In contrast to the local denition of Q, however, it is in general not convenient to optimize the energy Wcell of single cells individually as it is geometrically nonlinear and therefore, due to possible rotations, the corresponding minimizer for one cell might not be compatible with deformations dened on the whole domain. As a remedy we will introduce a mesoscopic localization technique and will consider `large cells' dened on a mesoscopic scale 3d−1 ε 3d . This main technical result is addressed in Section 3.2.

1.4 A specic model: The triangular lattice in two dimensions We present a planar model where the atoms in the reference conguration are given by the portion of a triangular lattice and only interact with their nearest neighbors. In Section 3.4 we will see that the model is admissible in the sense of Assumption 1.1.2. It serves as the most basic non-trivial example to which our theory applies. In particular, it (1) is frame indierent in its vector-valued arguments in more than one dimension, (2) gives rise to non-degenerate elastic bulk terms and (3) leads to surface contributions sensitive to the crack geometry with competing crystallographic lines. The goal is to perform a much more complete analysis of this model including a detailed characterization of low energy congurations and the identication of a limiting variational problem. Let L denote the rotated triangular lattice   1 √21 Z2 = {λ1 v1 + λ2 v2 : λ1 , λ2 ∈ Z}, L = RL 0 23 where RL ∈ SO(2) √ is some rotation and v1 , v2 are the lattice vectors v1 = RL e1 3 1 and v2= RL ( 2 e1 +  e ), respectively. Without loss of generality we assume that 2 2

RL =

cos φ − sin φ sin φ cos φ

for φ ∈ [0, π3 ). We collect the basic lattice vectors in the set

V = {v1 , v2 , v3 }, where v3 = v2 − v1 . The region Ω = (0, l) × (0, 1) ⊂ R2 , l > 0, is considered the macroscopic region occupied by the body under investigation, where for the sake of simplicity we set l1 = l and l2 = 1. As before, to avoid geometric artefacts, we will assume that l > √13 , so that it is possible for the body to completely break apart along lines parallel to Rv1 , Rv2 or Rv3 not passing through the left or right boundaries. In the reference conguration the positions of the specimen's atoms are given by the points of the scaled lattice Lε = εL that lie within Ω. The deformations 24

of our system are mappings y : Lε ∩ Ω → R2 and the energy associated to such a deformation y is assumed to be given by nearest neighbor interactions as   1 X |y(x) − y(x0 )| Eε (y) = W . (1.19) 2 x,x0 ∈L ∩Ω ε ε |x−x0 |=ε

Note that the scaling factor 1ε in the argument of W takes account of the scaling of the interatomic distances with ε. The pair interaction potential W : [0, ∞) → [0, ∞] is supposed to be of `Lennard-Jones-type' (cf. Remark 1.1.3): (i) W ≥ 0 and W (r) = 0 if and only if r = 1. (ii) W is continuous on [0, ∞) and C 2 in a neighborhood of 1 with α := W 00 (1) > 0. (iii) limr→∞ W (r) = β > 0. In order to obtain ne estimates on limiting energies and congurations we will also consider the following stronger versions of hypotheses (ii) and (iii): (ii') W is continuous on [0, ∞) and C 4 in a neighborhood of 1 with α := W 00 (1) > 0 and arbitrary α0 := W 000 (1). (iii') W (r) = β + O(r−2 ) as r → ∞, which is still satised, e.g., by the classical Lennard-Jones potential. In order to analyze the passage to the limit as ε → 0 it will be useful to interpolate and rewrite the energy as an integral functional. Let Cε be the set of equilateral triangles 4 ⊂ Ω of sidelength ε with vertices in Lε and dene S Ωε = 4∈Cε 4. By y˜ : Ωε → R2 we denote the interpolation of y , which is ane on each 4 ∈ Cε . The derivative of y˜ is denoted by ∇˜ y , whereas we write (y)4 for the (constant) value of the derivative on a triangle 4 ∈ Cε . Then (1.19) can be rewritten as X Eε (y) = W4 ((˜ y )4 ) + Eεboundary (y) 4∈Cε

4 =√ 3ε2

ˆ

(1.20)

W4 (∇˜ y ) dx +

Eεboundary (y),

Ωε

where

 1 W4 (F ) = W (|F v1 |) + W (|F v2 |) + W (|F (v3 )|) . (1.21) 2 √ (Note that |4| = 3ε2 /4.) Here the boundary term is the sum of pair interaction 0 )| 0 )| energies 41 W ( |y(x)−y(x ) or 12 W ( |y(x)−y(x ) over nearest neighbor pairs which form ε ε 25

the side of only one or non triangle in Cε , respectively. As above, in order to obtain nite and nontrivial energies in the limit ε → 0, we accordingly rescale Eε to Eε := εEε . We impose the same boundary conditions of uniaxial extension with possible alternatives as described in (1.12) and (1.13), where we set B1ε = {x ∈ Ω : x1 ≤ ε} and B2ε = {x ∈ Ω : x1 ≥ l − ε}. In the special case φ = 0 we will in addition assume that there is an upper bound R0 on the elongation of every atomic bond in a small ψ(ε)-neighborhood of the lateral boundaries of width ψ(ε) > 0 with ε  ψ(ε)  1:

|y(x) − y(x0 )| ≤ R0 ε if x1 , x01 ≤ ψ(ε) or x1 , x01 ≥ l − ψ(ε).

(1.22)

Without such an assumption, in the general low energy regime to be considered later, for φ = 0 the boundary values are not strong enough to prevent the specimen from breaking on the boundary into a large amount of completely separated components, rendering the system too sensitive to unphysical boundary eects. Conceivable alternative implementations of the boundary conditions as alluded to above will then result in energy changes of order O(ε). We will account for all such possibilities by characterizing not only energy minimizing congurations, but more generally all congurations which are energy minimizing up to an error term of order O(ε). Besides describing a basic experiment on elastic bodies, the assumption on the boundary values allows for a direct application of our results to the stability analysis of nanotubes: If the rotation RL and the length l are such that for a sequence εk → 0 the translated lattice εk L + (l, 0) concides with the original lattice εk L, we may view the system as an atomistic nanotube with macroscopic region 2πl S 1 × (0, 1). (Note that for small εk the bending energy contributions when rolling up (0, l) × (0, 1) into a cylinder are negligible as this mapping is an isometric immersion and thus innitesimally rigid.) Imposing periodic boundary conditions, for arbitrary l > 0 our system then models deformations of a nanotube subject to expansion of the diameter. We rst provide the basic energetic cleavage law for the triangular lattice. √ Theorem 1.4.1. Let l > √13 . Suppose aε / ε → a ∈ [0, ∞]. The limiting

minimal energy is given by

 lim inf {Eε (y) : y ∈ A(aε )} = min

ε→0

αl 2β √ a2 , sin(φ + π3 ) 3

 .

(1.23)

Although the theory of Section 1.3 applies, we have explicitly formulated the limiting minimal energy for our model problem since we want to give an independent proof as a rst approach to the general cleavage law (see Section 3.1). Indeed, the result can be shown in a comparatively elementary way by resorting to slicing methods and convexity estimates in combination with a suitable projection 26

technique. By way of contrast, the analogous result in arbitrary dimensions with general lattices and interaction potentials given in Theorem 1.3.1 is much more involved as it requires (1) new projection estimates for the size of cracks in the specimen, (2) a thorough analysis of all possible crack modes of a possible lattice unit cell and (3) full dimensional analysis of an anisotropic mesoscopic auxiliary problem in various regimes. √ In what follows, we specialize to sequences aε = εa for the sake of simplicity. If the assumptions (ii') and (iii') on W hold, we have the following sharp estimate on the discrete minimal energies up to error terms of the order of surface contributions.

Theorem 1.4.2. For ε small the discrete minimal energy is given by  inf Eε = min

 αl 2 [6α + 7α0 − 2(3α − α0 ) cos(6φ)]l √ 3 2β √ a + √ +O(ε). εa , sin(φ + π3 ) 3 27 3

Thus, while the zeroth order contributions in the elastic regime are isotropic, the higher order contributions as well as the fracture energy explicitly depend on the lattice orientation angle φ. A proof of this ne energy estimate will be given in Section 4.1. The results presented in this and the following section are content of the work [46]. (Parts of the results published in [46] were already obtained in [45].)

1.5 Limiting minimal congurations For notational convenience we let γ = max{|v1 · e2 |, |v2 · e2 |, |v3 · e2 |} and vγ ∈ V such that γ = |vγ · e2 |. We note that γ = sin(φ + π3 ) = |vγ · e2 | = vγ · e2 takes √ √ values in [ 3/2, 1] and that vγ is unique if φ 6= 0, i.e., γ > 3/2. Our analysis of the limiting minimal energy so far showed that in terms of the critical boundary displacement s √ 2 3β acrit = αγl the limit is attained for homogeneously deformed congurations if a ≤ acrit and for congurations cracked along lines parallel to Rvγ , if a ≥ acrit . However, it falls short of showing that in fact these congurations are the only possibilities to obtain asymptotically optimal energies. Indeed, in the special case that vγ is not unique, the limit is√also attained if the crack takes a serrated course parallel √ 3 T 1 to R( 2 , 2 ) or R(− 12 , 23 )T . Our next result shows that energy minimizing congurations converge to a homogeneous continuum deformation for subcritical boundary values, while in the supercritical case they converge to a continuum deformation which is completely 27

cracked and does not store elastic energy. If φ 6= 0 and hence vγ is unique, the crack path follows the optimal crystallographic line. For φ = 0 such a cleavage behavior fails in general. Nevertheless, we obtain an explicit characterization of all possible limiting crack shapes in this case as well in terms√of Lipschitz√curves whose tangent vector lies a.e. in the cone generated by (− 12 , 23 ) and ( 12 , 23 ). The basic idea behind our reasoning will be to `count' the number of `broken' springs, i.e. the springs intersected transversally by the crack path. We see that the springs broken by a crack line (p, 0) + Rvγ do not overlap in the projection onto the x2 -axis and the length of the projection of two adjacent broken springs . If in the generic equals εγ . This leads to a fracture energy of approximately 2β γ case φ 6= 0 we assume that the cleavage is not parallel to Rvγ we conclude that some springs in vγ direction must be broken, too. If we consider the adjacent triangles of such a spring and their neighbors we nd that the projection onto the x2 -axis of broken springs overlap. A careful analysis of this phenomenon then shows that every broken spring in vγ direction `costs' an additional energy , where P (γ) is the geometrical factor of ≈ 2εβ P (γ) γ

1 P (γ) = 2

√ 1− 3

! p 1 − γ2 . γ

(1.24)

√

(Note that P (γ) = 0 ⇔ γ = 23 ⇔ φ = 0 in accordance to the above considerations.) For the special case φ = 0 we provide a similar counting argument. In order to give a precise meaning to the convergence of discrete to continuum deformations, to each discrete deformation y : Lε → R2 we assign  as mentioned above  the ane interpolation y˜ on each triangle 4 ∈ Cε√ . Accordingly, to the rescaled discrete displacements u : Lε → R2 with y = id + εu (id denoting the identity mapping id(x) = x) we dene u ˜ to be its ane interpolation on each triangle 4 ∈ Cε . In the cracked regime we may of course only hope for a unique limiting deformation up to translation of the crack path. However, without an additional mild extra assumption on the admissible discrete congurations or their energy even this cannot hold true, as apart from the crack, parts of the specimen could ip their orientation and fold onto other parts on the body at zero energy. In order to avoid such unphysical behavior we add a frame indierent penalty term χ ≥ 0 to W4 with χ ≥ cχ > 0 in a neighborhood of O(2) \ SO(2) and χ ≡ 0 in a neighborhood of SO(2) and ∞, which in particular does not change the energy response in the linear elastic and in the fracture regime:

W4,χ (F ) = W4 (F ) + χ(F ).

(1.25)

For instance, an admissible choice for χ is the local orientation preserving condi-

28

tion in the elastic regime

( 0, if det(F ) > 0 or |F | > R, χ(F ) = ∞, if det(F ) ≤ 0 and |F | ≤ R,

(1.26)

for some threshold R  1. (Also 1 ≤ R = R(ε)  √1ε would be admissible.) We remark that such an innitely strong penalization of deformation gradients with non-positive determinant is widely used in the elastic models. Allowing for fracture, however, a penalization of orientation reversion between dierent cracked parts of the body is no longer physically justiable, whence we set χ = 0 for very large deformation gradients. We set ˆ 4 χ W4,χ (∇˜ y ) dx + εEεboundary (y), (1.27) Eε (y) = √ 3ε Ωε for y ∈ A(aε ). More generally than a sequence of minimizers we will consider sequences (yε ) of almost minimizers that satisfy

Eεχ (yε ) = inf{Eεχ (y) : y ∈ A(aε )} + O(ε).

(1.28)

For those deformations we will show in Section 4:

√

Theorem 1.5.1. Assume that W satises (i), (ii') and (iii'). Let√aε = εa, a 6= acrit and suppose (yε ) satises (1.28). Let uε such that yε = id + εuε . Then there exist u¯ε : Ω → R2 with |{x ∈ Ωε : u¯ε (x) 6= u˜ε (x)}| = O(ε) such that: (i) If a < acrit , then there is a sequence sε ∈ R such that k¯ uε − (0, sε ) − F a · kH 1 (Ω) → 0, 

 a 0 where F = 0 − a . 3 a

(ii) If a > acrit and φ 6= 0, then there exist sequences pε ∈ (0, l), sε , tε ∈ R such that (pε , 0) + Rvγ intersects both the segments (0, l) × {0} and (0, l) × {1} and, for the parts to the left and right of (pε , 0) + Rvγ (1)

Ω

Ω(2)

n o vγ ·e1 := x ∈ Ω : 0 < x1 < pε + vγ ·e2 x2 and n o ·e1 := x ∈ Ω : pε + vvγγ ·e x < x < l , 2 1 2

respectively, we have k¯ uε − (0, sε )kH 1 (Ω(1) ) + k¯ uε − (al, tε )kH 1 (Ω(2) ) → 0.

29

(iii) If a > acrit and φ = 0, then there exist sequences of Lipschitz functions gε : (0, 1) → (0, l) satisfying gε0 = ± √13 a.e. such that for the parts to the left and right of graph(gε ) Ω(1) [gε ] := {x ∈ Ω : 0 < x1 < gε (x2 )} and Ω(2) [gε ] := {x ∈ Ω : gε (x2 ) < x1 < l} ,

respectively, we have k¯ uε − (0, sε )kH 1 (Ω(1) [gε ]) + k¯ uε − (al, tε )kH 1 (Ω(2) [gε ]) → 0,

for suitable sequences sε , tε ∈ R. As a consequence, we obtain a complete characterization of limiting continuum displacements, when no mass leaks to innity.

Corollary 1.5.2. Under the assumptions and with the notation of Theorem 1.5.1, if supε kuε k∞ < ∞, up to passing to subsequences, u˜ε → u in measure where (i) if a < acrit , u(x) = F a x + (0, s) for some constant s ∈ R, ( (0, s), for x to the left of (p, 0) + Rvγ , (ii) if a > acrit and φ 6= 0, u(x) = (al, t), for x to the right of (p, 0) + Rvγ , for constants s, t ∈ R and p ∈ (0, l) such that (p, 0) + Rvγ intersects both the segments (0, l) × {0} and (0, l) × {1}, ( (0, s), if 0 < x1 < g(x2 ), (iii) if a > acrit and φ = 0, u(x) = (al, t), if g(x2 ) < x1 < l,

for a Lipschitz function g : (0, 1) → [0, l] with |g0 | ≤ s, t ∈ R.

√1 3

a.e. and constants

Conversely, for every u as given in the cases (i)-(iii) there is a minimizing sequence (yε ) satisfying (1.28) and u˜ε → u in measure. We close this section emphasizing that all the optimal congurations found in Theorem 1.5.1 and Corollary 1.5.2 by minimizing the energy without a priori assumptions show purely elastic behavior in the subcritical case and complete fracture in the supercritical regime. In particular, the elastic minimizer in (i) shows elongation a in e1 -direction and compression − a3 in the perpendicular e2 direction, a manifestation of the Poisson eect (with Poisson ratio 31 ), which cannot be derived in scalar valued models. On the other hand, the crack minimizer in (ii) for φ 6= 0 is broken parallel to Rvγ which proves that cleavage occurs along crystallographic lines, while we see that cleavage in the symmetric case φ = 0 in general fails. 30

1.6 Limiting variational problem We nally address the more general question if not only the minimal values or the minimizers but the whole energy functionals (1.19) converge to a continuum energy functional in a variational sense. We will analyze the limiting problem independently of its discrete approximations and will also discuss an application to fractured magnets in an external eld. As this analysis not only allows for the derivation of cleavage laws for brittle crystals, we will treat the problem in a slightly more general setting. The results announced here are proved in Section 5 and can be also found in [48]. The macroscopic region Ω ⊂ R2 occupied by the body is now supposed to be a bounded domain with Lipschitz boundary. The deformation are again mappings y : Lε ∩ Ω → R2 with the associated energy dened in (1.19). Our convergence result will be formulated in terms of rescaled displacement elds u = √1ε (y − id) and we write with a slight abuse of notation √ Eε (u) := Eε (y) = εEε (y) = εEε (id + εu). Likewise we consider the functionals Eεχ which arise from Eε by replacing W∆ by W∆,χ = W∆ + χ. We impose the following more general boundary conditions: ˜ ⊃ Ω is a bounded, open domain in R2 with Lipschitz boundary Assume that Ω ˜ of Ω. For (the continuous repredening the Dirichlet boundary ∂D Ω = ∂Ω ∩ Ω ˜ we dene the class of discrete displacements assuming sentative of) g ∈ W 1,∞ (Ω) the boundary value g on ∂D Ω as  ˜ → R2 : u(x) = g(x) for x ∈ Lε ∩ ΩD,ε , Ag = u : Lε ∩ Ω (1.29)

˜ : dist(x, ∂D Ω) ≤ ε}∪(Ω\Ω) ˜ where ΩD,ε := {x ∈ Ω . Note that in contrast to (1.12) the boundary values are formulated in terms of the displacement eld. Moreover, Eε (u), which in fact only depends on the restriction u|Ω , does not depend on the ˜ and on g| ˜ as long as the Dirichlet boundary ∂D Ω = particular choice of Ω Ω\Ω ˜ and the values of g on {x ∈ Ω : dist(x, ∂D Ω) < ε} remain unchanged. ∂Ω ∩ Ω ˜ with Similarly as before, we let C˜ε be the set of equilateral triangles 4 ⊂ Ω S 2 ˜ε = ˜ ε → R and u˜ : Ω ˜ ε → R2 we vertices in Lε and dene Ω ˜:Ω 4∈C˜ε 4. By y again denote the piecewise ane interpolation of y and u, respectively.

1.6.1 Convergence of the variational problems Our convergence analysis applies to discrete deformations which may elongate a number scaling with 1ε of springs very largely, leading to cracks of nite length in the continuum limit. √ On triangles not adjacent to such essentially broken springs, the defomations are ε-close to the identity mapping, so that the accordingly rescaled displacements are of bounded L2 -norm. Note that the rst of these assumptions can be inferred from suitable energy bounds. By way of example, 31

however, we see that this cannot be true for the displacement estimates in the bulk: The sequence of functionals (Eε )ε is not equicoercive.

Example 1.6.1.

Let B ⊂ Ω be an arbitrary ball. Assume that the specimen satisfying the boundary conditions is broken into the two parts B and Ω \ B , where the inner part is subject to a rotation R 6= Id so that

∇˜ yε (x) = R for x ∈ B. In particular, the energy of the conguration is of order 1. But for x ∈ B 1 |∇˜ uε (x)| = √ (R − Id) → ∞ for ε → 0. ε Thus, ∇˜ uε is not bounded in L1 and so uε does not converge. Nevertheless, it is interesting to investigate this regime in order to identify a corresponding continuum functional which describes the system in the realm of Grith models with linearized elasticity. In fact, on the one hand we will discuss two specic problems, our model of crystal cleavage and a model for fractured magnets, where boundary conditions or external elds break the rotational symmetry whence the sequence (Eεχ )ε satises suitable equicoercivity conditions. On the other hand, in the second part of this thesis we will show that one may establish equicoercivity in a certain sense for the brittle fracture models under consideration if one admits a generalized denition of the displacement eld. In Theorem 6.2.1 we show in the setting of continuum fracture mechanics that the example above essentially illustrates the only way that coercitvity may fail: The body breaks apart and in each connected component the deformation is near a dierent rigid motion. By dening the displacement eld on each component appropriately we see that one can establish a compactness result (see (6.12) below). Hereby, the aforementioned assumptions get justied a posteriori as indeed on each connected component the system essentially shows the above properties, in particular we obtain an L2 -bound for the rescaled displacement. Recall the denition and the main properties of the space SBV (Ω; R2 ), abbreviated as SBV (Ω) hereafter, in Section A.1. The sense in which discrete displacements are considered convergent to a limiting displacement in SBV is made precise in the following denition.

Denition 1.6.2. Suppose uε : Lε ∩ Ω˜ → R2 is a sequence of discrete displace˜ and write uε → u, ments. We say that uε converges to some u ∈ SBV 2 (Ω) if ˜ (i) χΩ˜ ε u˜ε → u in L1 (Ω)

and there exists a sequence Cε∗ ⊂ C˜ε with #Cε∗ ≤ of ε such that 32

C ε

for a constant C independent

(ii) k∇˜uε kL2 (Ω\∪ ≤ C. ˜ ∗ 4) 4∈Cε Consider the limiting functional ˆ ˆ X 1 2β 4 √ |v · νu | dH1 E(u) = √ Q(e(u)) dx + 3 Ω2 3 Ju v∈V


˜ , where e(u) = 1 ∇uT + ∇u denotes the symmetric part of for u ∈ SBV 2 (Ω) 2 the gradient. Q is the linearization of W4 around the identity matrix Id (see Lemma 2.4.2 for its explicit form). Observe that u is dened on the enlarged ˜ and therefore also jumps lying in Ω ˜ \ Ω (and thus particularly those lying set Ω on ∂D Ω) contribute to E(u). For a displacement eld u, which is the limit of a sequence (uε ) ⊂ Agε converging in the sense of Denition 1.6.2, we get u = g on ˜ Ω\Ω , where g = L1 - limε→0 gε . Consequently, if u|Ω does not attain the boundary condition g on the Dirichlet boundary ∂D Ω (in the sense of traces), this will be penalized in the energy E(u) as then H1 (Ju ∩ ∂D Ω) > 0. Moreover, as g by assumption is continuous, for any u ∈ Ag the jump set ˜ \ Ω, which shows that E(u) is in fact independent of the Ju does not intersect Ω ˜ and g| ˜ as long as ∂D Ω and g|∂ Ω remain unchanged. In particular choice of Ω D Ω\Ω Section 5.1 we prove the following Γ-convergence result (see [33] for an exhaustive treatment of Γ-convergence): ˜ with supε kgε k 1,∞ ˜ < +∞. If Theorem 1.6.3. (i) Let (gε )ε ⊂ W 1,∞ (Ω) W (Ω) (uε )ε is a sequence of discrete displacements with uε ∈ Agε and uε → u ∈ ˜ , then SBV 2 (Ω) lim inf Eε (uε ) ≥ E(u). ε→0

˜ and g ∈ W 1,∞ (Ω) ˜ with u = g on Ω ˜ \ Ω there is (ii) For every u ∈ SBV 2 (Ω) a sequence (uε )ε of discrete displacements such that uε ∈ Ag , uε → u ∈ ˜ and SBV 2 (Ω) lim Eεχ (uε ) = E(u).

ε→0

Note that the recovery sequence is obtained for the energy Eεχ which includes the frame indierent penalty term. The main idea will be to separate the energy into elastic and crack surface contributions by introducing a threshold such that triangles 4 with (y)4 beyond that threshold are considered as cracked and y˜ is modied there to a discontinuous function. The treatment of the elastic part draws ideas from [65] and [49]. To derive the crack energy, one could use a slicing technique, see, e.g., [17]. Although also possible in our framework, we follow a dierent approach here: We carefully construct crack shapes of discrete congurations in an explicit way which allows us to directly appeal to lower semicontinuity results for SBV functions in order to derive the main energy estimates. 33

In Example 1.6.1 we have seen that (Eε ) and (Eεχ ) are not equicoercive due to the frame indierence of W . We now add a term to Eε such that the sequence becomes equicoercive. Let m ˆ : R2×2 → S 1 be a function satisfying

m(RF ˆ ) = Rm(F ˆ ) for all F ∈ R2×2 , R ∈ SO(2),

m(Id) ˆ = e1 .

Moreover, we assume that m ˆ is C 2 ´in a neighborhood of SO(2) and R2×2 sym ⊂ 1 y ) with ker(Dm(Id)) ˆ . Let Fε (u) = Eε (u) + ε Ωε fκ (∇˜ ( κ(1 − e1 · m(F ˆ )), |F | ≤ T, fκ (F ) = (1.30) 0 else, for F ∈ R2×2 , where T, κ > 0. Likewise, we dene Fεχ . In Lemma 2.4.6 below we show that W∆,χ (F ) + fκ (F ) ≥ C|F − Id|2 for all F ∈ R2×2 with |F | ≤ T . This implies that the sequence (Fεχ )ε is equicoercive: Given a sequence of displacement elds (uε )ε with Fεχ (uε )+kuε k∞ ≤ C we nd a subsequence converging in the sense of Denition 1.6.2. Indeed, we get that #Cε∗ ≤ Cε , where Cε∗ := {∆ ∈ √ C˜ε : |(Id + ε˜ uε )∆ | > T }. By Lemma 2.4.6 we then get k∇˜ uε kL2 (Ω\∪ ≤C ˜ ∗ 4) 4∈Cε and therefore condition (ii) in Denition 1.6.2 is satised. By an SBV compactness theorem (see Theorem A.1.1) we then nd a (not relabeled) subsequence ˜ . This together with such that u ˜ε χΩ˜ ε \∪4∈C∗ 4 → u in L1 for some u ∈ SBV 2 (Ω) Sε kuε k∞ ≤ C and | 4∈Cε∗ 4| ≤ Cε implies that also condition (i) in Denition 1.6.2 holds with this function u. ˆ = D2 m Dene m ˆ 1 : R2×2 → [−1, 1] by m ˆ 1 = e1 · m ˆ and let Q ˆ 1 (Id) be the ˜ → Hessian at the identity. We introduce the limiting functional F : SBV 2 (Ω) [0, ∞) given by ˆ κ ˆ Q(∇u). F(u) = E(u) − 2 Ω We then obtain a Γ-convergence result similar to Theorem 1.6.3.

Theorem 1.6.4. The assertions of Theorem 1.6.3 remain true when Eε , Eεχ and E are replaced by Fε , Fεχ and F , respectively.

1.6.2 Analysis of a limiting variational problem We now analyze the limiting functional E for a rectangular slab Ω = (0, l) × (0, 1) with l ≥ √13 under uniaxial extension in e1 direction. We determine the minimizers and prove uniqueness up to translation of the specimen and the crack line for the boundary conditions

u1 = 0 for x1 = 0

and

u1 = al for x1 = l.

(1.31)

(More precisely: u ∈ SBV 2 ((−η, l + η) × (0, 1)) with u1 (x) = 0 for x ≤ 0 and u1 (x) = al for x ≥ l.) Note that we can investigate the limiting problem without any assumption on the second component of the boundary displacement. 34

Theorem 1.6.5. Let a 6= acrit . Then  min E(u) : u satises (1.31) = min



αl 2β √ a2 , γ 3

 .

All minimizers of E subject to (1.31) are of the form given in Corollary 1.5.2(i)(iii) depending on whether (i) a < acrit , (ii) a > acrit and φ 6= 0 or (iii) a > acrit and φ = 0. This theorem will be addressed in Section 5.2. An analogous result for isotropic, incompressible materials has been obtained recently by Mora-Corral [57]. Theorem 1.6.5 is an extension of this result to anisotropic, compressible brittle materials in the framework of linearized elasticity. Theorem 1.6.3 in combination with Theorem 1.6.5 gives a new perspective to ˜ = (−η, l + η) × (0, 1) the results presented in Section 1.4 and Section 1.5. Let Ω and dene for a ≥ 0  ˜ → R2 : A(a) = u =(u1 , u2 ) : Lε ∩ Ω u(x) = g(x) for x1 ≤ ε and x1 ≥ l − ε for some g ∈ G(a) ,

˜ : g1 (x) = 0 for x1 ≤ ε, g1 (x) = al for x1 ≥ l − ε}. where G(a) := {g ∈ W 1,∞ (Ω) Note that this is a slight variant of the boundary value problem investigated in Section 1.4. One implication of Theorem 1.5.1, Corollary 1.5.2 is that, under the tensile boundary conditions uε ∈ A(a), the requirement that uε be an almost energy minimizer satisfying (1.28) and supε kuε k∞ < ∞, guarantees the existence of a subsequence converging in the sense of Denition 1.6.2. (The convergence obtained in Theorem 1.5.1 is even stronger.) In particular, the sequence (Eεχ ) is mildly equicoercive. A fundamental theorem of Γ-convergence (see, e.g., [12, Theorem 1.21]) implies that such low energy sequences converge to limiting congurations given in Theorem 1.6.5 in the sense of Denition 1.6.2. Consequently, in this way we have re-derived the universal cleavage law given in Theorem 1.4.1 and the convergence result Corollary 1.5.2 (in the sense of Denition 1.6.2). In the second part of this thesis in Section 6.3 we will return to the investigation of a limiting cleavage law. Employing a compactness result presented in Section 6.2 we will see that the above analysis of minimal values and minimizers can be extended to the case of compression.

1.6.3 An application: Fractured magnets in an external eld In the above model we have seen that a mild equicoercivity of the sequence (Eεχ )ε is guaranteed by investigating a specic boundary value problem. We close this 35

section with an application to fractured magnets, where an external eld provides an even stronger equicoercivity condition. Assume that the material is a permanent magnet and let e1 be the magnetization direction. We suppose that there is a constitutive relation between ∇˜ y (x) and the local magnetization direction m(˜ ˆ y , x) ∈ S 1 of the deformed conguration y˜ at some point x ∈ Ω, which is of the form m(˜ ˆ y , x) = m(∇˜ ˆ y (x)) with m ˆ as dened in Section 1.6.1. Let Hext : R2 → R2 be an external√magnetic eld. The magnetic energy corresponding to the deformation y = id + εu is then given by ˆ 1 mag Eε (u) = − Hext · m(∇˜ ˆ y ), ε Ωε i.e. alignment of the magnetization direction with the external eld is energetically favored. The total energy of the system is given by

Eεtot = Eεχ + Eεmag . We now suppose that the external eld is homogeneous and satises without restriction Hext = κe1 for κ > 0. We then see that

κ Fε = Eεtot + |Ωε | ε with fκ as in (1.30) and corresponding Fε . By Theorem 1.6.4 we get that the renormalized functionals Fε Γ-converge to the renormalized total energy functot tional Eren = F . (Obviously, a conguration minimizes Eεtot if and only if it minimizes Fε .) tot ˜ . (u) for g ∈ W 1,∞ (Ω) We consider a boundary value problem minu∈Ag Eren Since the sequence (Fε )ε is equicoercive as discussed in Section 1.6.1, the theory of Γ-convergence implies limε→0 ( κ|Ωε ε | + minu∈Ag Eεtot (u)) = limε→0 minu∈Ag Fε (u) = tot (u) and also convergence of the corresponding (almost) minimizers minu∈Ag Eren tot in the sense of Denition 1.6.2 is of Fε , and hence Eεtot , to minimizers of Eren guaranteed. In this context, note that by a truncation argument taking g ∈ ˜ into account, we may indeed assume that a low energy sequence satises W 1,∞ (Ω) supε kuε kε < +∞.

36

Chapter 2 Preliminaries 2.1 Elementary properties of the cell energy Elastic energy We rst provide a lower bound for the cell energy. For that purpose, let V0 = Rd ⊗ (1, . . . 1) denote the subspace of innitesimal translations (x1 , . . . , x2d ) → (v, . . . , v), v ∈ Rd .

Lemma 2.1.1. For every T > 0 there is a constant C > 0 such that ¯ dist2 (G, SO(d)) ≤ CWcell (G)

for all G ∈ Rd×2d with G ⊥ V0 and |G| ≤ T . Proof. The proof is essentially contained in [63, Lemma 3.2] and relies on the

¯ growth assumptions on Wcell near SO(d) (see Assumption 1.1.2(iii)).  We now give a precise characterization of αA (see (1.16)). We may view a d(d+1) dˆ ˆ , symmetric matrix F = (fij ) ∈ Rd×d sym as a vector f = (f1 , . . . , fdˆ) ∈ R , d = 2 whose components are the entries fij with i ≤ j , numbered such that f1 = f11 . Then Qcell (F · Z) = f T Qf ˆ ˆ

d for some symmetric positive denite Q ∈ Rd× sym . For each r ∈ R there is conseˆ quently a unique f¯ ∈ Rd minimizing

f T Qf

subject to f1 = r

and a corresponding Lagrange multiplier λ ∈ R such that f¯T Q = λeT1 , i.e.

f¯ = λQ−1 e1 ,

37

ˆ

where e1 denotes the rst canonical unit vector in Rd . Multiplying with eT1 and using f¯1 = r, we nd that λ = eT Qr−1 e1 and thus 1

f¯ =

r eT1 Q−1 e1

Q−1 e1 .

For the minimal value we obtain

˜ Q(r) = f¯T Qf¯ =

r2 . eT1 Q−1 e1

ˆ denoting the (dˆ − 1) × (dˆ − 1) matrix obtained form Q by deleting With Q the rst row and the rst column and using that Q−1 = det1 Q cof Q and thus ˆ

Q)11 Q Q−1 e1 = det1 Q (cof Q)·1 and eT1 Q−1 e1 = (cof = det , this can alternatively be det Q det Q written as r2 det Q r det Q ˜ Q(r) = f¯ = (cof Q)·1 , . ˆ ˆ det Q det Q We summarize these observations in the following lemma:

Lemma 2.1.2. The reduced energy satises ˜ Q(r) = αA r2

with αA =

1 eT1 Q−1 e1

=

det Q . ˆ det Q

˜ ¯ For each r there exists a unique F¯ (r) ∈ Rd×d sym which satises Q(r) = Qcell (F (r)·Z) and f11 = r. F¯ (r) depends linearly on r.

Fracture energy In Theorem 1.3.1 we have seen that the limiting minimal fracture energy has the form (1.14). We now investigate this term in detail and determine the minimizers. We let P β(ν)|ν · ς| − |e1 · ς|βA (2.1) Λ(ς) := ν∈V |ς| and observe Λ(ς) ≥ 0 for all ς ∈ Rd \{0}. Note that the minimum in the denition (1.14) of βA and the minumum of Λ in (2.1) are attained on the compact set S d−1 = {ς ∈ Rd : |ς| = 1}. Moreover, we note that the minimizers in (1.14) are precisely the minimizers of Λ (with minimal value 0). They are obviously not perpendicular to e1 .

Lemma 2.1.3. The minimum (= 0) of Λ(ξ) in S d−1 is attained for some ξ ∈ P .

38

Remark 2.1.4.

A natural guess would be that in fact every minimizer of Λ lies in P . Surprisingly this turns out to be wrong in general. There are (nongeneric) models even leading to a continuum of optimal crack directions, as we will exemplify in Section 3.4 for a basic mass spring model in 2d. As a consequence, for such a model it is not possible to prove that in the fracture regime the body has to break apart along crystallographic hyperplanes.

Proof. For δ > 0 we dene Λδ (ς) = Λ(ς) + δ

|ς · e1 | . |ς|

Obviously Λδ attains its minimum with 0 < minξ∈Rd Λδ (ξ) ≤ δ . We show that for small δ if a ϕ ∈ S d−1 satises Λδ (ϕ) = minξ∈Rd Λδ (ξ) then ϕ ∈ P . We dene U = {ν ∈ V : ν · ϕ > 0} ∪ {e1 } and U0 = {ν ∈ V : ν · ϕ = 0}. Note that by (2.1) and (1.14) ϕ · e1 6= 0, so without loss of generality we may ˜ assume that ϕ · e1 > 0. For ν ∈ V \ {e1 } let β(ν) = 2β(ν). If e1 ∈ V , we set ˜ ˜ 1 ) = −βA + δ . If the claim β(e1 ) = 2β(e1 ) − βA + δ . Otherwise we only set β(e were false, then dim span U0 < d − 1. Therefore, we can choose some η ∈ Rd \ {0} such that η · ϕ = 0 and η · ν = 0 for all ν ∈ U0 . We now investigate the behavior of Λδ at ϕ in direction η . Using that ν · ϕ = ν · η = 0 for all ν ∈ U0 , for |t| suciently small we obtain P ˜ β(ν)ν · (ϕ + tη) . λ(t) := Λδ (ϕ + tη) = ν∈U |ϕ + tη| We dierentiate and obtain from η · ϕ = 0
P P ˜ ˜ ν∈U β(ν)ν · (ϕ + tη) (ϕ + tη) · η ν∈U β(ν)ν · η 0 λ (t) = − |ϕ + tη| |ϕ + tη|3 P P ˜ ˜ β(ν)ν ·η β(ν)ν · (ϕ + tη) = ν∈U − t|η|2 ν∈U . |ϕ + tη| |ϕ + tη|3 If λ0 (0) 6= 0 then ϕ is not a critical point of P Λδ which contradicts the above β(ν)ν·η 0 assumption. So we may assume that λ (0) = ν∈U|ϕ| = 0 and thus 0

2

P

λ (t) = −t|η|

˜ β(ν)ν ·ϕ , 3 |ϕ + tη|

ν∈U

leading to the contradiction

λ00 (0) = −|η|2 Λδ (ϕ) < 0.

39

Thus, we have shown that Λδ attains its minimum for some ϕ ∈ P for all δ > 0. As minξ∈Rd Λδ (ξ) ≤ δ , passing to the limit δ → 0 we obtain the claim.  We are now in a position to render more precisely the denition of the minimum length assumed in Section 1.2. Let X X M1 = min β(ν)|ν · ξ|, M2 = max β(ν)|ν · ξ|. ξ∈S d−1

ξ∈S d−1

ν∈V

ν∈V

It is not hard to see that M1 , M2 are independent of the particular rotation of the lattice. Then βA ≤ M2 and therefore the minimizer ξ ∈ S d−1 of (1.14) satises M1 |ξ · e1 | ≥ M . Consequently, an elementary argument shows that choosing C > 0 2 large enough independently of M1 , M2 , l2 , . . . , ld and setting

√ M2 L = L( AT A, Wcell , l2 , . . . , ld ) = C max{l2 , . . . , ld } M1

(2.2)

we nd that for specimens with l1 > L it is possible to completely break apart along hyperplanes not passing through the boundary parts B1 and B2 .

2.2 Interpolation In the following it will be useful to choose a particular interpolation y˜ of the lattice deformation y : Lε ∩ U → Rd for U ⊂ Ω open. We introduce a threshold value Cint ≥ 1 to be specied later and rst consider x) , Qε (¯  some cell 0 ◦ ¯ x), n Z ≤oCint . x¯ ∈ (Lε (U )) , where the lattice deformation satises diam ∇y(¯ d×2d Here for G = (g1 , . . . , g2d ) ∈ R and Z˜ ⊂ Z we dene diam G, Z˜ := n o max |gi − gj | : zi , zj ∈ Z˜ , so particularly we have

n o n o ¯ x), Z˜ = 1 max |y(¯ diam ∇y(¯ x + εzi ) − y(¯ x + εzj )| : zi , zj ∈ Z˜ , ε

(2.3)

¯ x) is given in (1.3). We will call cells with this property `intact cells' where ∇y(¯ and by Cε0 ⊂ (L0ε (U ))◦ we denote the set of their midpoints. The complement C¯ε0 := (L0ε (U ))◦ \ Cε0 labels the centers of cells we consider to be `broken'. We rst consider Qε (¯ x) for x¯ ∈ Cε0 . The interpolation we use was introduced in [65]. We repeat the procedure here for the sake of completeness. Consider the  d reference cell Q = A[− 21 , 12 )d with deformation y : A − 12 , 12 = Z → Rd . We rst interpolate linearly on the one-dimensional faces of Q, which are given by segments [zi , zj ], where zi −zj is parallel to one of the lattice vectors vn , n = 1, . . . , d. Subsequently we consider two-dimensional faces and dene a triangulation and interpolation as follows: Given a face co {zi1 , zi2 , zi3 , zi4 } with zi2 = zi1 + vn ,

zi3 = zi1 + vn + vm , 40

zi4 = zi1 + vm

we dene

1 ζ = (zi1 + . . . + zi4 ), 4

1 y(ζ) = (y(zi1 ) + . . . + y(zi4 )) 4  and interpolate linearly on each of the four triangles co zij , zij+1 , ζ for j = 1, . . . , 4 with the convention i5 = i1 . In general, having chosen a simplicical decomposition as well as corresponding linear interpolations on the faces of dimension n − 1 we decompose and interpolate on an n-dimensional face F = co {zi1 , . . . , zi2n } in the following way: Let n

2 1 X zi , ζ= n 2 j=1 j

n

2 1 X y(ζ) = n y(zij ). 2 j=1

We decompose F by the simplices co {w1 , . . . , wn , ζ}, where co {w1 , . . . , wn } is a simplex belonging to the decompostion of an (n−1)-dimensional face constructed in a previous step. We now interpolate linearly on these simplices. For cells lying at the boundary B1ε , B2ε we can repeat the above construction at least for the rst component y 1 : Let x ¯ ∈ L0ε (Ω)\(L0ε (Ω))◦ such that x¯ +εzi ∈ / Ω. This implies −lA ε ≤ (¯ x + εzi )1 ≤ 0 or l1 ≤ (¯ x + εzi )1 ≤ l1 + lA ε, respectively. Now let y 1 (¯ x + εzi ) = (1 + aε )(¯ x + εzi )1 . Thus, the rst component of Y (¯ x) = 1 (y1 , . . . , y2d ) is well dened and we may proceed as above to construct y˜ . We now concern ourselves with `broken cells' x ¯ ∈ C¯ε0 . Let y be a corresponding lattice deformation dened on Z . Recalling denition (2.3) we rst choose a ˙ = Z of the corners with partition Z1 ∪˙ . . . ∪Z 2   ¯ x), Zi ≤ #Zi Cint . diam ∇y(¯ 2d

¯ x); Zi , Zj ) > We note that this partition can be chosen in a way that εd(∇y(¯ 2−2d εCint for all sets Zi , Zj , i 6= j . Indeed, if there were z¯i ∈ Zi , z¯j ∈ Zj such ¯ x); Zi , Zj ) ≤ 2−2d εCint then that |y(¯ zi ) − y(¯ zj )| = εd(∇y(¯ |y(zi ) − y(zj )| ≤ |y(zi ) − y(¯ zi )| + |y(¯ zi ) − y(¯ zj )| + |y(zj ) − y(¯ zj )|   2 2 #(Zi ∪ Zj ) 2 (#Zi ) + (#Zj ) + 1 ≤ εC ≤ εCint int 22d 2d for all zi ∈ Zi , zj ∈ Zj and we could set Z˜ = Zi ∪ Zj . Clearly, the cardinality of this partition is at least two for every cell Qε (¯ x), x¯ ∈ C¯ε0 . Then, by Assumption 1.1.2(ii) and (iv) it is not hard to see that there is a constant C = C(Cint ) such that

¯ x)) ≥ C Wcell (∇y(¯

(2.4)

for x ¯ ∈ C¯ε0 . Note that C = C(Cint ) can be chosen independently of Cint for Cint ≥ 1. 41

We now each component i separately as follows.  choose the interpolation for ¯ x), Z ≤ Cint we dene y˜i as before for `intact cells' in C 0 \ C¯0 . Here If diami ∇y(¯ ε ε d for G = (g1 , . . . , g2d ) ∈ Rd×2 we dene similarly as in (2.3) diami {G, Z} := max {|(gj − gk ) · ei | : zj , zk ∈ Z} . Otherwise we set

y˜i (x) = y i (z1 ) + (x − z1 )i

on Qε (¯ x) and therefore ∇˜ y i = eTi . Consequently, |∇˜ y | ≤ CCint a.e. also on 0 broken cells Qε (¯ x), x¯ ∈ C¯ε . Finally having constructed the interpolation on all cells Qε (¯ x) we briey note that y˜ ∈ SBV (Ωε , Rd ), cf. Section A.1. For every interaction direction ν ∈ V we introduce a further interpolation y¯ν as follows. We choose vectors {vi1 , . . . , vid−1 } ⊂ {v1 , . . . , vd } such that ν, vi1 , . . . , vid−1 are linearly independent and for Dν = (ν, vi1 , . . . , vid−1 ) dene the lattice Gεν = εDν Zd partitioning Rd into cells of the form Qνε (λ) := εDν (λ + [0, 1)d ) for λ ∈ Zd . Note that Lε = Gεν . We describe the interpolation on the reference cell Qν = Dν [0, 1)d with deformation y : Dν {0, 1}d → Rd . If |y(ν) − y(0)| ≤ Cint ε we let

y¯ν (Dν x) = (1 − x1 )y(0) + x1 y(ν)

(2.5)

for x ∈ [0, 1)d . If |y(ν) − y(0)| > Cint ε we set y¯ν (Dν x) = y(0). Let Jy¯ν be the set of discontinuity points of y¯ν and denote by ∂ν Qν the two faces of ∂Qν which are not parallel to ν . Then y¯ν is typically discontinuous on ∂Qνε (λ) \ ∂ν Qνε (λ), λ ∈ Zd . This, however, will not aect our analysis. Essentially, we observe that Jy¯ν ∩ ∂ν Qνε (λ) 6= ∅ can only occur if there is some cube Qε (¯ x), x¯ ∈ C¯ε0 such that Qε (¯ x) ∩ Qνε (λ) 6= ∅. This is due to (2.5) and the denition of Cε0 . For later we compute the Hd−1 volume of ∂ν Qν . We rst choose νˆ ∈ Rd such that |ˆ ν | = 1 and νˆ · vij = 0 for j = 1, . . . , d − 1, i.e. νˆ is a unit normal vector to ν ∂ν Q . Then set ν¯ = |ν · νˆ|ˆ ν and obtain

| det(¯ ν , vi1 , . . . , vid−1 )| 1 d−1 | det Dν | det A H (∂ν Qν ) = = = . 2 |¯ ν| |ν · νˆ| |ν · νˆ|

(2.6)

We recall here some important properties of the interpolation y˜ on cells Qε (¯ x) , x¯ ∈ Cε0 being proved in [65] for the case p = 2. The extension of the results to general p are straightforward.

Lemma 2.2.1. Let y : Lε → Rd a lattice deformation, y˜ the corresponding linear interpolation. Then for every 1 ≤ p < ∞ there are constants c, C > 0 such that for every cell Q = Qε (¯x), x¯ ∈ Cε0 \ C¯ε0 we have ¯ Q , SO(d)) ¯ (i) c distp (∇y| ≤ ¯ Q |p ≤ (ii) c|∇y|

1 |Q|

´ Q

1 |Q|

´

Q

¯ Q , SO(d)) ¯ distp (∇˜ y , SO(d)) ≤ C distp (∇y|

¯ Q |p . |∇˜ y |p ≤ C|∇y| 42

The interpolation y˜ proves useful to show that in the continuum limit the discrete gradient reduces to a classical gradient (again cf. [65]).

Lemma 2.2.2. Let U ⊂ Ω, εk → 0 and a sequence yk : Lε (U ) → Rd with ¯ k * f in Lp , y˜k * y in W 1,p for some f ∈ Lp (U ), y ∈ W 1,p (U ), 1 ≤ p < ∞. ∇y Assume that C¯ε0 k = ∅ for all k. Then f = ∇y · Z. The following lemma shows that passing from y˜ to y¯ν , ν ∈ V , we do not change the limit.

Lemma 2.2.3. Let U ⊂ Ω, εk → 0 and yk : Lεk (U ) → Rd be a sequence of lattice deformations with #C¯ε0 k ≤ Cε−d+1 . Let y˜k , y¯ν,k be the corresponding k interpolations. Then passing to the limit εk → 0 we obtain y˜k − y¯ν,k → 0 in measure for all ν ∈ V . Proof. Let ν ∈ V and consider the sequences y˜k and y¯ν,k . By Dε0 k (U ) ⊂ L0εk we denote the midpoints x ¯ of all cells being either a broken cell itself or a neighbor of a broken cell, i.e. x ¯ ∈ C¯ε0 k or x¯ + εk ν ∈ C¯ε0 k for some ν ∈ V . Note that . #Dε0 k (U ) ≤ 3d #C¯ε0 k ≤ Cε−d+1 k We rst take a cell Qεk (¯ x) with x¯ ∈ / Dε0 k (U ) into account. By construction we have y˜k (¯ x + εzi ) = y¯ν,k (¯ x + εzi ) for a suitable zi ∈ Z . As |∇˜ yk |, |∇¯ ykν | ≤ CCint on Qεk (¯ x) we deduce that

yk (x) − y¯ν,k (x)| ≤ CCint εk sup |˜ x) x∈Qεk (¯

for all x ¯∈ / Dε0 k (U ). Noting that

X

|Qεk (¯ x)| ≤ Cεdk #Dε0 k (U ) ≤ Cεk → 0

¯ 0 (U ) x ¯∈D ε k

as εk → 0, we deduce that y˜k − y¯ν,k → 0 in measure for εk → 0.



2.3 An estimate on geodesic distances We now formulate a short lemma about the length of Lipschitz curves in sets W ⊂ Rd−1 of the form (3.7) introduced below: We estimate geodesic distances and the area swept by curves of given length emanating from a common point in terms of the area and surface of W . For this purpose, we dene distW (p, q) as the inmum of the length of Lipschitz curves in W connecting the points p, q ∈ W and let Hm denote the m-dimensional Hausdor measure.

43

Lemma 2.3.1. There are constants C, c, c0 > 0 (depending on D and D0 ) such ˜ , W ⊂ (0, 1)d−1 of the form (3.7) and ε small enough the following that for all W holds: (i) distW˜ (p, q) ≤ C(1 + ε(s−1)(d−3) ) for all p, q ∈ W (ii) For all p ∈ W , t ∈ (0, cε(s−1)(d−2) ) one has for ε small enough ˜ : dist ˜ (p, q) ≤ t}) ≥ c0 tε(1−s)(d−2) . Hd−1 ({q ∈ W W

Proof. We cover Ω˜ := (0, 1)d−1 up to a set of measure zero with the sets Cε (¯x) =

˜ ⊂ 1 Zd−1 . Also set Iε (W ˜)= x¯ + (0, 1l )d−1 , where l = d 1εˆ e, εˆ = ε1−s and x¯ ∈ Iε (Ω) l ˜ : Cε (¯ ˜ }. We let V˜ be the connected component of {¯ x ∈ Iε (Ω) x) ⊂ W [ ˜ Cε (¯ x) ⊂ W ˜) x ¯∈Iε (W

with largest Lebesgue measure. We note that for D0 suciently large W ⊂ V˜ and thus also V˜ satises condition (3.7) possibly passing to a larger D. Given ˜ V˜ for two points p, q ∈ Iε (U ) we denote the lattice geodesic distance of U = Ω, p and q in U , i.e. the length of the shortest polygonal path ΓU (p, q) := (x0 = p, x1 , . . . , xn = q) with xj ∈ Iε (U ) and xj+1 − xj = ± 1l ei for some i = 1, . . . , d − 1 connecting p and q , by dU (p, q). ˜ \ V˜ by V˜1 , . . . , V˜n and choose Iε (V˜i ) ⊂ Denote the connected components of Ω S ˜ such that V˜ i = x). It is easy to see that for (i) it suces to Iε (Ω) x ¯∈Iε (V˜i ) Cε (¯ 3−d show that dV˜ (p, q) ≤ C(1 + εˆ ) for all p, q ∈ Iε (V˜ ). Given p, q ∈ Iε (V˜ ) we rst note that dΩ˜ (p, q) ≤ d − 1. Let ΓΩ˜ (p, q) = (x0 , . . . , xm ) be a (non unique) ˜ ) for all j we are nished. shortest lattice path connecting p and q . If xj ∈ Iε (W Otherwise, for the local nature of the arguments we may assume that ΓΩ˜ (p, q) intersects exactly one V˜i . Let xj1 , xj2 ∈ Iε (V˜i ) be the rst and the last point in V˜i , i.e. xj ∈ / Iε (V˜i ) for j < j1 and j > j2 . Then it is elementary to see that

dV˜ (xj1 −1 , xj2 +1 ) ≤ Cld−3 Hd−2 (∂ V˜i ) ≤ C εˆ3−d Hd−2 (∂ V˜i ) for some C > 0 not depending on V˜ as the number of cubes at the boundary of V˜i can be bounded by Cld−2 Hd−2 (∂ V˜i )). Let ΓV˜ (xj1 −1 , xj2 +1 ) = (y0 , . . . , ym˜ ) be a shortest path. Then

(x0 , . . . , xj1 −1 , y1 , . . . , ym−1 , xj2 +1 , . . . , xm ) ˜ is a lattice path in V˜ connecting p and q which shows that dV˜ (p, q) ≤ m + m ˜ ≤ C + C εˆ3−d Hd−2 (∂ V˜i ) ≤ C + CDε(s−1)(d−3) . To show (ii) we let p ∈ W and t ∈ (0, cε(s−1)(d−2) ) for some small c > 0. Without restriction we may assume p ∈ Iε (V˜ ). If distW˜ (p, q) ≤ t for all q ∈ 44

V˜ the assertion is clear. Otherwise, there is some q ∈ Iε (V˜ ) with dV˜ (p, q) ≥ distV˜ (p, q) ≥ distW˜ (p, q) > 2t and a corresponding shortest path ΓV˜ (p, q) = (x0 = p, x1 , . . . , xm = q) with xi 6= xj for i 6= j and m ≥ m ¯ := d lt2 e. Now let U = Sm¯ ˜ j=0 Cε (xj ). Then it is not hard to see that U ⊂ {q ∈ V : distV˜ (p, q) ≤ t} for ε small enough and Hd−1 (U ) ≥ l1−d · c0 lt ≥ c0 ε(1−s)(d−2) t, as desired. 

2.4 Cell energy of the triangular lattice We close the preparatory chapter by collecting some elementary properties of the ˜ dened by cell energy W4 of the triangular lattice and the reduced energy W

˜ (r) = inf{W4 (F ) : eT F e1 = r}. W 1

(2.7)

Assume that W satises the assumptions (i), (ii) and (iii) given in Section 1.4.

Lemma 2.4.1. W4 is (i) frame indierent: W4 (QF ) = W4 (F ) for all F ∈ R2×2 , Q ∈ O(2), (ii) non-negative and satises W4 (F ) = 0 if and only if F ∈ O(2) and (iii) lim inf |F |→∞ W4 (F ) = lim inf |F |→∞ W4,χ (F ) = β . Proof. (i) is clear. For (ii) it suces to note that vF T F v = 1 for v ∈ V implies

that F T F = Id. As χ vanishes near ∞, (iii) can be seen by noting that if |F | → ∞, then for at least two vectors v ∈ V one has |F v| → ∞. Moreover, if |F | → ∞ with |F v1 | = 1, then W4 (F ) → β .  We compute the linearization about the identity matrix Id:

Lemma 2.4.2. Let F = Id + G for G ∈ R2×2 . Then for |G| small 1 W4 (F ) = Q(G) + o(|G|2 ), 2 
 g12 +g21 2 3α 2 2 where Q(G) = 16 3g11 + 3g22 + 2g11 g22 + 4 . 2

In particular, Q(G) only depends on the symmetric part GT + G /2 of G. Q is positive semidenite and thus convex on R2×2 and positive denite and strictly convex on the subspace R2×2 sym of symmetric matrices.


Proof. Let v ∈ V and G ∈ R2×2 small. We Taylor expand the contributions

W (|F v|) to the energy W4 :

 p T hv, (Id + G )(Id + G)vi W (|(Id + G)v|) = W  2 W 00 (1) GT + G = v, v + o(|G|2 ). 2 2 45

Now using the elementary identity

hv1 , Hv1 i2 + hv2 , Hv2 i2 + h(v2 − v1 ), H(v2 − v1 )i2
3 = 2 trace(H 2 ) + (trace H)2 8

(2.8)

for any symmetric matrix H ∈ R2×2 , we obtain by summing over v ∈ V  T 2 !  2 ! T 1 α 3 G +G G +G W4 (F ) = · · · 2 trace + trace + o(|G|2 ) 2 2 8 2 2

1 = Q(G) + o(|G|2 ). 2 2 2 + 2g22 + (g12 + g21 )2 ), Q is positive semidenite on R2×2 and As Q(G) ≥ 3α (2g11 16 2×2 positive denite on Rsym .  ˜ As a consequence, we have the following properties of the reduced energy W .

Lemma 2.4.3. The reduced energy satises ˜ (r) = 0 ⇐⇒ |r| ≤ 1. (i) W (ii) For r ≥ 1 one has ˜ (r) = W4 W



r 0

0 4−r 3



+ o((r − 1)2 ) =

α (r − 1)2 + o((r − 1)2 ). 4

˜ (r) = β . (iii) lim|r|→∞ W Proof. (i) If |r| ≤ 1, then one can choose Q ∈ SO(2) with eT1 Qe1 = r and

˜ (r) ≤ W4 (Q) = 0. If |r| > 1, then W ˜ (r) > 0 for otherwise there so 0 ≤ W 2×2 T would be a sequence Fk ∈ R with e1 Fk e1 = r and W4 (Fk ) → 0. But then dist(Fk , O(2)) → 0 by (ii) and (iii) of Lemma 2.4.1 and thus, up to subsequences, Fk → F ∈ O(2) with eT1 F e1 = r, which is impossible. (ii) This discussion shows that in fact for any η > 0 there exists δ > 0 such ˜ (r) → 0 as r & 1, we that W4 (F ) ≥ δ whenever dist(F, O(2)) ≥ η . Now since W ˜ (r)+δ obtain that, for suciently small r > 1 and δ > 0, any F with W4 (F ) < W T is contained in a small neighborhood of O(2) addition e1 F e1 = r holds, then . If in  in fact, F must be close to Id or to P =

1 0

0 . In particular, by continuity −1

˜ is attained for of W , the inmum on the right hand side in the denition of W those r. ˜ (r) = We now x such an r > 1 near 1 and choose F = Id + G such that W T W4 (F ) and e1 F e1 = r. As W4 is invariant under the reection P , we may without loss of generality assume that G is small. Then Lemma 2.4.2 yields  2 ! 3α g + g 12 21 2 2 3g11 + 3g22 + 2g11 g22 + 4 + o(|G|2 ). W4 (F ) = 32 2 46

Since g11 = r − 1 and W4 (F ) ≤ W4



r 0



0 1

= O((r − 1)2 ), by noting that

√ 2 2 +2g11 g22 = 38 (r−1)2 +( √13 g11 + 3g22 )2 we decduce from the minimality +3g22 3g11 property of F that g12 + g21 = o(r − 1) and g22 = − 13 g11 + o(r − 1) and F satises   FT + F r 0 = + o(r − 1) 0 4−r 2 3 with energy

 W4 (F ) = W4 =

FT + F 2



+ o((r − 1)2 )

α (r − 1)2 + o((r − 1)2 ). 4

˜ (r) ≥ β is immediate from Lemma 2.4.1(iii). Considering (iii) lim inf |r|→∞ W matrices F with F e1 = re1 and F vi = vi for i = 1 or i = 2 we see that also ˜ (r) ≤ β .  lim sup|r|→∞ W Under strengthened hypotheses on W we have the following expansion:

Lemma 2.4.4. If W in addition satises the assumptions (ii') and (iii'), then for r > 1 close to 1 we have  α(r − 1)2 1  0 0 ˜ + 6α + 7α − 2(3α − α ) cos(6φ) (r − 1)3 + O((r − 1)4 ), W (r) = 4 108   cos φ − sin φ where φ is such that RL = sin φ cos φ .

Proof. Let s = r − 1. By denition, ˜ (r) = min {W4 (F (s, x, y, z)) : x, y, z ∈ R} , W   1+s z+y where F (s, x, y, z) = z − y 1 + x . Due to the quadratic energy growth near √ SO(2), we need to minimize only over x, y, z with |x|, |z|, s|y| ≤ Cs for a constant C large enough. Indeed, as W4 (F (s, 0, 0, 0)) = O(s2 ), for a minimizer one has dist(F (s, x, y, z), O(2)) = O(s) by the subsequent Lemma 2.4.5(i) and without loss of generality dist(F (s, x, y, z), SO(2)) = O(s). We rst use that the absolute p value of the two rows and columns of F (s, x, y, z) is of order √ 1 + O(s). 2 2 Then (1 + s) + (z ± y) p= 1 + O(s), which implies |z ± y| = O( s) and so √ |z|, |y| = O( s), and also (1 + x)2 + (z ± y)2 = 1 + O(s), which then implies ±(1 + x) = 1 + O(s) and thus without loss of generality x = O(s). Finally using that the scalar product (1 + s)(z + y) + (1 + x)(z − y) = 2z + O(s3/2 ) of the two columns of F (s, x, y, z) in absolute value is also bounded by O(s), we obtain that |z| = O(s). 47

Set x = − 3s + sx1 , y = calculation gives

W4 (F (s, x, y, z)) =

√

sy1 , z = sz1 with |x1 |, |y1 |, |z1 | ≤ C . Explicit


α 8 + 3x21 + 8y12 + 12z12 + 6(x1 + y12 )2 s2 + O(s3 ). 32

Since α > 0, we thus obtain that this expression is minimized in x1 , √ y1 , z1 with √ √ x21 , y12 , z12 = O(s) and we may set x1 = sx2 , y1 = sy2 and z1 = sz2 with |x2 |, |y2 |, |z2 | ≤ C for some C > 0. Explicit expansion in powers of s then yields

W4 (F (s, x, y, z)) αs2 1  = + 48α + 56α0 − 16(3α − α0 ) cos(6φ) 4 864
 3 2 2 2 + 3α 81x2 + 72y2 + 108z2 s 
1 + 9αy 2 + α0 + (3α − α0 ) cos(6φ) x2 24  + 2(3α − α0 ) sin(6φ)z2 ) s7/2 + O(s4 )  1  αs2 + 6α + 7α0 − 2(3α − α0 ) cos(6φ) s3 = 4 108  √  3 αy22 s3 3α  2 √  3 9α 2 x2 + 2A sx2 s + + z2 + 2B sz2 s + O(s4 ) + 32 4 8 for A and B bounded uniformly in s and so

W4 (F (s, x, y, z))  1  αs2 + 6α + 7α0 − 2(3α − α0 ) cos(6φ) s3 = 4 108  √ 2 3 αy22 s3 3α  √ 2 3 9α + x2 + A s s + + z2 + B s s + O(s4 ). 32 4 8 Minimizing with respect to x2 , y2 and z2 we nally obtain that

  2 ˜ (1 + s) = αs + 1 6α + 7α0 − 2(3α − α0 ) cos(6φ) s3 + O(s4 ). W 4 108  The following lemma provides useful lower bounds for the energy W4 and the ˜. reduced energy W

Lemma 2.4.5. For all T > 1 one has: (i) There exists some c > 0 such that c dist2 (F, O(2)) ≤ W4 (F ) for all F ∈ R2×2 satisfying |F | ≤ T . 48

˜ (r) (ii) For δ > 0 small enough, there is a convex function V ≥ 0 with V (r) ≤ W 0 00 for r ≤ T , V (1) = 0 and such that the second derivative V+ (1) from the right at 1 exists and satises V+00 (1) = α2 − 2δ .

(iii) If in addition W satises assumptions (ii') and (iii'), then there exists a ˜ (r) ≤ V (r) + O((r − 1)4 ) for r ≤ T . convex function V ≥ 0 with V (r) ≤ W (iv) For ρ > 0 there is an increasing, subadditive function ψρ : [0, ∞) → (0, ∞) ˜ (r + 1) for all r ≥ 0 and ψ(r) = β for all which satises ψρ (r) − ρ ≤ W r ≥ cρ for some constant cρ only depending on ρ. Proof. (i) Let F√∈ R2×2 satisfying |F | ≤ T . By polar decomposition we nd R ∈

O(2) and U = F T F symmetric and non-negative denite such that F = RU . A short computation yields |U − Id| = dist(F, O(2)). Assume rst |U − Id| < η for η > 0 small enough. Since W4 (F ) is invariant under rotation and reection we obtain applying Lemma 2.4.2: 1 W4 (F ) = W4 (RT RU ) ≥ Q(U − Id) + o(|U − Id|2 ). 2

2×2 Noting that Q grows quadratically on Rsym (see Lemma 2.4.2) we obtain a constant c1 > 0 such that for |U − Id| < η

W4 (F ) ≥ c1 |U − Id|2 = c1 dist2 (F, O(2)). Consider the compact set M := {F ∈ R2×2 , dist(F, O(2)) ≥ η, |F | ≤ T }. W4 attains its minimum on M, which is strictly positive by Lemma 2.4.1(ii). This provides a second constant c2 > 0 such that for all F ∈ M

W4 (F ) ≥ c2 |U − Id|2 = c2 dist2 (F, O(2)). Taking c = min{c1 , c2 } yields the claim. (ii) We construct such a function directly applying Lemma 2.4.3.   for r ≤ 1, 0
α 2 V (r) = − δ (r − 1) for 1 ≤ r ≤ 1 + η, 4 
 α − δ η (2r − 2 − η) for r ≥ 1 + η, 4 when η > 0 is suciently small.   2 1 0 0 (iii) With f (r) := α(r−1) + 6α+7α −2(3α−α ) cos(6φ) (r−1)3 −C(r−1)4 4 108 for suciently large C , Lemma 2.4.4 shows that we can choose   for r ≤ 1, 0 V (r) = f (r) for 1 ≤ r ≤ 1 + η,   0 f (1 + η) + f (1 + η)(r − 1 − η) for r ≥ 1 + η, 49

when η > 0 is suciently small. (iv) We dene ( ¯ = ηr ψ(r) β

for 0 ≤ r ≤ βη , for r ≥ βη ,

˜ . Then we set ψ ρ (r) = for some η > 0 (depending on ρ) such that ψ¯ − ρ ≤ W ρ ρ ¯ + 1). As ψ is a concave function with ψ (0) > 0, it is subadditive. ψ(r  Recall the denition of fκ in (1.30). Finally, we provide a lower bound for W∆,χ + fκ which implies the equicoercivity of (Fεχ )ε . √ Lemma 2.4.6. Let T > 2. Then there are constants C1 , C2 > 0 such that for all F ∈ R2×2 with |F | ≤ T we obtain

i) |m(F ˆ ) − m(R(F ˆ ))| ≤ C1 |F − R(F )|2 , where R(F ) ∈ SO(2) is a solution of |F − R(F )| = minR∈SO(2) |F − R|, (ii) W∆,χ (F ) + fκ (F ) ≥ C2 |F − Id|2 . Proof. (i) Without restriction we may assume that |F − R(F )| is small as oth-

erwise the assertion is clear. So in particular, R(F ) is uniquely determined. 2×2 and R(F ) = Id. Indeed, once this Moreover, it suces to consider F ∈ Rsym is proved, we nd |m(F ˆ ) − m(R(F ˆ ))| = |R(F )m(R(F ˆ )T F ) − R(F )m(Id))| ˆ ≤ T 2 C|R(F ) F − Id| , as desired. 2×2 Let F ∈ R2×2 ˆ is sym , R(F ) = Id and set G = F − Id with G ∈ Rsym small. As m 2 C in a neighborhood of SO(2) we derive |m(F ˆ )−m(Id)| ˆ ≤ |Dm(Id) ˆ G|+C|G|2 = 2 2×2 ˆ . C|G| as Rsym ⊂ ker(Dm(Id)) (ii) By Lemma 2.4.5(i) the assertion is clear for all |F | ≤ T with c0 ≤ dist(F, O(2)) for c0 > 0 and C2 = C2 (c0 , T ) suciently small. Otherwise, we again apply Lemma 2.4.5(i) to obtain for c0 small enough

W∆,χ (F ) ≥ C dist2 (F, O(2)) + χ(F ) ≥ C dist2 (F, SO(2)) = C|F − R(F )|2 . 2 2 2 For convenience we write rij = eTi R(F )ej for i, j = 1, 2. As r12 = r21 = 1 − r11 2 2 we nd 1 − r11 = 1 − r11 + r11 (r11 − 1) = r12 + (1 − r11 )2 − (1 − r11 ). Thus, recalling m(R) ˆ = Re1 for all R ∈ SO(2) and applying (i) we get for 0 < c ≤ κ small enough

W∆,χ (F ) + fκ (F ) ≥ C|F − R(F )|2 + c(1 − e1 · m(R(F ˆ ))) + ce1 · (m(R(F ˆ )) − m(F ˆ )) ≥ C|F − R(F )|2 + c(1 − eT1 R(F )e1 ) − cC1 |F − R(F )|2 c c 2 C ≥ C2 |F − Id|2 , ≥ |F − R(F )|2 + (1 − r11 )2 + r12 2 2 2 as desired.

(2.9)

 50

Chapter 3 Limiting minimal energy and cleavage laws This chapter is devoted to the derivation of the limiting minimal energy. First, in Section 3.1 we give an independent proof for the triangular lattice. Section 3.2 contains the preliminary results concerning a mesoscopic localization technique and the proof of Theorem 1.3.1 is addressed in Section 3.3. Finally, in Section 3.4 we give some examples to which our cleavage law applies.

3.1 Warm up: Proof for the triangular lattice As a rst approach to the proof of the cleavage law we present an elementary proof for the planar model where the atoms in the reference conguration are given by a portion of a triangular lattice. We will rst establish a lower bound for the limiting minimal energy by considering slices of the form (0, l) × {x2 } for x2 ∈ (0, 1) and using the reduced energy dened in (2.7). In a second step we show that this bound is attained by either elastic deformations or by congurations with cleavage along a specic crystallographic line depending on the boundary displacements. We can classify (or `color') all triangles in Cε into two types, say `type one' and `type two', such that all triangles of the same type are translates of each other. Then only triangles of dierent type can share a common side. Denote (1) (2) the sets by Cε and Cε , respectively. Proof of Theorem 1.4.1. We rst show that the expression on the right hand side is a lower bound for the limiting minimal energy. For every deformation y ∈ A(aε ) we have by (1.20) and (1.21) ˆ
4 W4 ∇˜ y dx. Eε (y) ≥ √ 3ε Ωε ∩(0,l)×(ε,1−ε)

51

(1)

Let 0 < δ < α4 and choose R so large that W (r) > β − δ if r ≥ R. Dene C¯ε to be the set of those triangles 4 of type one for which at least one side in the deformed conguration y(4) is larger than 2Rε. By I ⊂ (ε, 1 − ε) we denote the set of those points x2 for which there exists x1 ∈ (0, l) such that (x1 , x2 ) lies in one of these triangles. We can then estimate the energy integral by splitting the x2 -integration into a rst part where x2 ∈ / I and a second part with x2 ∈ I . 1. If x2 ∈ / I , then all sidelengths of y(4) for a triangle 4 whose interior intersects the segment (0, l) × {x2 } are less or equal to 4Rε. This is clear for triangles of type one by construction. For triangles of type two it follows from the fact that the two sides of 4 intersecting (0, l)×{x2 } are also sides of triangles of type one and therefore bounded by 2Rε. The third side is thus less than 4Rε, too. It is elementary to see that for F ∈ R2×2

|eT1 F e1 | ≤ 8R,

if |vT F v| ≤ 4R for all v ∈ V.

(3.1)

Indeed, if λ1 , λ2 are the eigenvalues of 21 (F T + F ), then by (2.8) one has 34 (λ21 +
2 λ22 ) = 43 trace 12 (F T + F ) ≤ 3 · (4R)2 and thus |eT1 F e1 | ≤ max{|λ1 |, |λ2 |} ≤ 8R. Consequently, for almost every x2 ∈ / I we have eT1 ∇˜ y (x1 , x2 )e1 ≤ 8R for all x1 ∈ (0, l). ˜ (r) for r ≤ 8R By Lemma 2.4.5(ii) choose a convex function with V (r) ≤ W α x2 00 and V+ (1) = 2 − 2δ . For x2 ∈ (ε, 1 − ε) dene Ωε ⊂ (0, l) such that Ωxε 2 × {x2 } = Ωε ∩ (0, l) × {x2 }. Then for the rst part one obtains, if a < ∞, by convexity of V ˆ ˆ ˆ ˆ

4 4 √ W4 ∇˜ V eT1 ∇˜ y dx1 dx2 ≥ √ y e1 dx1 dx2 3ε (ε,1−ε)\I Ωxε 2 3ε (ε,1−ε)\I Ωxε 2 ˆ 4 ≥√ |Ωxε 2 |V (1 + aε ) dx2 3ε (ε,1−ε)\I 2 ≥ √ (1 − 2ε − |I|)(l − 2ε)(V+00 (1)a2ε + o(ε)) 3ε 2 → √ (1 − |I|)lV+00 (1)a2 (3.2) 3 as ε → 0. It is not hard to see that this asymptotic estimate remains true also for a = ∞. 2. On the other hand, the energy of the second part can be estimated by the (1) energy of all springs lying on the side of a triangle in C¯ε , which yields ˆ ˆ
4 √ (3.3) W4 ∇˜ y dx1 dx2 ≥ 2(β − δ)ε#C¯ε(1) , 3ε I Ωxε 2 as the length of at least two springs in each of these triangles is larger than Rε in the deformed conguration. Now the projection of any triangle onto the x2 -axis 52

(1) is an interval of length ε sin(φ + π3 ) = εγ , and so εγ#C¯ε ≥ |I|, i.e., ˆ ˆ
4 √ W4 ∇˜ y dx1 dx2 ≥ 2(β − δ)γ −1 |I|. 3ε I Ωxε 2

(3.4)

Summarizing (3.2) and (3.4) we nd

lim inf inf{Eε (y) : y ∈ A(aε )} ε→∞    2 α 2 −1 ≥ min √ − 2δ la (1 − |I|) + 2(β − δ)γ |I| : |I| ∈ [0, 1] 3 2    2 α 2 2(β − δ) = min √ − 2δ la , . γ 3 2 Now δ → 0 shows

 lim inf inf{Eε (y) : y ∈ A(aε )} ≥ min ε→∞

αl 2β √ a2 , γ 3

 .

This establishes the lower bound. It remains to prove that the right hand side in Theorem 1.4.1 is attained for some sequence of deformations. In order to do so, we consider two specic sequences of deformations. First, for a < ∞ let   1 + aε 0 el aε yε (x) = (Id + F )x = x. (3.5) 0 1 − a3ε By Lemma 2.4.2 we have that W4 (F ) = α4 a2ε + o(ε) and so

αl lim Eε (yεel ) = √ a2 ε→0 3 by (1.20). To dene y cr we choose any line (s, 0) + Rvγ intersecting both the segments (0, l) × {0} and (0, l) × {1} (as in Corollary 1.5.2). This is possible since l > √13 . Let a > 0 and set

( x yεcr (x) = x + aε le1

for x to the left of (s, 0) + Rvγ , for x to the right of (s, 0) + Rvγ

(3.6)

for atoms x with ε < x1 < l − ε. Except for a negligible contribution from the boundary layers, the energy of this conguration can be estimated as in Step 2 of the proof of the lower bound: It is given by the energy of springs intersecting (s, 0) + Rvγ , i.e., by the two springs lying on the boundary of the triangles of type one which are intersected by (s, 0) + Rvγ . These springs are elongated by a factor scaling with aε /ε, thus yielding a contribution β in the limit ε → 0.  53

3.2 Estimates on a mesoscopic cell 3.2.1 Mesoscopic localization The goal of this section is the derivation of a lower comparison potential on `large cells' dened on a mesoscopic scale εs with

s=

3d − 1 . 3d

We√dene the domain Uε = (0, εs λ) × (0, εs )d−1 for λ ∈ [λ0 , 2λ0 ], where λ0 ≥ L = L( AT A, Wcell , 1, . . . , 1). We consider the Bravais-lattice dened in Section 1.1 with a possible translation. For ρ ∈ A[0, 1)d set Lε,ρ = ερ + Lε . Let D, D0 > 0 ˜ ⊂ (0, 1)d−1 is connected with and suppose that W

1 ˜ ) ≤ 1, < Hd−1 (W 2

(3.7)

˜ ) < D, Hd−2 (∂ W

˜ : dist(x, ∂ W ˜ ) > D0 ε1−s } with such that the connected component W of {x ∈ W largest Lebesgue measure also satises (3.7) for ε small enough. We dene ∂W (L0ε,ρ (Uε )) = {¯ x ∈ L0ε,ρ : Qε (¯ x) ∩ ({0, εs λ} × εs W ) 6= ∅}.

(3.8)

Let y : Lε,ρ (Uε ) → Rd be the lattice deformations on Uε with corresponding energy

E(Uε , y) = εd−1

X

¯ x)) + 1 εd−1 Wcell (∇y(¯ 2 ◦

x ¯∈(L0ε,ρ (Uε ))

X

¯ x)). Wcell (∇y(¯

x ¯∈∂W (L0ε,ρ (Uε ))

The factor 21 takes account of the fact that in the proof of Theorem 1.3.1 half of the energy of the boundary cells will be assigned to each of the two adjacent mesoscopic cells. Let y˜ denote the interpolation for y dened in Section 2.2. For r ∈ R we will investigate the minimization problem of nding inf E(Uε , y) under certain boundary conditions given as follows. We dene the averaged boundary conditions ˆ   1 1 s 0 1 0 y ˜ (λε , x ) − y ˜ (0, x ) dx0 = λεs (1 + r). (3.9) εs(d−1) |W | εs W Here, x0 = (x2 , . . . , xd ), |W | = Hd−1 (W ) and y˜1 denotes the rst component of y˜. Moreover, we introduce the condition [
˜ = ∅, (3.10) Qε (¯ x) ∩ {0, εs λ} × εs W x ¯∈B¯ε0

54

˜ is of the form (3.7) and similar as in Section 2.2, B¯0 ⊂ L0 (Uε ) denotes where W ε ε,ρ  ∗ ∗ ¯ the cells, where diam1 ∇y(¯ x), Z > Cint for a xed Cint > 0 (which may dier from Cint to be chosen later). Note that in contrast to the denition of C¯ε0 we consider diam1 {·, Z} only instead of diam{·, Z}. We now concern ourselves with the minimization problem n o M (Uε , r) := inf E(Uε , y) : y satises (3.9) and (3.10) . Before we state the main theorem of this section we briey note that M (Uε , r) = 0 for −2 ≤ r ≤ 0 as the averaged boundary conditions may be satised by a suitable rotation of the specimen, i.e. y(x) = Rx for R ∈ SO(d).

Theorem 3.2.1. Let λ0 ≥ L and C2 > C1 > 0, C2 suciently large. Let δ > 0 small. Then for all W ⊂ (0, 1)d−1 as in (3.7) and ρ ∈ A[0, 1)d there is a function f : R × [εs λ0 , 2εs λ0 ] → R,

(r, λ) 7→ f (r, λ),

which for r ≤ max{ε(s−1)(d−3) , 1}C2 is convex in r and linear in λ and for ε small enough, independently of ρ and W , satises • for r ∈ R, λ ∈ [εs λ0 , 2εs λ0 ]: f (r, λ) ≤ M (Uε , r), √ • for r ∈ [0, C1 ε], λ ∈ [εs λ0 , 2εs λ0 ]:  α  r2 A f (r, λ) = ε λω(|W |) −δ 2 det A ε 1 ≤ M (Uε , r) ≤ f (r, λ) + 4εsd λ0 δ, ω(|W |) s(d−1)

(3.11)

• for r ≥ max{ε(s−1)(d−3) , 1}C2 , λ ∈ [εs λ0 , 2εs λ0 ]:   β A −δ det A 1 ≤ M (Uε , r) ≤ f (r, λ) + 2εs(d−1) δ ω(|W |)

f (r, λ) = εs(d−1) ω(|W |)

(3.12)

for a continuous function ω : [0, 1] → R with ω(1) = 1. The theorem shows that f (r,√λ) is a lower bound for M (Uε , r) which becomes sharp in the regimes r ∈ [0, C1 ε] and r ∈ [max{ε(s−1)(d−3) , 1}C2 , ∞) provided that |W | ≈ 1. The proof of Theorem 3.2.1 is essentially divided into three steps each of which dealing with one particular regime: The elastic regime (Lemma 3.2.2), the fracture regime (Lemma 3.2.4) and the one in between (Lemma 3.2.3). In addition, in the case d ≥ 4 we need a short additional argument in the intermediate 55

regime (Lemma 3.2.5). It will be convenient to rescale the system in order to obtain a problem on a macroscopic domain not depending on ε. Therefore, we ˆ = (0, λ) × (0, 1)d−1 , yˆ : Lεˆ,ρ (Uˆ ) → Rd and let εˆ = ε1−s = ε1/3d , U

X

ˆ Uˆ , yˆ) = εˆd ε−1 E(

ˆ )) x ¯∈(L0εˆ,ρ (U 2

X

= ε− 3

ˆ )) x ¯∈(L0εˆ,ρ (U

1 ¯ y (¯ Wcell (∇ˆ x)) + εˆd ε−1 2 ◦

1 2 ¯ y (¯ Wcell (∇ˆ x)) + ε− 3 2 ◦

X

¯ y (¯ Wcell (∇ˆ x))

ˆ )) x ¯∈∂W (L0εˆ,ρ (U

X

¯ y (¯ Wcell (∇ˆ x)),

ˆ )) x ¯∈∂W (L0εˆ,ρ (U

(3.13)

¯ y is dened as in (1.3) replacing ε by εˆ and ∂W (L0 (Uˆ )) as in (3.8) where ∇ˆ εˆ,ρ replacing εs by 1. The averaged boundary conditions now become ˆ   − y˜ˆ1 (λ, x0 ) − y˜ˆ1 (0, x0 ) dx0 = λ(1 + r), (3.14) W

´

where − denotes the averaged integral and the interpolation y˜ ˆ is dened as in Section 2.2. The condition (3.10) on the boundary cells reads as [
˜ =∅ Qεˆ(¯ x) ∩ {0, λ} × W (3.15) x ¯∈B¯ε0ˆ

and the minimum problem in the rescaled variant becomes n o ˆ ˆ ˆ ˆ M (U , r) := inf E(U , yˆ) : yˆ satises (3.14), (3.15) .

(3.16)

It is not hard to see that

ˆ (Uˆ , r) = ε 3d−1 ˆ (Uˆ , r). 3 M M (Uε , r) = εsd M

(3.17)

3.2.2 Estimates in the elastic regime ˆ (Uˆ , r) for r near zero. We rst determine M √ Lemma 3.2.2. For 0 ≤ r ≤ Cel ε the minimizing |W |

problem (3.16) satises

2 λαA r2 ˆ (Uˆ , r) ≤ λαA r + o(1) + o(1) ≤ M 2 det A ε 2 det A ε

(3.18)

for ε → 0 with αA as in (1.16). Here o(1) is independent of ρ ∈ A[0, 1)d , W ⊂ (0, 1)d−1 and λ ∈ [λ0 , 2λ0 ] and depends only on Cel .

56

Proof. In the following we drop the superscript ˆ· if no confusion arises. We rst show that

M (U, r) ≥ |W |

λαA r2 + o(1) 2 det A ε

(3.19)

for ε → 0, where o(1) only depends on Cel . We argue by contradiction. If the √ claim were false, there would exist a δ > 0, sequences εk → 0, Cel εk ≥ rk → 0, λk ∈ [λ0 , 2λ0 ], ρk ∈ A[0, 1)d , Wk ⊂ (0, 1)d−1 satisfying (3.7) as well as a sequence yk : Lεˆk ,ρk (Uk ) → Rd satisfying (3.14) with respect to rk , (3.15) and E(Uk , yk ) ≤ M (Uk , rk ) + k1 such that

E(Uk , yk ) ≤ |Wk |

λk αA rk2 − δ. 2 det A εk

(3.20)

Passing to a subsequence we may assume that ρk → ρ ∈ A[0, 1)d , λk → λ ∈ −1

[λ0 , 2λ0 ] and εk 2 rk → r ≥ 0. Moreover, as Hd−2 (∂Wk ) is uniformly bounded in k , we may assume that χWk → χW in measure for some W ⊂ (0, 1)d−1 with 1 ≤ |W | ≤ 1 by Theorem A.1.2. As discussed in Section 1.2, there is an obvious 2 choice for an elastic deformation, namely yk∗ (x) = (1 + rk )x for all x ∈ Lεˆk ,ρk (Uk ). We note that y˜k∗ satises (3.14) as this interpolation by construction is equal to the linear map yk∗ . It is elementary to see that E(Uk , yk∗ ) =


(1 + O(ˆ εk )) λk 1 C Wcell ((1 + rk )Z) ≤ Qcell (rk Z) + o(rk2 ) det A εk εk (3.21) √ C(Cel εk )2 ≤ ≤ C. εk

As usual we follow the convention of denoting dierent constants with the same −2

letter. As by (2.4) and (3.13) a broken cell contributes an energy of order εk 3 the comparison with (3.21) yields

C¯εˆ0 k = ∅

(3.22)

for k ∈ N suciently large. This together with (3.15) shows that y˜k is a continuous, piecewise linear interpolation on the set Vk , where [ [ Vk◦ = Qεˆk (¯ x), Vk = Vk◦ ∪ Qεˆk (¯ x). (3.23) x ¯∈(L0εˆ

k ,ρk

x ¯∈∂Wk (L0εˆ

(Uk ))◦

k ,ρk

(Uk ))

Applying Lemma 2.2.1(i) and Lemma 2.1.1 we obtain ˆ ˆ 2 ¯ k (¯ ¯ dist (∇˜ yk , SO(d)) ≤ C dist2 (∇y x), SO(d)) ≤ Cεk E(Uk , yk ) Vk

Vk

≤ Cεk (M (Uk , rk ) + k −1 ) ≤ Cεk (E(Uk , yk∗ ) + k −1 ) ≤ Cεk . 57

´ yk , SO(d)) we use the geometric rigidity result in In order to estimate Vk dist2 (∇˜ Theorem B.1. We nd a constant C = C(λ0 ) and rotations Rk ∈ SO(d) such that k∇˜ yk − Rk k2L2 (Vk ) ≤ Cεk .

(3.24)

In fact, C depends only on λ0 as all shapes Vk are related to (0, λ0 ) × (0, 1)d−1 through bi-Lipschitzian homeomorphisms with Lipschitz constants of both the homeomorphism itself and its inverse uniformly bounded in k , see Section B. Up to a not relabeled subsequence we may assume that Rk → R for some R ∈ SO(d). By Poincaré's inequality we obtain

k˜ yk − (Rk x + ck )k2L2 (Vk ) ≤ Cεk for suitable constants ck ∈ Rd . We let uk = Lemma 2.2.1(ii)

√1 (yk εk

(3.25)

− (Rk · +ck )) and obtain by

C ¯ k 2 2

∇u ≤ k∇˜ yk − Rk k2L2 (Vk ) ≤ C. L (Vk ) εk

(3.26)

From (3.24) and (3.25) we deduce that for a suitable subsequence (not relabeled) χVk u˜k * χU u and χVk ∇˜ uk * χU ∇u in L2 for some u ∈ H 1 (U, Rd ), where U = (0, λ) × (0, 1)d−1 . Then by (3.26) and possibly passing to a further subsequence d ¯ k * χU f in L2 . By (3.22) and we nd some f ∈ L2 (U, Rd×2 ) such that χVk ∇u ¯ k * χU ∇u · Z in L2 . Lemma 2.2.2 we obtain f = ∇u · Z , i.e. in particular χVk ∇u We now concern ourselves with the averaged boundary condition (3.14) for the displacement elds uk . We obtain ˆ yk1 (λk , x0 ) − y˜k1 (0, x0 )) dx0 λk (1 + rk ) = − (˜ Wk ˆ √ u1k (λk , x0 ) − u˜1k (0, x0 )) dx0 = (Rk )11 λk + εk − (˜ Wk

and therefore ˆ λk (1 − (Rk )11 ) rk − (˜ u1k (λk , x0 ) − u˜1k (0, x0 )) dx0 = λk √ + . √ εk εk Wk

(3.27)

We extend u ˜k ∈ H 1 (Vk , Rd ) to uˆk ∈ H 1 (Uk ∪ Vk , Rd ) such that kˆ uk kH 1 (Uk ∪Vk ) ≤ Ck˜ uk kH 1 (Vk ) , where C may be chosen independently of ε for the same reasoning as in (3.24). In particular, we note that u ˆ1k = u˜1k on {0, λk } × Wk . Dening u¯k ∈ H 1 (U, Rd ) by u¯k (x1 , x0 ) := uˆk ( λλk x1 , x0 ) and possibly passing to a further subsequence we may assume u ¯k * u weakly in H 1 (U ). Now choosing r∗ ∈ R such that the trace of u satises ˆ − (u1 (λ, x0 ) − u1 (0, x0 )) dx = λr∗ , (3.28) W

58

by the weak continuity of the trace operator, (3.27) yields −1

r∗ = lim εk 2 (rk + 1 − (Rk )11 ) ∈ [0, ∞). k→∞

(3.29)

For later use we remark that, since rk ≥ 0, the existence of this limit also implies that R11 = 1. We now derive a lower bound for the limiting energy. To this end, note d×2d that by Assumption 1.1.2 for n G ∈ R osmall we can write Wcell (Z + G) =

+ η(G) with sup η(G) : |G| ≤ ρ |G|2 ¯ k (x)|) and estimate χk (x) := χ[0,ε−1/4 ] (|∇u

→ 0 as ρ → 0. Furthermore, let

1 Q (G) 2 cell

k

X

E(Uk , yk ) ≥ εˆdk ε−1 k

x ¯∈(L0εˆ

k ,ρk

Wcell (Rk · Z +

x ¯∈(L0εˆ

ˆk

¯ k (¯ εk ∇u x))

(U ))◦

X

= εˆdk ε−1 k

√

Wcell (Z +

,ρk (U ))

√

¯ k (¯ x)) εk Rk−1 ∇u

◦

√ 1 ¯ k (x)) dx χk (x) Wcell (Z + εk Rk−1 ∇u εk det A Vk◦ ˆ   1 ¯ k (x)) + ε−1 η(√εk R−1 ∇u ¯ k (x)) dx. χk (x) Qcell (Rk−1 ∇u ≥ k k 2 det A Vk◦ ≥

The second term may be bounded by

√ ¯ 2 η( εk ∇uk ) ¯ χk |∇uk | √ ¯ 2 . | εk ∇uk | √

¯ k) ¯ k |2 is bounded in L1 and χk η(√ εk¯∇u Since χVk◦ χk |∇u → 0 uniformly, we deduce | εk ∇uk |2 √

¯ k) ¯ k |2 η(√ εk¯∇u that χVk◦ χk |∇u converges to zero in L1 as k → ∞. Consequently, | εk ∇uk |2

1 lim inf E(Uk , yk ) ≥ lim inf k→∞ k→∞ 2 det A

ˆ Vk◦

¯ k (x)) dx. Qcell (χk (x)Rk−1 ∇u

¯ k * As Rk → R and χk → 1 boundedly in measure, we obtain χVk◦ χk Rk−1 ∇u −1 2 χU R ∇u · Z weakly in L and thus ˆ 1 Qcell (R−1 ∇u(x) · Z) dx =: Elim (U, u). lim inf E(Uk , yk ) ≥ (3.30) k→∞ 2 det A U We now derive a lower bound for Elim (U, u) which will give a contradiction to (3.20) and thus (3.19) is proved. Since R11 = 1, we deduce that R1i = Ri1 = 0 for i = 2, . . . , d and therefore (R−1 ∇u(x))11 = (∇u(x))11 . Applying (1.15), Lemma 59

2.1.2 and (3.28) we obtain

Elim (U, u) ≥ ≥ ≥ =

ˆ 1 ˜ Q((∇u(x)) 11 ) dx 2 det A U ˆ ˆ λ |W | − αA (∂1 u1 (x1 , x0 ))2 dx1 dx0 2 det A W 0 ˆ ˆ λ 2 |W |αA  −1 ∂1 u1 (x1 , x0 ) dx1 dx0 λ λ − 2 det A W 0 |W |λαA ∗ 2 (r ) , 2 det A

(3.31)

where we have used Jensen's inequality. Recalling (3.29) we then obtain r∗ ≥ limk→∞ √rεkk = r and thus by (3.20), (3.30) and (3.31)

∞>

|W |λαA 2 |W |λαA 2 r − δ ≥ lim inf E(Uk , yk ) ≥ Elim (U, u) ≥ r , k→∞ 2 det A 2 det A

giving the desired contradiction. √ To see the upper bound in (3.18), for given 0 ≤ r ≤ Cel ε, ρ ∈ A[0, 1)d , λ ∈ [λ0 , 2λ0 ] we consider the deformation y : Lεˆ,ρ → Rd ,

y(x) = x + F¯ (r) x

(3.32)

with F¯ (r) as in Lemma 2.1.2. As F¯ (r) depends linearly on r we obtain

E(U, y) =

εˆd ε

X

εˆd Wcell (Z + F¯ (r) · Z) + 2ε ◦

ˆ )) x ¯∈(L0ε,ρ (U

X

Wcell (Z + F¯ (r) · Z)

ˆ )) x ¯∈∂W (L0ε,ρ (U

√ √ λ(1 + O(ˆ ε)) 1 Wcell (Z + ε F¯ (r/ ε) · Z) det A ε √ √ √ λ(1 + O(ˆ ε)) λ(1 + O(ˆ ε)) η( ε F¯ (r/ ε)) ¯ = Qcell (F (r/ ε) · Z) + , 2 det A 2 det A ε =

where η is as√before.√Note that the √ second term converges uniformly to zero as ¯ ε → 0 since ε F (r/ ε) ≤ Cr ≤ ε CCel . Thus, we obtain

E(U, y) =

λαA r2 + o(1). 2 det A ε

Since y˜ = y satises (3.14) and (3.15), this shows that M (U, r) ≤

3.2.3 Estimates in the intermediate regime ˆ (Uˆ , r) in an intermediate regime. We now determine M 60

λαA r2 2 det A ε

+ o(1). 

Lemma 3.2.3. Let Cmed,2 > 0, Cmed,1 > 0 suciently large, 1 < p < 34 . Then there is a constant C > 0 such that the minimizing problem (3.16) satises ˆ (Uˆ , r) ≥ C|W |λε− p2 rp M

(3.33)

√

ε Cmed,1 ≤ r ≤ Cmed,2 as ε → 0. The constant C is independent of ρ ∈ A[0, 1)d , W ⊂ (0, 1)d−1 and λ ∈ [λ0 , 2λ0 ].

for

Note that we only provide a lower bound which might not be sharp.

Proof. We follow the previous proof and only indicate the necessary changes. We

again drop the superscript ˆ· if no confusion arises. By Lemma 2.1.1 for a suitable constant c > 0 and some 1 < p < 34 the cell energy Wcell may be bounded from below by a function of the form ( p p √ ¯ ε1− 2 c χ{dist(G,SO(d))≥ ¯ ε} dist (G, SO(d)) G ⊥ V0 , |G| ≤ Cint , Vε (G) = Wcell (G) else. p

Then ε 2 r−p E(U, y) ≥ E(U, y; r), where  X p ¯ x)) + 1 E(U, y; r) := εˆd ε 2 −1 r−p Vε (∇y(¯ 2 0 ◦ x ¯∈(Lεˆ,ρ (U ))

X

 ¯ x)) . Vε (∇y(¯

x ¯∈∂W (L0εˆ,ρ (U ))

(3.34) We also note that

  p p ¯ Vε (G) ≥ ε1− 2 c distp (G, SO(d)) − ε2

(3.35)

d

for G ∈ Rd×2 with G ⊥ V0 and |G| ≤ Cint . We show that for suciently small ε

M(U, r) := inf {E(U, y; r) : y satises (3.14), (3.15)} ≥ C|W |λ

(3.36)

for some C > 0 and argue again by contradiction. If (3.36) were false, there √ would exist sequences εk → 0, Cmed,2 ≥ rk ≥ Cmed,1 εk , λk ∈ [λ0 , 2λ0 ], ρk ∈ A[0, 1)d , Wk ⊂ (0, 1)d−1 satisfying (3.7) as well as a sequence yk : Lεˆk ,ρk (Uk ) → Rd satisfying (3.14) with respect to rk , (3.15) and E(Uk , yk ; rk ) ≤ M(Uk , rk ) + k1 such that

E(Uk , yk ; rk ) ≤

|Wk |λk . k

(3.37)

As above we assume that ρk → ρ, λk → λ, χWk → χW in measure and rk → r up to subsequences. Plugging in the obvious choice for an elastic deformation yk∗ (x) = (1 + rk ) x, x ∈ Lεˆk ,ρk (Uk ) we see that p

E(Uk , yk∗ ; rk ) ≤ C εˆdk εk2

−1 −p rk

61

1− p2 p rk

εˆ−d k c εk

= C,

(3.38)

and thus as in (3.22) we deduce that C¯εˆ0 k = ∅ for all k suciently large since p

by (2.4) and (3.34) a broken cell contributes an energy of order rk−p εk2 p

− 23

≥

2

− −p Cmed,2 εk2 3

 1. Similarly as in the previous proof we obtain by using Lemma 2.2.1(i) and (3.35) ˆ ˆ p ¯ k (¯ ¯ dist (∇˜ yk , SO(d)) ≤ C distp (∇y x), SO(d)) Vk ˆVk p p −1 ¯ k (¯ x)) + C c εk2 |Vk | ≤C εk2 Vεk (∇y Vk

≤ Crkp E(Uk , yk∗ ; rk ) +

p Crkp + Cεk2 |Vk | k

√ and thus together with (3.38) and rk ≥ Cmed,1 εk ˆ distp (∇˜ yk , SO(d)) ≤ Crkp Vk

if C is suciently large. By geometric rigidity (see Theorem B.1) and Poincaré's inequality there are rotations Rk ∈ SO(d) and constants ck ∈ Rd such that ˆ p k∇˜ yk − Rk kLp (Vk ) ≤ C distp (∇˜ yk , SO(d)) ≤ Crkp (3.39) Vk

and Letting uk =

k˜ yk − (Rk · +ck )kpLp (Vk ) ≤ Crkp . 1 (yk rk

− (Rk · +ck )) we obtain by Lemma 2.2.1(ii)

C ¯ k p p

∇u ≤ p k∇˜ yk − Rk kpLp (Vk ) ≤ C L (Vk ) rk

(3.40)

and deduce that for a suitable subsequence (not relabeled) Rk → R and χVk u ˜k * 1 d d−1 χU u for some R ∈ SO(d) and u ∈ H (U, R ), where U = (0, λ) × (0, 1) . Then ¯ k * χU ∇u · Z in Lp possibly after as in the previous proof we derive χVk ∇u extracting a further subsequence. As before, in particular applying the argument in (3.27) and (3.28), we obtain that the limit function satises the constraint ˆ − (u1 (λ, x0 ) − u1 (0, x0 )) dx0 = λr∗ ∈ [0, ∞), W

where r∗ = limk→∞

1 (r rk k

+ 1 − (Rk )11 ) = 1 + limk→∞

62

1−(Rk )11 rk

≥ 1. Applying

(3.35), Lemma 2.2.1(i), (3.39) and (3.40) we compute p

|Vk |cεk2 lim inf E(Uk , yk ; rk ) + lim sup p k→∞ k→∞ rk det A ˆ c ¯ k (x), SO(d)) ¯ ≥ lim inf p distp (∇y dx k→∞ r det A V k k ˆ ˆ C p ¯ dist (∇˜ yk (x), SO(d)) dx ≥ C |∇u(x)|p dx ≥ lim inf p k→∞ r U k Vk for some C¯ > 0. We dene

ˆ

Elim (U, u; r ) = C¯ ∗

|∇u(x)|p dx U

and then the arguments in (3.31), in particular a slicing argument and Jensen's inequality, yield ¯ |λ(r∗ )p ≥ C|W ¯ |λ Elim (U, u; r∗ ) ≥ C|W since r∗ ≥ 1. Consequently, for Cmed,1 suciently large (independent of εk ) we derive C¯ lim inf E(Uk , yk ; rk ) ≥ |W |λ. k→∞ 2 In view of (3.37) this gives the desired contradiction. Thus, (3.36) holds and then by (3.34) the claim (3.33) follows. 

3.2.4 Estimates in the fracture regime ˆ (Uˆ , r) for large r. We now determine M

Lemma 3.2.4. Let λ0 ≥ L. For max{ˆε3−d , 1}Ccr ≤ r, Ccr suciently large, the minimizing problem (3.16) satises     β A ˆ (Uˆ , r) ≤ ε−s βA + o(1) (3.41) + o(1) ≤ M ε−s (|W | − C(1 − |W |)) det A det A

for ε → 0. Here C > 0 and o(1) are independent of ρ ∈ A[0, 1)d , W ⊂ (0, 1)d−1 and λ ∈ [λ0 , 2λ0 ]. Proof. We again drop the superscript ˆ· if no confusion arises. We rst show that εs M (U, r) ≥

|W |βA ˆ − |W |) + o(1) − C(1 det A

(3.42)

for ε → 0 and some xed Cˆ large enough. We again argue by contradiction. If the claim were false, there would exist a δ > 0, sequences εk → 0, max{ˆ εk3−d , 1}Ccr ≤ d d−1 rk , λk ∈ [λ0 , 2λ0 ], ρk ∈ A[0, 1) , Wk ⊂ (0, 1) satisfying (3.7) as well as a 63

sequence yk : Lεˆk ,ρk (Uk ) → Rd satisfying (3.14) with respect to rk , (3.15) and E(Uk , yk ) ≤ M (Uk , rk ) + k1 such that

|Wk |βA ˆ − |Wk |) − 2δ. − C(1 det A

εsk E(Uk , yk ) ≤

(3.43)

Up to choosing subsequences we may assume that ρk → ρ ∈ A[0, 1)d , λk → λ ∈ [λ0 , 2λ0 ] and Wk → W ⊂ (0, 1)d−1 . We again derive a rst upper bound of the minimal energy, now by testing with yk∗ (x) = x χ{x1 ≤λk /2} + (x + λk rk e1 ) χ{x1 >λk /2} . It is easy to see that only cells intersecting the set { λ2k } × (0, 1)d−1 give an energy contribution. As the quantity of these cells scales like εˆk−d+1 , by (1.9) we obtain −s E(Uk , yk∗ ) ≤ C εˆk−d+1 εˆdk ε−1 k C2 = Cεk C2

and thus εsk E(Uk , yk∗ ) ≤ C for all k ∈ N and some C large enough. Then, by (2.4) it is not hard to see that there is some C˜ such that

#C¯εˆ0 k ≤ C˜ εˆ−d+1 , k

(3.44)

where C˜ can be chosen independently of Cint ≥ 1. We now choose Cint large ∗ enough (depending on δ and possibly larger than the xed Cint ) such that for d×2d every partition Z = Z1 ∪ . . . ∪ Zn and G ∈ R with diam {G, Zi } ≤ Cint and mini,j d(G; Zi , Zj ) ≥ 2−2d Cint (see Section 2.2) we obtain (cf. Assumption 1.1.2(iv))

Wcell (G) −

n X

δ 1 X X X β(zs , zt ) − . 2 1≤i,j≤n z ∈Z z ∈Z C˜ s i t j

W Zi (G[Zi ]) ≥

i=1

(3.45)

i6=j

We obtain from Lemma 2.1.1 and Lemma 2.2.1(i), X X 1 ¯ k (¯ ¯ k (¯ εsk E(Uk , yk ) ≥ εˆd−1 Wcell (∇y x)) + εˆd−1 Wcell (∇y x)) k k 2 x ¯∈Fεˆ0 \C¯ε0ˆ x ¯∈C¯ε0ˆ k k k X ˆ ≥ C εˆ−1 dist2 (∇˜ yk , SO(d)), k x ¯∈Fεˆ0 \C¯ε0ˆ k

Qεˆk (¯ x)

k

where Fεˆ0 k = Cεˆ0 k ∪ ∂Wk (L0εˆk ,ρk (Uk )). Note that dist(F, B√d (0)) ≤ dist(F, SO(d)) for all F ∈ Rd×d , where B√d (0) ⊂ Rd×d denotes the ball centered at 0 with radius √ d. Therefore, with Vk as in (3.23) and recalling the construction of y˜ in Section 2.2 with uniformly bounded ∇˜ y we derive ˆ X ˆ 2 dist (∇˜ yk , B√ (0)) ≤ dist2 (∇˜ yk , SO(d)) + CC 2 εˆd #C¯0 int k

d

Vk

x ¯∈Fεˆ0 \C¯ε0ˆ k

≤

Qεˆk (¯ x)

εˆk

k

C εˆk εsk E(Uk , yk ) 64

+ C εˆk ≤ C εˆk → 0

(3.46)

for εk → 0. In the following we only consider the rst component w ˜k := y˜k1 of the deformations. For η > 0 we enlarge the set (0, λk ) × Wk and dene Wkη = ((−η, λk + η) × Wk ) ∪ Uk . We extend w˜k to Wkη by w˜k (x1 , x0 ) = w˜k (0, x0 ) + x1 e1 for −η < x1 ≤ 0 and w ˜k (x1 , x0 ) = w ˜k (λk , x0 ) + (x1 − λk )e1 for λk ≤ x1 ≤ λk + η . η We note that Jw˜k ∩ Wk ⊂ Uk by (3.15). Due to the boundary condition (3.14) there are points qk1 ∈ {0} × Wk and qk2 ∈ {λk } × Wk such that (3.47)

ε3−d , 1}Ccr ). |w˜k (qk1 ) − w˜k (qk2 )| ≥ λk (1 + max{ˆ

Due to Lemma 2.3.1(i) and condition (3.15) there is a constant C = C(D) such that ∗ sup{|w˜k (j, s) − w˜k (j, t)| : s, t ∈ Wk } ≤ C(D)(1 + εˆ3−d )Cint for j = 0, λk . Choosing Ccr suciently large this together with (3.47) shows

inf{|w˜k (p) − w˜k (q)| : p, q ∈ Wkη , p · e1 = 0, q · e1 = λk } ≥

λk max{ˆ ε3−d , 1}Ccr . 2

Let Ukη = (−η, λk + η) × (0, 1)d−1 . For M = λ02Ccr we now introduce the truncated function u ˜k : Ukη → R dened by  u˜k (x) := max min{(w˜k (x) − w˜k (0, x0 )), M }, −M on Wkη and zero elsewhere, where x0 = (x2 , . . . , xd ). Then it is not hard to see that u ˜k (0, x0 ) = 0, |˜ uk (λk , x0 )| = M for x0 ∈ Wk a.e. and thus

|˜ uk (x1 , x0 )| ≤ η, |˜ uk (λk − x1 , x0 )| ≥ M − η for x1 ∈ (−η, 0), x0 ∈ Wk a.e. (3.48) ∗ a.e. on Wkη . The Moreover, |∇˜ uk | ≤ |∇w˜k | + |∇x0 w˜k (0, x0 )| ≤ |∇w˜k | + CCint lattice deformation corresponding to u ˜k is denoted by uk , i.e. uk = u˜k |Lεˆk ,ρk (Uk ) . η Keeping in mind that Wk is open, it is elementary to see that by truncation of the function no further discontinuity points arise, i.e. Ju˜k ∩ Wkη ⊂ Jw˜k ∩ Uk . Moreover, by (3.7) we obtain

Hd−1 (Ju˜k \ Wkη ) ≤ 2(Dη + 1 − |Wk |).

(3.49)

We now show that we can nd a weakly converging subsequence of (˜ uk )k . To see this, we rst note that by (3.44) and (3.49) there is some C > 0 such that Hd−1 (Ju˜k ) ≤ Hd−1 (Jw˜k ) +Hd−1 (Ju˜k \Wkη ) ≤ C for all k ∈ N. Moreover k∇˜ uk k∞ ≤ ∗ ∗ CCint + k∇w˜k k∞ ≤ C(Cint + Cint ) and k˜ uk k∞ ≤ M for all k ∈ N by construction. Now applying Theorem A.1.1 we deduce that there is some u ∈ SBV (U η ) such that up to a subsequence (not relabeled) u ˜k → u in the sense of (A.4) and a.e., η d−1 where U = (−η, λ + η) × (0, 1) . By (3.48) for M large enough with respect to η the limit function satises

ess inf{|u(p) − u(q)| : p ∈ (−η, 0) × W, q ∈ (λ, λ + η) × W } ≥ 65

M . 2

(3.50)

∗ Note that the above compactness theorem implies k∇uk∞ ≤ C(Cint + Cint ). We now improve this bound by showing that k∇uk∞ ≤ T for some T√> 0 large ∗ enough independent of δ (recall that Cint may depend on δ ). Let R = d + CCint for some C > 0 suciently large. Then (3.46) yields ˆ ˆ 2 dist (∇˜ uk , BR (0)) ≤ dist2 (∇˜ yk , B√d (0)) ≤ C εˆk → 0, Vk

Vk

for εk → 0. Consequently, we get ˆ ˆ 2 dist (∇u, BR (0)) ≤ lim inf k→∞

U

dist2 (∇˜ uk , BR (0)) = 0,

Vk

where we used the convexity of dist2 (·, BR (0)). Therefore, |∇u| ≤ R a.e. in U and ∗ by the extension of y˜k to Wkη we get |∇u| ≤ CCint a.e. on U η \ U . Consequently, choosing T > 0 suciently large we obtain

|∇u| ≤ T a.e. in U η .

(3.51)

We now concern ourselves with the energy contribution of the broken cells C¯εˆ0 k . For all ν ∈ V we let y¯ν,k : Uk → Rd , w ¯ν,k : Uk → R and u¯ν,k : Ukη → R be the interpolations introduced in Section 2.2. Recalling the construction of the interpolations we obtain by (1.10) and (3.45) X ¯ k (¯ εsk E(Uk , yk ) ≥ εˆd−1 Wcell (∇y x)) k x ¯∈C¯ε0ˆ

≥

Xˆ ν∈V

where Γ(ν) =

k

Jy¯ν,k ∩Γ(ν)

εˆkd−1 ¯0 δ β(ν) dHd−1 − εˆd−1 k #Cεˆk ˜ , 1 d−1 ν H (∂ν Qεˆk ) C 2

∂ν Qνεˆk (λ). Then by (2.6) and (3.44) we get Xˆ β(ν) s |ν · ξy¯ν,k | dHd−1 − δ. εk E(Uk , yk ) ≥ det A ν∈V Jy¯ν,k

S

λ∈Zd

Applying (3.49) it is not hard to see that there is some Γk,ν with Hd−1 (Γk,ν ) ≤ C(Dη + 1 − |Wk |) such that Ju¯ν,k ⊂ Jw¯ν,k ∪ Γk,ν ⊂ Jy¯ν,k ∪ Γk,ν . Furthermore, the normals ξy¯ν,k and ξu¯ν,k coincide on Ju¯ν,k ∩ Jy¯ν,k . Therefore, we derive

δ + C(Dη + 1 − |Wk |) + εsk E(Uk , yk ) Xˆ β(ν) ≥ |ν · ξu¯ν,k | dHd−1 =: ES (Uk , uk ). det A ν∈V Ju¯ν,k With the notation introduced in Section A.1 we get using Theorem A.1.5 ˆ 1 X ES (Uk , uk ) ≥ #Ju¯ν,s β(ν) dHd−1 (s). ν,k det A ν∈V Πν 66

(3.52)

Then by the equiboundedness of ES (Uk , uk ) and Fatou's lemma we deduce that lim inf k→∞ #Ju¯ν,s < +∞ for a.e. s ∈ Πν and all ν ∈ V . As u¯ν,k and ∇¯ uν,k are ν,k ν,s uniformly bounded, by Theorem A.1.1 and Lemma 2.2.3 u ¯ν,k converges (up to a ν,s ν subsequence) to u in the sense of (A.4) for a.e. s ∈ Π . In particular, we get

lim inf #Juν,s ≥ #Juν,s . ν ν,k k→∞

Applying Fatou's lemma and the slicing theorem once more we then derive ˆ 1 X #Juν,s β(ν) dHd−1 (s) lim inf ES (Uk , uk ) ≥ k→∞ det A ν∈V Πν ˆ X (3.53) 1 d−1 β(ν)|ν · ξu | dH =: ES,lim (U, u). = det A Ju ν∈V By (1.14) and slicing in e1 -direction we get ˆ 1 ES,lim (U, u) ≥ βA |e1 · ξu | dHd−1 det A Ju ˆ 1 βA #Jue1 ,s dHd−1 (s). = det A (0,1)d−1 We now choose M = λ02Ccr suciently large (independently of δ ) such that M ≥ 4T λ. Then due to (3.50) and (3.51) it is not hard to see that #Jue1 ,s ≥ 1 for |βA a.e. s ∈ W and therefore ES,lim (U, u) ≥ |W . Letting η → 0 and choosing Cˆ det A suciently large we now conclude by (3.43), (3.52) and (3.53):

∞>

|W |βA ˆ − |W |) ≥ lim inf ES (Uk , yk ) − δ − 2δ ≥ lim inf εsk E(Uk , yk ) + C(1 k→∞ k→∞ det A |W |βA ≥ ES,lim (U, u) − δ ≥ − δ. det A

This gives the desired contradiction. To see the upper bound in (3.41) we choose ξ ∈ S d−1 such that (1.14) is minimized and dene the hyperplane Π = {x ∈ Rd : x · ξ = c} for a suitable c ∈ R such that Π ∩ U ⊂ {δ ≤ x1 ≤ λ − δ} for some δ > 0. We set ( x, x · ξ ≤ c, y(x) = (3.54) x + rλe1 , x · ξ > c. The energy corresponding to the deformation y is given by the bonds intersecting Π. These springs, associated to the lattice directions ν ∈ V , are elongated by a factor scaling with r/ˆ ε and yield a contribution β(ν) in the limit ε → 0 by (1.10).

67

As the projection in ν -direction onto the hyperplane {x · ξ = c} of the face ∂ν Qν has Hd−1 -volume 1 1 d−1 εˆd−1 det A |ν · νˆ| εˆd−1 det A ν ν = H (∂ν Q ) · νˆ ν = 2 |ν| |ν · νˆ| |ν · ξ| |ν · ξ| · ξ |ν| (see (2.6)) it is not hard so see that

 1  |ν · ξ| + O εˆd−1 det A|e1 · ξ| εˆd−2

(3.55)

springs in ν -direction are broken. This yields the energy  β  A ε−s + o(1) + O(ˆ ε2 ε−1 ), det A for ε → 0 as desired.



3.2.5 Estimates in a second intermediate regime We provide an additional lemma needed in the case d ≥ 4. ∗ ∗ Lemma 3.2.5. Let d ≥ 4, Cmed,2 > 0 and Cmed,1 > 0 suciently large. Then there is a constant C > 0 such that the minimization problem (3.16) satises

ˆ (Uˆ , r) ≥ Cεd−2+s(1−d) r M ∗ ∗ for Cmed,1 ≤ r ≤ Cmed,2 εˆ3−d as ε → 0. The constant C is independent of ρ ∈ ˜ ⊂ (0, 1)d−1 and λ ∈ [λ0 , 2λ0 ]. A[0, 1)d , W ∗ Proof. The superscript ˆ· is again dropped where no confusion arises. Let Cmed,2 > ∗ ˜ , λ and r with C ∗ ˆ3−d be given and consider a 0. Let ρ, W med,1 ≤ r ≤ Cmed,2 ε deformation y : Lεˆ,ρ (U ) → Rd satisfying (3.14) with respect to r, (3.15) and E(U, y) ≤ 2M (U, r). Due to (3.14) there is a q ∈ W such that |˜ y 1 (λ, q) − y˜1 (0, q)| ≥ λ(1 + r). Applying Lemma 2.3.1(ii) for t = λ(1+r) and (3.15) we nd ∗ 4CCint ˜ with Hd−1 (V ) ≥ c0 λ0 rε(1−s)(d−2) such that a set V ⊂ W

|˜ y (λ, q) − y˜(0, q)| ≥

∗ λ0 Cmed,1 λ(1 + r) ≥ 2 2

for all q ∈ V . Fix Cint as dened in Section 2.2 and S recall k∇˜ y k∞ ≤ CCint . ∗ Choose Cmed,1 large enough such that ((0, λ) × {q}) ∩ x¯∈C¯0 Qεˆ(¯ x) 6= ∅ for all ε ˆ d−1 q ∈ V . As the orthogonal projection of a cell onto {0} × R has Hd−1 -measure smaller than C εˆd−1 we deduce

#C¯εˆ0 ≥ Cεs−1 r 68

2

for some C > 0. As every broken cell provides at least the energy Cε− 3 = Cεd(1−s)−1 by (2.4) we derive

M (U, r) ≥ Cεd(1−s)−1 #C¯εˆ0 ≥ Cεd−2+s(1−d) r.  ˆ Proof of Theorem 3.2.1. We begin to construct such a function f : R×[λ0 , 2λ0 ] → ˆ (Uˆ , r). Let ω(|W |) = |W |(1 − C(1 − |W |)) with R for the rescaled problem M the constant C of Lemma 3.2.4. For δ > 0 small we set fˆ(r, λ) = −δλω(|W |) for r ≤ 0 and for C1 > 0 suciently large we dene   α r2 A −δ fˆ(r, λ) = ω(|W |)λ 2 det A ε √ for r ∈ [0, C1 √ε], λ ∈ [λ0 ,√2λ0 ]. Choose the ane function√g : R → R satisfying λω(|W |)g(C1 ε) = fˆ(C1 ε, λ) and λω(|W |)g 0 = ∂r fˆ(C1 ε, λ). For t > C1 C p suciently large we let h(r) = Cε− 2 rp − t for r ≥ Cmed,1 with Cmed,1 as in Lemma 3.2.3, so that there is a (unique) intersection point of the graphs of g √ 0 0 and h, (¯ rt , g(¯ rt )) = (¯ rt , h(¯ rt )), for which h√(¯ rt ) ≥ g . Note that r¯t ∼ ε. Then ˆ we set f (r, λ) = λω(|W |)g(r) for r ∈ [C1 ε, r¯t ] and fˆ(r, λ) = λω(|W |)h(r) for r ∈ [¯ rt , max{ˆ ε3−d , 1}C2 ] for C2 > 0 large enough. Finally, we let  β  A fˆ(r, λ) = ε−s ω(|W |) −δ det A ∗ for r ≥ max{ˆ ε3−d , 1}C2 . In the case d ≥ 4 we observe that for Cmed,1 ≤ r ≤ 3−d C2 εˆ we have p

Cω(|W |)λε− 2 rp ≤ Cεd−2+s(1−d) r ≤ Cε−s εˆ ≤ Cε−s for some p > 1 small enough. Consequently, by Lemmas 3.2.2, 3.2.3, 3.2.4, 3.2.5 it is not hard to see that fˆ is convex for r ≤ max{ˆ ε3−d , 1}C2 and satises ˆ (Uˆ , ·) for ε small enough independently of ρ ∈ A[0, 1)d and W . Moreover, fˆ ≤ M we obtain 1 ˆ (Uˆ , r) ≤ fˆ(r, λ) + 4λ0 δ fˆ(r, λ) ≤ M ω(|W |) √ for r ∈ [0, C1 ε] and

ˆ (Uˆ , r) ≤ fˆ(r, λ) ≤ M

1 fˆ(r, λ) + 2ε−s δ ω(|W |)

for r ≥ max{ˆ ε3−d , 1}C2 . To nish the proof it suces to recall M (Uε , r) = sd ˆ ˆ ε M (U , r) by (3.17) and to set f (r, λ) = εsd fˆ(r, ε−s λ) for all r ∈ R and λ ∈ εs [λ0 , 2λ0 ].  69

3.3 Proof of the cleavage law We are now in a position to prove the main theorem about the limiting minimal energy. Proof of Theorem 1.3.1. Let y ∈ A(aε ). We partition (0, l2 ) × . . . × (0, ld ) up to a set of size O(εs ) with sets Vi , i ∈ I , which are translates of the cube εs (0, 1)d−1 . ∗ > 0 we denote the set Furthermore, we set Vid = (0, l1 ) × Vi for all i ∈ I . For Cint 0 ¯ of broken cells by Bε as dened at the beginning of Section 3.2. We let

n x) ⊂ Vid } > I¯ := i ∈ I : #{¯ x ∈ B¯ε0 : Qε (¯

o 2βA ε(1−s)(d−1) C¯ det A

∗ ¯ int ) as in (2.4). Then for i ∈ I¯ we estimate with C¯ = C(C

εd−1

X

¯ x)) ≥ εd−1 Wcell (∇y(¯

x ¯∈(L0ε (Vid ))◦

s(d−1) 2βA 2βA ¯=ε . C (1−s)(d−1) det A ε C¯ det A

(3.56)

√ Now consider some Vi for i ∈ I \ I¯. For λ0 ≥ L( AT A, Wcell , 1, . . . , 1) we partition Vid into sets of the form U1 = (u0 , u1 ) × Vi , . . . , Un = (un−1 , un ) × Vi , where u0 = lA ε, un = l1 − lA ε and uj − uj−1 ∈ εs [λ0 , 2λ0 ] for all j = 1, . . . , n. This can and will be done so that N (uj ) := #T (uj ) = min #T (¯ u), u ¯∈J(uj )

(3.57)

for all j = 1, . . . , n − 1, where

T (s) = {¯ x ∈ B¯ε0 : Qε (¯ x) ⊂ Vid and Qε (¯ x) ∩ ({s} × Vi ) 6= ∅} s

s

and J(uj ) = [uj − ε 2λ0 , uj ] or [uj , uj + ε 2λ0 ]. Moreover, we have N (u0 ) = N (un ) = 0 due to the boundary conditions (1.12). We now show that n X j=0

N (uj ) ≤

Cε(s−1)(d−2) . λ0

(3.58)

We cover J(uj ) × Vi with translates of (0, εlA ) × (0, εs )d−1 , where lA is as dened in (1.11). As every cell is contained in at most two of these translates we derive

j λ εs k 0 #{¯ x ∈ B¯ε0 : Qε (¯ x) ⊂ Vid and Qε (¯ x) ∩ (J(uj ) × Vi ) 6= ∅} ≥ N (uj ) 4lA ε for j = 1, . . . , n − 1 due to the construction (3.57). Summing over j , we nd n X j=0

N (uj ) ≤

Cε Cε1−s ε(1−s)(1−d) Cε(s−1)(d−2) d ¯0 : Qε (¯ #{¯ x ∈ B x ) ⊂ V } ≤ = ε i εs λ0 λ0 λ0 70

since i ∈ I \ I¯. Note that the estimate (3.58) relies only on the fact that i ∈ I \ I¯ but is independent ε. Sn−1 Sof the particular set Vi , the deformation y andd−1 Let Ti = j=1 x¯∈T (uj ) Qε (¯ x) and Si = π1 Ti , where π1 Ti ⊂ R denotes the set which arises from Ti by orthogonal projection onto {0} × Vi and cancellation of the rst component. Using (3.58) we nd

H

d−2

(∂Si ) ≤

n−1 X

N (uj )H

d−2

d−2

(∂ π1 Qε ) ≤ Cε

n−1 X

s(d−2) N (uj ) ≤ Cλ−1 . 0 ε

j=1

j=1

Choose λ0 so large that Hd−2 (∂Si ) ≤ δεs(d−2) . Let Vi,ε = {x ∈ Vi : dist(x, ∂Vi ) ≥ Cε} with C so big that π1 Qε (¯ x) ∩ Vi,ε = ∅ whenever Qε (¯ x) 6⊂ Vid . By the isoperimetric inequality we deduce that there is a unique connected component ˜ i of Vi,ε \ Si satisfying |W ˜ i | := Hd−1 (W ˜ i ) ≥ (1 − Cε1−s − Cδ d−1 d−2 )εs(d−1) , where C W ˜ i) ≤ is a constant only depending on the dimension. Moreover, we have Hd−2 (∂ W s(d−2) ˜ and so we see that for δ small enough Wi is of the form (3.7) (after Cε rescaling by ε−s ). Furthermore, by a similar argument (e.g. by enlarging the cubes which form Ti ) we nd that
˜ i : dist(x, ∂ W ˜ i ) = D0 ε and dist(x, ∂Vi,ε ) 6= D0 ε} ≤ Cδεs(d−2) . Hd−2 {x ∈ W

˜ i as described in (3.7) (replacing Consequently, we dene Wi corresponding to W ε1−s by ε due to the dierent scaling) and obtain |Wi | := Hd−1 (Wi ) ≥ (1−Cε1−s − d−1 Cδ d−2 )εs(d−1) and Hd−2 (∂Wi ) ≤ Cεs(d−2) for some possibly larger constant C . Clearly, Wi is of the form (3.7). The sets Uj dened above correspond to Uε ˜ i satisfy considered in Section 3.2 up to a translation. In particular, the sets W condition (3.10) due to the construction of Si . We dene ˆ   1 1 0 1 0 − y˜ (uj , x ) − y˜ (uj−1 , x ) dx0 rj := −1 + uj − uj−1 Wi for j = 1, . . . , n. Note that this denition is meaningful as y˜1 is dened on all of (0, l1 ) × Wi (see Section 2.2). As y ∈ A(aε ) it is not hard to see that n X j=1

ˆ   1 0 1 0 (uj − uj−1 ) rj = −(l1 − 2lA ε) + − y˜ (l1 − lA ε, x ) − y˜ (lA ε, x ) dx0 Wi

= −(l1 − 2lA ε) + l1 − 2lA ε + (l1 − 2lA ε)aε = (l1 − 2lA ε)aε . We dene Wid = (0, l1 ) × Wi and the energy X ¯ x)). Eεi (y) := εd−1 Wcell (∇y(¯ x ¯∈(L0ε (Vid ))◦

71

√ For C1 ≥ 2acrit ε, C2 > 0 suciently large and for δ > 0 as before choose f as in Theorem 3.2.1. Then for ε small enough Eεi (y)

≥ε

d−1

n  X j=1

≥

n X

X x ¯∈(L0ε (Uj ))

M (Uj , rj ) ≥

j=1

¯ x)) + 1 Wcell (∇y(¯ 2 ◦

n X

X

 ¯ x)) Wcell (∇y(¯

x ¯∈∂Wi (L0ε (Uj ))

f (rj , uj − uj−1 ).

j=1

Here we observe that due to the construction of the sets Wi we have ∂Wi (L0ε (Uj )) ⊂ (L0ε (Vid ))◦ for all j = 1, . . . , n, i ∈ I . If there is
some j such that rj ≥ βA ˜ (|Wi |) det C2 max{1, ε(s−1)(3−d) } then Eεi (y) ≥ ω − δ by (3.12), where ω ˜ (·) = A s(d−1) −s(d−1) s(d−1) ε ω(ε ·) (note that now |Wi | ∼ ε ). Otherwise all rj lie in the regime, where f is convex in r and linear in λ. We then compute using Jensen's inequality n X uj − uj−1 f (rj , λ0 εs ) ≥ f (rj , uj − uj−1 ) = s λ0 ε j=1 j=1 Pn  Pn  j=1 (uj − uj−1 ) rj j=1 uj − uj−1 s Pn ≥ f , λ0 ε λ 0 εs j=1 uj − uj−1  l1 − 2lA ε  s f a , λ ε , = ε 0 λ 0 εs √ whence for aε ≥ 2acrit ε, due to the monotonicity of f , also  √ l1 − 2lA ε  s Eεi (y) ≥ f 2a ε, λ ε crit 0 λ0 εs   β   (l − 2l ε)α 4a2 A 1 A A crit − (l1 − 2lA ε)δ ≥ ω ˜ (|Wi |) −δ =ω ˜ (|Wi |) 2 det A det A

Eεi (y)

n X

by (3.11), where the last√inequality holds for δ small enough. Repeating the calculation for aε ≤ 2acrit ε and using (3.11) yields n (l − 2l ε)α a2 o βA 1 A A ε Eεi (y) ≥ ω − (l1 − 2lA ε)δ, −δ . ˜ (|Wi |)mε := ω ˜ (|Wi |) min 2ε det A det A P Using that Eε (y) ≥ i∈I Eεi (y) and ω ˜ (|Wi |) ≥ σ(δ)εs(d−1) , where σ(δ) = min{ω(s) : d−1

1−Cδ d−2 ≤ s ≤ 1} ≤ 1 for all i ∈ I and recalling (3.56) we get for δ small enough   εs(d−1) 2βA ¯ σ(δ)εs(d−1) mε lim inf inf{Eε (y) : y ∈ A(aε )} ≥ lim inf #I¯ + #(I \ I) ε→0 ε→0 det A ≥ lim inf #I σ(δ)εs(d−1) mε ε→0

≥ σ(δ)

d Y

lj min

j=2

72

n l α a2 o βA 1 A − l1 δ, −δ , 2 det A det A

√ as aε / ε → a. Letting δ → 0 shows Qd lim inf inf{Eε (y) : y ∈ A(aε )} ≥

j=2 lj

min

n1

o l1 αA a2 , βA .

det A 2 Here we used that limδ→1 σ(δ) = 1. It remains to prove that the right hand side in Theorem 1.3.1 is attained for some sequence of deformations. This essentially follows from the sharpness of the estimates (3.11) and (3.12). In particular, as in (3.32) for a < ∞ we consider y el (x) = x + F¯ (aε ) x, x ∈ Lε ∩ Ω, (3.59) ε→0

ε

and as in the proof of Lemma 3.2.2 it is not hard to see that

lim Eε (yεel ) =

ε→0

d Y j=1

d

lj

 a 2 Y αA αA ε lim a2 . = lj 2 det A ε→0 ε 2 det A j=1

For we proceed as in (3.54): We choose ξ such that (1.14) is satised. Due to the assumption l1 ≥ L it is possible to dene a hyperplane Π = {x ∈ Rd : x·ξ = c} such that Π ∩ Ω ⊂ Ω \ (B1ε ∪ B2ε ). We let ( x, x · ξ < c, yεcr (x) = x ∈ Lε ∩ Ω. (3.60) x + l1 aε e1 , x · ξ > c,

yεcr

Again counting the quantity of broken springs as in (3.55) we derive limε→0 Eε (yεcr ) = d Q βA lj det .  A j=2

3.4 Examples: mass-spring models In the following we examine several mass-spring models to which the above results apply. We calculate the constants αA , βA explicitly and thus we can provide the limiting energy of Theorem 1.3.1 as well as the critical value of boundary displacements acrit. Moreover, we specify minimizing congurations and discuss their behavior depending on the properties of the cell energy. Note that the cell energies under consideration which consist of pair interac¯ . Similarly as in Section 1.5 we have tion energies are typically minimized on O(d) to introduce a frame indierent penalty term to avoid unphysical behavior and to satisfy Assumption 1.1.2. Choose some χ ≥ 0 which vanishes in a neighborhood ¯ ¯ \ SO(d) ¯ of SO(d) and ∞ and satises χ ≥ cχ > 0 in a neighborhood of O(d) . For example, as in (1.26) we may set ( ¯ x)| ≥ R y ) > 0 a.e. on Q(¯ x) or |∇y(¯ ¯ x)) = 0, if det(∇˜ χ(∇y(¯ ∞ otherwise for some R  1. The penalty term does not change the energy in the elastic and fracture regime. 73

3.4.1 Triangular lattices with NN interaction We rst concern ourselves with the triangular lattice. Although the cleavage law was presented in detail in Section 1.4, we include this model here for the sake of completeness and briey indicate that it ts into framework of Section 1.1.  the 1  1 √2 Recall Ω = (0, l) × (0, 1) and L = AZ2 = Tφ Z2 for φ ∈ [0, π3 ), where 3 0 2   cos φ − sin φ Tφ = . For a deformation y : Lε ∩ Ω → R2 let sin φ cos φ

Eε (y) =

 |y(x) − y(x0 )|  ε X W +ε 2 x,x0 ∈L ε ε |x−x0 |=ε

X

¯ x)), χ(∇y(¯

x ¯∈(L0ε (Ω))◦

where W : [0, ∞) → [0, ∞) satises the assumptions (i), (ii) and (iii) for α := W 00 (1) > 0 and lim > 0. Denoting the i-th column of G by Gi and  r→∞ W (r) = β  −1 1 1 −1 letting Z = A 21 the cell energy can be written as −1 −1 1 1

Wcell (G) =

1 W (|G2 − G1 |) + W (|G3 − G2 |) + W (|G4 − G3 |) 2
+ W (|G1 − G4 |) + 2W (|G4 − G2 |) + χ(G) .

Note that in Lemma 2.4.1 and Lemma 2.4.2 we have shown that this is an admissible cell energy in the sense of Assumption 1.1.2. We compute   3 1 0 3α  1 3 0 Q= 8 0 0 2 and therefore αA = α . √ 3 T 1 Tφ (− 2 , 2 ) we get

With ν1φ = Tφ (1, 0)T , ν2φ = Tφ ( 12 ,

√

3 T ) 2

and ν3φ =

√ |νiφ · ς| 3β = |e1 · ς| sin(φ + π3 )

P3 βA = min1 β ς∈S

i=1

and then we re-derive

√ n1 2 3β o 2 Elim (a) = √ min lαa , . 2 sin(φ + π3 ) 3

3.4.2 Square lattices with NN and NNN interaction The behavior in the elastic regime of the following two dimensional model comprising nearest and next to nearest neighbor atomic interactions was treated by 74

Friesecke and Theil in [50]. We let Ω = (0, l1 ) × (0, l2 ), set L = AZ2 = Tφ Z2 for φ ∈ [0, π2 ) and

Eε (y) =

 |y(x) − y(x0 )|  ε X W1 2 x,x0 ∈L ε ε |x−x0 |=ε

+

ε 2

X

W2

x,x0 ∈L√ ε |x−x0 |= 2ε

 |y(x) − y(x0 )|  X ¯ x)) √ +ε χ(∇y(¯ 2ε x ¯∈(L0 (Ω))◦ ε

for deformations y : Lε ∩ Ω → R2 and potentials W1 , W2 as above with α1 , β1 and α2 , β2 , respectively. The associated cell energy is given by  |G − G |  1 X 1 X i √ j + χ(G) Wcell (G) = W2 W1 (|Gi − Gj |) + 4 2 √ 2 |z −z |=1 i

|zi −zj |= 2

j

In [50] it is shown that the cell energy is admissible. We calculate   2α1 + α2 α2 0 1 2α1 + α2 0  Tφ∗ , Q = Tφ∗T  α2 2 0 0 2α2 with

√  c2 s2 −√ 2cs Tφ∗ = √s2 c2 2cs  , √ 2 2cs − 2cs c − s2 

c := cos φ, s := sin φ.

An elementary computation then shows

αA =

2α1 α2 (α1 + α2 ) . 2α1 α2 + 4α12 c2 s2 + α22 (c2 − s2 )2

Letting ν1φ = Tφ e1 , ν2φ = Tφ e2 , ν3φ = Tφ (e1 + e2 ) and ν4φ = Tφ (e1 − e2 ) and γ1 = max{c, s}, γ2 = c + s we obtain for the fracture constant

β¯1 (|ν1φ · ς| + |ν2φ · ς|) + β¯2 (|ν3φ · ς| + |ν4φ · ς|) βA = min1 ς∈S |e1 · ς| n β + 2β 2β + 2β o 1 2 1 2 = min , . γ1 γ2 ς Here we used that it suces to minimize over the set P = { |ς| : ς = νiφ , i = 1, . . . , 4} ⊂ S 1 . Below the critical value acrit energetically optimal congurations are given by functions of the form (3.59) with ! aε 0 F¯ (aε ) = . (α21 −α21 )cs(c2 −s2 ) −α22 (c2 −s2 )2 −4α21 c2 s2 a a 2α1 α2 +α2 (c2 −s2 )2 +4α2 c2 s2 ε 2α1 α2 +α2 (c2 −s2 )2 +4α2 c2 s2 ε 2

1

2

75

1

In particular, the congurations show the Poisson-eect and in the case that α1 6= α2 and φ ∈ (0, π2 ) \ { π4 } also shear eects occur. Limiting minimal congurations beyond critical loading are given by deformations of the form (3.60), where the normal ξ to the hyperplane Π is an element of {νiφ : i = 1, . . . 4}. While in the previous example the cleavage direction was determined only by the geometry of the problem (i.e. by φ), it now depends also on the ratio of β1 , β2 . We note that here for every φ ∈ (0, π2 ) \ { π4 } by choosing the specic values β1 = 1 and β2 = 12 max{cot φ, tan φ} − 12 the minimum in the expression for βA is attained at ν2φ and ν3φ , respectively, at ν1φ and ν4φ . As a consequence, unlike for the triangular lattice in the previous example, also for general lattice orientations there may be deformations with almost optimal energy whose rescaled displacements in the continuum limit have a serrated jump set.

3.4.3 Cubic lattices with NN and NNN interaction We consider the following three dimensional model with nearest and next nearest interactions in the domain Ω = (0, l1 )×(0, l2 )×(0, l3 ). We let L = AZ3 = Tφ,ψ Z3 , where    cos ψ − sin ψ 0 1 0 0 π Tφ,ψ =  sin ψ cos ψ 0 0 cos φ − sin φ for φ, ψ ∈ [0, ). 2 0 0 1 0 sin φ cos φ We let

Eε (y) =

 |y(x) − y(x0 )|  ε2 X W1 2 x,x0 ∈L ε ε |x−x0 |=ε

+

ε2 2

X x,x0 ∈L√ ε |x−x0 |= 2ε

W2

 |y(x) − y(x0 )|  X ¯ x)) √ + ε2 χ(∇y(¯ 2ε 0 ◦ x ¯∈(L (Ω)) ε

for deformations y : Lε ∩ Ω → R2 and potentials W1 , W2 as above with α1 , β1 and α2 , β2 , respectively. The associated cell energy is given by  |G − G |  1 X 1 X i √ j + χ(G). W1 (|Gi − Gj |) + Wcell (G) = W2 8 4 √ 2 |zi −zj |=1

|zi −zj |= 2

In [63] it has been shown that the cell energy is admissible. As mentary computation shows  2α1 + 2α2 α2 α2 0 0  α 2α + 2α α 0 0 2 1 2 2   1 α2 α2 2α1 + 2α2 0 0 Q = Tψ∗T Tφ∗T   0 0 0 2α2 0 2  0 0 0 0 2α2 0 0 0 0 0 76

before, an ele-

 0 0   0   Tφ∗ Tψ∗ , 0   0  2α2

where 

Tφ∗ =

1 0  0  0  0 0

0 c21 s21 0 0 c1 s1

0 s21 c21 0 0 −c1 s1

0 0 0 c1 s1 0

0 0 0 −s1 c1 0

 0 −2c1 s1   2c1 s1  , T ∗ ψ 0   0  c21 − s21

c22  s22   0  c2 s2   0 0 

=

s22 c22 0 −c2 s2 0 0

0 −2c2 s2 0 2c2 s2 1 0 0 c22 − s22 0 0 0 0

0 0 0 0 c2 s2

 0 0   0  , 0   −s2  c2

with the abbreviations c1 = cos φ, c2 = cos ψ , s1 = sin φ and s2 = sin ψ . Applying Lemma 2.1.2 we then obtain

αA =

α2 (2α1 + α2 )2 (α1 + 2α2 ) . 8α13 c22 s22 + 2α1 α22 (4 − c22 s22 ) + 4α12 α2 (4c22 s22 − 1) + α23 (3 − 4c22 s22 )

In particular, αA is independent of c1 and s1 . We let V1φ,ψ = Tφ,ψ {e1 , e2 , e3 }, V2φ,ψ = Tφ,ψ {e1 + e2 , e1 − e2 , e1 + e3 , e1 − e3 , e2 + e3 , e2 − e3 } as well as γ1 = max{|c2 |, |c1 s2 |, |s1 s2 |}, γ2 = max{|c2 ± c1 s2 |, |c2 ± s1 s2 |, |c1 s2 ± s1 s2 |}, γ3 = max{|c2 ± c1 s2 ± s1 s2 |}, γ4 = max{|2c2 ± c1 s2 ± s1 s2 |, |c2 ± 2c1 s2 ± s1 s2 |, |c2 ± c1 s2 ± 2s1 s2 |}. One can show that

P = {ς/|ς| : ς = Tφ,ψ e1 , i = 1, 2, 3} ∪ {ς/|ς| : ς = Tφ,ψ (e1 ± e2 ± e3 )} ∪ {ς/|ς| : ς = Tφ,ψ (e1 ± e2 ± e3 ± ei ), i = 1, 2, 3}. Then

P βA = min1

ν∈V1φ,ψ

β1 |ν · ς| +

P

ν∈V2φ,ψ

β2 |ν · ς|

|e1 · ς| n β + 4β 2β + 6β 3β + 6β 4β + 10β o 1 2 1 2 1 2 1 2 , , , = min . γ1 γ2 γ3 γ4 ς∈S

77

Chapter 4 Limiting minimal energy congurations This chapter is devoted to the proofs of Theorem 1.5.1 and Corollary 1.5.2. First, in Section 4.1 we derive a ne estimate on the limiting minimal energy (see Theorem 1.4.2). This will be an essential ingredient to analyze the number and position of broken triangles in more detail (see Section 4.2). Finally, the main convergence results for almost minimizers are addressed in Section 4.3.

4.1 Fine estimates on the limiting minimal energy We rst prove Theorem 1.4.2 about the sharp estimate on the discrete minimal energies. Assume that W in addition satises assumptions (ii'), (iii') and recall (1) (2) the denition of Cε and Cε in Section 3.1. In order to investigate a deformation (1) y again we let C¯ε and C¯ε denote the set of triangles 4 (of type one respectively) for which at least one side in y(4) is larger than 2Rε, where now the threshold value R > 1 is chosen in such a way that cR := inf{W (r) : r ≥ R} ≥ β2 . According to Lemma 2.4.5(iii) we may choose a convex function V such that ˜ (r) ≤ V (r) + O((r − 1)4 ) for r ≤ 8R. 0 ≤ V (r) ≤ W (4.1) As in (3.1) we observe that |eT1 (y)4 e1 | is bounded by 8R on triangles with bond length not exceeding 4Rε and thus lies in the convex regime of ´ V . Moreover, we nd that every triangle in C¯ε provides at least the energy √43ε 4 W4 (∇˜ y ) ≥ cR ε . a−η For given 0 < η < a we also dene Rε,η = √ε as a threshold for triangles we consider `essentially broken':  C¯ε,η = 4 ∈ C¯ε , |∇yε v| > Rε,η for at least two v ∈ V . (4.2) The minimal energy contribution of all the springs on such a triangle in C¯ε,η is given by   a−η η 2β ε := 2 inf W (r) : r ≥ √ ε = (2β + O(ε))ε ε 79

by the assumption (iii') on W . By I ⊂ (ε, 1 − ε) we denote the set of points x2 for which the segment (0, l) × {x2 } intersects a broken triangle (of type one) in (1) C¯ε . In addition, we say x2 ∈ I η ⊂ I if one of the intersected triangles lies in (1) C¯ε,η ∩ C¯ε . With these preparations we can now proceed to prove Theorem 1.4.2: Proof of Theorem 1.4.2. Let Eε (y) = inf Eε + O(ε). Inspired by (3.2) and (3.3) we establish a lower bound for the energies additionally taking the set I \ I η into account. Since the sidelength of any triangle whose interior intersects (0, l) × (I \ I η ) is bounded by 4Rε,η , we nd

|eT1 ∇˜ y (x1 , x2 ) e1 | ≤ 8Rε,η for all (x1 , x2 ) ∈ (0, l) × (I \ I η ) as in (3.1). Let k = k(x2 ) count the number of triangles in C¯ε on the S slice (0, l) × {x2 }, x2 ∈ I \ I η , and dene C¯εx2 ⊂ (0, l) such that ((0, l) × {x2 }) ∩ 4∈C¯ε 4 = C¯εx2 × {x2 }. Then ˆ eT1 ∇˜ y (x1 , x2 ) e1 dx1 ≤ 8kεRε,η . (4.3) x C¯ε 2

and so

ˆ

√ eT1 ∇˜ y (x1 , x2 ) e1 ≥ (1 + εa)(l + O(ε)) − 8kεRε,η x x Ωε 2 \C¯ε 2    √ √ 8k(a − η) + O( ε) = 1+ ε a− l. l ´ 1 k(x2 ) dx2 , a convexity argument as in the proof of Since #(C¯ε \ C¯ε,η ) ≥ εγ I\I η Theorem 1.4.1
on slices (0, l) × {x2 } with x2 ∈ (ε, 1 − ε) \ I and on the unbroken part Ωεx2 \ C¯εx2 × {x2 } of slices with x2 in I \ I η then shows that Eε (y) ≥

√ 4(l − 2ε) 2β η η √ V (1 + εa)(1 − 2ε − |I|) + Gη,ε |I \ I η | + |I | + O(ε), γ 3ε (4.4)

where

 Gη,ε = min k∈N

4l √ V 3ε

    √ √ 8k(a − η) kcR . 1+ ε a− + O( ε) + l γ

We note that this minimum exists and can be taken over 1 ≤ k ≤ K0 for some K0 ∈ N large enough and independent of η as kcγR → ∞ for k → ∞. We choose 0 < η < a large enough such that !  2 lα 2 αl 8k kcR √ a < min √ a − (a − η) + . 1≤k≤K0 l γ 3 3 80

√ ˜ (1 + √εr) + Recalling that, by (4.1) and Lemma 2.4.3, √4l3ε V (1 + εr) = √4l3ε W O(ε) → √lα3 r2 uniformly in r on bounded sets in R, we see that thus Gη,ε exceeds √ the elastic term √4l3ε V (1 + εa) for ε suciently small. So from (4.4) we obtain √ 4l 2β η η |I | + O(ε). (4.5) Eε (y) ≥ √ V (1 + εa)(1 − 2ε − |I η |) + γ 3ε √ As √4l3ε V (1 + εa) → √lα3 a2 and β η → β for all η > 0, for ε small enough we √ thus obtain by minimizing over |I η | ∈ [0, 1] that inf Eε ≥ √43 εl V (1 + εa)(1 − ˜ (1 + √εa) + O(ε) or inf Eε ≥ 2β+O(ε) (1 − 2ε) = 2β + O(ε), 2ε) + O(ε) = √43 εl W γ γ respectively, depending on a. Applying (3.5) and (3.6) we then get indeed √ 4 l ˜ inf Eε = √ W (1 + εa) + O(ε) 3ε

or

inf Eε =

respectively. The claim now follows from Lemma 2.4.4.

2β + O(ε), γ

(4.6)



Remark 4.1.1.

From the proof of Theorem 1.4.2, especially taking (3.5) and (3.6) into account, it follows that Theorem 1.4.2 still holds if Eε is replaced by Eεχ .

4.2 Sharp estimates on the number of the broken triangles √ Throughout this section we will assume that aε = εa, yε is a sequence of deformations satisfying (1.28) and uε = √1ε (yε −id) are the corresponding rescaled displacements. Moreover, we suppose that the threshold value R is chosen as above Equation (4.1) implying cR ≥ β2 and that C¯ε is dened accordingly. For a rescaled displacement u ˜ we denote by Dµ ⊂ (ε, 1 − ε) for µ > 0 the (1) set of x2 such that there is precisely one triangle 4x2 ∈ C¯ε with int(4x2 ) ∩ ((0, l) × {x2 }) 6= ∅ and ˆ u(x1 , x2 )e1 dx1 ≤ lµ. (4.7) eT1 ∇˜ x

x

Ωε 2 \C¯ε 2

(for the denition of C¯εx2 recall (4.3).) Note that Dµ ⊂ I η for µ small enough: For x2 ∈ Dµ we have ˆ √ eT1 ∇˜ y (x1 , x2 )e1 dx1 ≥ εl(a − µ) + O(ε) x C¯ε 2

and using the arguments in (3.1) we see that for given η (not too small) we can choose µ small enough such that 4x2 ∈ C¯ε,η and thus x2 ∈ I η . We also dene 81

µ C¯ε,η ⊂ C¯ε,η as the set of those essentially broken triangles 4 for which there exists some x2 ∈ Dµ such that int (4) ∩ ((0, l) × {x2 }) 6= ∅. The projection of a triangle √ 3 ⊥ 4 onto the linear subspace spanned by vγ is an interval of length 2 ε. We denote the center of this interval by m4 . The following lemmas give sharp estimates on the number of broken triangles and their position.

Lemma 4.2.1. Let a < acrit and suppose u˜ε is a minimizing sequence satisfying Eε (id +

√

εuε ) = inf Eε + O(ε).

Then ε#C¯ε = O(ε). Proof. Using (4.4), (4.6) and Lemma 2.4.5(iii) we nd √ 4l ˜ Eε (yε ) = √ W (1 + εa) + O(ε) 3ε   √ 4(l − 2ε) ˜ 2β η W (1 + εa)(1 − 2ε − |I|) + min Gη,ε , ≥ √ |I| + O(ε) γ 3ε   √ 2β η 4l ˜ = √ W (1 + εa)(1 − |I|) + min Gη,ε , |I| + O(ε). γ 3ε An elementary computation yields, whenever ε is small enough,   −1  √ 2β η 4l ˜ |I| ≤ min Gη,ε , · O(ε) − √ W (1 + εa) γ 3ε    −1 αl 2 2β − √ a + o(1) · O(ε) = O(ε). = min Gη,ε , γ 3 (The argument leading to (4.5) together with a < acrit shows that the term in parentheses is bounded from below by √ a positive constant independent of ε). 4l ˜ √ Then the elastic energy is 3ε W (1 + εa) + O(ε) and consequently, the crack energy coming from triangles in C¯ε is of order O(ε). As every broken triangle in (1) C¯ε provides at least energy εcR we conclude ε#C¯ε = O(ε). But then, possibly (2) after replacing R by 2R, also ε#C¯ε = O(ε) as those triangles are neighbors of broken triangles of type 1. 

Lemma 4.2.2. Let a > acrit , φ 6= 0 and suppose u˜ε is a minimizing sequence satisfying √ Eε (id +

εuε ) = inf Eε + O(ε).

Then |I η | =
1 − O(ε) for 0 < η < a. Furthermore, for µ suciently small, µ ε# C¯ε \ C¯ε,η = O(ε) and  µ sup |m41 − m42 | : 41 , 42 ∈ C¯ε,η = O(ε). 82

Proof. Without loss of generality we choose η suciently large such that by (4.5) and (4.6) we get

Eε (yε ) =

√ 2β 4l ˜ 2β η η + O(ε) ≥ √ W |I |. (1 + εa)(1 − 2ε − |I η |) + γ γ 3ε

So for ε small enough we obtain  −1 αl 2 2β η 1 − |I | ≤ √ a + o(1) − · O(ε) = O(ε) γ 3 since a > acrit . Consequently, the crack energy from triangles in C¯ε,η is given by 2β + O(ε) and thus the energy contribution from C¯ε \ C¯ε,η is of order O(ε). γ
As in the proof of Lemma 4.2.1 we nd ε# C¯ε \ C¯ε,η = O(ε). Let kη (x2 ) and (1) (1) kηC (x2 ) count the number of triangles in C¯ε,η ∩ C¯ε and (C¯ε \ C¯ε,η ) ∩ C¯ε intersected by (0, l) × {x2 }, respectively. We dissect I η \ Dµ into two disjoint sets: By (1) D1 ⊂ I η \ Dµ we denote the set where we nd more than one triangle 4x2 ∈ C¯ε with int(4x2 ) ∩ ((0, l) × {x2 }) 6= ∅. The complement D2 is the set where (4.7) does not hold. Using a convexity argument for x2 ∈ D2 we obtain


2β + O(ε) ≥ 2βε(#C¯ε,η ∩ C¯ε(1) ) + 2cR ε# (C¯ε \ C¯ε,η ) ∩ C¯ε(1) γ ˆ 1−ε ˆ 4 +√ W4 (∇˜ y ) dx1 dx2 x ¯εx2 3ε ε Ωε 2 \C ˆ ˆ   αl 2cR 2β 1−ε kη (x2 ) dx2 + kηC (x2 ) dx2 + √ µ2 + o(1) |D2 | ≥ γ ε γ D1 3   2β η 2cR αl ≥ |I | + |D1 | + √ µ2 + o(1) |D2 | γ γ 3   2β η 2cR αl 2 ≥ |I | + min , √ µ + o(1) |I η \ Dµ |. γ γ 3 It follows |I η \ Dµ | = O(ε) and |Dµ | = 1 − O(ε), whence the crack
energy from 2β µ µ ¯ ¯ ¯ triangles in Cε,η is given by γ + O(ε) and then also ε# Cε \ Cε,η = O(ε). µ Finally, we concern ourselves with the projected distance of triangles in C¯ε,η . We rst note that it suces to show  µ sup |m41 − m42 | : 41 , 42 ∈ C¯ε,η ∩ C¯ε(1) = O(ε) (2) µ ˜ ∈ C¯µ ∩ C¯ε(1) with since for a suitable η˜ ≥ η for any 4 ∈ C¯ε,η ∩ C¯ε there is a 4 ε,˜ η |m4 − m4˜ | ≤ ε. Let x2 , z2 ∈ Dµ , x2 < z2 with z2 − x2 ≤ Cε and |m41 − m42 | > 0 (1) for the corresponding broken triangles 41 , 42 ∈ C¯ε . We may assume if a triangle intersects (0, l) × {z2 } or (0, l) × {x2 } then its interior does so, too. Denote by d¯ = γ −1 |m41 −m42 | the distances of the centers in vγ -projection onto the x1 -axis.

83

Let x1 , z1 ∈ (0, l) be the points on the slices (0, l) × {x2 } and (0, l) × {z2 } satisfying πvγ⊥ (x1 , x2 ) = m41 and πvγ⊥ (z1 , z2 ) = m42 , respectively, where πvγ⊥ denotes the orthogonal projection onto the linear subspace spanned by vγ⊥ . Let w = e1 · vγ |x2 − z2 |/γ . Then the vγ -projection of z = (z1 , z2 ) onto the x2 -slice is given by (˜ z1 , x2 ) with z˜1 = z1 − w. Then d¯ = |x1 − z˜1 | and without restriction we may assume x ˜1 . √1 > z 3ε Let sε = 4γ . We now consider the area bounded by the parallelogram with corners (˜ z1 + sε , x2 ), (x1 − sε ,√x2 ), (z1 + d¯ − sε , z2 ), (z1 + sε , z2 ). It is covered ¯ 2γ d by √ − 1 stripes of width 23 ε in vγ -direction consisting of lattice triangles 3ε intersecting the parallelogram, the rst √ of these stripes touching 41 , the last one touching 42 (note that if γ d¯ = 23 ε the parallelogram is degenerated to a segment). For the intermediate stripes (4.7) shows that √ and y1 (t, x2 ) ≤ t + εlµ ∀ t < x1 − sε √ y1 (t, z2 ) ≥ t + εl(a − µ) ∀ t > z1 + sε .

¯ ε < t < x1 −sε lie in the bottom This shows that if (t, x2 ) and (t+w, z2 ), x1 − d+s and top triangles of some intermediate stripe, respectively, which are unbroken by construction of Dµ , then √ √ |y(t + w, z2 ) − y(t, x2 )| ≥ y1 (t + w, z2 ) − y1 (t, x2 ) ≥ w + εl(a − 2µ) ∼ ε. ¯

2γ d Consider the √ atomic chains in vγ direction that lie on the boundary of these 3ε √ stripes. They are of length γ −1 (z2 −x2 )+O(ε) ≤ Cε  ε. So there is a constant c > 0 such that each of these chains contains at least one spring elongated by a factor of more than √cε . By passing, if necessary, to a lower threshold η˜ ≥ η , we obtain that the triangles sharing such a spring are broken and additionally one neighbor of each. As broken triangles for such springs on neighboring chains might overlap, we only consider every second atom chain and denote the set of type one triangles adjacent to such a spring on atom chains of odd numbers by (1) C¯vγ (41 , 42 ). We note that √ (41 , 42 ). (4.8) γ d¯ ≤ 3ε#C¯v(1) γ

The projection onto the x2 -axis of the spring in vγ -direction is an interval J of length γε. Counting broken springs, it is elementary to see that the energy ´ 4 contribution √3ε (ε,l−ε)×J W4 (∇˜ yε ) of the part of these broken triangles that lies in the stripe (0, l) × J is bounded from below by

2ε(1 + P (γ))β η˜,

(4.9)

where P (γ) is the projection coecient from (1.24) satisfying P (1) = 12 and in √ particular P (γ) = 0 if and only if γ = 23 . On the other hand, the energy within 84

stripes (0, l) × J 0 when J 0 is the projection of an arbitrary broken triangle is still bounded from below by 2εβ η˜. µ ¯(1) such that Now let 4i , i = 1, . . . , Mε , denote all triangles 4 in C¯ε,˜ η ∩ Cε (i)

(i)

there exists x2 ∈ Dµ with (0, l) × {x2 } intersecting with the interior of 4. The (1) (M ) numbering shall be chosen so as to satisfy x2 < . . . < x2 ε . As 1 − |Dµ | = O(ε), (i+1) (i) there exists a constant C > 0 such that x2 − x2 < Cε, i = 1, . . . , Mε − 1. (i ) (i ) (i) (i) We dene the subset {x2 j }j=1,...Nε of {x2 }i=1,...,Mε such that x2 = x2 j for a j = 1, . . . Nε if and only if |m4i − m4i+1 | > 0. According to our previous S ε ¯(1) (1) considerations, if Ivη˜γ is the projection of C¯vγ := N j=1 Cvγ (4ij , 4ij +1 ) onto the x2 -axis, then |Ivη˜γ | ≤ γε#C¯v(1) . (4.10) γ As before using (4.9) and (4.10) we see that the total energy is greater or equal to

#C¯v(1) 2ε(1 + P (γ))β η˜ + |I η˜ \ Ivη˜γ | γ = |I η˜| ≥

2β η˜ + O(ε) γ

η˜ 2β η˜ η˜ ¯(1) εβ η˜ − |I η˜ | 2β + O(ε) + 2#C¯v(1) εP (γ)β + 2# C vγ vγ γ γ γ

2β + 2#C¯v(1) εP (γ)β η˜ + O(ε), γ γ

(1) (1) (1) and so #C¯vγ = O(1). As every 4 ∈ C¯vγ is in at most two dierent C¯vγ (4ij , 4ij +1 ), P ε ¯(1) this also yields N j=1 #Cvγ (4ij , 4ij +1 ) = O(1). Applying (4.8) we nd that

O(1) =

Nε X j=1

#C¯v(1) (4ij , 4ij +1 ) ≥ γ

Nε Nε X γ d¯ij cX √ ≥ |m4ij − m4ij +1 | ε 3ε j=1 j=1

for a constant c > 0, when d¯i = γ −1 |m4i − m4i+1 |. This concludes the proof.  The above Lemmas 4.2.1 and 4.2.2 show that for a sequence of almost minimizers (˜ yε ) satisfying (1.28), the number #C¯ε of largely deformed triangles is bounded independently of ε for a < acrit , while in the supercritical case for φ 6= 0 there are two subsets

Ω(1) ε := {x ∈ Ωε : 0 ≤ x1 ≤ pε − cε + (vγ · e1 )x2 } , Ω(2) ε := {x ∈ Ωε : pε + cε + (vγ · e1 )x2 ≤ x1 ≤ l} ,

(4.11)

c > 0 independent of ε and pε to be chosen appropriately, such that the number (1) (2) of triangles in C¯ε intersecting Ωε ∪ Ωε is bounded uniformly in ε. We √ recall that the last claim in Lemma 4.2.2 does not hold if vγ is not unique (γ = 23 ). Indeed, (1) if P (γ) vanishes, we cannot conlude that #C¯vγ = O(1) in the above proof. In 85

this case we do not expect that√ the essential√part of the broken triangles lies in in a small stripe parallel to R( 12 , 23 )T or R( 21 , 23 )T as we √ have already seen that the crack can take a serrated course. Nevertheless, if γ = 23 (or equivalently φ = 0) one can show that up to a number being uniformly bounded in ε the broken triangles C¯ε lie in a stripe around the graph of a Lipschitz function. Recall that ψ(ε) is the width of the lateral boundaries (see (1.22)).

Lemma 4.2.3. Let u˜ε be a minimizing sequence satisfying Eε (id +

√

εuε ) = inf Eε + O(ε).

Let a > acrit and φ = 0. Then there exist Lipschitz functions gε : (0, 1) → µ )= (ψ(ε), l−ψ(ε)) with |gε0 | = √13 a.e. such that for µ suciently small ε#(C¯ε \ C¯ε,η O(ε) and (4.12)

µ 4 ⊂ {(x, y) ∈ Ω : g (y) − Cε ≤ x ≤ g (y) + Cε} , ∪4∈C¯ε,η ε ε

for some C > 0 independent of gε and ε. Proof. We√have v2 = ( 21 ,

√

√

3 T ) , 2

√

v3 := v2 − v1 = (− 12 , 23 )T and v2⊥ = (− 23 , 21 )T , v3⊥ = −( 23 , 12 )T . By Lemma 4.2.2 we immediately get |I η | = 1 − O(ε) and µ ε#(C¯ε \ C¯ε,η ) = O(ε) for µ suciently small recalling that these properties were derived independently of the choice of γ . Similarly as before we note that after ˜ ∈ C¯µ ∩ C¯ε(1) . We passing to a suitable η˜ ≥ η it suces to show the claim for 4 ε,˜ η (1) µ ¯ ¯ estimate the dierence of broken triangles Cε,η ∩ Cε projected onto the linear the projection of some triangle subspaces spanned by v2⊥ and v3⊥ . We recall that √ 3 4 on these subspaces are intervals of length 2 ε and denote the centers of the (2) (3) intervals by m4 and m4 , respectively. Let x2 , z2 ∈ Dµ , x2 < z2 with z2 − x2 ≤ Cε and (2)

(2)

(4.13)

(3)

(3)

(4.14)

(m41 − m42 ) · v2⊥ > 0 or

(m41 − m42 ) · v3⊥ < 0

(1) for the corresponding broken triangles 41 , 42 ∈ C¯ε . Without restriction we treat the case (4.13). As in the proof of Lemma 4.2.2 we may assume if a triangle intersects (0, l) × {z2 } or (0, l) × {x2 } then its interior does so, too. Denote by (i) (i) d¯(i) = √23 |m41 − m42 |, i = 2, 3, the distances of the centers in vi -projection onto the x1 -axis. (i) (i) (i) (i) (i) (i) Let x1 , z1 ∈ (0, l) such that πvi⊥ (x1 , x2 ) = m41 and πvi⊥ (z1 , z2 ) = m42 , respectively, where πvi⊥ denotes the orthogonal projection onto the linear subspace spanned by vi⊥ , i = 2, 3. Let w(2) = √13 |x2 − z2 | and w(3) = − √13 |x2 − z2 |. Then

86

(i)

(i)

the vi -projection of z (i) = (z1 , z2 ) onto the x2 -slice is given by (˜ z1 , x2 ) with (i) (i) (i) (i) (i) (i) ¯ z˜1 = z1 − w for i = 2, 3. We note that d = |x1 − z˜1 |. Taking (4.13) into (2) (2) (3) account we obtain x1 < z˜1 < z˜1 . Let sε = 2ε . As in the previous proof we consider areas bounded by parallel(i) ograms. For i = 2, 3, let P (i) be the parallelogram with corners (x1 + sε , x2 ), ¯(i) (i) (i) (i) (˜ z1 − sε , x2 ), (z√1 − sε , z2 ), (z1 − d¯(i) + sε , z2 ). They are covered by dε − 1 stripes of width 23 ε in vi -direction, respectively (note that P (2) can again be de¯(2) generated l to amsegment l if d m = ε). It is not hard to see that both parallelograms 2|z2 −x2 | √ 3ε

√

2|z2 −x2 | √ 3ε

+ 1 stripes of width 23 ε in e1 -direction, where the stripes at the top and at the bottom are only partially covered (the exact number depends of the precise location of the slices (0, l) × {x2 } and (0, l) × {z2 }). We denote the number of these covered stripes by N (41 , 42 ) and the orthogonal projection onto the x2 -axis by I(41 , 42 ). Setting cover

or

2 (i) (i) n(i) (41 , 42 ) = √ |m41 − m42 | 3ε

(4.15)

it is elementary to see that d¯(2) = n(2) (41 , 42 )ε and d¯(3) = (n(2) (41 , 42 ) + N (41 , 42 ) − 1)ε. ¯i Following the lines of the previous proof we see that each of the dε atomic chains in vi direction lying on the boundary of the stripes which cover P (i) , contains at least one spring elongated by a factor of more than √cε . Consequently, on the N (41 , 42 ) stripes in e1 -direction we have at least 2n(2) (41 , 42 )+ N (41 , 42 ) − 1 > N (41 , 42√) broken springs orientated in v2 or v3 direction. Let J be an interval of length 23 ε such that the stripe (0, l) × J consists of lattice triangles. It is elementary to see that if two broken springs in v2 and v3 lie in the ¯ stripe at least three ´ triangles are broken, i.e. lie in the set Cε,˜η . Thus, the energy 4 contribution √3ε (ε,l−ε)×J W4 (∇˜ yε ) of the stripe can be bounded from below by η˜ 3εβ . More generally, if on a stripe there are k ∈ N broken springs in v2 and v3 the energy contribution is at least (k + 1)εβ η˜. On the other hand, we recall that on an arbitrary stripe (0, l)×J 0 consisting of lattice triangles the energy is always bounded from below by 2εβ η˜. Consequently, we derive that ´ in the above situation the energy contribution of the N (41 , 42 ) yε ) is bounded from below by stripes √43ε (ε,l−ε)×I(41 ,42 ) W4 (∇˜

N (41 , 42 )εβ η˜ + (2n(2) (41 , 42 ) + N (41 , 42 ) − 1)εβ η˜

(4.16)

and note that the energy contribution of N (41 , 42 ) stripes is always bounded from below by 2N (41 , 42 )εβ η˜. µ ¯(1) such that Now let 4i , i = 1, . . . , Mε , denote all triangles 4 in C¯ε,˜ η ∩ Cε (i)

(i)

there exists x2 ∈ Dµ with (0, l) × {x2 } intersecting with the interior of 4. The (1) (M ) numbering shall be chosen so as to satisfy x2 < . . . < x2 ε . As 1 − |Dµ | = O(ε), 87

(i+1)

(i)

there exists a constant C > 0 such that x2 − x2 < Cε, i = 1, . . . , Mε − 1. (ij ) (i ) (i) (i) We dene the subset {x2 }j=1,...Nε of {x2 }i=1,...,Mε such that x2 = x2 j for a j = 1, . . . Nε if and only if m4i and m4i+1 satisfy (4.13) or (4.14). We let p(j) = 2 ε or p(j) = 3 if (4.13) or (4.14) holds, respectively. Let I´ε = ∪N j=1 I(4ij , 4ij +1 ). yε ) can Taking (4.16) into account the energy contribution √43ε (ε,l−ε)×Iε W4 (∇˜ be bounded from below by N

ε 4 1X √ |Iε |εβ η˜ + (2n(p(j)) (4ij , 4ij +1 ) − 1)εβ η˜ + O(ε). 2 3ε i=1

The factor 21 accounts for the possibility that two adjacent intervals I(4ij , 4ij +1 ), I(4ij+1 , 4ij+1 +1 ) may overlap. We thus see that the total energy is greater or equal to Nε 4 η˜ 1 X √ β + (2n(p(j)) (4ij , 4ij +1 ) − 1)εβ η˜ + O(ε) 2 i=1 3 PNε (p(j)) (4ij , 4ij +1 ) = O(1). We now construct the function gε : and so j=1 n (1)

(2)

(0, 1) → (0, l). For i ∈ Mε let Mi and Mi be the orthogonal projections of the center of 4i onto the x1 and x2 -axis, respectively. If i ∈ / Nε set (1)

(1)

M − Mi g˜˜ε = i+1 (2) (2) Mi+1 − Mi (2)

(2)

on the interval [Mi , Mi+1 ]. Now let g˜ be the Lipschitz function satisfying (2) (1) gε0 | ≤ √13 on g˜ε (M1 ) = M1 and g˜ε0 = g˜˜ε . By construction it is easy to see that |˜ (2)

(2)

gε0 k∞ ≤ [M1 , MNε ]. We extend g˜ε arbitrarily to (0, 1) such that k˜ we have Nε X (p(j)) (p(j)) |m4i − m4i +1 | = O(ε), j

√1 . 3

By (4.15)

j

j=1

and then is not hard to see that there is some C > 0 independent of g˜ε and ε such that (4.12) holds. Recalling (1.22) it remains to choose gε : (0, 1) → (ψ(ε), l − ψ(ε)) with gε0 = ± √13 a.e. and kgε − g˜ε k∞ ≤ Cε.  We conclude that for φ = 0 in the supercritical case there are two subsets

Ω(1) gε := {x ∈ Ωε : 0 ≤ x1 ≤ gε (x2 ) − cε} , Ω(2) gε := {x ∈ Ωε : cε + gε (x2 ) ≤ x1 ≤ l} ,

(4.17)

where gε is chosen appropriately as in Lemma 4.2.3 and c > 0 independent of (1) (2) ε, such that the number of triangles in C¯ε intersecting Ωgε ∪ Ωgε is bounded ¯ = ψ(ε) − cε  1 one has uniformly in ε. Note that with ε  ψ(ε)
¯ (0, ψ(ε)) × (0, 1) ∩ Ωε ⊂ Ω(1) gε , (4.18)
¯ (l − ψ(ε), l) × (0, 1) ∩ Ωε ⊂ Ω(2) , gε

88

(1)

(2)

so that, in particular, Ωgε and Ωgε are connected.

4.3 Convergence of almost minimizers As a further preparation we shows that broken triangles can be `healed'. In order to treat the dierent cases simultaneously in the following we will call these sets the `good set'   for a < acrit , Ωε (1) (2) Ωgood = Ωε ∪ Ωε for a > acrit , φ 6= 0 and   (1) (2) Ωgε ∪ Ωgε for a > acrit , φ = 0, (i)

(i)

with Ωε and Ωgε , i = 1, 2, as dened in (4.11) and (4.17).

Lemma 4.3.1. Suppose y˜ε is a minimizing sequence satisfying Eε (yε ) = inf Eε + O(ε). There exists y¯ε ∈ W 1,∞ (Ωgood ; R2 ) with ∇¯ yε bounded in L∞ (Ωgood ) uniformly in ε such that |{x ∈ Ωgood : y¯ε (x) 6= y˜ε (x)}| = O(ε2 )

and

ˆ

ˆ 2

dist2 (∇˜ yε , SO(2)) dx.

dist (∇¯ yε (x), SO(2)) dx ≤ C S Ωgood \ 4∈C¯ε 4

Ωgood

Proof. For notational convenience we drop the subscript ε in the following proof.

By Lemmas 4.2.1, 4.2.2 and 4.2.3 we can partition the area covered by the (closed) triangles in C¯ intersecting Ωgood into connected components C1 , . . . , CN such that [ ˙ N, 4 = C1 ∪˙ . . . ∪C ¯ 4∈C:4∩Ω good 6=∅

where N is bounded uniformly in ε. Then the maximal diameter of each set Ci is bounded by a term O(ε). For each i, the largest connected component Di of the complement Ωgood \ Ci lying in the same component of Ωgood is unique (with area of the order 1 while all the other components of the complement are of size O(ε2 )). Let Vi be the union of triangles whose interior is contained in Di that touch the boundary of Ci . We now proceed to dene y¯ by modifying y˜ on all the triangles not contained in Di , successively for i = 1, . . . , N . For each i this modication is done iteratively on triangles 4 which share at least one side with a triangle that has been modied previously or with a triangle lying in Vi in such a way that y¯ is continuous along such sides and y¯|4 is ane and minimizes dist((¯ y )4 , SO(2)). 89

In order to estimate dist(∇¯ y , SO(2)) we will use the geometric rigidity result in Theorem B.1 and recall that the constant is invariant under rescaling of the domain. For later use we mention that if dist2 (∇f (x), SO(2)) is equiintegrable, then R can be chosen in such a way that also |∇f (x) − R|2 is equiintegrable, cf. [28]. Consider a single step in the modication process, when y˜ is modied to y¯ on 4, and let U be the union of triangles that have been modied previously or lie in Vi . By Theorem B.1, there is a rotation R ∈ SO(2) such that ˆ ˆ 2 |∇¯ y (x) − R| dx ≤ C dist2 (∇¯ y (x), SO(2)) dx U

U

holds. Since ∇¯ y is piecewise constant, this means X X |(¯ y )40 − R|2 ≤ C dist2 ((¯ y )40 , SO(2)). 40 ⊂U

40 ⊂U

It is not hard to see that there exists an extension w of y¯ from U to U ∪ 4 such that X |(w)4 − R|2 ≤ C |(¯ y )40 − R|2 . 40 ⊂U

(If there is only one side of 4 on the boundary of U , say adjacent to 40 ⊂ U , then one can take w with (w)4 = (¯ y )40 . If at least two sides, say in v1 and v2 direction, are shared by triangles 41 , 42 ⊂ U , respectively, then these sides have a common corner and the unique extension w satises (w)4 vi = (¯ y )4i vi = Rvi + ((¯ y )4i − R)vi , i = 1, 2.) Now by construction of y¯ on 4 we see that X |(¯ y )40 − R|2 dist2 ((¯ y )4 , SO(2)) ≤ C 40 ⊂U

and so

ˆ

ˆ 2

dist2 (∇¯ y (x), SO(2)) dx.

dist (∇¯ y (x), SO(2)) dx ≤ C U ∪4

U

Iterating this estimate we nally arrive at ˆ ˆ 2 dist (∇¯ y (x), SO(2)) dx ≤ C

dist2 (∇˜ y , SO(2)) dx.

S Ωgood \ i Ci

Ωgood

Here the constant C can be chosen independently of ε. This is due to the facts that the number of modication steps is bounded uniformly in ε and  after rescaling the shapes U with 1ε  there is also only a uniformly bounded number of shapes U involved in the previous rigidity estimates. Moreover, each triangle is covered by no more than three of the sets Vi . 90

The uniform boundedness of the number of modication steps also shows that |{x ∈ Ωgood : y¯(x) 6= y˜(x)}| = O(ε2 ) and, by denition of C¯ and construction of y¯, that k∇¯ y kL∞ (Ωgood ) = O(1).  Note that up to a set of small size y¯ε satises the same boundary conditions as (i) (i) y˜ε on the lateral boundary. More precisely, there are Γε ⊂ (0, 1), |Γε | = O(ε), (1) i = 1, 2, such that y¯ε and y˜ε coincide on (0, ε) × ((0, 1) \ Γε ) and (l − ε, l) × (2) ((0, 1)\Γε ). With these boundary conditions and the geometric rigidity estimate in Theorem B.1 we can now derive strong convergence results for y¯ε and even the corresponding rescaled displacement u ¯ε = √1ε (¯ yε − id) on Ωgood . We rst consider the supercritical case and treat the cases φ 6= 0 and φ = 0 separately.

4.3.1 The supercritical case Lemma 4.3.2. If a > acrit and φ 6= 0, then there exist sequences sε , tε ∈ R such that k¯ uε − (0, sε )kH 1 (Ω(1) + k¯ uε − (al, tε )kH 1 (Ω(2) → 0. ε ) ε )

Proof. We again drop the subscript ε. By applying the geometric rigidity estimate

in Theorem B.1 to Ω(1) and to Ω(2) , we obtain rotations R(1) , R(2) ∈ SO(2) such that

k∇¯ y − R(i) kL2 (Ωε(i) ) ≤ Ck dist(∇¯ y , SO(2))kL2 (Ω(i) , ε )

i = 1, 2.

(4.19)

Here C can be chosen independently of ε as all the possible shapes of Ω(i) are related through bi-Lipschitzian homeomorphisms with Lipschitz constants of both the homeomorphism itself and its inverse bounded uniformly in ε, see Section B. Now using that ∇¯ y is uniformly bounded in L∞ , we obtain from Lemmas 4.3.1 and 2.4.5(i) 2 X

ˆ k∇¯ y−

R(i) k2L2 (Ω(i) ) ε

dist2 (∇˜ y , SO(2)) dx

≤C S Ωgood \ 4∈C¯ε 4

ˆ

i=1

dist2 (∇˜ y , O(2)) + χ(∇˜ y ) dx

≤C S Ωgood \ 4∈C¯ε 4

ˆ ≤C

W4,χ (∇˜ y ) dx. S Ωgood \ 4∈C¯ε 4

But, as seen before, ˆ ˆ 4 4 χ √ W4,χ (∇˜ y ) dx ≤ E (y) − √ W4,χ (∇˜ y ) dx 3ε Ωgood \S4∈C¯ε 4 3ε S4∈C¯ε 4 2β η η ≤ inf E χ + O(ε) − |I | = O(ε), γ 91

where the last step followed from Lemma 4.2.2 and (4.6), and so 2 X

k∇¯ y − R(i) k2L2 (Ω(i) ) = O(ε2 ). ε

i=1

By Poincaré's inequality we then deduce that there are ζ (i) ∈ R2 such that 2 X

k¯ y − R(i) · −ζ (i) kH 1 (Ω(i) = O(ε). ε )

(4.20)

i=1 (i)

We extend y¯ as an H 1 -function from Ωε to Ω(i) (as dened in Theorem 1.5.1), (1) i = 1, 2, such that (4.20) still holds and y¯1 (0, x2 ) = 0 for x2 ∈ (0, 1) \ Γε , (2) y¯1 (l, x2 ) = l(1 + aε ) for x2 ∈ (0, 1) \ Γε . The trace theorem for Sobolev functions with x1 = 0 or x1 = l according to i = 1 and i = 2, respectively, gives 2 X

k¯ y (x1 , ·) − R(i) (x1 , ·) − ζ (i) kL2 (0,1) = O(ε).

i=1

In particular, setting ζ˜(1) = ζ (1) and ζ˜(2) = ζ (2) − laε e1 , the rst components satisfy 2 X

(i)

(i)

(i)

kx1 − R11 x1 − R12 · −ζ˜1 kL2 ((0,1)\Γ(i) = O(ε). ε )

(4.21)

i=1

But then also the constant function       1 1 (i) (i) (i) (i) (i) (i) (i) ˜ ˜ R = x1 − R11 x1 − R12 · − − ζ1 − x1 − R11 x1 − R12 · −ζ1 2 12 2 (i)

(i)

(i)

is of order ε in L2 (( 12 , 1) \ (Γε ∪ (Γε + 12 ))) and thus |R12 | ≤ Cε. An elementary argument now yields

|R(i) − Id| = O(ε)

or

|R(i) + Id| = O(ε).

It is not hard to see that |R(i) − Id| = O(ε) as otherwise, e.g. for i = 1, on the set T = {4 ∈ Cε : 4 ⊂ (0, ε) × (0, 1)} we get, due to the boundary conditions, ˆ ˆ 2 (1) 2 O(ε ) = |∇¯ y−R | ≥ |1 + aε + 1|2 + O(ε2 ) ≥ Cε, T

T

which is clearly impossible. Returning to (4.21) and (4.20), it now follows that (i) |ζ˜1 | = O(ε) and then √ k¯ u − (0, sε )kH 1 (Ω(1) ) + k¯ u − (al, tε )kH 1 (Ω(2) ) = O( ε), ε

where sε =

(1) √1 ζ ε 2

and tε =

ε

(2) √1 ζ . ε 2

 92

Lemma 4.3.3. If a > acrit and φ = 0, then there exist sequences sε , tε ∈ R and Lipschitz functions gε as in Lemma 4.2.3 such that k¯ uε − (0, sε )kH 1 (Ω(1) + k¯ uε − (al, tε )kH 1 (Ω(2) → 0. g ) g ) ε

ε

Proof. Without restriction we only estimate y¯ (again dropping subscripts ε) on (1)

Ωgε . We rst note that we may not simply proceed as in (4.19) and (4.20) as, (1) due to a possibly complicated shape of the set Ωgε , the corresponding constants cannot be controlled. As a remedy, we n claim that we can nd oa partition (Tj )j , (1)

(1)

j = 1, . . . , Mε of Ωgε of the form Tj = x ∈ Ωgε : tj−1 ≤ x2 ≤ tj for suitable tj ∈ [0, 1], j = 0, . . . , Mε with t0 = 0 and tMε = 1 such that the Tj are related through bi-Lipschitzian homeomorphism with uniformly bounded Lipschitz constants to cubes of sidelength dj = tj − tj−1 ≥ ψ(ε)  ε. We will show this at the end of the proof. Recalling that the constant in (B.1) is invariant under rescaling of the domain and repeating the above arguments in (4.20) we now obtain R(j) ∈ SO(2) and ξ (j) ∈ R2 , j = 1, . . . , Mε , such that Mε X

2

∇¯ y − R(j) L2 (Tj ) = O(ε2 ) and

Mε X

j=1

j=1



y¯ − R(j) · −ξ j 2 2 d−2 = O(ε2 ). j L (Tj )

Let T˜j = (tj−1 , tj ) for j = 1, . . . , Mε and T ∗ = rescaling argument and the trace theorem yield Mε X

SMε

j=1 (tj−1

+

dj , tj ). 2

A standard



y¯(0, ·) − R(j) (0, ·) − ξ j 2 2 ˜ = O(ε2 ). d−1 j L (Tj )

(4.22)

j=1 (1)

(1)

d

Similarly as above we calculate the norm in L2 (T ∗ \ (Γε ∪ (Γε + 2j ))) on the trace {x1 = 0} of the piecewise constant function       dj (j) dj (j) (j) (j) (j) (j) (j) R12 = x1 − R11 x1 − R12 · − − ξ1 − x1 − R11 x1 − R12 · −ξ1 2 2

PMε 2 (i) 2 2 and now nd that j=1 dj |R12 | = O(ε ). Consequently, noting that dj ≥ ψ(ε)  ε for all j = 1, . . . , Mε and proceeding as before, we obtain Mε X

d2j |R(j) − Id|2 = O(ε2 )

j=1

so that Mε X

y¯ − id − ξ j 2 1 = O(ε2 ). H (Tj ) j=1

93

(4.23)

Due to the boundary conditions, (4.22) and (4.23) yield and therefore

PMε

j=1

d2j |ξ1j |2 = O(ε2 )

Mε X

y¯ − id − (0, ξ2j ) 2 1 = O(ε2 ). H (Tj )

(4.24)

j=1 (1)

¯ We dene the stripe S = (0, ψ(ε)) × (0, 1) and note that S ∩ Ωε ⊂ Ωgε by (4.18). From Poincaré's inequality we obtain a ζ ∈ R2 such that k¯ y − id − ζk2H 1 (S) ≤ C k∇¯ y − Idk2L2 (S) = O(ε2 ).

(4.25)

Note that the constant C can be chosen independently of the length of S , i.e. independently of ε. Applying (4.24) we may suppose ζ = (0, ζ2 ). ¯ Moreover, by (4.24) and (4.25) there is some ρε ∈ (0, ψ(ε)) such that the trace on the slice Γ = {ρε } × (0, 1) satises

ˆ

O(ε2 ) |¯ y − id − (0, ζ2 )|2 = ¯ ψ(ε) Γ

and

Mε X

O(ε2 )

y¯ − id − (0, ξ2j ) 2 2 = . L (Γ∩Tj ) ¯ ψ(ε) j=1

We compare the trace on Γ and deduce from dj = H1 (Γ ∩ Tj ) Mε X

Mε   X

2

y¯ − id − ξ j 2 2 dj |ζ − ξ | ≤ C + k¯ y − id − ζk 2 L (Γ∩Tj ) L (Γ∩Tj )

j=1

j 2

j=1 2

O(ε ) O(ε2 ) . = ¯ = ψ(ε) ψ(ε) Thus, returning to (4.24) we conclude

k¯ y − id −

(0, ζ2 )k2H 1 (Ω(1) ) gε

Mε Mε X X

j 2

d2j |ξj − ζ|2 y¯ − id − ξ H 1 (Tj ) + C ≤C j=1

j=1

≤ O(ε2 ) + C

Mε X

dj |ξj − ζ|2 =

j=1

and nally

k¯ u − (0, sε )k2H 1 (Ω(1) ) = gε

(2)

O(ε2 ) ψ(ε)

O(ε) →0 ψ(ε)

for ε → 0, where sε = √1ε ζ2 . For Ωgε we proceed likewise. To nish the proof it suces to show the existence of a partition (Tj )j with the (1) above properties. Recall that Ωgε = {x ∈ Ω : 0 < x1 < g(x2 ) − cε} and kg 0 k∞ =

94

√1 , 3

g ≥ ψ(ε). Let r0 = 0 and dene r1 , . . . , rMε ∈ (0, 1) inductively by setting rj+1 = rj + g(rj ), so that rMε + g(rMε ) ≥ 1. Now setting ( (1) {x ∈ Ωgε : rj−1 ≤ x2 ≤ rj } for 1 ≤ j ≤ Mε − 1, Tj = (1) {x ∈ Ωgε : rMε −1 ≤ x2 ≤ 1 for j = Mε , it is not hard so see that every Tj is related to λ(0, 1)2 for a suitable λ through some bi-Lipschitzian homeomorphism with uniformly bounded Lipschitz constants. By construction, tj − tj−1 ≥ g(tj ) ≥ ψ(ε)  ε for j = 1, . . . , Mε . 

4.3.2 The subcritical case Strong convergence in the subcritical case can be shown along the lines of the proofs of the main linearization results in [64] and [65]. We include a simplied proof adapted to the present situation here for the sake of completeness.

Lemma 4.3.4. If a < acrit , then there is a sequence sε ∈ R such that k¯ uε − (0, sε ) − F a ·kH 1 (Ωgood ) → 0.   a 0 where F = 0 − a . 3 a

Proof. We again drop subscripts ε if no confusion arises. With the help of the geometric rigidity estimate (Theorem B.1) we nd by arguing as in the proof of Lemma 4.3.2 that ˆ 2 W4,χ (∇˜ y ) dx = O(ε) k∇¯ y − RkL2 (Ωε ) ≤ C S Ωε \

4∈C¯ε

4

for a suitable rotation R ∈ SO(2) with

√ |R ± Id| = O( ε)

(4.26)

and

√ k¯ y ± id − ζkH 1 (Ωε ) = O( ε) √ for some ζ ∈ R2 with ζ1 = O( ε) and thus, due to the boundary conditions, k¯ u − (0, ζ2 )kH 1 (Ωε ) = O(1). In particular, u ¯ε − (ζε )2 e2 converges  up to passing to a subsequence  weakly. It T now suces to prove that ke(¯ uε ) − F a kL2 (Ωε ) → 0, where e(u) = (∇u) 2 +∇u denotes the symmetrized gradient, for then the assertion follows from Korn's inequality. 95

√ 1 Vε,χ (F ) = Vε (F ) + 1ε χ(Id + √ To this end, we let Vε (F )1 = 2ε W4 (Id + εF ) and 1 εF ), so that Vε,χ (F ) → 2 D W4 (Id)[F, F ] = 2 Q(F ) uniformly on compact subsets of R2×2 . Then by frame indierence (see Lemma 2.4.1) q  √ √ √ (Id + εF )T (Id + εF ) W4,χ (Id + εF ) = W4,χ  T  (4.27) F +F 1 √ = εVε,χ + √ f ( εF ) 2 ε p T with f (F ) = (Id + F )T (Id + F )−Id− F 2+F , so that |f (F )| ≤ C min{|F |, |F |2 }. Then by Lemma 2.4.5(i) and (4.27), Vε,χ satises   T √ √ 1 √ c 1 F +F + √ f ( εF ) ≥ dist2 (Id + εF, O(2)) + χ(Id + εF ) Vε,χ 2 ε ε ε √ c 2 ≥ dist (Id + εF, SO(2)) ε 2 q √ √ c ≥ (Id + εF )T (Id + εF ) − Id ε T 2 F + F √ 1 = c + √ f ( εF ) . (4.28) 2 ε √ T In the sequel we set Aε (F ) = F 2+F + √1ε f ( εF ). Choose convex functions ψk : R2×2 → R with linear growth at innity such that ψ1 ≤ ψ2 ≤ . . . and ψk (F ) → 12 Q(F ) uniformly on compact subsets of R2×2 . The previous quadratic estimate on Vε,χ (Aε (F )) from below and the fact that Vε,χ → 21 Q uniformly on compacts then shows that we can also choose δ > 0 and a sequence rk → ∞ such that 1 Vε,χ (Aε (F )) − δχ{|Aε (F )|≥rk } |Aε (F )|2 ≥ ψk (Aε (F )) − , k whenever ε (depending on k ) is suciently small. With (4.27) we now obtain that ˆ ˆ 1 W4,χ (¯ y ) dx = Vε,χ (Aε (∇¯ u)) dx ε Ωε Ωε ˆ ˆ 1 ≥ ψk (Aε (∇¯ u)) dx + δ χ{|Aε (∇¯u)|≥rk } |Aε (∇¯ u)|2 dx − . k Ωε Ωε √ √ As ψk has linear growth at innity and √1ε f ( ε∇¯ uε ) ≤ C min{|∇¯ uε |, ε|∇¯ uε |2 }, ∇¯ uε bounded in L2 , by splitting the integration into two parts according to |∇¯ uε | ≤ M or |∇¯ uε | > M and eventually sending M to innity, we nd ˆ ˆ lim inf ψk (Aε (∇¯ uε )) dx = lim inf ψk (e(¯ uε )) dx. ε→0

ε→0

Ωε

96

Ωε

When u ¯ε − (ζε )2 e2 * u in H 1 , by Theorem 1.4.1 it then follows that ˆ αla2 4 √ = lim √ Vε,χ (Aε (∇¯ uε )) dx ε→0 3 3 Ωε ˆ 4 χ{dist(x,∂Ω)≥k−1 } ψk (e(¯ uε )) dx ≥ lim inf √ ε→0 3 Ω ˆ 4δ 4 + lim sup √ χ{|Aε (∇¯uε )|≥rk } |Aε (∇¯ uε )|2 dx − √ . ε→0 3 Ωε 3k Using that by convexity of ψk the rst term on the right hand side is lower semicontinuous in ∇¯ uε and that χ{dist(·,∂Ω)≥k−1 } ψk → 12 Q monotonically, we nally nd by letting k → ∞ ˆ αla2 2 √ ≥√ Q (e(u)) 3 3 Ω ˆ (4.29) 4δ 2 √ χ{|Aε (∇¯uε )|≥rk } |Aε (∇¯ uε )| dx. + lim lim sup k→∞ ε→0 3 Ωε ´ A slicing and convexity argument similar to (3.2) now shows that √23 Ω Q(e(w)) ≥ 2 αla √ 3

for all w ∈ H 1 subject to w1 (0, x2 ) = 0 and w1 (l, x2 ) = al and thus

4δ lim lim sup √ k→∞ ε→0 3

ˆ χ{|Aε (∇¯uε )|≥rk } |Aε (∇¯ uε )|2 dx = 0, Ωε

or, in other words, |Aε (∇¯ uε )|2 is equiintegrable. By the estimate |Vε,χ (F )| = √ | 1ε W4,χ (Id + εF )| ≤ C(1 + |F |2 ), (4.28) shows that also

c dist2 (∇¯ yε , SO(2)) ≤ Vε,χ (Aε (∇¯ uε )) ε is equiintegrable, so that by the discussion in the proof of Lemma 4.3.1 in fact√we may assume that 1ε k∇¯ yε − Rk2L2 (Ωε ) is equiintegrable, too, and |R − Id| = O( ε) by (4.26). But then also |∇¯ uε |2 is equiintegrable and this together with (4.29) yields ˆ ˆ 2 2 αla2 lim √ Q(e(¯ uε )) = √ Q(e(u)) = √ . ε→0 3 Ωε 3 Ω 3

97

For some δ > 0 small enough we nally obtain that ˆ αla2 2 √ =√ Q(F a ) dx 3 3 Ω  ˆ 2 √ = inf Q(e(w)) − δ|e(w) − F a |2 dx : 3 Ω



1

w ∈ H (Ω), w(0, x2 ) = 0, w(l, x2 ) = al ˆ 2 ≤ lim inf √ Q(e(¯ uε )) − δ|e(¯ uε ) − F a |2 dx ε→0 3 Ωε 2 αla = √ − δ lim sup ke(¯ uε ) − F a k2L2 (Ωε ) ε→0 3 and therefore limε→0 ke(¯ uε ) − F a k2L2 (Ωε ) = 0 indeed.



4.3.3 Proof of the main limiting result After all these preparatory lemmas, the proof of our main limiting result Theorem 1.5.1 is now straightforward. Proof of Theorem 1.5.1. Choose sε as in Lemmas 4.3.4 if a < acrit , pε , sε and tε as in (4.11) and Lemma 4.3.2 if a > acrit and φ 6= 0 and nally gε and sε and tε as in Lemma 4.3.3 if a > acrit and φ = 0. By Lemmas 4.3.4, 4.3.2 and 4.3.3, u ¯ε (i) (i) can be extended as an H 1 -function from Ωε to Ω, Ωε to Ω(i) , i = 1, 2, or Ωgε to Ω(i) [gε ], i = 1, 2, respectively, such that still, respectively,

k¯ uε − (0, sε ) − F a ·kH 1 (Ω) → 0,

(4.30)

k¯ uε − (0, sε )kH 1 (Ω(1) ) + k¯ uε − (al, tε )kH 1 (Ω(2) ) → 0,

(4.31)

k¯ uε − (0, sε )kH 1 (Ω(1) [gε ]) + k¯ uε − (al, tε )kH 1 (Ω(2) [gε ]) → 0.

(4.32)

This completes the proof as by Lemma 4.3.1 we also still have |{x ∈ Ωε : u ¯ε (x) 6= u˜ε (x)}| = O(ε).  Finally, we give the proof of Corollary 1.5.2. Proof of Corollary 1.5.2 . First, let (yε ) be a minimizing sequence satisfying (1.28). Then by Theorem 1.5.1 we obtain (4.30), (4.31) or (4.32), respectively. Taking the condition supε kuε k∞ < ∞ into account, in the cases (i) and (ii) we get supε |sε | < ∞ and supε |sε |, supε |tε | < ∞ such that, passing to subsequences, we obtain sε → s and sε → s, tε → t, pε → p, respectively, for suitable constants s, t ∈ R, p ∈ (0, l). In (iii) we rst note that up to subsequences gε converges uniformly to some Lipschitz function g : (0, 1) → [0, l] satisfying |g 0 | ≤ √13 a.e. Then using again the uniform bound supε kuε k∞ < ∞ we get constants s, t such that sε → s and tε → t up to subsequences. It follows that u ˜ε → u as given in (i), (ii) and (iii), respectively. 98

Conversely, we assume that u is given as in Corollary 1.5.2 and show that there is a minimizing sequence (yε ) satisfying (1.28) with u ˜ε → u in measure. For (i) and (ii) this is obvious by the proof of Theorem 1.4.1 taking the congurations in (3.5) and (3.6) up to suitable translations. For given u in (iii) with corresponding function g and constants s, t we approximate g : (0, 1) → [0, l] uniformly by √ 3ε Lipschitz functions gε : (0, 1) → (0, l) being ane on intervals of length 2 with gε0 = ± √13 a.e. We set

( √ if 0 < x1 < gε (x2 ), x + (0, εs), yε (x) = √ x + (aε l, εt), if gε (x2 ) < x1 < l, so that u ˜ε = yε√−id → u in measure. As in the proof of Theorem 1.4.1, except for ε negligible contributions of the boundary layers, Eεχ (yε ) is given by the energy of the springs intersected transversally by graph(gε ). These springs are elongated by a factor scaling with √1ε yielding a contribution εβ in the limit. It is elementary to see that on every stripe in e1 direction of length 4β springs, and consequently Eεχ (yε ) → √ . 3

99

√

3ε 2

the graph intersects two 

Chapter 5 The limiting variational problem The rst part of this chapter is devoted to the the Γ-convergence result for Eε . Afterwards we will investigate the limiting variational problem.

5.1 Convergence of the variational problems Recall the denition of the sets Cε and C˜ε in Section 1.6. As a further preparation we modify the interpolation y˜ on triangles with large deformation: We x a threshold explicitly as R = 7 and let C¯ε ⊂ C˜ε be the set of those triangles where |(˜ y )4 | > R. By denition of the boundary values in (1.29) we nd C¯ε ⊂ Cε for ε small enough. We introduce another interpolation y 0 which leaves y˜ unchanged on 4 ∈ C˜ε \ C¯ε and replaces y˜ on 4 ∈ C¯ε by a discontinuous function with constant derivative satisfying |(y 0 )4 | ≤ R. In fact, by introducing jumps we achieve a ˜ ε ). release of the elastic energy. Note that y 0 ∈ SBV (Ω More precisely, observe that on 4 ∈ C¯ε we have |(˜ y )4 v| ≥ 2 for at least two springs v ∈ V . Indeed, using the elementary identity (2.8) we nd that |F | > 7 implies

X v∈V

|F v|4 =

3 3 hv, F T F vi2 ≥ (trace(F T F ))2 = |F |4 8 8 v∈V

X

4

and so maxv∈V |F v|4 > 78 > 44 . Hence, |F v| > 4 for at least one v ∈ V and at least two springs are elongated by a factor larger than 2. For m = 2, 3 let C¯ε,m ⊂ C¯ε be the set of triangles where |(˜ y )4 v| ≥ 2 holds for exactly m springs v ∈ V . For i, j, k = 1, 2, 3 pairwise distinct let hi denote the segment between the centers of the sides in vj and vk direction and dene the set Vi = hj ∪ hk . ˜ ε ). On 4 ∈ C˜ε \ C¯ε we simply set y 0 = y˜. On We now construct y 0 ∈ SBV 2 (Ω 4 ∈ C¯ε,2 , assuming |(˜ y )4 vi | ≤ 2, we choose y 0 such that ∇y 0 assumes the constant value (y 0 )4 on 4 with (y 0 )4 vi = (˜ y )4 vi and |(y 0 )4 v| = 1 for v ∈ V \ {vi }. Moreover, we ask that y 0 = y˜ at the three vertices and on the side orientated in vi direction. This can and will be done in such a way that y 0 is continuous 101

on int(4) \ hi . We note that the denition of (y 0 )4 is unique up to a reection, unless (˜ y )4 vi = 0. We may and will assume that

dist ((y 0 )4 , SO(2)) ≤ dist ((y 0 )4 , O(2) \ SO(2)) .

(5.1)

For 4 ∈ C¯ε,3 we set (y 0 )4 = Id and y 0 = y˜ at the three vertices such that y 0 is continuous on int(4) \ Vi for some i ∈ {1, 2, 3}. Here, the index i can be taken arbitrarily at rst. However, in what follows it will also be necessary to use the following unambiguously dened `variants' of y 0 : If on every 4 ∈ C¯ε,3 the set Vi is chosen as the jump set of y 0 we denote this interpolation explicitly as yV0 i . We dene the interpolation u0 for the rescaled displacement eld by u0 = ¯ε √1 (y 0 − id). We note that by construction also on an edge [p, q] ⊂ ∂4 for 4 ∈ C ε jumps may occur. There, however, the jump height |[u0ε ]| can be bounded by

√ 1 |[u0ε ](x)| ≤ ε k∇u0ε k∞ ≤ ε · cε− 2 = c ε

(5.2)

for a constant c > 0 independent of ε and x ∈ [p, q]. This holds since the interpolations are continuous at the vertices. The following lemma shows that we may pass from u ˜ε to u0ε without changing the limit.

Lemma 5.1.1. If uε → u in the sense of Denition 1.6.2 and Eε (uε ) is uniformly ˜ and H1 (Ju0 ) is ˜ , χ ˜ ∇u0ε * ∇u in L2 (Ω) bounded, then χΩ˜ ε u0ε → u in L1 (Ω) Ωε ε uniformly bounded. Proof. We rst note that there is some M > 0 such that #C¯ε ≤

M ε

(5.3)

for all ε > 0. To see this, we just recall that every triangle 4 ∈ C¯ε provides at least the energy ε inf {W (r) : r ≥ 2}. In fact we may assume that Cε∗ = C¯ε in yε )4 − Id| ≤ √Cε and so Denition 1.6.2 as for ∆ ∈ Cε∗ \ C¯ε we have |(˜ uε )4 | ≤ √Cε |(˜

k∇˜ uε kL2 (Ω˜ ε \∪4∈C¯

ε

4)

≤ k∇˜ uε kL2 (Ω˜ ε \∪4∈C∗ 4) + k∇˜ uε kL2 (∪4∈Cε∗ \C¯ε 4) ε ! 21 √ 2 3ε C ≤ C + #(Cε∗ \ C¯ε ) ≤ C. · 4 ε

It follows that χΩ˜ ε ∇u0ε is bounded uniformly in L2 and, in particular, equiintegrable. Finally, the jump lengths H1 (Ju0ε ) are readily seen to be bounded by Cε#C¯ε ≤ C . But then Ambrosio's compactness Theorem for GSBV (see Theorem ˜ . A.1.3 or [4, Theorem 2.2]) shows that indeed χΩ˜ ε ∇u0ε * ∇u in L2 (Ω) 

102

5.1.1 The Γ-lim inf -inequality With the above preparations at hand, we may now prove the Γ-lim inf -inequality in Theorem 1.6.3. ˜ with supε kgε k 1,∞ ˜ < +∞ Proof of Theorem 1.6.3(i). Let (gε )ε ∈ W 1,∞ (Ω) W (Ω) 2 ˜ 2 ˜ be given. Let u ∈ SBV (Ω) and consider a sequence uε ⊂ SBV (Ωε ) with uε ∈ Agε converging to u in SBV 2 in the sense of Denition 1.6.2. We split up the energy into bulk and crack parts neglecting the contribution εEεboundary from the boundary layers: X X Eε (uε ) ≥ ε W4 ((˜ yε )4 ) + ε W4 ((˜ yε )4 ) 4∈Cε \C¯ε

4 =√ 3ε

4∈C¯ε

ˆ

W4 Id +

√

X

X
ε∇u0ε + ε

Ωε

4∈C¯ε

v∈V, |(˜ yε )4 v|>2

1 W (|(˜ yε )4 v|) 2

(5.4)

=: Eεelastic (uε ) + Eεcrack (uε ). We note that by contruction of the interpolation u0ε we may take the integral over Ωε . As both parts separate completely in the limit, we discuss them individually.

Elastic energy. We rst concern ourselves with the of the energy. We n elastic part o recall W4 (Id + G) = 21 Q(G) + ω(G) with sup

χ[0,ε−1/4 ) (|∇u0ε (x)|).

Let χε (x) := r2 Q(F ). We compute

Eεelastic (uε )

4 ≥√ 3

: |F | ≤ ρ

Note that for F ∈ R

2×2

ˆ

 χε (x)

Ωε

ω(F ) |F |2

→ 0 as ρ → 0.

, r > 0 one has Q(rF ) =


1 1 √ 0 0 Q(∇uε ) + ω ε∇uε (x) dx. 2 ε

The second term of the integral can be bounded by √ 0 0 2 ω ( ε∇uε ) χε |∇uε | √ . | ε∇u0ε |2 Since is bounded in L and follows that

∇u0ε

2

lim inf ε→0

Eεelastic (uε )

√ ω ( ε∇u0ε ) √ χε | ε∇u0 |2 ε

converges uniformly to 0 as ε → 0 it

ˆ 4 1 ≥ lim inf √ χε (x) Q(∇u0ε (x)) dx ε→0 2 3 Ωε ˆ 4 1 ≥ lim inf √ Q(χΩε χε (x)∇u0ε (x)) dx. ε→0 3 Ω2

By assumption χΩε ∇u0ε * ∇u weakly in L2 . As χε → 1 boundedly in measure on Ω, it follows χΩε χε ∇u0ε * u weakly in L2 (Ω). By lower semicontinuity (Q 103

is convex by Lemma 2.4.2) we conclude recalling that Q only depends on the symmetric part of the gradient: ˆ 4 1 elastic lim inf Eε (uε ) ≥ √ Q(e(u(x))) dx. ε→0 3 Ω2

Crack energy. By construction the functions u0ε have jumps on destroyed triangles

4 ∈ C¯ε . We now write the energy of such a triangle in terms of the jump height [u] = u+ − u− . We rst concern ourselves with a triangle 4 ∈ C¯ε,3 . For the variant u0ε,Vi , i = 1, 2, 3 we consider the springs in vj , vk direction for j, k 6= i. Thus, we compute √ 0 ] = εvj + ε[u0ε,Vi ]hk , (5.5) ε(˜ yε )4 vj = ε(yε0 )4 vj + [yε,V i hk where [u0ε,Vi ]hk denotes the jump height on the set hk . Here and in the following equations, the same holds true if we interchange the roles of j and k . We claim that 1 1 0 (5.6) |(˜ yε )4 vj | ≥ ε 4 √ [uε,Vi ]hk + 1. ε 1

Indeed, for | √1ε [u0ε,Vi ]hk | ≤ ε− 4 this is clear since |(˜ yε )4 vj | ≥ 2. Otherwise, applying (5.5) we compute for ε small enough: 1 0 1 0 |(˜ yε )4 vj | = √ [uε,Vi ]hk + vj ≥ √ [uε,Vi ]hk − 1 ε ε   1 1 1 1 ≥ ε 4 √ [u0ε,Vi ]hk + 1 − ε 4 ε− 4 − 1 ε 1 1 1 1 1 0 0 − = ε 4 √ [uε,Vi ]hk − 2 + ε 4 ≥ ε 4 √ [uε,Vi ]hk + 1. ε ε Let ρ > 0 suciently small. Applying Lemma 2.4.5(iv) there is an increasing subadditive function ψ ρ with ψ ρ (r − 1) − ρ ≤ W (r) for r ≥ 1. We dene ψ˜ρ = ψ ρ − ρ. The monotonicity of ψ ρ and (5.6) yield   ρ ρ − 14 0 ˜ ˜ W (|(˜ yε )4 vj |) ≥ ψ (|(˜ yε )4 vj | − 1) ≥ ψ ε [uε,Vi ]hk . (5.7) Now for 4 ∈ C¯ε,3 we may estimate the energy as follows: 3

W4 ((˜ yε )4 ) =

1X W (|(˜ yε )4 vl |) 2 l=1

3   1 o 1 X n ˜ρ  − 1 0 ρ −4 0 ˜ 4 ≥ ψ ε |[uε,Vi ]hk | + ψ ε |[uε,Vi ]hj | =: W4,3 ((˜ yε )4 ) , 4 i=1

104

(i)

where i, j, k = 1, 2, 3 are pairwise distinct. With νu = νu0ε,V we can also write i

3

1 2 2 X W4,3 ((˜ yε )4 ) = · · √ 4 ε 3 i=1

ˆ hj ∪hk

 1 
ψ˜ρ ε− 4 |[u0ε,Vi ]| |vj · νu(i) | + |vk · νu(i) | dH1 .

The factors in front occur since H1 (hj ) √= 2ε and, letting νj be a normal of hj , one has |νj · vj | = 0 and |νj · vk | = 23 . Consequently, dening φρi (r, ν) = ψ ρ (r) (|vj · ν| + |vk · ν|) and φ˜ρi (r, ν) = ψ˜ρ (r) (|vj · ν| + |vk · ν|), respectively, we get 3 ˆ 1 1 X W4,3 ((˜ yε )4 ) = √ φ˜ρi (ε− 4 |[u0ε,Vi ]|, νu(i) ) dH1 3ε i=1 Ju0 ∩int(4) ε,Vi

on every 4 ∈ C¯ε,3 . For 4 ∈ C¯ε,2 we proceed analogously. Assuming |(˜ y ε ) 4 vi | ≤ 2 we compute for the springs in vj , vk direction (abbreviated by vj,k ) as in (5.5) √ (5.8) ε(˜ yε )4 vj,k = ε(yε0 )4 vj,k + ε[u0ε ]hi . Note that in this case we do not have to take a special variant of u0ε into account. Repeating the steps (5.6) and (5.7) we nd   1 1 0 ρ −4 ˜ (W (|(˜ yε )4 vj |) + W (|(˜ yε )4 vk |)) ≥ ψ ε |[uε ]hi | =: W4,2 ((˜ yε )4 ) . 2 √

Noting that |vj · νi | = |vk · νi | = 23 , |vi · νi | = 0 and that every of these terms occurs twice in the sum of the right hand side of the following formula, it is not hard to see that this energy satises the same integral representation formula as W4,3 : 3

1 X W4,2 ((˜ yε )4 ) = √ 3ε i=1

ˆ Ju0

ε,Vi

∩int(4)

1 φ˜ρi (ε− 4 |[u0ε,Vi ]|, νu(i) ) dH1 .

(i)

(Recall that the interpolation variant u0ε,Vi and its crack normal νu do not depend on i on 4 ∈ C¯ε,2 .) Let σ > 0. Note that C¯ε ⊂ Cε for ε suciently small as supε kgε kW 1,∞ (Ω) ˜ < +∞. Thus, the crack energy can be estimated by ˆ 1 1 X ρ crack Eε (uε ) ≥ √ φ˜ρi (ε− 4 |[u0ε,Vi ]|, νu(i) ) dH1 − Eε,∪∂4 (˜ yε ) 3 i Ju0 ∩Ω˜ ε ˆ ε,Vi
1 X ρ ≥√ φρi (σ −1 |[u0ε,Vi ]|, νu(i) ) − 2ρ dH1 − Eε,∪∂4 (˜ yε ) , ˜ 3 i Ju0 ∩Ωε ε,Vi

ρ where Eε,∪∂4 (˜ yε ) compensates for the extra contribution provided by jumps lying on the boundary of some 4 ∈ C¯ε . We will show that this term vanishes in the limit.

105

Now by construction the φρi (r, ν), i = 1, 2, 3, are products of a positive, increasing and concave function in r and a norm in ν . Moreover, u0ε and its variants converge to u in L1 with ∇u0ε bounded in L2 and thus equiintegrable. By Ambrosio's lower semicontinuity Theorem [4, Theorem 3.7] we obtain ˆ X 1 ρ crack lim inf Eε (uε ) ≥ √ φρi (σ −1 |[u]|, νu ) dH1 − CM ρ − lim sup Eε,∪∂4 (˜ yε ) , ε→0 ε→0 3 Ju i where we used that supε H1 (Ju0ε ) ≤ CM for a constant C > 0 by (5.3). We recall that ψ ρ (r) → β for r → ∞. In the limit σ → 0 this yields ˆ X 1 ρ crack lim inf Eε (uε ) ≥ √ 2β |v · νu | dH1 − CM ρ − lim sup Eε,∪∂4 (˜ yε ) . (5.9) ε→0 ε→0 3 Ju v∈V Taking (5.2) and (5.3) into account we compute  n o  1 Xˆ 1 1 lim sup |ψ˜ρ ε− 4 |[u0ε ]| | ≤ lim CM sup |ψ ρ (r) − ρ| : r ≤ ε− 4 · cε 2 ε→0

ε→0

∂4

4∈C¯ε

= CM ρ. ρ ˜ ρ for some C˜ > 0. We nally let ρ → 0 This proves lim supε |Eε,∪∂4 (˜ yε ) | ≤ CM in (5.9). This nishes the proof of (i).  We now prove the Γ-lim inf -inequality in Theorem 1.6.4. Proof of Theorem 1.6.4, rst part. Following the proof of Theorem 1.6.3(i) it suces to show ˆ ˆ κ 1 0 ˆ χε fκ (∇yε ) ≥ − Q(∇u), lim inf ε→0 ε Ω 2 Ω ε

ˆ = D2 m where Q ˆ 1 (Id). Let u0ε = √1ε (yε0 − id). With a slight abuse of notation we set e(F √ ) = 12 (F T + F ) and a(F ) = F − e(F ) for matrices F ∈ R2×2 . Let 2×2 F = Id + εG for around the identity matrix yields √ G ∈ R . Linearization 2 dist(F, SO(2)) = ε|e(G)| + εO(|G| ). It is not hard to see that this implies √ (5.10) R(F ) = Id + εa(G) + εO(|G|2 ), where R(F ) ∈ SO(2) is dened as in Lemma 2.4.6. As m(Id) ˆ = e1 and e(G) ∈ ker(Dm(Id)) ˆ , we nd by expanding m ˆ1

m ˆ 1 (F ) = 1 + with sup

n

ω(H) |H|2

√

√ εˆ εDm ˆ 1 (Id)a(G) + Q(G) + ω( εG) 2

o

: |H| ≤ ρ → 0 as ρ → 0. 106

(5.11)

We concern ourselves with the term Dm ˆ 1 (Id)a(G). Recall ˆ )) − √ that |m(R(F m(F ˆ )| ≤ C|R(F ) − F |2 by Lemma 2.4.6(i). For F = Id + εG this implies by (5.10)

m(F ˆ ) − m(Id) ˆ √ ε→0 ε √ √ m(R(F ˆ )) − e1 √ + O( ε) = lim e1 · a(G)e1 + O( ε) = 0. = lim e1 · ε→0 ε→0 ε

Dm ˆ 1 (Id)a(G) = e1 · Dm(Id)G ˆ = lim e1 ·

√ ˆ In particular, (5.11) then implies 0 ≤ 1ε fκ (F ) = − κ2 Q(G) − 1ε ω( εG) and thus ˆ is positive semidenite. We proceed exactly as in the proof of Theorem −Q 1.6.3(i) and conclude ˆ ˆ κ  1 κ √ 0 0 0 ˆ χε fκ (∇yε ) ≥ lim inf − χε Q(∇u ω( ε∇u ) ) + lim inf ε ε ε→0 ε Ω ε→0 2 ε Ωε ε ˆ κ ˆ Q(∇u). ≥− 2 Ω 

5.1.2 Recovery sequences It remains to construct recovery sequences in order to complete the proof of Theorem 1.6.3.

Proof of Theorem 1.6.3(ii).

The basic tool for the proof of the Γ-limsup-inequality is the density result ˜ R2 ) with u = g on Ω ˜ \ Ω. Without given in Theorem A.1.7. Let u ∈ SBV 2 (Ω, ∞ ˜ 2 restriction we can assume u ∈ L (Ω, R ) as this hypothesis may be dropped by applying a truncation argument and taking Q(F ) ≤ C|F |2 into account. In fact, it suces to provide a recovery sequence for an approximation un dened in Theorem A.1.7. Although our notion of convergence in Denition 1.6.2 is not given in terms of a specic metric, similarly to a general density result in the theory of Γ-convergence this can be seen by a diagonal sequence argument. The crucial point is that due to (5.12) below we may assume that for ε suciently small (depending on n)

#Cε∗ = #Dε ≤

CH1 (Ju ) CH1 (Jun ) ≤ , ε ε

where C is independent of n and ε. If (un,ε )ε is a recovery for un , one may therefore pass to a diagonal sequence which is a recovery sequence for u, in particular converging to u the sense of Denition 1.6.2. For simplicity write u instead of un in what follows. 107

Let δ > 0 and dene Juδ = {x ∈ Ju , |[u](x)| ≥ δ}. Since |[u]| is Lipschitz continuous on Ju , it cannot oscillate innitely often between values ≤ δ and values ≥ 2δ on a single segment. Consequently, there is a nite number Nuδ of disjoint subsegments S1 , . . . , SNuδ in Ju such that |[u]| < 2δ on every Sj and S uδ 1 2δ |[u]| > δ on Ju \(S1 ∪. . .∪SNuδ ). Note that H1 ( N i=1 Si ) ≤ H (Ju \Ju ) =: ρ(δ) → 0 for δ →P 0. We cover S1 , . . . , SNuδ by pairwise disjoint rectangles Q1 , . . . QNuδ which satisfy j H1 (∂Qi ) + |Qi | ≤ Cρ(δ). It is not hard to see that |u(x) − u(y)| ≤ ˜ . CH1 (∂Qi ) + 2δ for x, y ∈ Qj as ∇u ∈ L∞ (Ω) ˜ \ SNuδ Qj and dene We modify u on the rectangles Qi : Let uδ = u on Ω i=1 uδ = cj on Qj for cj ∈ R2 in such a way that Juδ = Juδδ up to an H1 -negligible ˜ , ∇u ∈ L∞ (Ω) ˜ we nd uδ → u in L1 (Ω) ˜ and ∇uδ → ∇u in set. As u ∈ L∞ (Ω) 2 ˜ 1 L (Ω). Moreover, we have H (Ju ∆Juδ ) ≤ Cρ(δ) → 0 for δ → 0. Consequently, it suces to establish a recovery sequence for a function u ∈ ˜ with u = g in a neighborhood of Ω ˜ \ Ω and Ju = Juδ for some δ > 0. Note W(Ω) after the above modication the segments of Ju might not be pairwise disjoint. √ ˜ We dene uε (x) = u(x) for x ∈ Lε ∩ Ω and let yε (x) = id + εuε (x). Clearly ˜ ε. ˜ε , u0ε we again denote the interpolations on Ω we have uε ∈ Agε for all ε. By u Up to considering a translation of u of order ε, we may assume that Ju ∩ Lε = ∅. Let Dε be the sets of triangles where Ju crosses at least one side of the triangle. Then

#Dε ≤

CH1 (Ju ) + CNu ε

(5.12)

˜ R2 ) and ε, where Nu denotes the for a constant C > 0 independent of u ∈ W(Ω, (smallest) number of segments whose union gives Ju . From now on for the local nature of the arguments we may assume that Ju consists of one segment only. Indeed, if Ju consists of segments S1 , . . . , SNu , which are possibly not disjoint, the number of triangles ∆ ∈ C˜ε with 4 ∩ Si1 ∩ Si2 6= ∅ for 1 ≤ ii < i2 ≤ Nu scales like Nu and therefore their energy contribution is negligible in the limit. We show C¯ε = Dε for ε small enough. Let 4 ∈ Dε . We see that, if Ju = Juδ crosses a spring v at point x∗ , say, then a computation similar as in (5.8) together with ∇u ∈ L∞ shows 1 δ (5.13) |(˜ yε )4 v| = √ [u(x∗ )] + O(1) ≥ √ + O(1). ε ε Thus, 4 ∈ C¯ε for ε small enough. On the other hand, if√we assume 4 ∈ / Dε , then for at least two springs v ∈ V we have |(˜ yε )4 v| ≤ 1 + ε k∇uk∞ < 2 for ε small enough leading to 4 ∈ / C¯ε . We claim that k∇u0ε kL∞ (Ω) (5.14) ˜ ≤ C. 108

This is clear for 4 ∈ / Dε = C¯ε as ∇u ∈ L∞ . For 4 ∈ C¯ε,3 it follows by construction. √ For 4 ∈ C¯ε,2 there is a v ∈ V such that (yε0 )4 v = (˜ yε )4 v = v + O( ε). By Lemma 2.4.5(i) and (5.1) we get a rotation Rε ∈ SO(2) such that

|Rε − (yε0 )4 |2 = dist2 ((yε0 )4 , SO(2)) = dist2 ((yε0 )4 , O(2)) ≤ CW4 ((yε0 )4 ) = O(ε). √ This yields |(yε0 )4 − Id| = O( ε) and thus |(u0ε )4 | = O(1). ˜ε → u in L1 as u and thus every u˜ε is bounded uniformly We note that χΩ˜ ε u ˜ε \ S in L∞ and, u being Lipschitz away from J , u ˜ → u uniformly on Ω u ε 4∈Dε 4, S ∗ where | 4∈Dε 4| ≤ Cε. Letting Cε = Dε this shows that uε → u in the sense of Denition 1.6.2 recalling (5.12) and the fact that |(˜ uε )4 | = O(1) for 4 ∈ / Dε . We next establish an even stronger convergence of the derivatives. Consider ∇˜ uε on triangles in Cε \ Dε . As ∇u is Lipschitz there, the oscillation on such a triangle, 0 0 osc4 ε (∇u) := sup {|∇u(x) − ∇u(x )|, x, x ∈ 4}, tends to zero uniformly (i.e., not depending on the choice of the triangle). We thus obtain ˆ ˆ 2 2 (osc4 k∇˜ uε − ∇uk∞ ≤ ε (∇u)) → 0 ˜ ε \∪4∈D 4 Ω ε

˜ ε \∪4∈D 4 Ω ε

˜ . Note that in for ε → 0, so that even χΩ˜ ε \∪4∈Dε 4 ∇˜ uε → ∇u strongly in L2 (Ω) ˜ . Indeed, recall #Dε ≤ Cε−1 by (5.12). Using (5.14) fact χΩ˜ ε ∇u0ε → ∇u in L2 (Ω) on the set of broken triangles we then get ˆ ¯ ε ε2 → 0 |∇u0ε − ∇u|2 ≤ C#D S 4∈Dε

4

for ε → 0. We now split up the energy in bulk and surface parts ˆ 1 χ elastic crack Eε (uε ) = Eε (uε ) + Eε (uε ) + O(ε) + χ(∇˜ yε ) ε Ωε

(5.15)

as dened in (5.4). Note that indeed the contribution εEεboundary is of order O(ε) ˜ and Ju ⊂ Ω since u = g in a neighborhood of Ω ˜ \ Ω. We as ∇u ∈ L∞ (Ω) ´ 1 rst observe that ε Ωε χ(∇˜ yε ) = 0 for ε small enough. Indeed, for ∆ ∈ C¯ε this follows from (5.13). For ∆ ∈ / Dε it suces to recall |(˜ uε )4 | = O(1) which implies that (˜ uε )4 is near SO(2). Repeating the steps in the elastic energy estimate in (i), applying χΩε ∇u0ε → ∇u strongly in L2 (Ω), (5.14) and Q(F ) ≤ C|F |2 for a constant C > 0 we conclude that ˆ 4 1 elastic lim sup Eε (uε ) = √ Q(e(u(x))) dx. (5.16) ε→0 3 Ω2 It is elementary to see that Ju crosses

2|νu · v| H1 (Ju ) √ + O(1) 3ε 109

(5.17)

springs in v-direction for v ∈ V , where νu is a normal to the segment Ju . Recalling (5.13), the crack energy may be estimated by

lim sup Eεcrack (uε ) ε→0

n o 2 X 1 ≤ lim sup H1 (Ju ) sup W (r) : r ≥ δε− 2 + O(1) √ |νu · v| + O(ε) ε→0 3 v∈V 2 X = H1 (Ju ) β √ |νu · v|. 3 v∈V This together with (5.15) and (5.16) shows that uε is a recovery sequence for u.  χ Finally, we construct recovery sequences for the functionals Fε to conclude the proof of Theorem 1.6.4. Proof of Theorem 1.6.4, second part. Following the proof of Theorem 1.6.3(ii) it suces to show ˆ ˆ 1 κ ˆ lim fκ (∇˜ yε ) = − Q(∇u). ε→0 ε Ω 2 Ω ε ´ f (∇˜ yε ) = 0 for ε small First, by (5.13) and the denition of fκ we get S ∆ κ ∆∈Dε

enough. For ∆ ∈ / Dε we yε )∆ = (∇yε0 )∆ and thus we nd fκ ((∇˜ yε )∆ ) = √ have0 (∇˜ κ ˆ 0 −ε Q((∇u )∆ ) − κω( ε∇(u )∆ ) by (5.11). We obtain 2

ε

1 ε

ε

ˆ fκ (∇˜ yε ) = Ωε

1 ε

ˆ S Ωε \ ∆∈Dε ∆

κ ≤− 2

ˆ

S Ωε \ ∆∈Dε

fκ (∇yε0 ) C 0 ˆ Q(∇u ε) + ε ∆

ˆ

√ ω( ε∇u0ε ).

Ωε

√ Using (5.14) and the denition of ω we observe 1ε kω( ε∇u0ε )k∞ → 0 for ε → 0. This together with strong convergence χΩε ∇u0ε → ∇u in L2 (Ω) shows ˆ ˆ 1 κ ˆ lim sup fκ (∇˜ yε ) ≤ − Q(∇u). ε Ωε 2 Ω ε→0 

5.2 Analysis of the limiting variational problem We nally give the proof of Theorem 1.6.5 determining the minimizers of the limiting functional E . An analogous result for isotropic energy functionals has been obtained in [57]. We thus do not repeat all the steps of the proof provided in [57] but rather concentrate on the additional arguments necessary to handle anisotropic surface contributions. 110

Proof of Theorem 1.6.5. We rst for the energy E√. To P establish a lower bound 1 this end, we begin to estimate |v · ν| for ν ∈ S . We recall that γ ∈ [ 23 , 1] v∈V √ and dene P : [ 23 , 1] × S 1 → [0, ∞) by   √ √  1 − √3 1−γ 2 |v · ν|, 3 γ > , γ γ 2 P (γ, ν) = √ √  max 3|e2 · ν| − |e1 · ν|, 0 , γ = 23 . √

As vγ is unique for γ > 23 , the function P is well dened. In the generic case, √ i.e. for γ > 23 , an elementary computation yields p X √ ⊥ √ 1 1 − γ2 |v · ν| ≥ |vγ · ν| + 3|vγ · ν| = |vγ · ν| + 3 ± e1 · ν ± vγ · ν γ γ v∈V √ 3 ≥ |e1 · ν| + P (γ, ν) γ √ ⊥ P v = ± for ν ∈ S 1 . In the rst step we used that 3vγ . In the v∈V\{vγ } √

√

special case φ = 0 ⇔ γ = 23 , i.e. v1 = e1 , v2,3 = ± 21 e1 + 23 e2 we obtain √ P P 3|e2 · ν| for |ν2 | > 21 and v∈V |v · ν| = 2|e1 · ν| for v∈V |v · ν| = |e1 · ν| + |ν2 | ≤ 21 , ν ∈ S 1 . Consequently, it is not hard to see that

√ 3 |e1 · ν| + P (γ, ν) |v · ν| ≥ γ v∈V

X √

3 . 2

also holds for γ =

4 E(u) ≥ √ 3

ˆ Ω

Thus, we get

1 Q(e(u(x))) dx + 2

ˆ Ju

2β 2β |e1 · νu | + √ P (γ, νu ) dH1 . γ 3

By Lemma 2.4.2 we obtain min{Q(F ) : eT1 F e1 = r} = slicing method (see Theorem A.1.5) we get

ˆ

1

ˆ

E(u) ≥ 0

0

l


2 2β α √ eT1 ∇u(x1 , x2 )e1 dx1 + #S x2 (u) γ 3

α 2 r . 2



Then using the

dx2 + E γ (u), (5.18)

where #S x2 denotes the number of jumps on a slice (0, l) × {x2 } and ˆ 2β γ √ dH1 . E (u) = 3P (γ, νu ) Ju In case #S x2 (u) ≥ 1, the inner integral in (5.18) is obviously bounded from below by 2β . If #S x2 (u) = 0, by applyig Jensen's inequality we nd that this term is γ 111

bounded from below by αla2 due to the boundary conditions. We thus obtain  αla 2 inf E ≥ min √3 , 2β . On the other hand, it is straighforward to check that γ

E(uel ) = αla2 and E(ucr ) = 2β , which shows that uel is a minimizer for a < acrit γ cr and u is a minimizer for a > acrit . It remains to prove uniqueness: (i) Let a < acrit and u be a minimizer of E . Since E(u) = E(uel ) we infer from (5.18) that u has no jump on a.e. slice (0, l) × {x2 } and satises eT1 ∇u e1 = a a.e. by the imposed boundary values and strict convexity of the mapping t 7→ t2 on [0, ∞). Thus, if Ju 6= ∅, a crack normal must satisfy νu = ±e2 H1 -a.e. Taking √ E γ (u) and the fact that P (γ, e2 ) > 0 for γ ∈ [ 23 , 1] into account, we then may assume Ju = ∅ up to an H1 negligible set, i.e., u ∈ H 1 (Ω). We nd u1 (x1 , x2 ) = ax1 +f (x2 ) a.e. for a suitable function f , and the boundary condition u1 (0, x2 ) = 0 yields f = 0 a.e. In particular, eT1 ∇u e2 = 0 a.e. Applying strict convexity of Q on symmetric matrices (Lemma 2.4.2) we now observe eT2 ∇u e2 = − a3 and eT1 ∇u e2 + eT2 ∇u e1 = 0 a.e. So the derivative has the form   a 0 ∇u(x) = 0 − a for a.e. x. 3

Since Ω is connected, we conclude u(x) = (0, s) + F a x = uel (x) a.e. (ii) Let a > acrit , φ 6= 0 and u be a minimizer of E . We again consider the lower bound (5.18) for the energy E and now obtain that on a.e. slice (0, l) × {x2 } a minimizer u has precisely one jump and that eT1 ∇u e1 = 0 a.e. Now Lemma 2.4.2 shows that ∇u is antisymmetric a.e. As a consequence, the linearized rigidity estimate for SBD functions of Chambolle, Giacomini and Ponsiglione [25] yields that there is a Caccioppoli partition (Ei ) of Ω such that X [ u(x) = (Ai x + bi )χEi and Ju = ∂ ∗ Ei ∩ Ω, i

i

where ATi = −Ai ∈ R2×2 and bi ∈ R2 . (See Section A.2 for the denition and basic properties of Caccioppoli partitions.) As E γ (u) = 0, we also note that νu ⊥ vγ a.e. on Ju . Following the arguments in [57], in particular using regularity results for boundary curves of sets of nite perimeter and exhausting the sets ∂ ∗ Ei with Jordan curves, we nd that [ Ju = ∂ ∗ Ei ∩ Ω ⊂ (p, 0) + Rvγ i

for some p such that (p, 0) + Rvγ intersects both segments (0, l) × {0} and (0, l) × {1}. We thus obtain that (Ei ) consists of only two sets: E1 to the left and E2 to the right of (p, 0) + Rvγ , say. Due to the boundary conditions we conclude that A1 = A2 = 0 and b1 = (0, s), b2 = (al, t) for suitable s, t ∈ R. (iii) Let a > acrit , φ = 0 and u be a minimizer of E . We follow the lines of the √ proof in (ii). The only dierence is that E γ (u) = 0 now implies that |νu · e1 | ≥ 23 112

a.e. and then arguing similarly as before we obtain

Ju ⊂ g((0, 1)) up to an H1 -negligible set, where g : (0, 1) → [0, l] is a Lipschitz function with |g 0 | ≤ √13 a.e. We now conclude as in (ii). 

113

Part II A quantitative geometric rigidity result in SBD and the derivation of linearized models from nonlinear Grith energies

115

Chapter 6 The model and main results The aim in Section 1.6 was the investigation of the convergence of energies in brittle fracture of the form (1.19). We have already seen that the analysis involves a simultaneous passage from discrete-to-continuum and from nonlinear to linearized elastic energies. It turned out that in the derivation of a small strain limit one has to face major diculties concerning coercivity of the functionals due to the frame indierence of the energy density. The main goal of this part is the analysis of the passage from nonlinear to linearized Grith models in a general framework. For the sake of simplicity and to avoid further complicacies of technical nature we treat the problem in a continuum setting in two dimensions. Let Ω ⊂ R2 open, bounded with Lipschitz boundary. Recall the properties of the space SBV (Ω, R2 ), frequently abbreviated as SBV (Ω) hereafter, in Section A.1. For M > 0 we dene n o SBVM (Ω) = y ∈ SBV (Ω, R2 ) : kyk∞ + k∇yk∞ ≤ M, H1 (Jy ) < +∞ . (6.1) Here M may be chosen arbitrarily large (but xed) and therefore the constraint kyk∞ + k∇yk∞ ≤ M is not a real restriction as we are interested in the small displacement regime in the regions of the domain where elastic behavior occurs. The uniform bound on the absolute continuous part of the gradient is natural when dealing with discrete energies where the corresponding deformations are piecewise ane on cells of microscopic size (see e.g. [17] or the construction of the interpolation in Section 5.1). Moreover, the uniform bound on the function is assumed only to simplify the exposition and may be dropped. Let W : R2×2 → [0, ∞) be a frame-indierent stored energy density with W (F ) = 0 i F ∈ SO(2). Assume that W is continuous, C 3 in a neighborhood of SO(2) and scales quadratically at SO(2) in the direction perpendicular to innitesimal rotations. In other words, we have W (F ) ≥ c dist2 (F, SO(2)) for all F ∈ R2×2 and a positive constant c. We briey note that we can also treat inhomogeneous materials where the energy density has the form W : Ω × R2×2 → [0, ∞). Moreover, it suces to assume W ∈ C 2,α , where C 2,α is the Hölder space 117

with exponent α > 0. For ε > 0 dene the Grith-energy Eε : SBVM (Ω) → [0, ∞) by ˆ 1 Eε (y) = W (∇y(x)) dx + H1 (Jy ). (6.2) ε Ω We denote the small parameter occurring in the energy by ε to remind of functionals of the form (1.20). Having the application to discrete systems in mind, we will sometimes refer to ε as the `atomic length scale'. For later we also introduce a relaxed energy functional. For ρ > 0, ε > 0 and U ⊂ Ω dene fερ (x) = min{ √xερ , 1} and ˆ ˆ 1 ρ Eε (y, U ) = W (∇y(x)) dx + fερ (|[y](x)|) dH1 (x). (6.3) ε U Jy ∩U Clearly, we have Eερ (y, U ) ≤ Eε (y) for all y ∈ SBVM (Ω) and U ⊂ Ω.

6.1 Rigidity estimates We rst concern ourselves with the question if the functionals Eε can be related to a limiting functional for ε → 0. We observe that for congurations with uniform bounded energy Eε (yε ) the absolute continuous part of the gradient satises ∇yε ≈ SO(2) as the stored energy density is frame-indierent and minimized on SO(2). Assuming that yε → y in L1 , one can show that ∇y ∈ SO(2) a.e. applying lower semicontinuity results for SBV functions (see [56]) and the fact that the quasiconvex envelope of W is minimized exactly on SO(2) (see [67]). A classical result due to Liouville states that a smooth function y satisfying the constraint ∇y ∈ SO(2) is a rigid motion. In the theory of fracture mechanics global rigidity can fail if the crack disconnects the body. More precisely, Chambolle, Giacomini and Ponsiglione have proven that for congurations which do not store elastic energy (i.e. ∇y ∈ SO(2) a.e.) and have nite Grith energy (i.e. H1 (Jy ) < +∞) the only way that rigidity may fail is that the body is divided into at most countably many parts each of which subject to a dierent rigid motion (see [25]). Consequently, the limit of the sequence Eε (in the sense of Γ-convergence) is given by the functional which is nite for piecewise rigid motions and measures the segmentation energy which is necessary to disconnect the body. The exact statement is formulated in Corollary 6.3.2 as a direct consequence of our main Γ-convergence result in Theorem 6.3.1. To obtain a better understanding of the problem it is interesting to pass to rescaled congurations and to derive a limiting linearized energy as it was performed in [36] in the framework of nonlinear elasticity theory. The main ingredient in that analysis is a quantitative rigidity result due to Friesecke, James and Müller (see Theorem B.1). Extending the classical Liouville results it states 118

that, loosely speaking, if the deformation gradient is close to SO(2) (in L2 ) then it is in fact close to one single rotation R ∈ SO(2) (in L2 ). The rst goal of this part of the thesis is to `combine' the rigidity results of the pure elastic and pure brittle regime in order to derive a rigidity estimate for general Grith functionals (6.2) where both energy forms are coexistent. Recall the notion of a Caccioppoli partition in Section A.2 and the denition of the perimeter P (E, Ω) of a set E ⊂ R2 in Ω (see (A.7)). Let Ωρ = {x ∈ Ω : dist(x, ∂Ω) > Cρ} for ρ > 0 and for some suciently large constant C .

Theorem 6.1.1. Let Ω ⊂ R2 open, bounded with Lipschitz boundary. Let M > 0 and 0 < η, ρ  1. Then there is a constant C = C(Ω, M, η) and a universal c > 0 such that the following holds for ε > 0 small enough: ´ For each y ∈ SBVM (Ω, R2 ) with H1 (Jy ) ≤ M and Ω dist2 (∇y, SO(2)) ≤ M ε, there is an open set Ωy with |Ω \ Ωy | ≤ Cρ, a modication yˆ ∈ SBVcM (Ω, R2 ) with kˆy − yk2L2 (Ωy ) + k∇ˆy − ∇yk2L2 (Ωy ) ≤ Cερ and Eερ (ˆ y , Ωρ ) ≤ Eε (y) + Cρ

(6.4)

with P the following properties: We nd a Caccioppoli partition P = (Pj )j of Ωρ with j P (Pj , Ωρ ) ≤ C and for each Pj a corresponding rigid motion Rj x + cj , Rj ∈ SO(2) and cj ∈ R2 , such that the function u : Ω → R2 dened by for x ∈ Pj for x ∈ Ω \ Ωρ

( yˆ(x) − (Rj x + cj ) u(x) := 0

(6.5)

satises the estimates (i) H1 (Ju ) ≤ C, X ˆ (iii) ke(RjT ∇u)k2L2 (Pj ) ≤ Cε, j

ˆ , where e(G) = for some constant Cˆ = C(ρ)

ˆ (ii) kuk2L2 (Ωρ ) ≤ Cε, ˆ 1−η (iv) k∇uk2L2 (Ωρ ) ≤ Cε G+GT 2

(6.6)

for all G ∈ R2×2 .

This result will be addressed in Section 9. We remark that estimate (6.6) might be wrong without allowing for a small modication of the deformation as we show by way of example in Section 7.1. Moreover, we get a suciently strong bound only for the symmetric part of the gradient (see (iii)) which is not surprising due to the fact that there is no analogue of Korn's inequality in SBV. However, there is at least a weaker bound on the total absolutely continuous part of the gradient (see (iv)) which will essentially be needed to derive the Γconvergence result. We emphasize that also (ii) is highly nontrivial as Poincaré's inequality cannot be applied due to the presence of discontinuity points. In Section 10 we show that the qualitative piecewise rigidity result in two dimensions can be obtained as a corollary of Theorem 6.1.1. 119

Corollary 6.1.2. [Chambolle, Giacomini, Ponsiglione] Let y ∈ SBV (Ω, R2 ) such that H1 (Jy ) < +∞ and ∇y ∈ SO(2) a.e. Then y is a collection of an at most countable family of rigid deformations, i.e., there exists a Caccioppoli partition P = (Pj )j subordinated to Jy such that y(x) =

X j

(Rj x + cj )χPj (x),

where Rj ∈ SO(2) and cj ∈ R2 . There is also a linearized version of Theorem 6.1.1 which can be interpreted 2×2 as a `piecewise Korn-Poincaré-inequality in SBD'. Let R2×2 : AT = skew = {A ∈ R −A} be the set of skew symmetric matrices. Set ˆ ˆ 1 ρ V (e(∇u)(x)) dx + fερ (|[u]|) dH1 (6.7) Fε (y, U ) = ε U Ju ∩U for a coercive quadratic form V , i.e. V (G) ≥ c|G|2 for c > 0 and G ∈ R2×2 sym . 0 0 Furthermore, dene Fε = Fε (·, Ω), where fε ≡ 1. For the denition of the space SBD we refer to Section A.1.

Theorem 6.1.3. Let Ω ⊂ R2 open, bounded with Lipschitz boundary. Let M > 0, and 0 < ρ  1. Then there is a constant C = C(Ω, M ) such that for ε > 0 small enough the following holds: For each u ∈ SBD2 (Ω, R2 ) ∩ L2 (Ω, R2 ) with H1 (Ju ) ≤ M and ˆ

|e(∇u)(x)|2 dx ≤ M ε, Ω

there is an open set Ωu with |Ω \ Ωu | ≤ Cρ, a modication uˆ : Ω → R2 with kˆ u − uk2L2 (Ωu ) + ke(∇ˆ u − ∇u)k2L2 (Ωu ) ≤ Cρε and Fερ (ˆ u, Ωρ ) ≤ Fε (u) + Cρ

with P the following properties: We nd a Caccioppoli partition P = (Pj )j of Ωρ with j P (Pj , Ωρ ) ≤ C and for each Pj a corresponding innitesimal rigid motion 2 1 Aj x + cj , Aj ∈ R2×2 ˆ ) ≤ C and skew and cj ∈ R , such that H (Ju (i) ke(∇ˆ u)k2L2 (Ωρ ) ≤ Cε, (ii)

X j

ˆ kˆ u − (Aj · −cj )k2L2 (Pj ) ≤ Cε.

(6.8)

ˆ . for some constant Cˆ = C(ρ) The main technical results to prove the linearized version are addressed in Section 8 where we establish a local Korn-Poincaré-type inequality. The estimates in the linearized regime are crucial for the derivation of the nonlinear rigidity result in Theorem 6.1.1 which can be compared with the fact that in the proof of the geometric rigidity result in nonlinear elasticity (Theorem B.1) the usage of Korn's inequality is essential. 120

6.2 Compactness For a given (ordered) Caccioppoli partition P = (Pj )j of Ω let n o X 2 2 R(P) = T : Ω → R : T (x) = χPj (Rj x + cj ), Rj ∈ SO(2), cj ∈ R (6.9) j

be the set of corresponding piecewise rigid motions. Likewise we dene the set of piecewise innitesimal rigid motions, denoted by A(P), replacing Rj ∈ SO(2) by Aj ∈ R2×2 skew . Moreover, we dene the triples  D := (u, P, T ) : u ∈ SBV (Ω), P C.-partition of Ω, T ∈ R(P) ,  D∞ := (u, P, T ) : P C.-partition of Ω, T ∈ R(P), (∇T )T u ∈ GSBD2 (Ω, R2 ) . The space GSBD2 (Ω, R2 ), abbreviated by GSBD2 (Ω) hereafter, generalizes the denition of the space SBD(Ω, R2 ) based on certain slicing properties, see Section A.1. We now formulate the main compactness theorem.

Theorem 6.2.1. Let Ω ⊂ R2 open, bounded with Lipschitz boundary. Let M > 0 and εk → 0. If Eεk (yk ) ≤ C for a sequence yk ∈ SBVM (Ω), then there exists a subsequence (not relabeled) such that the following holds: There are triples (uk , P k , Tk ) ∈ D, where P k = (Pjk )j and (i) uk (x) − εk (yk (x) − Tk (x)) → 0 a.e., ˆ ˆ √ 1 1 T W (Id + εk ∇Tk ∇uk ) ≤ W (∇yk ) + o(1) (ii) εk Ω εk Ω −1/2

(6.10)

for εk → 0, such that we nd a limiting triple (u, P, T ) ∈ D∞ with in measure for all j ∈ N, (ii) Tk → T in L2 (Ω), ∇Tk → ∇T in L2 (Ω),

(i) χPjk → χPj

(6.11)

for k → ∞. Moreover, we get (i) uk → u

a.e. in Ω,

e(∇TkT ∇uk )

(ii)

* e(∇T T ∇u)

(iii) k∇uk kL∞ (Ω) ≤

2×2 weakly in L2 (Ω, Rsym ),

(6.12)

−1/8 Cεk ,

for k → ∞ and for the surface energy we obtain lim inf H1 (Jyk ) ≥ k→∞

where ∂P :=

S

j

1X P (Pj , Ω) + H1 (Ju \ ∂P ), j 2

∂ ∗ Pj . 121

(6.13)

Here ∂ ∗ denotes the essential boundary (see (A.8)). If we drop the condition kyk∞ ≤ M in the denition of SBVM (Ω), then (6.11) only holds for the derivatives of the piecewise rigid motions. In the following we say a triple (uk , P k , Tk ) ∈ D converges to (u, P, T ) ∈ D∞ and write (uk , P k , Tk ) → (u, P, T ) if (6.10)-(6.13) are satised. Of course, the partition of the limiting conguration is not unique as the following simple example shows.

Example 6.2.2. ( 12 , 1)

and

Consider Ω = (0, 1) × (0, 1), Ω1 = (0, 1) × (0, 12 ), Ω2 = (0, 1) ×

√ yk = idχΩ1 + (id + a εk )χΩ2

for a ∈ R2 . Then possible alternatives are e.g. (1) P 1 = Ω with R11 x + c11 = id √ or (2) P12 = Ω1 , P22 = Ω2 with R12 x + c21 = id and R22 x + c22 = id + a εk . Letting

√ 1 1 u1k = εk − 2 yk − id and u2k = εk − 2 yk − idχΩ1 − (id + a εk )χΩ2 we obtain in the limit εk → 0 two dierent congurations:

u1 = 0 · χΩ1 + aχΩ2 , P11 = Ω, u2 = 0, P12 = Ω1 , P22 = Ω2 . Clearly, we can equally well consider an example where we vary the rotations, e.g.    √  √ 1 0 cos a εk sin a εk √ √ x χΩ2 (x) yk (x) = x χΩ1 (x) + − sin a εk cos a εk 0 1

for a ∈ R. We now introduce a special subclass of partitions in which uniqueness will be guaranteed. The above example already shows that dierent partitions are not equivalent in the sense that they may contain a dierent `amount of information'. Note that on the various elements of the partition the conguration u is dened separately and the dierent pieces of the domain are not `aware of each other'. In particular, the possible discontinuities of u on ∂P do not have any physically reasonable interpretation. On the contrary, in the rst example where we did not split up the domain, we gain the jump height as an additional information. The observation that coarser partitions provide more information about the behavior at the jump set motivates the denition of the coarsest partition.

Denition 6.2.3.

Let (yk )k be a given (sub-)sequence as in Theorem 6.2.1.

(i) We say a partition P of Ω is admissible for (yk )k and write P ∈ ZP ((yk )k ) if there are triples (uk , P k , Tk ) ∈ D for k ∈ N as well as u, T such that (u, P, T ) ∈ D∞ and (6.10)-(6.13) hold. (ii) We say a piecewise rigid motion T is admissible for (yk )k and P writing T ∈ ZT ((yk )k , P) if there are triples (uk , P k , Tk ) ∈ D for k ∈ N as well as u such that (u, P, T ) ∈ D∞ and (6.10)-(6.13) hold. 122

(iii) We say a conguration u is admissible for (yk )k and P and write u ∈ Zu ((yk )k , P) if there are triples (uk , P k , Tk ) ∈ D for k ∈ N as well as T such that (u, P, T ) ∈ D∞ and (6.10)-(6.13) hold. (iv) We say a partition P of Ω is a coarsest partition for (yk )k if the following holds: The partition is admissible, i.e. P ∈ ZP ((yk )k ), and for all P admissible u ∈ Zu ((yk )k , P) the corresponding piecewise rigid motions Tk = j (Rjk x+ ckj )χPjk given by (iii) satisfy

|Rjk1 − Rjk2 | + |ckj1 − ckj2 | →∞ √ εk

(6.14)

for all j1 , j2 ∈ N, j1 6= j2 and k → ∞. In Lemma 10.2.2 below we nd an equivalent characterization of coarsest partitions being the maximal elements of the partial order on the sets of admissible partitions which is induced by subordination. Loosely speaking, the above denition particularly implies that given a coarsest partition a region of the domain is partitioned into dierent sets (Pj )j if and only if the jump height of the approximating sequence uk tends to innity on (∂ ∗ Pj )j . Recall the denition of the piecewise innitesimal rigid motions A(P) in (6.9). We now obtain a unique characterization of the limiting conguration up to piecewise innitesimal rigid motions.

Theorem 6.2.4. Let εk → 0 be given. Let Eεk (yk ) ≤ C for a sequence yk ∈ SBVM (Ω) and let (ykn )n∈N be a subsequence for which the assertion of Theorem 6.2.1 holds. Then we have the following: (i) There is a unique T ∈ ZT ((ykn )n , P) for all P ∈ ZP ((ykn )n ). (ii) There is a unique coarsest partition P¯ of Ω. ¯ all possible limiting congurations are of the (iii) Given some u ∈ Zu ((ykn )n , P) ¯ form u + ∇T A(P), i.e. the limiting conguration is determined uniquely up to piecewise innitesimal rigid motions.

Remark 6.2.5. Although not stated explicitly, for problems of the form (6.7) we can derive a compactness result similar to Theorem 6.2.1. This allows to solve more general variational problems for fracture mechanics in the realm of linearized elasticity. For technical reasons dealing with energy functionals with the main energy term ˆ |e(∇u)(x)|2 dx + H1 (Ju ) (6.15) Ω

123

often an a priori L∞ bound is imposed in the literature (see e.g. [8, 26, 66] or Section 1.6.1) such that compactness results applied. Possible ´ in SBD can be 1 alternatives are to add a term of the form Ju |[u] νu | dH giving control over the jump height (see e.g. [8]). Recently, the space of generalized functions of bounded deformation was introduced to overcome this diculty. In this framework it suces to assume an L1 bound on the function u similarly as in the compactness results for GSBV, i.e. variational problems for energy functionals of the form (6.15) with an additional term kukL1 (Ω) are treatable. In fact, in many situations such a lower order term is present, see e.g. [4, 54]. However, there are also applications where the existence of lower order terms can not be expected such as the work at hand which deals with the passage to rescaled congurations. Moreover, in a wide class of problems arising from discrete energies one typically does not have an L1 bound for the functions as the energies only depend on the relative distance of the material points. The aforementioned result sheds a new light on this problem. Exploiting condition (6.8)(ii) we may derive a compactness result for energies (6.15) without any extra term by subtracting suitable innitesimal rigid motions on a partition of the domain.

6.3

Γ-convergence and application to cleavage laws

We now show that the energies Eε converge to a Grith functional with linearized elastic energy. Let Q = D2 W (Id) be the Hessian of the stored energy density W at the identity. Dene E : D∞ → [0, ∞) by ˆ 1X 1 Q(e(∇T T ∇u)) + P (Pj , Ω) + H1 (Ju \ ∂P ), (6.16) E(u, P, T ) = j 2 Ω 2 S where as before P = (Pj )j and ∂P = j ∂ ∗ Pj . The surface energy of the limiting functional has two parts. We call the left part segmentation energy and the right part inner crack energy. Recall that we say (uk , P k , Tk ) → (u, P, T ) if (6.10)(6.13) hold.

Theorem 6.3.1. Let εk → 0. Then Eεk Γ-converge to E with respect to the convergence given in Theorem 6.2.1, i.e. (i) Γ − lim inf inequality: For all (u, P, T ) ∈ D∞ and for all sequences (yk )k ⊂ SBVM (Ω) and corresponding (uk , P k , Tk ) ∈ D as given in Theorem 6.2.1 such that (uk , P k , Tk ) → (u, P, T ) we have lim inf Eεk (yk ) ≥ E(u, P, T ). k→∞

124

(ii) Existence of recovery sequences: For every (u, P, T ) ∈ D∞ with u ∈ L2 (Ω) we nd a sequence (yk )k ⊂ SBVM (Ω) and corresponding (uk , P k , Tk ) ∈ D such that (uk , P k , Tk ) → (u, P, T ) and lim Eεk (yk ) = E(u, P, T ).

k→∞

As a direct consequence we get that the Γ-limit is given by the segmentation energy if we do not pass to rescaled congurations.

Corollary 6.3.2. Let εk → 0. Then Eεk Γ-converge to Eseg with respect to the L1 (Ω)-convergence, where ( P 1 Eseg (y) =

2

j

P (Pj , Ω) y = T ∈ R(P) for a Caccioppoli partition P,

+∞

else.

Finally, as an application of the above results we return to the investigation of cleavage laws (see Sections 1.3, 1.4, 1.6.2). We consider a special boundary value problem of uniaxial compression/extension. Let Ω = (0, l) × (0, 1), Ω0 = (−η, l + η) × (0, 1) for l > 0, η > 0 and for aε ∈ R dene

A(aε ) := {y ∈ SBVM (Ω0 ) : y1 (x) = (1 + aε )x1 for x1 ≤ 0 or x1 ≥ l}. As usual in the theory of SBV functions the boundary values have to be imposed in small neighborhoods of the boundary. In what follows the elastic part of the energy (6.2) still only depends on y|Ω , whereas the surface energy is given by H1 (Jy ) with Jy ⊂ Ω0 . In particular, jumps on {0, l} × (0, 1) contribute to the energy Eε (y) (compare also the discussion before Theorem 1.6.3). The present problem in the framework of continuum fracture mechanics with isotropic surface energies is a slightly simplied model of the version considered in Section 1.6.2. detQ ˆ As a preparation we dene α = det ˆ , where Q arises from Q by deleting the Q 2×2 rst row and column (see Lemma 2.1.2). Moreover, let F a ∈ Rsym be the unique T a a T matrix such that e1 F e1 = a and Q(F ) = inf{Q(F ) : e1 F e1 = a} = αa2 . We recall that the proof of the result in Theorem 1.3.1 and the special case in Theorem 1.4.1 fundamentally relied on the application of certain slicing techniques which were not suitable to treat the case of compression. Having general compactness and Γ-convergence results we can now complete the picture found in Section 1.6.2 by extending the results to the case of uniaxial compression. √ Theorem 6.3.3. Suppose aε / ε → a ∈ [−∞, ∞]. The limiting minimal energy

is given by

lim inf{Eε (y) : y ∈ A(aε )} = min

ε→0

q

n1 2

o αla , 1 . 2

(6.17)

Let acrit := 2αl . For every sequence (yε )ε of almost minimizers, up to passing to subsequences, we get ε−1/2 (yε (x) − x) → u(x) for a.e. x ∈ Ω, where 125

(i) if |a| < acrit , u(x) = (0, s) + F a x for s ∈ R, ( (0, s) x1 < p, (ii) if |a| > acrit , u(x) = for s, t ∈ R, p ∈ (0, l). (la, t) x1 > p,

6.4 Overview of the proof As the proof of the rigidity result is very long and technical, we present here a short overview and highlight the principal strategy. The rst idea is to replace the highly nonlinear problem of Theorem 6.1.1 by a linearized version (cf. Theorem 6.1.3) which is easier to treat since (1) the estimate only involves the function itself and not its derivative and (2) the set of innitesimal rigid motions is a linear space in contrast to SO(2). This estimate will be an essential ingredient to establish the general nonlinear result afterwards.

6.4.1 Korn-Poincaré-type inequality Essentially, (6.8)(ii) can be seen as a kind of Korn-Poincaré inequality. The classical Korn-Poincaré inequality in BD states that there is a constant C depending only on the domain Ω ⊂ R2 such that

ku − P ukL2 (Ω) ≤ C|Eu|(Ω)

(6.18)

for all u ∈ BD(Ω, R2 ), where P is a linear projection onto the space of innitesimal rigid motions and |Eu| denotes the total variation of the symmetrized distributional derivative (see Theorem B.4). Assuming that the linearized elastic energy is given by ke(∇u)k2L2 (Ω) ∼ ε, the aim is to show that the right hand side in (6.18) is of order ε. Clearly, this cannot be inferred directly by energy bounds since we only have control over the size of the discontinuity set which typically satises H1√ (Ju ) ∼ 1. The fundamental idea is to show that the jump height satises [u] ∼ ε in a suitable sense, whence the classical Korn-Poincaré inequality can be applied to obtain the desired result in (6.8)(ii). However, it turns√out that in general there is no hope to control the jump heights in terms of ε. This can already seen by rather simple congurations where the jump set is given by the boundary of balls (cf. the example in [5] demonstrating that BD does not embed into BV). Counterexamples can even be constructed in a way such that the domain is not disconnected by the jump set into dierent connected components as we show in Section 7.1. The main strategy of the work at hand is to nd bounds on the jump heights after a suitable modication of the jump set and the displacement eld whose total energy almost coincides with the original energy. An exact statement of the Korn-Poincaré-type inequality can be found in Theorem 8.0.1 below. 126

By a density argument we can assume that the jump set is contained in a nite number of rectangle boundaries. (We will call these sets boundary components or cracks in the following.) These boundaries will be altered during an iterative procedure. Clearly, we have to assure that in this process the length of the boundary components does not increase too much. To this end, it is convenient to measure the length of the jump set by a convex combination of the Hausdormeasure H1 and the `diameter' of a crack given by p (6.19) |Γ|∞ = |π1 Γ|2 + |π2 Γ|2 , where Γ denotes the boundary component and π1 , π2 the orthogonal projections onto the coordinate axes. One of the advantages of | · |∞ in contrast to H1 is that due to the strict convexity of | · |∞ it is often energetically favorable if dierent cracks are combined to one larger boundary component leading to a simplication of the jump set. Nevertheless, during the modication process it cannot be avoided that additional cracks are added near original ones. To keep track of this amplication, the boundary components have to be assigned with a weight which indicates if (or: how much) this crack has already been `used' to introduce another discontinuity set. Now a further diculty arises from the the fact that during each iteration step of the modication these weights have to be carefully adjusted (see Section 8.2). The overall aim of the modication is to assure that in a small neighborhood of a boundary component Γ the energy can be controlled. Indeed, it turns out that if the elastic energy exceeds ε|Γ|∞ or the size of the jump set in a neighborhood is much larger than |Γ|∞ , then it is energetically favorable to replace the crack by a larger rectangle and to replace the function u in the interior of the rectangle by an innitesimal rigid motion (see proof of Theorem 8.4.2a),b)). Moreover, the modication of the jump set occurs not only due to energetic but also due to geometrical reasons. Exploiting the properties of | · |∞ we can nd a ner characterization of the cracks in the neighborhood, e.g. one can show that there are at most two cracks whose size is comparable to |Γ|∞ (see Corollary 8.3.4). Furthermore, we can always nd small stripes in the neighborhood which do not intersect the jump set (see Lemma 8.3.5). Having these properties for the neighborhood of a rectangle Γ and assuming p that for all smaller cracks Γl we have already established that [u] ∼ |Γl |∞ ε, the main technical issue is to derive a trace estimate on the boundary of Γ. Then replacing the function u by an appropriate innitesimal rigid motion in the p interior of the rectangle we will indeed obtain [u] ∼ |Γ|∞ ε on the boundary of Γ. Consequently, the assertion can be proved using an algorithm which iteratively changes the jump set and determines the trace at boundary components once the required conditions in a neighborhood are fullled (see Theorem 8.4.2). Obviously one expects that the crack opening of small cracks is generically small. In our framework this heuristically follows by a rescaling argument in 127

(6.18) and the observation that after modication the energy in a neighborhood is bounded by ∼ |Γ|∞ ε. However, a rigorous investigation of the trace on the boundary of Γ is very subtle as due to the iterative application of the arguments the involved constants might become arbitrarily large. The proof will be carried out in several steps. In the rst step we assume that in the neighborhood N of Γ only small cracks Γl are present. This indeed induces pthat |Eu|(N ) is suciently small as on each Γl we have already shown [u] ∼ |Γl |∞ ε. In general, the idea is to construct thin long paths in N which avoid cracks being to large. We then rst measure the distance of the function from an innitesimal rigid motion only on this path and may apply this result to estimate the distance in the whole set N afterwards (see Section 8.5.2). A major drawback of such a technique is that the constant in (6.18) crucially depends on the domain and explodes for sets getting arbitrarily thin. Consequently, in this context we have to carry out a careful quantitative analysis how the constant in (6.18) depends on the shape of the domain (see Section 7.1). It turns out, however, that the paths in general cannot be selected in a way such that they only intersect suciently small cracks. Nevertheless, it can be shown that boundary components being too large for a direct application of the above ideas occupy only a comparably small region. In this region we then do not use the Korn-Poincaré inequality (Theorem B.4), but circumvent the estimation of the surface energy by a slicing technique. Indeed, by the modication procedure alluded to above we always nd small stripes in the neighborhood which do not intersect the jump set at all. The assertion then follows as this exceptional set can then taken arbitrarily small by an iterative application of the slicing method (see Section 8.5.3). We briey note that such a technique is only employable as we treat a linear problem. This is one of the main reasons why the derivation of Theorem 6.1.3 is easier than the proof of Theorem 6.1.1. (Compare also [25], where the treatment of the linearized version is remarkably easier than the nonlinear problem due to the applicability of a slicing method.) Finally, one has to face the problem that there are (at most) two other cracks Γ1 , Γ2 intersecting N being larger than Γ. In particular, (6.18) cannot be directly applied since no estimate of the jump heights at Γ1 and Γ2 is available. However, the result can also be established in this case if the elastic and surface energy in the two areas close to Γ, Γ1 and Γ, Γ2 is suciently small (see Section 8.5.5). In fact, such a smallness assumption can always be inferred by a careful modication of the crack set (see proof of Theorem 8.4.2c) and Section 8.3.2). Finally, we remark that the result crucially depends on the application of a suitable L2 - trace theorem for SBV functions (see Lemma 7.2) which can be established in our framework because of the suciently regular jump set. Moreover, it is essential that there are at most two large cracks in a neighborhood. Already with three or four cracks the congurations might be signicantly less rigid. 128

6.4.2 SBD-rigidity The main estimates in the rigidity result (see (6.6)) do not only provide bounds for the function itself but also for the derivative. The key point is the derivation of an estimate for the symmetric part of the gradient. Using the expansion

|e(RT (∇y − Id))|2 = dist2 (∇y, SO(2)) + O(|∇y − R|4 )

(6.20)

and recalling that k dist(∇y, SO(2))k2L2 (Ω) ∼ ε we see that it suces to establish an estimate of fourth order. Indeed, also in the proof of the geometric rigidity result in nonlinear elasticity (see [49]) one rst derives a bound for k∇y − Rk4L4 (Ω) to control the symmetric part. The control over the full gradient is then obtained by Korn's inequality. Clearly, in our framework the result in Theorem B.1 cannot be applied due to the presence of cracks, in particular Ω \ Jy will generically not be a Lipschitz set. Therefore, by a density argument we again rst assume that the jump set is contained in a nite number of rectangle boundaries. A careful quantitative analysis shows that the constant in (B.1) depends on the quotient of the diameter of the domain, denoted by k , and the minimal distance of two cracks, denoted by s. In particular, C = C(k/s) ∼ 1 if k ∼ s. Provided that ks is not too large, the principal strategy will be to show that possibly after a modication we get k∇y − Rk2L∞ (Ω) ≤ (C(k/s))−1 which then gives

ke(RT (∇y − Id))k2L2 (Ω) ≤ ε + (C(k/s))−1 k∇y − Rk2L2 (Ω) ≤ Cε

(6.21)

by (6.20) and Theorem B.1. Of course, in general we cannot suppose that ks is not large. Moreover, a global rigidity result may fail due to the separation of the domain by the jump set. Consequently, we will apply the presented ideas on a ne partition of the Lipschitz domain Ω consisting of squares with diameter k . This local result will be used to modify the jump set such that the minimal distance of each pair of cracks increases. Then we can repeat the arguments for a larger k . The idea is that after an iterative application of the arguments we obtain an estimate for k ≈ ρ which then will provide rigid motions on the connected components of the domain (see (6.5)) with the desired properties. In Section 9.2 we construct piecewise constant SO(2)-valued mappings approximating the deformation gradient. In each square Q of diameter k we may assume that the elastic energy is bounded by ∼ εk as otherwise it would be energetically favorable to introduce jumps at the boundary of the square and to replace the deformation in the interior by a rigid motion. (The same technique has been used in the proof of the Korn-Poincaé inequality.) Similarly as in [49] we pass to the harmonic part of the deformation (denoted by yˆ) and obtain by the mean value property −2 k∇ˆ y − RQ k2L∞ (Q) y − RQ k2L2 (Q) ˆ ≤ Ck k∇ˆ

≤ C(k/s)k −2 k dist(∇y, SO(2))k2L2 (Q) ≤ C(k/s)k −1 ε 129

(6.22)

ˆ ⊂ Q is a slightly smaller square. Consefor a suitable RQ ∈ SO(2), where Q ε quently, if we can assure that k ≤ (C(k/s))−2 we obtain the desired L∞ -bound which allows to derive an estimate of the form (6.21). We note that for this argument we at least have to assume that k  ε which will be denoted as the `superatomistic regime' (recall the discussion about the signication of ε after (6.2)). In the subsequent Section 9.2.2 we show that not only the distance of the derivative from a piecewise rigid motion can be controlled but also the distance of the function itself. On the one hand this is essential for (6.6), on the other hand such an estimate is crucial for establishing a modication of the deformation and the jump set. The main idea is to apply the Korn-Poincaré-type inequality on the T function RQ y−id. Major diculties arise from the facts that the rotation RQ may vary from one square to another and that the inequality derived in Section 8 only provides a local estimate (see formulation in Theorem 8.4.8). Consequently, the arguments have to be repeated for several shifted copies of the ne partition (see Lemma 9.2.3). Moreover, the projections PQ onto the the space of innitesimal rigid motions (see (6.18)) have to be combined with the rotations RQ in a suitable way to obtain appropriate rigid motions, which do not vary too much on adjacent squares (see Lemma 9.2.5). Having an approximation of the deformation by piecewise rigid motions dened on squares with diameter k , we then are able to modify the function such that the minimal distance s˜ of two cracks of the new conguration satises s˜ ∼ k (see Lemma 9.3.1). Now we can repeat the above procedure for some larger k˜ ˜ s))−2 is guaranteed and we can repeat the arguments in such that ε/k˜ ≤ (C(k/˜ (6.22). The strategy is to end up with k ≈ ρ after a nite number of iterations. As the number of iteration steps is not bounded but grows logarithmically with 1ε we have to assure that in each step the surface and the elastic energy do not increase to much. The crucial point is that during the iteration process the coarseness of the partition k grows much faster than the stored elastic energy ε such that the argument in (6.22) may be repeated. The details are given in Theorem 9.4.3. Having an estimate for k ≈ ρ it is then not hard to establish the desired result up to a small exceptional set (see Theorem 9.4.2). Clearly, we cannot assume that initially s ≥ ε. In this case the argument in (6.22) can typically not be applied. As a remedy we do not employ the geometric rigidity result directly but rst approximate the deformation in each square by an H 1 -function, where the distance can be measured by the curl of ∇y . (see Theorem A.1.9 below which was one of the essential ingredients to prove the qualitative result in [25].) We address this problem in Lemma 9.2.2 and subsequently we show that we may modify the conguration such that s˜ ≥ ε (see Theorem 9.4.4). Finally, by a density argument we can approximate each SBV function by a conguration where the jump set is contained in a nite number of rectangle boundaries (see proof of Theorem 9.4.1). Observe that standard density results 130

as [31] cannot be applied directly in our framework since in general an L∞ bound for the derivative is not preserved. The problem can be circumvented by using a dierent approximation introduced in [26] at cost of a non exact approximation of the jump set, which suces for our purposes. The rigidity result, which we then have established, only holds up to a small exceptional set as due to the modication of the jump set the deformation might not be dened in the interior of certain rectangles. We emphasize that such an estimate is not enough to obtain good compactness and convergence results, in particular for the convergence of the surface energy further diculties arise. Therefore, we eventually have to construct a suitable extension to the whole domain. A major challenge is to determine the surface energy correctly, at least for the relaxed functional (6.3). This problem is addressed in Section 9.5. For small cracks a good extension is already provided by the Korn-Poincaré inequality. Near large cracks we dene the extension as a piecewise constant rigid motion such that the jump heights on the new jump sets are suciently small (see the proof of Theorem 6.1.1). Consequently, the length of these jumps may possibly be much larger than H1 (Jy ), but due to the small jump height their contribution to (6.3) is considerably small. Whereas the diameter dened in (6.19) was very convenient for the derivation of the linearized estimate, it does not provide the right surface energy for large cracks. Thus, for large boundary components, in particular for the boundary ∂P of the partition (Pj )j , we have to construct an appropriate jump set consisting of Jordan curves which provides the correct crack energy up to a small error (see Lemma 9.5.1).

6.4.3 Compactness and Γ-convergence The SBD-rigidity estimate turns out to be the fundamental ingredient to derive compactness and Γ-convergence results for functionals of the form (6.2). Theorem 6.2.1 essentially relies on a diagonal sequence argument for ε, ρ → 0, where ρ denotes the `error' of the modication obtained by Theorem 6.1.1. The convergence of the partitions and the corresponding rigid motions is based on compactness theorems for Caccioppoli partitions and piecewise constant functions (see Section A.2). A major challenge for the compactness of the rescaled congurations is the fact that the constant in (6.6) depends on ρ and explodes for ρ → 0. For the symmetric part of the gradient this problem can be bypassed by taking (6.20), (6.6)(iv) and Eεk (yεk ) ≤ C into account, which shows that the constant may be chosen independently of ρ. For the function itself, however, the problem is more subtle since a uniform bound cannot be inferred by energies bounds. In particular, generically the limiting congurations are not in L2 , but only nite almost everywhere. The strategy to establish the latter assertion is to show that for xed ε the functions (uρε )ρ essentially coincide in a certain sense on the bulk part of the domain (see Lemma 10.1.2) if one chooses the rigid motions in (6.5) 131

in an appropriate way. Afterwards, by a careful analysis we can derive that such a property is preserved in the limit ε → 0 (see the proof of Theorem 6.2.1). To establish (6.13) one has to separate the eects arising form the segmentation energy and the inner crack energy. This can be done by employing a structure theorem for Caccioppoli partitions (see Theorem A.2.1). Moreover, the estimate is rst carried out in terms of the relaxed functionals (6.3) and drawing ideas from Section 5.1 we conclude that it is also satised for E . In Theorem 6.2.4 the fundamental point is the proof of existence and uniqueness of the coarsest partition. Uniqueness follows from the observation that under the assumption that there are two dierent coarsest partitions one always can nd an even coarser partition. Existence is a more challenging problem. We rst give an alternative characterization and identify coarsest partitions as the maximal elements of the partial order on the set of admissible partitions which is induced by subordination. We then show that each chain of the partial order has an upper bound repeating some arguments of the main compactness result. Consequently, the claim is inferred by an application of Zorn's lemma. Finally, the identication of the limiting congurations follows by measuring the dierence of admissible triples and taking the linearization formula (6.20) into account. Eventually, the Γ-convergence result is straightforward since the main estimates, in particular the estimates for the surface energy, were provided in Theorem 6.2.1. To establish Theorem 6.3.3 we follow the proof of Theorem 1.6.5.

132

Chapter 7 Preliminaries In this short preparatory chapter we establish a trace theorem and analyze how the constants of the geometric rigidity result and the Korn-Poincaré-inequality depend on the shape of the domain.

7.1 Geometric rigidity and Korn: Dependence on the set shape In general, the constants of the inequalities stated in Section B depend crucially on the set shape. This will be discussed in detail in this section. As an introductory example we consider the deection of a thin elastic beam.

Example 7.1.1. Let U = (0, 1)×(0, δ) and let y : U → R2 be given by y(x1 , x2 ) = (x2 + 1)(sin(x1 ), cos(x1 )). Then



 (x2 + 1) cos(x1 ) sin(x1 ) ∇y(x1 , x2 ) = −(x2 + 1) sin(x1 ) cos(x1 ) p and therefore dist2 (∇y, SO(2)) = | ∇y T ∇y − Id|2 = x22 , i.e. k dist(∇y, SO(2))k2L2 (U ) = 13 δ 3 .   cos φ sin φ Let Rφ ∈ SO(2), Rφ = for φ ∈ [0, 2π]. Then |∇y(x) − R|2 ≥ − sin φ cos φ | sin(x1 ) −´sin φ|2 + | cos(x1 ) − cos φ|2 . It is not hard to see that it exists a C > 0 1 such that 0 |∇y(x)−R|2 dx1 ≥ C for all φ ∈ [0, 2π] and x2 ∈ (0, δ). We conclude that C k∇y − Rk2L2 (U ) ≥ Cδ ≥ 2 k dist(∇y, SO(2))k2L2 (U ) δ for all R ∈ SO(2). A similar argument shows ky − (R · +c)k2L2 (U ) ≥ Cδ ≥

C k dist(∇y, SO(2))k2L2 (U ) δ2

133

for all R ∈ SO(2) and c ∈ R2 . Similar examples can be constructed in the linearized framework for the KornPoincaré inequality given in Theorem B.4. As a direct consequence we get that the estimate (6.6) might be wrong without allowing for a small modication of the deformation.

Example 7.1.2. Let ε > 0.

Assume without restriction that the set U = (0, 1) × 1 (0, ε 3 ) considered above satises U ⊂ Ω. Dene y : Ω → R2 by y(x) = id + e2 for x ∈ Ω\U and y(x) = (x2 +1)(sin(x1 ), cos(x1 )) for x ∈ U . Then y ∈ SBV 2 (Ω) with 1 1 Jy = (0, 1) × {0, ε 3 } ∪ {1} × (0, ε 3 ) and k dist(∇y, SO(2))k2L2 (Ω) = 3ε . However, for all R ∈ SO(2) and c ∈ R2 we have 1

1

k∇y − Rk2L2 (Ω) ≥ Cε 3 ,

ky − (R · +c)k2L2 (Ω) ≥ Cε 3 .

Although omitted here, a similar estimate can be derived for the symmetric part of the gradient. For s > 0 we partition R2 up to a set of measure zero into squares Qs (p) = p + s(−1, 1)2 for p ∈ I s := s(1, 1) + 2sZ2 . Let ◦ o n [ Qs (p) : I ⊂ I s . (7.1) U s := U ⊂ R2 : U = p∈I

Here the superscript ◦ denotes the interior of a set. In order to quantify how the constants in Theorem B.1 and Theorem B.4 depend on the set shape we will estimate the variation from a square Qs (a) to a neighboring square Qs (b), b = a + 2sν for ν = ±ei , i = 1, 2. We rst introduce some further notation. For y ∈ SBV 2 (U ) with U ∈ U s and for R ∈ SO(2) we set uR = RT y − id, where id is the identity function. On a cube Qs (p) ⊂ U we dene for shorthand (we drop the integration variable if no confusion arises) ˆ ˆ 2 γ(p) = dist (∇y, SO(2)), ER (p) = |¯ eR (∇y)| + |Dj y|(Qs (p) ∩ U ). Qs (p)

Qs (p)

Here the symmetric part of the gradient is dened by

e¯R (∇y) := e(∇uR ) =

RT ∇y + (∇y)T R − Id, 2

(7.2)

where Id denotes the identity matrix. Moreover, for subsets V ⊂ U , V ∈ U s we write ˆ X γ(V ) = γ(p), ER (V ) = |¯ eR (∇y)| + |Dj y|(V ), (7.3) s p∈I (V )

V

where I s (V ) := {p ∈ I s : Qs (p) ⊂ V }. 134

We rst assume y ∈ H 1 (U ) and proceed similarly as in [49]. Applying Theorem B.1 we obtain R(a), R(b) ∈ SO(2) such that ˆ |∇y − R(p)|2 ≤ Cγ(p) for p = a, b. (7.4) Qs (p)

Likewise on the rectangle Qs (a, b) := (Qs (a) ∪ Qs (b))◦ we obtain R(a, b) ∈ SO(2) such that ˆ ˆ 2 |∇y − R(a, b)| dx ≤ C dist2 (∇y, SO(2)) ≤ C(γ(a) + γ(b)). Qs (a,b)

Qs (a,b)

Combining these estimates we see |Qs (p)||R(p) − R(a, b)|2 ≤ C(γ(a) + γ(b)) for p = a, b and therefore (7.5)

s2 |R(a) − R(b)|2 ≤ C(γ(a) + γ(b)).

More general, we consider a dierence quotient with two arbitrary points a, b ∈ I s (U ). We assume that there is a path ξ = (ξ0 , . . . , ξm ) such that

ξ1 = a, ξm = b, ξj − ξj−1 = ±2sei for some i = 1, 2, ∀j = 2, . . . , m.

(7.6)

Then iteratively applying the above estimate (7.5) we obtain Xm s2 |R(a) − R(b)|2 dx ≤ Cm γ(ξj ).

(7.7)

j=1

Now we concern ourselves with the Korn-Poincaré inequality. Assume that y ∈ SBV 2 (U ) and recall uR = RT y − id for R ∈ SO(2). For later purposes, we consider more general rectangles and derive the dierence of the deformation on adjacent squares as a special case. Let b1 , b2 ∈ R2 , and Bi = bi + (−li , li ) × (−mi , mi ) ∈ U s for i = 1, 2, where we assume without restriction that l1 ≥ m1 > 0, l2 ≥ m2 > 0. Suppose that there is a point b12 ∈ B1 ∩ B2 . 2×2 For given A1 , A2 , A12 ∈ R2×2 : GT = −G} and c1 , c2 , c12 ∈ R2 skew = {G ∈ R 2 we set Ei := kuR − (Ai · +ci )kL2 (Bi ) for i = 1, 2 and suppose that (7.8)

kuR − (A12 · +c12 )k2L2 (B1 ∪B2 ) ≤ C(E1 + E2 ). As above this implies

k(Ai − A12 ) · +(ci − c12 )k2L2 (Bi ) ≤ C(E1 + E2 )

for i = 1, 2.

We let Bi− = bi + (−li , 0) × (−mi , mi ), Bi+ = bi + (0, li ) × (−mi , mi ) and for shorthand we write Aˆi = Ai − A12 , cˆi = ci − c12 . We then derive

ci k2L2 (B − ) |Bi | li2 |Aˆi |2 ≤ 4kAˆi li e1 k2L2 (B − ) = 4kAˆi (· + li e1 ) + cˆi − Aˆi · −ˆ i

i

≤ 8kAˆi · +ˆ ci k2L2 (B + ) + 8kAˆi · +ˆ ci k2L2 (B − ) ≤ C(E1 + E2 ) i

i

135

(7.9)

for i = 1, 2 and therefore

|B1 ∪ B2 |(l1 + l2 )2 |A1 − A2 |2 ≤ Cκ(E1 + E2 ),

(7.10)


|B1 ∪B2 | l1 +l2 2 where κ = min . Observe that in the rst equality we essentially used |B | min l j j j j the skew symmetry. Since |y − b12 | ≤ C(l1 + l2 ) for all y ∈ B1 ∪ B2 we likewise compute |B1 ∪ B2 | ˆ kAi · +ˆ ci k2L2 (Bi ) + C|B1 ∪ B2 |(l1 + l2 )2 |Aˆi |2 |B1 ∪ B2 ||Aˆi b12 + cˆi |2 ≤ C |Bi | ≤ Cκ(E1 + E2 ) for i = 1, 2. Employing the triangle inequality we then deduce

|B1 ∪ B2 ||(A2 − A1 ) b12 + c2 − c1 |2 ≤ Cκ(E1 + E2 ).

(7.11)

Consider Z ⊂ B1 ∪ B2 , Z ∈ U s . Similar arguments yield by (7.10)

k(A2 − A1 ) · +c2 − c1 k2L2 (Z) ≤ Ck(A2 − A1 )b12 + c2 − c1 k2L2 (Z) + C|Z| maxj lj2 |A1 − A2 |2 |Z| ≤C κ(E1 + E2 ) |B1 ∪ B2 |

(7.12)

and therefore by the triangle inequality

kuR − (A1 · +c1 )k2L2 (B2 ∩Z) ≤ CkuR − (A2 · +c2 )k2L2 (B2 ∩Z) +C

|Z| κ(E1 + E2 ). |B1 ∪ B2 |

(7.13)

In particular, employing Z = B1 ∪ B2 and recalling (7.8) we nd

kuR − (A1 · +c1 )k2L2 (B1 ∪B2 ) ≤ Cκ(E1 + E2 ).

(7.14)

Before we treat the case of two adjacent squares we observe that in the above estimates the constants may be rened in the case that B1 ⊂ B2 under additional assumptions on the energies. Let δ ≥ Csl2−1 . Let B2 = (−l2 , l2 ) × (−s, s) ∈ U s and B1 ⊂ B2 , B1 ∈ U s a general set such that |B2 \ B1 | ≤ δ|B2 |. In particular, this implies that the diameter of each connected component of B2 \ B1 is smaller than Cδl2 . Moreover, we assume that for all Z ⊂ B2 , Z ∈ U s one has

kuR − (Ai · +ci )k2L2 (Bi ∩Z) ≤ |Bi ∩ Z||B2 |−1 Hi

(7.15)

for H1 , H2 ≥ 0. Arguing similarly as in (7.10) we nd |A1 −A2 |2 ≤ C|B2 |−1 l2−2 (H1 + H2 ). (Observe that the connectedness of B1 is not necessary. Moreover, the estimate can also be derived if B2 consists of several connected components.) We 136

write A˜ = A1 − A2 and c˜ = c1 − c2 for shorthand. Let b0 ∈ B1 and Q ⊂ B1 be the square containing b0 . Applying a scaled version of Young's inequality and using s ≤ Cδl2 we compute

|Q||A˜ b0 + c˜|2 = kA˜ b0 + c˜k2L2 (Q) ≤ (1 + δ)kA˜ · +˜ ck2L2 (Q) + (1 + 1δ )kA˜ (· − b0 )k2L2 (Q) ≤ (1 + δ)2 kuR − (A1 · +c1 )k2L2 (Q) + Cδ kuR − (A2 · +c2 )k2L2 (Q) + Cδ |Q|s2 |B2 |−1 l2−2 (H1 + H2 ) ≤ (1 + Cδ)|B2 |−1 |Q|H1 + Cδ |B2 |−1 |Q|H2 + C|B2 |−1 |Q|δ(H1 + H2 ) ≤ (1 + Cδ)|B2 |−1 |Q|H1 + Cδ |B2 |−1 |Q|H2 . Consider some connected Z ⊂ B2 \ B1 , Z ∈ U s , and observe that we nd some b0 ∈ B1 such that |x − b0 | ≤ Cδl2 . Then repeating the above calculation, again employing Young's inequality, we derive

˜2 kA˜ · +˜ ck2L2 (Z) ≤ (1 + δ)|Z||A˜ b0 + c˜|2 + (1 + 1δ )|Z| max |x − b0 |2 |A| −1

≤ (1 + Cδ)|B2 | |Z|H1 +

x∈Z −1 C |B2 | |Z|H2 . δ

Then Young's inequality yields

kuR − (A1 · +c1 )k2L2 (Z) ≤ (1 + δ)kA˜ · +˜ ck2L2 (Z) + (1 + 1δ )kuR − (A2 · +c2 )k2L2 (Z) ≤ (1 + Cδ)|B2 |−1 |Z|H1 + Cδ |B2 |−1 |Z|H2 .

(7.16)

Finally, it is not hard to see that (7.16) holds for all Z ⊂ B2 , Z ∈ U s . Now assume the special case that B1 = Qs (a) and B2 = Qs (b), b = a + 2sν for ν = ±ei , i = 1, 2. Then by Theorem B.4 we obtain A(a), A(b), A(a, b) ∈ R2×2 skew and c(a), c(b), c(a, b) such that

kuR − (A(p) · +c(p))kL2 (Qs (p)) ≤ CER (p) for p = a, b, kuR − (A(a, b) · +c(a, b))kL2 (Qs (a,b)) ≤ CER (a, b),

(7.17)

where for shorthand ER (a, b) = ER (Qs (a, b)). As in this case κ = 8, (7.10) and (7.12) for Z = Qs (a, b) yield

s2 |A(a) − A(b)| ≤ CER (a, b), k(A(b) − A(a)) · +c(b) − c(a)kL2 (Qs (a,b)) ≤ CER (a, b). Similarly as in (7.6) we now consider a dierence quotient with two arbitrary points a, b ∈ I s (U ) connected by a path ξ = (ξ1 , . . . , ξm ). Iterative application of

137

the last estimate yields

k(A(b) − A(a)) · +c(b) − c(a))kL2 (Qs (b)) Xm ≤ k(A(ξj ) − A(ξj−1 )) · +c(ξj ) − c(ξj−1 ))kL2 (Qs (b)) j=2 Xm ≤ k(A(ξj ) − A(ξj−1 )) · +c(ξj ) − c(ξj−1 ))kL2 (Qs (ξj )) j=2 Xm + 2s|(A(ξj ) − A(ξj−1 )) (b − ξj )| j=2 Xm Xm ≤C ER (ξj ) + 2s|A(ξj ) − A(ξj−1 )|m2s j=1 j=2 Xm ≤ Cm ER (ξj , ξj−1 )

(7.18)

j=2

and therefore

kuR − (A(a) · +c(a))k2L2 (Qs (b)) ≤ 2kuR − (A(b) · +c(b))k2L2 (Qs (b)) + 2k(A(b) − A(a)) · +c(b) − c(a))k2L2 (Qs (b))  Xm 2 Xm ≤ Cm2 ER (ξj , ξj−1 ) ≤ Cm3 j=2

(7.19)


2

ER (ξj , ξj−1 ) .

j=2

In the last step we have used Hölder's inequality. We now apply (7.7) and (7.19) to derive a rst weak rigidity result and a Korn-Poincaré-type inequality, respectively.

Lemma 7.1.3. Let µ, s > 0 such that l := µs−1 ∈ N. Then there is a constant C > 0 independent of µ, s such that for all connected sets U ∈ U s , U ⊂ (−µ, µ)2 , the following holds: (i) For all y ∈ H 1 (U ) there is a rotation R ∈ SO(2) such that ˆ

ˆ

ˆ

−2

2

2

2

|∇y − R| ≤ C(s |U |)

dist (∇y, SO(2)) ≤ Cl U

U

dist2 (∇y, SO(2)).

4 U

2×2 (ii) For all y ∈ SBV 2 (U ) and all rotations R ∈ SO(2) there is an A ∈ Rskew and c ∈ R2 such that

ˆ

|uR (x) − (A x + c)|2 dx ≤ C(s−2 |U |)3 (ER (U ))2 . U

´

(iii) More precisely, for all V ⊂ U , V ∈ U s one has (−V stands for

1 |V |

ˆ |U |− |uR (x) − (A x + c)|2 dx ≤ C(s−2 |U |)3 (ER (U ))2 . V

138

´

)

Proof. We rst show (i). The second inequality is obvious as |U | ≤ 4µ2 . To see the rst inequality we x p0 ∈ I s (U ) and consider an arbitrary p ∈ I s (U ). As U is connected there is a path ξ = (ξ1 = p0 , . . . , ξm = p) with m ≤ |U |(2s)−2 . We rst apply (7.4) on each square and then by (7.7) we obtain ˆ Xm γ(ξj ) ≤ C|U |s−2 γ(U ). |R(p) − R(p0 )|2 ≤ C|U |s−2 j=1

Qs (p)

Then setting R = R(p0 ) and summing over all p ∈ I s (U ) we derive ˆ ˆ   X 2 2 2 |∇y − R| ≤ C |∇y − R(p)| + |R(p) − R(p0 )| s p∈I (U )

U

≤C

X p∈I s (U ) −2 2

Qs (p)

(γ(p) + |U |s−2 γ(U )) ≤ C#I s (U ) |U |s−2 γ(U )

≤ C(|U |s ) γ(U ). Property (ii) can be proved similarly using (7.19) instead of (7.7). Finally, (iii) is a direct consequence of (ii) since we may replace #I s (U ) by #I s (V ) if we only integrate over the set V . 

Remark 7.1.4.

(i) The fact that we covered the sets U with squares is not essential. Recalling how we derived (7.5) and (7.10) we could equally well cover U with rectangles Ri = ti +(−ai , ai )×(−bi , bi ), where c1 ai ≤ bi ≤ c2 ai and c1 s ≤ bi ≤ c2 s for constants 0 < c1 < c2 . The constants in (7.4) and (7.17), respectively, only depend on c1 , c2 as all the possible shapes are related to (−s, s)2 through bi-Lipschitzian homeomorphisms with Lipschitz constants of both the homeomorphism itself and its inverse bounded (see Section B).

(ii) Let U = (0, 1) × (0, δ). If we choose s = 2δ , Lemma 7.1.3(i) provides a constant ∼ δ −2 . Example 7.1.1 shows that this estimate is sharp in the sense that the exponent of δ cannot be improved. (iii) The argumentation developed in (7.8)-(7.14) remains true if we pass from the linearized to the nonlinear setting, i.e. replacing R2×2 skew by SO(2). This follows from the fact that we essentially use the property that the matrices satisfy |Ae1 | = |Ae2 | (cf. (7.9)). Likewise, we can estimate the dierence of rigid motions on chains similarly as in (7.19). This will be used in Section 9. (iv) Following the above arguments we nd that in Lemma 7.1.3(i) one can replace p = 2 by any 1 < p < ∞ replacing l4 suitably by l2p . (v) In view of the proof of (i),(ii), in the choice of R and A x + c we have the freedom to select any of the rotations or innitesimal rigid motions which are given on each square Qs (p) ⊂ U by application of (7.4) or (7.17), respectively. 139

7.2 A trace theorem in SBV2 .

By the trace theorem for BD functions (Theorem B.5) one can control the L1 norm of the function on the boundary. In our framework we may establish a trace theorem in L2 for SBV2 functions if the jump set is suciently regular: Let Qµ = (−µ, µ)2 and recall the denition of SBV 2 (Qµ ) in Section A.1. We T suppose that some y ∈ SBV 2 (Qµ ) or S the corresponding uR = R y − id, R ∈ SO(2), respectively, satises JuR = j Γj ∩ Qµ , where Γi = ∂Ri for rectangles Ri = (ai1 , ai2 ) × (bi1 , bi2 ) ⊂ R2 (note that for the application we have in mind we do not require that the rectangles are subsets of Qµ .). Clearly, as uR ∈ H 1 (Qµ \ JuR ) the trace is well dened in L2 . More precisely, we have the following statement.

Lemma 7.2.1. Let R ∈ SO(2) andSµ > 0. There is a constant C > 0 such that for all uR ∈ SBV 2 (Qµ ) with JuR = nj=1 Γj ∩ Qµ , where Γj = ∂Rj , one has ˆ |uR |2 dH1 ≤ Cµke(∇uR )k2L2 (Qµ ) + Cµ kuR k2L2 (Qµ ) ∂Qµ ˆ  Xn Xn  1 −1 2 1 1 (H (Γj )) |[uR ]| dH . +C H (Γj ) j=1

j=1

(7.20)

Γj ∩Qµ

Proof. Let Qµ = (−µ, µ)2 and uR ∈ SBV 2 (Qµ ) with JuR =

Sn

Γj ∩ Qµ . In what follows we drop the subscripts µ and R for notational convenience. First by approximation of Sobolev functions on Lipschitz sets (see, e.g., [40, Section S 4.2]) we may assume that uR |Rj is smooth for j = 0, . . . , n, where R0 = Q \ nj=1 Rj . We only consider the part ∂ 0 Q = (−µ, µ)×{µ} of the boundary. Let πx = {x}×R and compute for the second component u2 by a slicing argument in e2 -direction: ˆ µˆ µ ˆ µ ˆ µ ˆ µ 2 2 |u2 (x, µ) − u2 (x, y)| dx dy = D u (x, t) dt dx dy 2 2 −µ −µ −µ −µ y ˆ µˆ µ ˆ µ  X 2  2 ≤C µ |∂2 u2 (x, t)| dt + |[u](z)| dx dy −µ

−µ

−µ

ˆ

j=1

z∈Ju ∩πx µ

≤ Cµ2 ke(∇u)k2L2 (Q) + Cµ −µ

 X

2 |[u](z)| dx.

z∈Ju ∩πx

In the second step we have used Hölder's inequality. We now estimate the term on the right side. As Γj is a rectangle, except for two x-values there are exactly two points t1j , t2j ∈ R such that Γj ∩ πx = {(x, t1j ), (x, t2j )} if Γj ∩ πx 6= ∅ . We write P |Γj |H = H1 (Γj ), |S|H = j H1 (Γj ) for shorthand. Letting zjk,x = (x, tkj ) ∈ R2 and setting |[u](zjk,x )| = 0 if zjk,x ∈ / Q ∩ Γj , we then obtain by the discrete version

140

of Jensen's inequality ˆ µ X ˆ 2 |[u](z)| dx = 4 −µ

µ

2 |Γj |H k,x |S|H |[u](zj )| dx j k=1,2 2|S|H |Γj |H −µ ˆ µX X |S|H 2 |Γj |H  |[u](zjk,x )| ≤4 dx j k=1,2 2|S|H |Γj |H −µ ˆ  X  −1 |[u]|2 dH1 . ≤ 2|S|H |Γj |H

z∈Ju ∩πx

X X

j

Γj ∩Q

Consequently, letting E be the right hand side of (7.20) we derive ˆ ˆ ˆ  C µ µ 2 1 |u2 | dH ≤ |u2 (x, µ) − u2 (x, y)|2 dx dy + kuk2L2 (Q) ≤ CE. µ ∂0Q −µ −µ The same argument with slicing in the directions ξ1 = √1 (−1, −1) yields 2

√1 (1, −1) 2

and ξ1 =

ˆ

ˆ 2

|u · ξ2 |2 dH1 ≤ CE,

1

|u · ξ1 | dH ≤ CE, ∂20 Q

∂10 Q

where ∂10 Q = (−µ, 0) × {µ} and ∂20 Q = (0, µ) × {µ}. The claim now follows by combination of the previous estimates. 

141

Chapter 8 A Korn-Poincaré-type inequality This section is devoted to the derivation of a Korn-Poincaré-type inequality which roughly√ speaking emerges from ´the inequality in Theorem B.4 by replacing j |E uR |(Ω) by εH1 (JuR ), where ε ∼ Ω |e(∇uR )|2 . The√main strategy will be to show that the jump height on JuR is at most of order O( ε) from which the claim will follow by application of Theorem B.4. The examples in Section 7.1, however, show that this might not be possible in general. Therefore, in Sections 8.2, 8.3 we rst introduce a suitable modication scheme to alter the jump set. Section 8.5 then contains the main technical estimates for the analysis of the jump height. In Section 8.4 we combine the previously established results and present an algorithm which iteratively modies the jump set such that the estimates on the jump height may be applied. We rst formulate the main result of this chapter. For µ > 0 let Qµ = (−µ, µ)2 and by diam(R) denote the diameter of a rectangle R ⊂ Qµ .

Theorem 8.0.1. Let ε > 0 and h∗ > 0 suciently small. Then there is a constant C = C(h∗ ) and a universal constant c¯ > 0 such that for all u ∈ SBD2 (Qµ , R2 )∩L2 (Qµ , R2 ) the following holds: We get pairwise disjoint, paraxial rectangles R1 , . . . , Rn with Xn j=1

diam(Rj ) ≤ (1 + c¯h∗ ) H1 (Ju ) + ε−1 ke(∇u)k2L2 (Qµ )




such that for E := nj=1 Rj and the square Q˜ = (−˜µ, µ˜)2 with µ˜ = max{µ − P P 2 j diam(Rj ), 0} we have |E| ≤ c¯( j diam(Rj ))2 and S

ku(x) − (A x + c)k2L2 (Q\E) ≤ Cµ2 ke(∇u)k2L2 (Qµ ) + Cµ2 εH1 (Ju ) ˜ 2 for some A ∈ R2×2 skew and c ∈ R .

This Korn-Poincaré-type estimate is in the spirit of the Poincaré inequality in SBV due to De Giorgi, Carriero, Leaci (see [39]) with the dierence that we do not truncate the function (which is forbidden in the SBD framework), but 143

S provide an exceptional set E = j Rj , where the estimate does not hold. This set is associated to the parts of Qµ being detached from the bulk part of Qµ by Ju . In contrast to the recently established estimate in [24], Theorem 8.0.1 provides an exceptional set with a rather simple geometry. Most notably we have control over H1 (∂E) which allows to apply compactness results for GSBD functions (see Theorem A.1.3). Moreover, for h∗  1 and ε  ke(∇u)k2L2 (Qµ ) (H1 (Ju ))−1 we obtain a ne estimate on the sum of the diameter of the rectangles which will be essential in the energy estimates in Chapter 9.

8.1 Preparations Let Qµ = (−µ, µ)2 and recall the denition of U s in (7.1). We will concern ourselves with subsets V ⊂ Qµ of the form [m V s := {V ⊂ Qµ : V = Qµ \ Xi , Xi ∈ U s , Xi pairwise disjoint} (8.1) i=1

for s > 0. Note that each set in V ∈ V s coincides with a set U ∈ U s up to subtracting a set of zero Lebesgue measure, i.e. U ⊂ V , L2 (V \ U ) = 0. The essential dierence of V and the corresponding U concerns the connected components of the complements Qµ \ V and Qµ \ U . Observe that one may Smˆ ˆ S ˆ ˆ ˆ ), e.g. by have Qµ \ m i=1 Xi with (X1 , . . . , Xm ) 6= (X1 , . . . , Xm i=1 Xi = Qµ \ combinationS of dierent sets (see Figure regard Smˆ ˆ 8.1). In such a case we will m s V1 = Qµ \ i=1 Xi and V2 = Qµ \ i=1 Xi as dierent elements of V . For the whole chapter we will tacitly assume that all considered sets are elements of V s for some small, xed s > 0. ˆ4 X5 = X ˆ2 X2 ⊂ X ˆ1 X1 = X ˆ2 X3 ⊂ X V

ˆ3 X4 = X

Figure 8.1: S The square Qµ with S a subset V . The set V has two representations

and V2 = Qµ \ 4i=1 Xˆ i , where Xˆ 2 = X2 ∪ X3 , which are regarded as dierent elements of V s . S The corresponding set U ∈ U s arises from V by subtracting the black boundary lines 5i=1 ∂Xi . V1 = Qµ \

5 i=1 Xi

144

Let W ∈ V s and arrange the components X1 , . . . , Xm of the complement such that ∂Xi ⊂ Qµ for 1 ≤ i ≤ n and ∂Xi ∩ ∂Qµ 6= ∅ otherwise. Dene Γi (W ) = ∂Xi for i = 1, . . . , n. In theS following we will often refer to these sets as boundary components. Note that ni=1 Γi (W ) might not cover ∂W ∩Qµ completely if n < m. We frequently drop the subscript and write Γ(W ) or just Γ if no confusion arises. In addition to the Hausdor-measure |Γ|H = H1 (Γ) (we will use pboth notations) we dene the `diameter' of a boundary component by |Γ|∞ := |π1 Γ|2 + |π2 Γ|2 , where π1 , π2 denote the orthogonal projections onto the coordinate axes. Note that by denition of V s (in contrast to the denition of U s ) two components in (Γi )i might not be disjoint. Therefore, we choose an (arbitrary) order (Γi )ni=1 = (Γi (W ))ni=1 of the boundary components of W , introduce [ Θi = Θi (W ) = Γi \ Γj (8.2) j 0 to be specied below we set

|Θ|∗ = h∗ |Θ|H + (1 − h∗ )|Θ|∞ .

(8.3)

For sets W ∈ V s we then dene

kW kZ =

Xn j=1

|Θj (W )|Z

for Z = H, ∞, ∗. Note that kW k∞ , kW kH and thus also kW k∗ are independent of the specic order which we have chosen in (8.2). Indeed, for kW k∞ this S is clear as |Θi |∞ = |Γi |∞ , for kW kH it follows from the fact that kW kH = H1 ( ni=1 Γi ). Before we introduce the modication procedure we collect some elementary properties of | · |∗ .

Lemma 8.1.1. Let W ⊂ Qµ . Let Γ = Γ(W ) be a boundary component with Γ = ∂X and let Θ ⊂ Γ be the corresponding set dened in (8.2). Moreover, let V ∈ U s be a rectangle with V ∩ X 6= ∅. Suppose that h∗ is suciently small. Then (i) |Γ|∗ ≥ |∂R(Γ)|∗ if Γ is connected, where R(Γ) denotes the smallest (closed) rectangle such that Γ ⊂ R(Γ), (ii) |Θ|∗ = |Γ|∗ ⇔ |Θ|H = |Γ|H , (iii) |∂(X \ V )|∞ ≤ |Θ|∞ and |Θ \ V |H ≤ |Θ|H , 145

(iv) |∂(V ∪ X)|∗ ≤ |∂V |∗ + |Γ|∗ , (v) |∂(V ∪ X)|∗ ≥ |∂R(V ∪ X)|∗ if V ∪ X is connected, where R(V ∪ X) denotes the smallest rectangle such that V ∪ X ⊂ R(V ∪ X). Now assume that Γ = ∂R for a rectangle R ∈ U s . Then (vi)

√1 |Γ|H 2

≤ 2|Γ|∞ ≤ |Γ|H ,

(vii) |∂(V ∪ R)|∗ ≤ |∂V |∗ + 12 |Γ|∗ provided that Γ \ V is not connected and |Γ|∞ ≤ c|∂V |∞ for a constant c > 0 suciently small. Proof. If Γ is connected, we obtain |Γ|H ≥ |∂R(Γ)|H and |Γ|∞ = |∂R(Γ)|∞ . This

yields (i) and likewise we obtain (v). Assertions (ii)-(iv) follow directly from the denition of | · |∗ , where in (ii) we particularly use |Θ|∞ = |Γ|∞ . Claim (vi) is elementary. To see (vii) we assume without restriction V = (−a, a) × (0, b) and π1 Γ = (−d, d) with d > a as well as π2 Γ ⊂ (0, b). An elementary calculation p 2 ≤ b + 2d ≤ |∂V |∞ + 14 |Γ|∞ . Here we yields |∂(V ∪ R)|∞ = (2d)2 + b2 ≤ b + (2d) 2b 4 used that 4d ≤ b for c small enough. As |∂(V ∪ R)|H ≤ |∂V |H + |Γ|H the claim now follows from (8.3) and (vi) if we choose h∗ small enough.  The properties stated here will be exploited frequently and we will not always refer to this lemma. One method of the modication procedure below will be the `combination' of dierent boundary components by adding additional sets to the original boundary (see case c) in the proof of Theorem 8.4.2). To keep track of the components we already `used' to modify the boundary, it is convenient to introduce a weight ωmin ≤ ω(Γj ) ≤ 1 for all Γj = Γj (W ) with 21 ≤ ωmin < 1 to be specied below. We dene |Θj |Z,ω = ω(Γj )|Θj |Z and likewise a weighted version of k·kZ by setting X kW kZ,ω := ω(Γj )|Θj |Z (8.4) j

for Z = H, ∞, ∗. For Z = ∗ we write for shorthand |·|ω = |·|∗,ω and k·kω = k·k∗,ω . We briey note that in contrast to k · k∗ , the value of (8.4) depends on the order given in (8.2) and therefore we will always consider a specic order of the boundary components in the following.

8.2 Modication of sets For λ ≥ 0 and xed small υ > 0 let Wλs ⊂ V s be the subset consisting of the sets W ∈ V s with a corresponding weight ω and an ordering of the boundary

146

components (Γi )ni=1 such that the following properties are satised:

(i) (ii) (iii) (iv) (v)

Θi ⊂ ∂Ri , Γi ⊂ Ri for a rectangle Ri −1 |∂Ri |∗ ≤ ωmin ω(Γi )|Θi |∗ Ri \ Xj is connected for all j = 1, . . . , n ω(Γi ) = 1 Γi = Θi = ∂Ri for a rectangle Ri

∀ ∀ ∀ ∀ ∀

Γi Γi Γi Γi Γi

: : : : :

ω(Γi ) < 1, ω(Γi ) < 1, ω(Γi ) < 1, (8.5) |Γi |∞ ≥ 19υλ, ω(Γi ) = 1.

Observe that (iv),(v) imply that boundary components larger than 19υλ are always rectangular and pairwise disjoint. In particular, W0s consists of the sets where all boundary components are rectangular. By an elementary argumentation taking (8.3), (8.5)(i),(ii) into account and recalling ωmin ≥ 21 , h∗  1, we observe

|Γi |∞ ≤ |∂Ri |∞ ≤ C|Γi |∞

∀ Γi : ω(Γi ) < 1,

(8.6)

i.e. the diameter of Γi and the corresponding rectangle Ri are comparable. Sm Consider a set W = Qµ \ i=1 Xi ∈ Wλs , λ ≥ 0, and a rectangle V ∈ U s with |∂V |∞ ≥ λ and V ⊂ Qµ . We dene the modication [m ˜i, ˜ = Qµ \ X (8.7) W i=0

˜ 0 = V . We observe that W ˜ = (W \V )∪ ˜ i = Xi \V for i = 1, . . . , m and X where X 2 ˜ ∂V (as a subset of R ). Therefore, for shorthand we will write W = (W \ V ) ∪ ∂V to indicate the element of V s which is given by (8.7). ˜ : First let Γ0 (W ˜ ) = ∂V We have the following boundary components of W (it is convenient to start with index 0) and for j ≥ 1 we have by construction ˜ ) = ∂(Xj \V ). Observe that some boundary components may be empty and Γj (W ˜ ))n˜ for n therefore reordering the indices we let (Γj (W ˜ ≤ n be the nonempty j=1 ˜ boundary components. Clearly, for each Γj (W ), j ≥ 1, there is exactly one ˜ ) = ∂(Xi \ V ). (This mapping is corresponding ∂Xij = Γij (W ) such that Γj (W j ˜ injective.) We order the components of W such that 1 ≤ j1 < j2 if and only if ij1 < ij2 , i.e. we preserve the ordering of W . ˜ ) = ∂V We now dene the corresponding subsets as in (8.2) and obtain Θ0 (W ˜ ) = Θi (W ) \ V for j ≥ 1. Moreover, we choose the same correas well as Θj (W j ˜ ) with ω(Γj (W ˜ )) < 1 sponding rectangles as given for W by (8.5)(i), i.e. for Γj (W ˜ we dene Rj (W ) = Rij (W ). From now on for notational convenience we may assume that ij = j for all ˜ )) = 1 and for j ≥ 1 j ≥ 1. We obtain the following `new' weights: Set ω(Γ0 (W

˜ )) = ω(Γj (W

o if ω(Γj (W )) = 1, else.

( 1 min

n

|Θj (W )|∗ ˜ )|∗ ω(Γj (W )), 1 |Θj (W

147

(8.8)

We note that

˜ )) ≥ ω(Γj (W )) and ω(Γj (W ˜ ))|Θj (W ˜ )|∗ ≤ ω(Γj (W ))|Θj (W )|∗ (8.9) ω(Γj (W ˜ )|∗ ≤ |Θj (W )|∗ . This for all j ≥ 1. To see this, it suces to show |Θj (W follows from Lemma 8.1.1(iii) and the observation that by construction (recall in ˜ ) = ∂(Xj \ V ) and Θj (W ˜ ) = Θj (W ) \ V . particular (8.2)) we have Γj (W s ˜ Note that W might not be an element of Wλ . We now show, however, that ˜ W can be modied to a set in Wλs . ˜ = (W \ V ) ∪ ∂V for a rectangle Lemma 8.2.1. Let λ ≥ 0 and W ∈ Wλs . Let W V ∈ U s with |∂V |∞ ≥ λ and V ⊂ Qµ . Then there is another rectangle V 0 ∈ U s with V ⊂ V 0 ⊂ Qµ such that U := (W \ V 0 ) ∪ ∂V 0 ∈ Wλs and ˜ kω , kU kω ≤ kW

(8.10)

˜ , U we adjusted the weights as in (8.8). where for both sets W

Proof. Without restriction we can assume V ∩ W 6= ∅ as otherwise there is

˜ clearly satises (8.5)(i),(iv). ( Recall nothing to show. We rst see that W that in (i) we take the same rectangles as for the boundary components of W .) ˜ ) with ω(Γj (W ˜ )) < To see (8.5)(ii) it suces to note that for a given Θj (W ˜ ˜ 1, (8.8) implies ω(Γj (W ))|Θj (W )|∗ = ω(Γj (W ))|Θj (W )|∗ . Possibly (8.5)(iii) or ˜ ), i = 1, . . . , k , with ω(Γj (W ˜ )) = 1 such (8.5)(v) are violated, i.e. there are Γji (W i ˜ ˜ ˜ ˜ ), that Γji (W ) is not rectangular or Γji (W ) 6= Θji (W ) or there are sets Θji (W i = k + 1, . . . , l, such that for the corresponding rectangles Rij given by (8.5)(i) one has that Rji \ X is disconnected for a suitable component X . Note that ˜ ) 6= ∅ for i = 1, . . . , l as W ∈ W s . So it remains to modify W ˜ ∂V ∩ Γji (W λ iteratively to obtain a set satisfying (8.5)(iii) and (8.5)(v). ˜ and V0 = V . Assume Wi = (W \ Vi ) ∪ ∂Vi ⊂ W ˜ has been Set W0 = W constructed, where Vi ∈ U s is a rectangle with V ⊂ Vi . Moreover, suppose that (8.10) holds replacing U by Wi and that Wi satises (8.5)(i),(ii),(iv) and Γ(Wi ) ∩ ∂Vi 6= ∅

for all Γ(Wi ) ∈ Fi

(8.11)

for the boundary component ∂Vi = Γ0 (Wi ) with ω(Γ0 (Wi )) = 1. Here Fi = Fi1 ∪ Fi2 , where Fi1 denotes the set of the not rectangular boundary components Γ(Wi ) with ω(Γ(Wi )) = 1 and Fi2 denotes the set of boundary components for which the corresponding rectangle is disconnected. Observe that |∂Vi |∞ ≥ λ as |∂V |∞ ≥ λ. If now Wi ∈ Wλs (i.e Fi1 = Fi2 = ∅), we stop and set U = Wi . ˆ ∈ Fi . If Γ ˆ ∈ Fi1 we let Vi+1 ∈ U s be the smallest (closed) Otherwise, we choose Γ ˆ . By Lemma 8.1.1(v) we get |∂Vi+1 |∗ ≤ |∂(Vi ∪ X)| ˆ ∗, rectangle containing Vi and Γ ˆ is the component of Qµ \Wi corresponding to Γ ˆ . Now by Lemma 8.1.1(iv) where X and (8.5)(v) we obtain

ˆ ∗ ≤ |∂Vi |∗ + |Γ| ˆ ∗ = |∂Vi |ω + |Θ| ˆ ω |∂Vi+1 |∗ ≤ |∂(Vi ∪ X)| 148

ˆ ⊂Γ ˆ as given by (8.2). If Γ ˆ ∈ F 2 \ F 1 we let Vi+1 ∈ U s be the smallest for Θ i i ˆ , where R ˆ is the corresponding rectangle given by rectangle containing Vi and ∂ R ˆ ∗ ≤ 2|Θ| ˆ ω . Moreover, the fact (8.5)(i). By (8.5)(ii) and ωmin ≥ 12 we derive |∂ R| ˆ i that (8.5)(iii) holds for W , is violated for Wi and Wi = (W \Vi )∪∂Vi implies R\V ˆ ∞ ≤ 19υ|∂Vi |∞ by (8.5)(iv) and thus |∂ R| ˆ ∞ ≤ Cυ|∂Vi |∞ is disconnected. As |Γ| by (8.6), Lemma 8.1.1(v),(vii) then yields for υ suciently small ˆ ∗ ≤ |∂Vi |∗ + 1 |∂ R| ˆ ∗ ≤ |∂Vi |ω + |Θ| ˆ ω. |∂Vi+1 |∗ ≤ |∂(Vi ∪ R)| 2 Let Wi+1 = (Wi \ Vi+1 ) ∪ ∂Vi+1 (recall (8.7)) and adjust the weights of the boundary components of Wi+1 as in (8.8). Recall that for all Γj (Wi+1 ) with ˆ. Γj (Wi+1 ) 6= ∂Vi+1 we nd a (unique) corresponding Γj (Wi ) with Γj (Wi ) 6= ∂Vi , Γ By (8.9) we then derive X kWi+1 kω = |∂Vi+1 |∗ + ω(Γj (Wi+1 ))|Θj (Wi+1 )|∗ Γj (Wi+1 )6=∂Vi+1 (8.12) X ˆ ω+ ω(Γ (W ))|Θ (W )| = kW k . ≤ |∂Vi |ω + |Θ| j i j i ∗ i ω ˆ Γj (Wi )6=∂Vi ,Γ

Consequently, (8.10) still holds and arguing as before we see that Wi+1 satises (8.5)(i),(ii),(iv). Moreover, (8.5) (iii),(v) can only be violated if (8.11) holds with Fi+1 6= ∅. We now continue with iteration step i + 1 and observe that after a nite number of steps i∗ we nd a rectangle Vi∗ ⊃ V and a set Wi∗ = (W \ Vi∗ ) ∪ ∂Vi∗ ∈ Wλs as in each step the number of boundary components decreases. Dene V 0 = Vi∗ and U = (W \ V 0 ) ∪ ∂V 0 . Note that U and Wi∗ coincide as sets in R2 , but the weights have been obtained in a dierent way. Therefore, to see (8.10) it remains to show ω(Γj (Wi∗ )) = ω(Γj (U )) for all boundary components Γj . For Γj = ∂V 0 it suces to recall that ω(∂V 0 ) = ω(∂Vi ) = 1 for all 1 ≤ i ≤ i∗ . If Γj ∩ ∂V 0 = ∅ it follows from the fact that Γj has not been changed during the modication procedure. Otherwise, as Wi∗ ∈ Wλs and thus boundary components of Wi∗ with weight 1 are pairwise disjoint (see (8.5)(v)), we know that ω(Γj (Wi∗ )) < 1. Let Θj ⊂ Γj as given in ˜ ) be the component corresponding to Θj . Then by iterative (8.2) and let Θj (W ˜ )) ≤ ω(Γj (Wi∗ )) < 1 and thus using iteratively application of (8.9) we get ω(Γj (W (8.8) we nd

˜ ))|Θj (W ˜ )|∗ . ω(Γj (Wi∗ ))|Θj |∗ = ω(Γj (Wi∗ −1 ))|Θj (Wi∗ −1 )|∗ = . . . = ω(Γj (W ˜ )| |Θ (W

∗ j ˜ )) = ω(Γj (W Consequently, again employing (8.8) we derive ω(Γj (Wi∗ )) = |Θ j |∗ ω(Γj (U )), as desired.  As a direct consequence of the above result, we get that sets in V s can be modied such that the boundary components have rectangular form.

149

Corollary 8.2.2. Let W ∈ V s with connected boundary components. Then there is a subset U ⊂ W such that |W \ U | ≤ ckU k2∞ for some c > 0 and all boundary components of U are rectangular and pairwise disjoint. Moreover, we have kU k∗ ≤ kW k∗ . In particular, if we introduce a weight ω corresponding to U by ω(Γj (U )) = 1 for all j and dene an (arbitrary) ordering of the boundary components we obtain U ∈ W0s . Proof. We follow the lines of the previous proof. Set W0 = W and assume Wi ⊂ W has been constructed with kWi k∗ ≤ kW k∗ . If Wi ∈ W0s we stop, otherwise we nd a component Γ = Γ(Wi ) which is not rectangular. Let Wi+1 = (Wi \ R(Γ)) ∪ ∂R(Γ), where R(Γ) is the smallest closed rectangle which contains Γ and all components Γj with Γj ∩ Γ 6= ∅. Using Lemma 8.1.1(i) we clearly have |∂R(Γ)|∗ ≤ |Γ|∗ . As in the previous proof, in particular by (8.12), we then get kWi+1 k∗ ≤ kWi k∗ ≤ kW k∗ . We now continue with iteration step i + 1 and note that we nd the desired set U after a nite number of iterations. Let (Γi )i be the boundary components of U with corresponding sets (Xi )i . It S is elementary to see thatPW \ U ⊂ iP Xi and thus by the isoperimetric inequality  we conclude |W \ U | ≤ i |Xi | ≤ C i |Γi |2 ≤ CkU k2∞ .

8.3 Neighborhoods of boundary components Consider W ∈ Wλs , λ ≥ 0. In this section we concern ourselves with neighborhoods of a boundary component Γ = Γ(W ) with ω(Γ) = 1 and |Γ|∞ ≥ λ. This implies that Γ has rectangular shape by (8.5)(v). We begin with a rectangular neighborhood and show that essentially the neighborhood can contain at most two other `large' boundary components. Afterwards we will introduce a dodecagonal neighborhood. The main condition which will allow us to investigate properties of the neighborhoods will be the following minimality condition for k · kω : We require

˜ kω ≥ kW kω kW

for all rectangles V ∈ U s with Γ ⊂ V ⊂ Qµ ,

(8.13)

˜ = (W \ V ) ∪ ∂V (recall (8.7)) and the weights are adjusted as in (8.8). where W In Section 8.4 we will see that (8.13) is one of the necessary conditions such that a trace estimate on Γ can be established (see Theorem 8.4.1). On the contrary, ˜ , we will show that it is convenient to replace W if (8.13) is violated for some W ˜ (see case a) in the proof of Theorem 8.4.2). by W Without restriction let Γ = ∂X with X = (−l1 , l1 ) × (−l2 , l2 ) for 0 < l2 ≤ l1 and l1 , l2 ∈ sN.

150

8.3.1 Rectangular neighborhood This section is devoted to the denition and properties of rectangular neighborhoods of Γ. As the technical proofs in this part are in principle not relevant to understand the proof of the main result in Section 8.4, they may be omitted on rst reading. The essential points in this section are the denition of the neighborhood N t (Γ) (cf. Figure 8.2), the choice of the size of the neighborhoods (see (8.14) and (8.26)) and the properties that the length of ∂W in N t (Γ) can be controlled (see Lemma 8.3.1) as well as that there are at most two other `large' boundary components (see Corollary 8.3.4 and Figure 8.3). Moreover, Lemma 8.3.5 shows that up to two small exceptional sets one can nd a covering of the neighborhood (see Figure 8.4) such that on each element the projection k · kπ (see (8.18)) can be controlled which will be essential for a slicing argument in the proof of Theorem 8.4.1. For t ∈ sN with t  l1 we set

N t (Γ) := (−t − l1 , l1 + t) × (−t − l2 , l2 + t) \ X, t Nj,± (Γ) := N t (Γ) ∩ {±xj ≥ lj } for j = 1, 2. (in the following we will use ± for shorthand if something holds for sets with index + and −.) We drop Γ in the brackets if no confusion arises. t up to a set of measure 0 with disjoint translates of a `quasi We cover N2,± ˜ t t t square' (0, t˜) × (0, t), tt ≈ 1. If l2 ≥ 2t we cover N1,± \ (N2,− ∪ N2,+ ) with 1 t translates of the rectangle (0, t) × (0, a) with 2 t ≤ a ≤ t. By E±,± we denote the four squares in the corners whose boundaries contain the points (±l1 , ±l2 ), t by itself, i.e. by a translate of the respectively. For l2 < 2t we cover each N1,± rectangle (0, t) × (0, 2t + 2l2 ). For convenience we will often refer to these sets as `squares' in the following. We number the squares by Qt0 , Qt1 , . . . , Qtn = Qt0 such that Qtj ∩ Qtj+1 6= ∅ for j = 0, . . . , n − 1 and let J t = {Qt1 , . . . , Qtn }. For shorthand we dene τ¯ = υ|Γ|∞ for 0 < υ  1 and we will assume that (possibly by passing to a smaller s)

τ¯ = υ|Γ|∞ ∈ sN

and

τ¯  s.

(8.14)

This assures that all the neighborhoods we consider below can be chosen as elements of U s . Let (Γj )j = (Γj (W ))j be the boundary components of W and (Θj )j the corresponding subsets dened by (8.2). Let (Rj )j be the associated rectangles as given in (8.5)(i) and (8.5)(v), respectively. We will always add a subscript to avoid a mix up with Γ.

151

N t (Γ)

t (Γ) N1,−

Γj1

Γ

X

Θj2

Γj3

Figure 8.2: Neigborhood N t (Γ) with other small boundary components. The part t (Γ) is colored in dark grey. N1,−

Lemma 8.3.1. Let λ ≥ 0 and c > 0. Let W ∈ Wλs and let Γ be a boundary component with ω(Γ) = 1 and |Γ|∞ ≥ λ. Assume that (8.13) holds. Then there is a constant C = C(c) such that (i) |∂W ∩ N t |H ≤ C ht∗ t

(ii) |Γj ∩ N |H ≤

C ht∗

for all t ≥ c¯τ , for all t ∈ sN and all Γj with ω(Γj ) = 1.

Proof. (i) Let V = (−l1 − tˆ, l1 + tˆ)×(−l2 − tˆ, l2 + tˆ) ∈ U s , where tˆ = 2 max{t, 19C τ¯}

˜ = (W \V )∪∂V and adjust the weights with the constant C from (8.6). Dene W as in (8.8). It is not hard to see that |∂V |∗ ≤ |Γ|∗ + 8tˆ. Let F be the set of boundary components having nonempty intersection with N t and let G ⊂ F be the subset satisfying ω(Γj (W )) = 1 for Γj (W ) ∈ G . By (8.5)(iv) and (8.6) we nd Θj (W ) ⊂ ∂Rj ⊂ V for Γj (W ) ∈ F \ G . Recall ˜ ) ∈ F with Γj (W ˜ ) 6= ∂V we nd a that due to the choice of tˆ for all Γj (W (unique) corresponding Γj (W ) ∈ G with Γj (W ) 6= ∂V, Γ. (Without restriction ˜ )|∗ ≤ we take the same index.) For Γj (W ) ∈ G it is elementary to see that |Θj (W |Γj (W )|∗ − h∗ |Γj (W ) ∩ V |H . Consequently, using ω(∂V ) = ω(Γ) = 1 we derive X ˜ kω = |∂V |∗ + ˜ ))|Θj (W ˜ )|∗ kW ω(Γj (W ˜ )6=∂V Γj (W X ≤ |Γ|ω + 8tˆ + (|Γj (W )|∗ − h∗ |Γj (W ) ∩ V |H ) Γj (W )∈G X + ω(Γj (W ))|Θj (W )|∗ Γj (W )∈F / X X ≤ kW kω + 8tˆ − h∗ |Γj (W ) ∩ V |H − ω(Γj (W ))|Θj (W )|∗ Γj (W )∈F \G

Γj (W )∈G

≤ kW kω + 8tˆ − ωmin h∗ |∂W ∩ N |H . t

152

For the components not being in F we proceeded as in (8.12). Since kW kω ≤ ˜ kω by condition (8.13) and ωmin ≥ 1 , we nd |∂W ∩ N t |H ≤ C tˆ ≤ C t , kW 2 h∗ h∗ where in the last step we used t ≥ c¯ τ. (ii) We argue as in (i) with the dierence that we set tˆ = t and F = G = {Γj }. Then repeating the above calculation we obtain

˜ kω ≤ kW kω + 8tˆ − h∗ |Γj ∩ N t |H , kW where for all other components we proceeded as in (8.12). We conclude by em˜ kω . ploying kW kω ≤ kW  t We now analyze the components intersecting N more precisely. In particular, we will show that at most two large boundary components lie in the neighborhood of Γ (see Corollary 8.3.4). The properties can be established by exploiting elementary geometric arguments and essential ideas of the procedure are exemplarily illustrated in Figure 8.3.

V

Γ3

V

Γ

Γ1 Γ2

Γ Γ1

Γ2

N t (Γ)

N t (Γ)

Figure 8.3: The left picture shows three boundary components Γ1 , Γ2 , Γ3 intersecting

˜ N t (Γ). Below we argue that such a conguration violates (8.13) for W P = (W \ V ) ∪ ∂V , where V is dotted rectangle. Indeed, one might have P |∂V |H > |Γ|H + k=1,2,3 |Γk |H , but we can show that Pone always has |∂V |∞ < |Γ|∞ + k=1,2,3 |Γk |∞ , whereby we obtain |∂V |∗ < |Γ|∗ + k=1,2,3 |Γk |∗ for h∗ suciently small. Likewise, we can control the position of the at most two large components Γ1 , Γ2 in N t (Γ): A conguration depicted on the right, where Γ1 , Γ2 do not intersect opposite parts of N t (Γ), violates (8.13) for the dotted rectangle V .

We rst introduce a coarser covering of N t : LetSc¯ τ ≤ t
≤ C τ¯. Let Y t be the k t ◦ union of connected sets Y having the form Y = for Qti ∈ J t . Cover i=j Qi j t each set N2,± with seven sets Y2,± such that j ¯ |Y2,± | ≥ Ct|Γ| ∞,

1 ¯ Ct|Γ|∞ 8

j+1 j ¯ | ≤ 41 Ct|Γ| ≤ |Y2,± ∩ Y2,± ∞

(8.15)

t for a constant C¯ > 0. If l2 ≥ l21 we proceed likewise for N1,± passing possibly l t 1 to a smaller constant C¯ . If l2 < 2 we cover N1,± by itself. Denote the covering

153

t by C t = C t (Γ) = {Y1t , . . . Ymt } and order the sets in a way that Yit ∩ Yi+1 6= ∅ for t t all i = 1, . . . , m, where by convention Yi = Yi modm . In particular, (8.15) implies Yitmodm ∩ Yjtmodm = ∅ for |i − j| ≥ 2. This construction implies that for υ suciently small

Rj ∩ Yit 6= ∅

⇒

t Rj ∩ Yi+l = ∅ for |l| ≥ 3

(8.16)

for all Rj and i = 1, . . . , n. To see this, we rst observe that (8.17)

|∂Rj ∩ N t |H ≤ Cth−1 ∗ .

Indeed, if |Γj |∞ < 19¯ τ , we obtain |∂Rj |∞ ≤ C τ¯ by (8.6) and thus |∂Rj |H ≤ √ 2 2|Rj |∞ ≤ Ct. Otherwise, recalling |Γ|∞ ≥ λ, by (8.5)(iv),(v) we have ∂Rj = Γj and thus employing Lemma 8.3.1(ii) we get |∂Rj ∩ N t |H ≤ Cth−1 ∗ . t ¯ ∞ for some |l| ≥ 3, (8.16) follows as |∂Rj ∩ N t |H ≤ If now dist(Yit , Yi+l ) ≥ C|Γ| ¯ Cth−1 ∗  C|Γ|∞ for υ small enough (depending on h∗ ). t ¯ ∞ . This is only possible in ) ≤ C|Γ| On the other hand, suppose dist(Yit , Yi+l l1 t t t \ (N1,− ∪ N1,+ ), the case l2 ≤ 2 if (up to interchanging + and −) Yit ⊂ N2,+ t t t t t t Yi+l ⊂ N2,− and dist(Yi , N1,± ) ≥ c|Γ|∞ or dist(Yi+l , N1,± ) ≥ c|Γ|∞ . Now assume ¯ that (8.16) was wrong. Then by (8.15) and |∂Rj ∩ N t |H ≤ Cth−1 ∗  C|Γ|∞ this t t t would imply Rj ∩ N2,± 6= ∅ and Rj ∩ (N1,− ∪ N1,+ ) = ∅. But then we would get that Rj \ X is not connected which contradicts (8.5)(iii). 7 Y2,+ z }| {

|

|{z} 3 4 Y2,+ ∩ Y2,+

{z } 2 Y2,+

Γ

X

Y¯it }|

z |

{z Yit

{ }

Figure 8.4: On the upper left side of the neigborhood N t (Γ) one can see elements of

the partition C t (Γ) (which are not necessarily of the same size). The sets where two elements overlap are striped. In the lower part an element Yit and the corresponding enlarged set Y¯it are highlighted. For Y ⊂ N t we set R(Y ) = {Rj : Rj ∩ Y 6= ∅} and dene

|∂Rj |π = min{|∂Rj |∞ , t − maxi=1,2 dist(πi Rj , πi Γ)} 154

(8.18)

for all Rj ∈ R(N t ).PIt is obvious that |∂Rj |π ≤ |∂Rj |∞ . For a set Y ⊂ N t we then dene kY kπ = Rj ∈R(Y ) |∂Rj |π . The projection k · kπ is one essential object we will need to apply a slicing argument in the investigation of the jump heights in Section 8.5.

Remark 8.3.2.

We have already introduced the (small) parameters h∗ , 1 − ωmin , υ . In the following sections we will additionally consider q, r. The subsequent lemmas will hold if we choose the involved parameters suciently small. To avoid confusion about the relation of the dierent parameters, we state at this point that the parameters can be chosen in the order h∗ , q, 1 − ωmin , r, υ . In what follows, we will not always repeat the relation of the parameters for convenience. We now show that we can control k · kπ in a suitable way. For that purpose, for a set Yit ∈ C t we dene [ t Y¯it = Yi+l . |l|≤1

Lemma 8.3.3. Let λ ≥ 0, W ∈ Wλs . Let Γ be a boundary component with ω(Γ) = 1 and |Γ|∞ ≥ λ. Assume that (8.13) holds. Let c¯ τ ≤ t ≤ C τ¯. If we choose h∗ , υ and 1 − ωmin small enough, there are two sets Y 1 , Y 2 ∈ C t such that 19 t for all Y t ∈ C t with Y t ∩ (Y¯ 1 ∪ Y¯ 2 ) = ∅. kY t kπ ≤ 20 19 t t Additionally, if kY 1 kπ , kY 2 kπ ≥ 20 t, then Y¯ 1 ∪ Y¯ 2 intersects both N1,+ and N1,− t t t t . or both N2,+ and N2,− . If l2 ≤ l21 , then Y¯ 1 ∪ Y¯ 2 intersects N1,+ and N1,− We briey remark that by similar arguments the additional statement can also be proved without the extra assumption kY 1 kπ , kY 2 kπ ≥ 19 t. We omit the 20 proof of this fact here as we will not need it in the following. Proof. For convenience we drop the superscript t in the following proof. We proceed in two steps: In a) we rst show that it is not possible that there are three sets Y 1 , Y 2 , Y 3 ∈ t for k, l = 1, 2, 3. Provided that C such that Y¯ k ∩ Y l = ∅ if k 6= l and kY k kπ > 19 20 a) is proven we can then select the two desired sets Y 1 , Y 2 as follows: 19 (1) If kY kπ ≤ 20 t for all Y ∈ C , we can choose arbitrary sets Y 1 , Y 2 satisfying the additional condition. Otherwise, we can assume that there is some Y ∗ with kY ∗ kπ > 19 t. 20 (2) If kY kπ ≤ 19 t for all Y ∈ C t with Y ∩ Y¯ ∗ = ∅, we set Y 1 = Y ∗ and choose 20 Y 2 arbitrarily such that the additional condition holds. (3) Otherwise, we set Y 1 = Y ∗ and choose Y 2 with kY 2 kπ > 19 t and Y 2 ∩ Y¯ ∗ = 20 19 t ∅. Now a) indeed shows that kY kπ ≤ 20 t for all Y ∈ C with Y ∩ (Y¯ 1 ∪ Y¯ 2 ) = ∅. In step b) we concern ourselves with the additional assertions on the position ¯ of Y 1 ∪ Y¯ 2 in case (3). a) Suppose that there are three sets Y 1 , Y 2 , Y 3 ∈ C such that Y¯ k ∩ Y l = ∅ if 19 k 6= l and kY k kπ > 20 t for k, l = 1, 2, 3. First note that the assumption implies 155

that if e.g. Y 1 = Yi , then Y 2 , Y 3 ∈ / {YS i−2 , . . . , Yi+2 }. Let V be the smallest ˜ = (W \ V ) ∪ ∂V rectangle containing Γ and the sets R := 3k=1 R(Y k ). Dene W ˜ (recall (8.7)). Similarly as in (8.12) we intend to estimate kW kω . To this end, we have to control the dierence of |∂Rj |∗ and |Θj |ω for Rj ∈ R. By (8.5)(ii),(iv),(v) and (8.6) we have ( ωmin |∂Rj |∗ |∂Rj |∞ ≤ 19Cυλ, |Θj |ω = ω(Γj )|Θj |∗ ≥ |∂Rj |∗ else, with the P constant C from (8.6). ForPnotational convenience we dene kW kω,R = P kW kω + Rj ∈R (|∂Rj |∗ − |Θj |ω ) = Rj ∈R |Θ | + |∂R j ω j |∗ . We get / Rj ∈R

kW kω,R ≤ kW kω + ≤ kW kω + ≤ kW kω +

X

(ω −1 − 1)|Θj |ω

min Rj ∈R, |∂Rj |∞ 0 small enough such that (i) The covering {Yˆ1 , . . . , Yˆk } of N \ (K1 ∪ K2 ) consisting of the connected components of {Y \ (K1 ∪ K2 ) : Y ∈ C}, satises kYˆi kπ ≤ 19 τ for all 20 i = 1, . . . , k . (ii) Γi ∩ N ⊂ K1 ∪ K2 for all components Γi with |Γi |∞ ≥ 19¯τ . Proof. By Lemma 8.3.3 we obtain that there are two sets Y 1 , Y 2 ∈ C with

19 τ for Y ∈ C with Y ∩ (Y¯ 1 ∪ Y¯ 2 ) = ∅. We Y¯ 1 ∩ Y¯ 2 = ∅ such that kY kπ ≤ 20 only construct the set K1 . Choose Yi = Y 1 and set Sl = Yi+l ∈ C for |l| ≤ 3. In 19 τ for l = −3, 3. Set particular, we have Sl ∩ S0 6= ∅ for l = −1, 1 and kSl kπ ≤ 20 S2 S = l=−2 Sl . Arguing as in (8.24) or (8.25) for t = τ , respectively, depending on whether S is contained in at most two adjacent parts of the neighborhood or S intersects three (possible for l2 ≤ l21 ), we derive |∂V |∞ ≤ |Γ|∞ + P parts of the neighborhood 19 1 Rj ∈R(S) |∂Rj |∞ +( 10 − 100 )τ −kSkπ , where V is the smallest rectangle containing Γ and R(S). (Note that in the above calculation we possibly have to repeat the translation argument indicated at the end of the proof of Lemma 8.3.5.) Thus, arguing as in the proof of Lemma 8.3.3, in particular taking (8.19) and condition (8.13) into account, we nd

˜ kω − kW kω ≤ (1 − h∗ ) 19 τ − (1 − h∗ )kSkπ 0 ≤ kW 10

(8.27)

˜ = (W \ V ) ∪ ∂V . We now construct for h∗ , 1 − ωmin small enough, where W the set K1 and the corresponding (at most) two connected components T1 , T2 of S \ K1 by distinction of the two following cases: a) If there is some Rj with |∂Rj |π ≥ 19 τ we choose K1 ∈ Y as the smallest set 20 such that Rj ∩ N ⊂ K1 . Then the (at most) two connected components T1 , T2 of 2 S \ K1 satisfy kTi kπ ≤ 19 τ by (8.27). Using (8.17) we derive that |K1 | ≤ C hτ ∗ , as 20 desired. S 0 b) Otherwise, we choose K1 as follows. Assume S = ( ni=1 Qi )◦ for Qi ∈ J and S τ and let k ∈ {0, . . . , n0 } be the index (if existent) such that k( ki=1 Qi )◦ kπ ≤ 19 20 S Sk+1 S k l 19 k( i=1 Qi )◦ kπ > 20 τ . Now dene T1 = ( i=1 Qi )◦ and choose K1 = ( i=k+1 Qi )◦ 2 for l large enough such that |K1 | ≥ c¯hτ ∗ . Finally, let T2 = S \ (T1 ∪ K1 ) and 19 observe that for c¯ large enough also kT2 kπ ≤ 20 τ by (8.27) and (8.17) since each rectangle can intersect at most one of the sets T1 , T2 . 160

Let Sl1 , Sl2 be the connected components of Sl \K1 for l = −2, −1, 0, 1, 2. Both cases a),b) above imply kSli \ K1 kπ ≤ maxk=1,2 kTk kπ ≤ 19 τ for l = −2, . . . , 2, 20 i = 1, 2, which gives assertion (i). Assertion (ii) follows from the construction of the set K1 and denition (8.26). Indeed, if Γi ∩ N 6= ∅, then Γi ∩ N τ /20 τ and then Γi ∩ N ⊂ Y¯ 1 ∪ Y¯ 2 . and thus recalling (8.18) we nd |Γi |π ≥ 19 20 Finally, dist(K1 , K2 ) ≥ c|Γ|∞ follows directly from the fact that in the case τ the set Y¯ 1 ∪ Y¯ 2 intersects both N1,+ and N1,− or both N2,+ kY 1 kπ , kY 2 kπ ≥ 19 20 and N2,− (N1,± if l2 ≤ l21 ). 

8.3.2 Dodecagonal neighborhood We now introduce neighborhoods of Γ which in general have dodecagonal shape and dier from N t (Γ) near the corners of Γ. These neighborhoods will be essential in the modication algorithm below (see Section 8.4.2) as we have to treat the modication near the corners of a boundary component with special care. For t > 0 we dene [ −1 ˆ t (Γ) = M {x ∈ N t (Γ) : |xi + li | ≥ qh−1 (8.28) ∗ t, |xi − li | ≥ qh∗ t} i=1,2

for q  1 to be specied below. Moreover, for ˜l = l1 + min{t, q −1 h∗ l2 } let

ˆ t (Γ) ∪ Γ ∪ (˜l, 0) ∪ (−˜l, 0)) ∩ N t (Γ), M t (Γ) := co(M

(8.29)

ˆ t (Γ) where co(·) denotes the convex hull of a set. Observe that M t (Γ) ⊃ M ˆ t (Γ) dier by some triangles. Moreover, the shape of M t (Γ) and that M t (Γ), M is dodecagonal for l2 > qh−1 ∗ t and decagonal otherwise, cf. Figure 8.7. For ˆ = M ˆ τ (Γ) for a choice of τ satisfying shorthand we write M = M τ (Γ) and M (8.26). For later reference we also dene t t Mkt (Γ) = M t (Γ) ∩ (Nk,+ (Γ) ∪ Nk,− (Γ))

for k = 1, 2.

(8.30)

Recall the denition in (8.14). Let K1 , K2 ∈ Y be the sets constructed in Lemma 8.3.5. Let Γm = Γm (W ) be another boundary component satisfying 2 Γm ∩ K 6= ∅ for some K ∈ {K1 , K2 } and |Γm |∞ ≥ qh∗τ¯ with q given in (8.28). For q large enough we have |Γm |∞ ≥ 19¯ τ and thus ω(Γm ) = 1 by (8.5)(iv). Moreover, (8.26) implies that Γm is one of the (at most) two rectangular boundary components given by Corollary 8.3.4. By the choice in (8.26), K is constructed in case a) of the proof of Lemma 8.3.5 and therefore it is not hard to see that K is contained in one of the sets Nj,± , j = 1, 2. Let Xm ∈ U s be the corresponding component of Qµ \ W . We now treat two dierent cases depending on whether K is near a corner of Γ or not: ˆ 6= ∅. As K is contained in one of the sets Nj,± , j = 1, 2 we (I) Assume K ∩ M 2 assume e.g. K ⊂ N1,− . As |K| ≤ C hτ ∗ by Lemma 8.3.5, we nd |π2 Γm | ≤ C hτ∗ and 161

thus |π1 Γm |  |π2 Γm |. Consequently, for q suciently large we have |π1 Γm |  τ¯ which implies (8.31)

Γm ∩ {−l1 − 21¯ τ } × R 6= ∅.

Let Q1 , Q2 ∈ J be the neighboring squares of K , i.e. Qi ∩K = ∅ and ∂K ∩∂Qi 6= ∅ for i = 1, 2. Let Ψ = (Q1 ∪ K ∪ Q2 \ Xm )◦ and observe that Ψ ⊂ N1,− as ˆ 6= ∅. By (8.31) the set Ψ = Ψ1 ∪Ψ2 ∪Ψ3 decomposes into three rectangles, K ∩M where (up to translation and sets of measure zero) Ψ1 = (0, τ ) × (0, τ + a1 ), ˆ and Ψ3 = (0, τ ) × (0, τ + a3 ) for − 1 τ ≤ a1 , a3 ≤ τ . (Recall Ψ2 = (0, ψ) × (0, ψ) 2 the construction of K in the proof of Lemma 8.3.5 a).) Furthermore, let

Φ = {x ∈ Qµ : dist(x, Ψ) ≤ 20¯ τ }. Before we go on with case (II) we state two observations. We say that two sets are C -Lipschitz equivalent if they are related through a bi-Lipschitzian homeomorphism with Lipschitz constants of both the homeomorphism itself and its inverse bounded by C . M t (Γ) Γ Ψ23 Γ1m

Ψ22

X

Ψ21 ˆ1 Ψ 3

ˆ1 Ψ 2 @

Γ2m ˆ1 Ψ 1

Figure 8.5: Neigborhood M t (Γ) with two other boundary components Γ1m , Γ2m (the

1 , X 2 are striped) and corresponding neighborhoods Ψ ˆ 1 and Ψ2 . interiors Xm m

Lemma 8.3.6. Let Γ, Γm with ω(Γm ) = ω(Γ) = 1 and |Γ|∞ ≥ λ, |Γm |∞ ≥ q2 hτ¯∗ be given. In the situation of (I) the following holds: (i) Let V ∈ U s be the smallest rectangle containing X and Xm . Then Φ ⊂ V . (ii) ψˆ ≤ C hψ∗ . In particular, there is a suitable set Ψ2 ⊂ Ψ∗2 ⊂ Ψ such that each set Ψ1 , Ψ∗2 , Ψ3 is C(h∗ )-Lipschitz equivalent to a square. 162

21¯ τ 21¯ τ 21¯ τ Proof. (i) As K ∩ Mˆ 6= ∅ we get that Φ ⊂ N ∗ := N1,− \ (N2,+ ∪ N2,− ) if we again

choose q large enough. By (8.31) we have ∂N ∗ ∩ Γm 6= ∅. Therefore, the smallest rectangle V containing Γ and Γm satises N ∗ ⊂ V which gives the assertion. (ii) By Lemma 8.3.1(ii) we obtain ψˆ ≤ |Γm ∩ N 2ψ |H ≤ C hψ∗ . If also ψˆ ≥ hC∗ ψ we set Ψ∗2 = Ψ2 , otherwise we choose some Ψ∗2 ⊃ Ψ2 with |π2 Ψ∗2 | = ψ .  τ ˆ = ∅. (i) We rst treat the case l2  (II) Assume now K ∩ M and similarly h∗ as in (I) suppose without restriction that K ⊂ N1,− . Again let Q1 , Q2 be the ˆ = (Q1 ∪ K ∪ Q2 \ Xm )◦ . If Qj ⊂ N1,− for neighboring squares of K and set Ψ ˆ decomposes as before in (I). j = 1, 2 the set Ψ Otherwise, we may assume that e.g. Q1 ⊂ N2,− \ N1,− . Observe that then 2 ˆ K1 , Q2 ⊂ N1,− as |K1 | ≤ C hτ ∗ and l2  hτ∗ . As indicated in Figure 8.5, the set Ψ ˆ 1, Ψ ˆ 2, Ψ ˆ 3 , where (up to translation and sets of measure contains three rectangles Ψ ˆ and Ψ ˆ 1 = (0, τ + ψ) × (0, τ ), Ψ ˆ 2 = (0, ψ) × (0, ψ) ˆ 3 = (0, τ ) × (0, τ + a3 ) zero) Ψ ˆ for 0 ≤ a3 ≤ τ . Note that ψ = 0 is possible and that an argumentation as in Lemma 8.3.6 yields ψˆ ≤ C hψ∗ . Now let

ˆ j \ (M 21¯τ (Γ) ∪ M 21¯τm (Γm )), j = 1, 2, 3, Ψj = Ψ

Ψ=

 [3 j=1

Ψj

◦

,

where τ¯m = υ|Γm |∞ . Furthermore, let Φ = {x ∈ Qµ : dist(x, Ψ) ≤ 20¯ τ }. (ii) We nally treat the case that l2 is small with respect to l1 (i.e. l2 ≤ C hτ∗ ) which particularly implies that M 21¯τ (Γ) is decagonal. Suppose without restriction that K ⊂ N1,− . If K ∩ N2,+ = ∅ or K ∩ N2,− = ∅ we may proceed as before in ˆ ⊃Ψ ˆ 1 ∪Ψ ˆ 2 ∪Ψ ˆ 3 contains three rectangles, where (up to (II)(i). Otherwise, the set Ψ ˆ 1 = (0, τ +ψ)×(0, τ ), Ψ ˆ 2 = (0, ψ)×(0, 2l2 ) translation and sets of measure zero) Ψ ˆ and Ψ3 = (0, τ +ψ)×(0, τ ) (cf. Figure 8.6). The same argumentation as in Lemma ˆ j \ M 21¯τ (Γ) for j = 1, 2, 3. Observe that 8.3.6(ii) yields 2l2 ≤ C hψ∗ . We let Ψj = Ψ in contrast to case (II)(i) we only subtract the set M 21¯τ (Γ). We now have the following properties.

Lemma 8.3.7. Let Γ, Γm with ω(Γm ) = ω(Γ) = 1 and |Γ|∞ ≥ λ, |Γm |∞ ≥ q2 hτ¯∗ be given. In the situation of (II) the following holds: (i) Let V ∈ U s be the smallest rectangle containing X and Xm . Then we have Φ ∩ {x : x1 ≥ −l1 − ψ} ∩ M 21¯τm (Γm ) ⊂ V . (ii) In the cases (II)(i),(ii) we have ψˆ ≤ C hψ∗ and 2l2 ≤ C hψ∗ , respectively. Moreover, there is a suitable set Ψ2 ⊂ Ψ∗2 ⊂ Ψ such that each set Ψ1 , Ψ∗2 , Ψ3 is C(h∗ )-Lipschitz equivalent to a square. Proof. (i) It suces to note that {x : x2 ≥ −l1 − ψ} ∩ M 21¯τm (Γm ) ⊂ [−l1 − ψ, ∞) × π2 Γm and π1 Φ ⊂ (−∞, l1 ] (cf. Figure 8.6).

163

(ii) The bounds on ψˆ and l2 were already discussed above. As in the proof of ˆ∗ ⊃ Ψ ˆ 2 such that Ψ ˆ ∗ is C(h∗ )-Lipschitz equivaLemma 8.3.6(ii) we can choose Ψ 2 2 ˆ ∗2 \ M 21¯τ (Γ), ˆ ∗2 \ (M 21¯τ (Γ) ∪ M 21¯τm (Γm )) or Ψ∗2 = Ψ lent to a square. Let Ψ∗2 = Ψ respectively, depending on the cases (II)(i) and (II)(ii). For q suciently large in (8.28) it is elementary to see that Ψ1 , Ψ∗2 , Ψ3 are C(h∗ )-Lipschitz equivalent to a square.  M221¯τm (Γm ) Θl1 X Xm

| {z } ψ Ψ

Θl2

M121¯τm (Γm )

Figure 8.6: Sketch of Ψ (grey) in the case (II)(ii), where only parts of the boundary

components Γ,Γm are depicted. In particular M 21¯τm (Γm )∩Ψ 6= ∅ and M 21¯τ (Γ)∩Ψ = ∅. Also note that M 21¯τm (Γm ) is dodecagonal, whereas M 21¯τ (Γ) is decagonal. Moreover, for later reference (see proof of Lemma 8.4.5) we have also drawn two boundary components Θl1 , Θl2 ⊂ M 21¯τm (Γm ) in dashed lines.

8.4 Proof of the Korn-Poincaré-inequality This section is devoted to the main proof of Theorem 8.0.1. We concern ourselves with functions y ∈ H 1 (W ) on W ∈ V s , W ⊂ Qµ (recall (7.1), (8.1)). In the following we will again omit to write V s . Let R ∈ SO(2) and dene u = uR = RT y − id. For shorthand we set α(U ) = k¯ eR (∇y)k2L2 (U ) = ke(∇u)k2L2 (U ) for U ⊂ W (cf. (7.2)). As a further preparation, we dene H(W ) ⊃ W ∈ V s as the `variant of W without holes'. Arrange the components X1 , . . . , Xm such that ∂Xi ⊂ Qµ for 1 ≤ i ≤ n and ∂Xi ∩ ∂Qµ 6= ∅ otherwise. We set [n H(W ) = W ∪ Xj . (8.32) j=1

The main idea will be to analyze the trace of u at the boundary components. Therefore, we will have to change the set W iteratively. We rst introduce further conditions for the neighborhood of a boundary component which allows us to apply a trace estimate. Then we present the main modication algorithm. Afterwards, the proof of Theorem 8.0.1 will be straightforward by employing Theorem B.4. 164

8.4.1 Conditions for boundary components and trace estimate Recall denition (8.5) and assume that in an iteration step Wi ∈ Wλs for λ ≥ 0 with the corresponding weight ω and a specic ordering of the boundary components (Γ(Wi )j )nj=1 is given. Consider Γ = Γ(Wi ) with |Γ|∞ ≥ λ and recall that ˆ = N 2ˆτ (Γ), where Γ = Θ is rectangular by (8.5)(v). Let N

τˆ = q 2 τ¯h∗−1 = q 2 υh−1 ∗ |Γ|∞  |Γ|∞

(8.33)

with q from (8.28) and τ¯ as dened in (8.14). Recall that τˆ is the least length of boundary components considered in Section 8.3.2. The latter inequality holds if we choose υ suciently small with respect to q . For ε > 0 and for D = D(h∗ ) suciently large we require

ˆ ∩ Wi ) + ε|∂Wi ∩ N ˆ |H ≤ Dεˆ α(N τ.

(8.34)

Moreover, let Ψj and ψ j , j = 1, 2, be dened as in Section 8.3.2 (I),(II) corresponding to the sets Kj , j = 1, 2, provided by Lemma 8.3.5. We introduce the condition

α(Ψj ∩ Wi ) + ε|∂Wi ∩ Ψj |H ≤ D(1 − ωmin )−1 εψ j

(8.35)

for j = 1, 2, where D = D(h∗ ) as in (8.34). For η ≥ 0 we let Tη (Wi ) be the set of Γl (Wi ) satisfying |Γl (Wi )|∞ ≤ η and

N 2ˆτl (∂Rl ) ⊂ H(Wi ), where τˆl = q 2 υ|Γl |∞ h−1 ∗ (cf. (8.33)) and Rl is the corresponding rectangle given in (8.5)(i) or (8.5)(v), respectively. Moreover, recalling again the denition of Θl (Wi ) ⊂ Γl (Wi ) = ∂Xl in (8.2) we dene [ Θl (Wi ), (8.36) Sλ (Wi ) = Γl (Wi )∈Sλ (Wi )

where Sλ (Wi ) = {Γl : |Γl |∞ > λ} ∪ {Γl : ω(Γl ) = 1, N 2ˆτl (∂Rl ) 6⊂ H(Wi )}. (Note that by (8.5)(iv) all components of Sλ (Wi ) have weight 1.) 2 We assume that for all Γl (Wi ) ∈ Tτˆ (Wi ) there S are Al ∈ R2×2 skew , cl ∈ R such that for the extension u ¯ ∈ SBV (Wτˆ ), Wτˆ := Wi ∪ Γl (Wi )∈Tτˆ (Wi ) Xl , dened by ( Al x + cl x ∈ Xl for Γl (Wi ) ∈ Tτˆ (Wi ) u¯(x) = (8.37) u(x) else, we have the trace estimate ˆ ˆ 2 1 |¯ u(x) − (Al x + cl )| dH (x) = Θl (Wi )

≤ 165

|[¯ u](x)|2 dH1 (x)

Θl (Wi ) ε4 C∗ |Θl (Wi )|2∗

υ

(8.38)

for some C∗ = C∗ (h∗ ) > 0 suciently large. (The left hand side has to be understood as the trace of u ¯|Wτˆ \Xl on Θl (Wi ).) We now state that under suitable conditions also Γ = Γ(Wi ) with |Γ|∞ ≥ λ satises an estimate similar to (8.38).

Theorem 8.4.1. Let υ, h∗ , ε, ωmin > 0 and λ > 0. Then there is a constant ˆ ∗ ) > 0 such that for υ suciently small (depending on h∗ and ωmin ) the Cˆ = C(h following holds: For all Wi ∈ Wλs , for all u ∈ H 1 (Wi ) and boundary components Γ = Γ(Wi ) with |Γ|∞ ≥ λ such that (8.13), (8.34), (8.35), N 2ˆτ (Γ) ⊂ H(Wi ) hold and (8.38) is satised for Tτˆ (Wi ) one has (in the sense of traces) ˆ

C∗  ε ˆ |¯ u(x) − (A x − c)| dH (x) ≤ C + |Γ|2∗ 4 2 υ Γ 2

1



(8.39)

2 for suitable A ∈ R2×2 skew , c ∈ R .

As the proof of this assertion is very technical and involves several steps we postpone it to Section 8.5.

8.4.2 Modication algorithm We now show that we may modify the set W iteratively such that successively we nd a component Γ which satises the conditions (8.13), S (8.34), (8.35) and (8.38) such that Theorem 8.4.1 can be applied. Dene W s = λ≥0 Wλs .

Theorem 8.4.2. Let ε > 0 and h∗ ≥ σ > 0 suciently small. Let C1 = C1 (σ, h∗ ) ≥ 1 large, 0 < C2 = C2 (σ, h∗ ) < 1 small enough and let c > 0 be a universal constant. For all W ∈ V s with connected boundary components and u ∈ H 1 (W ) there is a set U ∈ W C2 s with |U \ W | = 0 and an extension u¯ dened by ( A l x + cl u¯(x) = u(x)

x ∈ Xl

else,

for all Γl (U ) with N 2ˆτl (∂Rl ) ⊂ H(U ),

(8.40)

such that for all Γl (U ) with N 2ˆτl (∂Rl ) ⊂ H(U ) ˆ

|[¯ u](x)|2 dH1 (x) ≤ C1 ε|Θl (U )|2∗ .

(8.41)

Θl (U )

Moreover, one has |W \ U | ≤ ckU k2∞ and εkU k∗ + α(U ) ≤ (1 + σ)(εkW k∗ + α(W )).

Remark 8.4.3.

(8.42)

In the proofs of Theorem 8.4.2 and Theorem 8.4.1 we will see that the constants Ci = Ci (σ, h∗ ) have polynomial growth in σ : We nd z ∈ N large enough such that C1 (σ, h∗ ) ≤ C(h∗ )σ −z and C2 (σ, h∗ ) ≥ C(h∗ )σ z . 166

The proof relies on an iterative modication procedure. First choose (8.43)

C2 = Cυ

for C small enough and consider W ∈ V s as an element of V C2 s such that (8.14) is satised for all boundary components Γl (W ). From now on we will always tacitly assume that all involved sets lie in V C2 s and write Wλ instead of WλC2 s . In the proof below we will show that C2 is in fact a constant only depending on h∗ and σ. ˆ , where W ˆ is the modication constructed in Corollary 8.2.2. We set W0 = W Choosing an ordering of the boundary components and setting ω(Γj (W0 )) = 1 for all j we obtain W0 ∈ W0 . Moreover, we let λ0 = 0, B00 = ∅. Assume that λ0 ≤ . . . ≤ λi and that Wi ⊂ . . . ⊂ W0 , Wj ∈ Wλj , are given (the inclusion holds up to sets of negligible L2 -measure) as well as {Bkj : k = 0, . . . , j} for j = 0, . . . , i. In each iteration step the sets Bkj , k = 0, . . . , j , will describe the set where we already `used' the `energy lying in the set' to modify W . Suppose that in an iteration step i the following conditions are satised:

εkWi kω + α(Wi ) ≤ εkW k∗ + α(W ) + h∗ (1 − ωmin )

Xi j=0

α(Bji )

(8.44)

as well as

(i) Each x ∈ Qµ lies in at most two dierent Bji1 , Bji2 and each x ∈ Wi lies in at most one Bji , j, j1 , j2 ∈ {0, . . . , i},

(ii) Either Θl (Wi ) ⊂ Bji for some 0 ≤ j ≤ i or [i  (8.45) Γl (Wi ) ∈ Gi := Γl : Γl ∩ Bji = ∅, ω(Γl ) = 1 for all Γl (Wi ), j=0

ηi

(iii) Each Bji with Bji ∩ Wi 6= ∅, satises Bji ∩ Wi ⊂ Mk l (Γl (Wi )), for some Γl (Wi ) ∈ Gi and k ∈ {1, 2}, j = 0, . . . , i. Here ηli := 21υ min{|Γl (Wi )|∞ , λi } = min{21¯ τl , 21υλi } and the neighborhood Mk was dened in (8.30). Moreover, recalling (8.36) we suppose

α(N τˆl (Γl (Wi )) ∩ Wi ) + ε|N τˆl (Γl (Wi )) ∩ (∂Wi \ Sλi (Wi ))|H ≤ Dεˆ τl for all Γl (Wi ) ∈ Gi ∩ Tλi (Wi )

(8.46)

where D is dened as in (8.34). Furthermore, recalling (8.37) we assume that there is an extension u ¯i ∈ SBV 2 (Wλi ) such that all boundary components Γl (Wi ) ∈ Tλi (Wi ) satisfy

ˆ |¯ ui (x) − (Al x − cl )| dH ≤ Cˆ 2

Θl (Wi )

1

i   X 2 n ω(Γl (Wi ))2 ε |Θl (Wi )|2∗ (8.47) 2 4 3 ω ˆ i (Γl (Wi )) υ n=0

167

2 for Al ∈ R2×2 ˆ i (Γl (Wi )) := 1 − 1−ω2min #{j = 0, . . . , i : Θl (Wi ) ⊂ skew , cl ∈ R , where ω Bji } and Cˆ is the constant from (8.39). In particular, this implies that (8.38) ˆ −2 Pi (2/3)n as ω is satised if we replace C∗ by Cω ˆ i (Γl (Wi )) ≥ ωmin (see min n=0 (8.45)(i), (ii)). Finally, we assume

(i) ω(Γl (Wi )) ≥ ω ˆ i (Γl (Wi )), ∀ Γl , (8.48) −1 (ii) |∂Rl (Wi )|∗ ≤ ω(Γl (Wi ))(ˆ ωi (Γl (Wi ))) |Θl (Wi )|∗ , ∀ Γl : ω(Γl ) < 1. The second condition is a renement of (8.5)(ii). Recall that ∂Wi ∩ Qµ ⊂ Wi by denition (see (8.1)). This particularly implies that each Θl (Wi ) is contained in at most one set Bji (see (8.45)(i),(ii)). Before we give the proof of Theorem 8.4.2 we rst observe that the above stated properties are preserved under modication.

Lemma 8.4.4. Let ε > 0 and λ ≥ 0. Let Wi ∈ Wλ , u¯i and {Bji : j = 0, . . . , i} be given such that (8.44)-(8.48) hold for Wi , u¯i and λ (replace λi by λ). For a ˜ i = (Wi \ V ) ∪ ∂V and assume that rectangle V ⊂ Qµ with |∂V |∞ > λ, let W (recall (8.7), (8.8)) ˜ i kω + α(W ˜ i ) ≤ εkWi kω + α(Wi ). εkW

Let Wi+1 ∈ Wλ be the set given by Lemma 8.2.1 and dene Bji+1 = Bji \ Sλ (Wi+1 ) i+1 for j = 0, . . . , i (recall (8.36)) and Bi+1 = ∅ as well as u¯i+1 = u¯i . Then (8.44)(8.48) hold for Wi+1 , u ¯i+1 and λ (replace λi+1 by λ). ˜ = (Wi \ V ) ∪ ∂V for some rectangle V with |V |∞ > λ and choose Proof. Let W

V 0 ⊃ V such that Wi+1 := (Wi \ V 0 ) ∪ ∂V 0 ∈ Wλ as in Lemma 8.2.1. Then by assumption and (8.10) we have εkWi+1 kω + α(Wi+1 ) ≤ εkWi kω + α(Wi ), where we adjust the weights as described in (8.8). As |Bji \ Bji+1 | = 0 for all j = 0, . . . , i, (8.44) is trivially satised. Clearly, (8.45)(i) still holds as Bji+1 ⊂ Bji for all j = 0, . . . , i. Moreover, S i+1 Γl (Wi+1 ) ∩ i+1 = ∅ for all Γl (Wi+1 ) ∈ Sλ (Wi+1 ) by denition. Consej=0 Bj quently, Sλ (Wi+1 ) ⊂ Gi+1 and to conrm (8.45)(ii) it suces to consider the components not lying in Sλ (Wi+1 ). Let Γl (Wi+1 ) ∈ / Sλ (Wi+1 ). Using that ∂V 0 ∈ Sλ (Wi+1 ) and arguing similarly as in Section 8.2 (see remark after (8.7)) we nd a (unique) corresponding Γl (Wi ) (for notational convenience we use the same index) such that Θl (Wi+1 ) = Θl (Wi ) \ V 0 . If Γl (Wi ) ∈ Gi we immediately get Γl (Wi+1 ) ∈ Gi+1 by (8.9) and the fact that the sets (Bji )ij=1 do not become larger. Consequently, we can assume that Θl (Wi ) ⊂ Bji for some j and it now remains to show Θl (Wi+1 ) ⊂ Bji+1 . To see this, it suces to observe Θl (Wi+1 ) = Θl (Wi ) \ V 0 ⊂ Θl (Wi ) and Θl (Wi+1 ) ∩ Sλ (Wi+1 ) = ∅, where the latter holds as the sets (Θj (Wi+1 ))j are pairwise disjoint (see (8.2)). Due to the modication procedure (see the construction of V 0 in the proof of Lemma 8.2.1), for all Γl (Wi ) ∈ Gi we nd a Γj (Wi+1 ) ∈ Gi+1 such that Γl (Wi ) ⊂ 168

Xj (Wi+1 ), where ∂Xj (Wi+1 ) = Γj (Wi+1 )). In fact, one can choose either Γl (Wi ) ηi

itself or ∂V 0 . (Note that both are elements of Gi+1 .) Therefore, Mk l (Γl (Wi )) ⊂ η i+1

Mk j (Γj (Wi+1 )) for k = 1, 2 and thus condition (8.45)(iii) holds. To conrm (8.48)(i) we rst observe that ω ˆ i+1 (∂V 0 ) = ω(∂V 0 ) = 1. Moreover, we see that ω ˆ i+1 (Γl (Wi+1 )) = ω ˆ i (Γl (Wi )) for all Γl (Wi+1 ) 6= ∂V 0 , where Γl (Wi ) is the unique corresponding component. In fact, ω ˆ i+1 (Γl (Wi+1 )) ≥ ω ˆ i (Γl (Wi )) i+1 i+1 follows immediately from the denition of (Bj )j=1 and the reverse inequality follows from the observation that Θl (Wi ) ⊂ Bji implies Θl (Wi+1 ) ⊂ Bji+1 (see above). Now this together with the fact that ω(Γl (Wi+1 )) ≥ ω(Γl (Wi )) (see (8.9)) yields (8.48)(i). We now show that (8.47) remains true. Similarly as before we nd for all Γl (Wi+1 ) ∈ Tλ (Wi+1 ) a (unique) corresponding Γl (Wi ). If Γl (Wi+1 ) ∩ ∂V 0 = ∅, then Θl (Wi+1 ) = Θl (Wi ) and there is nothing to show since Γl (Wi ) ∈ Tλ (Wi ) due to the fact that |Γl (Wi )|∞ = |Γl (Wi+1 )|∞ ≤ λ and ω(Γl (Wi )), ω ˆ i (Γl (Wi )) remain unchanged. Otherwise, ω(Γl (Wi+1 )) < 1 by (8.5)(v) and thus ω(Γl (Wi+1 ))|Θl (Wi+1 ))|∗ = ω(Γl (Wi ))|Θl (Wi )|∗

(8.49)

by (8.8), which together with the fact that u ¯i+1 = u¯i and ω ˆ i+1 (Γl (Wi+1 )) = ω ˆ i (Γl (Wi )) implies (8.47). To see that |Γl (Wi )|∞ ≤ λ also holds in this case (and thus Γl (Wi ) ∈ Tλ (Wi )) we note that |Γl (Wi+1 )|∞ ≤ 19υλ by (8.5)(iv) and therefore (8.49) together with (8.5)(i) and (8.6) implies |Γl (Wi )|∞ ≤ λ for υ small enough. Observe that by the same argument as in (8.49) property (8.48)(ii) is satised. (Recall that in the modication procedure we never change the rectangles ∂Rl .) Finally, (8.46) holds. Indeed, for a given Γl (Wi+1 ) ∈ Tλ (Wi+1 )∩Gi+1 we deduce Γl (Wi+1 ) ∩ ∂V 0 = ∅ by (8.5)(v) and thus Γl (Wi+1 ) = Γl (Wi ), where Γl (Wi ) is the corresponding component of Wi . The assertion now follows from the i-th iteration step of (8.46). In fact, for the left part it suces to recall Wi+1 ⊂ Wi . For the right part we note Sλ (Wi+1 ) = ∂V 0 ∪ (Sλ (Wi ) ∩ ∂Wi+1 ) ⊃ Sλ (Wi ) ∩ ∂Wi+1 (again recall that we did not change the rectangles ∂Rl ) and ∂Wi+1 \ ∂Wi ⊂ ∂V 0 ⊂ Sλ (Wi+1 ) which then yields ∂Wi+1 \ Sλ (Wi+1 ) ⊂ ∂Wi \ Sλ (Wi ) by an elementary computation.  We are now in a position to prove Theorem 8.4.2. Proof of Theorem 8.4.2. Using Corollary 8.2.2 we rst see that (8.44)-(8.48) hold ˆ and λ0 = 0, B 0 = ∅. Assume that Wi ∈ Wλ , λi , {B i : j = 0, . . . , i} for W0 = W 0 j i and u ¯i have already been constructed and that (8.44)-(8.48) hold. If now all Γl (Wi ) with N 2ˆτl (∂Rl (Wi )) ⊂ H(Wi ) satisfy |Γl (Wi )|∞ ≤ λi we stop Wi . We observe that in this case (8.41) holds for C1 = P∞and set nU−2= −4 ˆ C n=0 (2/3) ωmin υ by (8.47). Otherwise, there is some smallest Γ = Γ(Wi ) with respect to | · |∞ satisfying |Γ|∞ > λi and N 2ˆτ (Γ) ⊂ H(Wi ). To simplify the exposition, we will suppose that the choice of Γ is unique. At the end of the proof 169

we briey indicate the necessary changes if there are several components of the same size. q 3 Choose ωmin ≥ . We observe that Tτˆ ⊂ Tλi for τˆ as dened in (8.33). 4 Indeed, for τˆ ≤ λi it is obvious and for τˆ > λi it follows from the choice of Γ with respect . Thus, by (8.47) we get that (8.38) is satised replacing C∗ by Pi to | · |∞ 4 ˆ n C n=0 (2/3) . If Γ additionally fullls (8.13), (8.34) and (8.35), we may apply 3 Theorem 8.4.1. Therefore, recalling ω(Γ) = 1 we get that for suitable A ∈ R2×2 skew , 2 c∈R ˆ  1 4 Xi  2 n  ε |Γ|2∗ |¯ ui (x) − (Ax + c)|2 dx ≤ Cˆ + · Cˆ 4 n=0 2 3 3 υ Γ Xi+1  2 n ω(Γ)2 ε ≤ Cˆ |Γ|2 . n=0 3 ω ˆ (Γ)2 υ 4 ∗ Thus, (8.47) holds, as desired. We dene u ¯i+1 (x) = A x + c for x ∈ X and u¯i+1 = u¯i else, where ∂X = Γ. Moreover, we set Wi+1 = Wi , λi+1 = |Γ|∞ , Bji+1 = Bji for i+1 j = 0, . . . , i and Bi+1 = ∅. Clearly, (8.44)-(8.48) still hold due to choice of Γ with respect to |·|∞ . In particular, for (8.47) we note that Tλi+1 (Wi+1 ) = Tλi (Wi )∪{Γ}. Likewise, (8.46) is fullled by (8.34) and the fact that Sλi+1 (Wi+1 ) = Sλi (Wi ) (recall (8.36)). Moreover, (8.45)(iii) still holds as λi+1 ≥ λi . As also (8.5) is satised for λi+1 , we get Wi+1 ∈ Wλi+1 . We continue with the next iteration step. Otherwise (a) (8.13), (b) (8.34) or (c) (8.35) is violated.

˜ i = (Wi \ V ) ∪ ∂V , we get In case (a) we nd some V % Γ such that setting W ˜ kWi kω ≤ kWi kω and therefore ˜ i kω + α(W ˜ i ) ≤ εkWi kω + α(Wi ). εkW Here we adjusted the weights as in (8.8). Let λi+1 = |Γ|∞ . It is not hard to see that Wi ∈ Wλi+1 satises (8.44)-(8.48) also for λi+1 . In fact, (8.47) follows from the choice of Γ with respect to |·|∞ and the fact that |∂V |∞ > λi+1 . For the other properties we may argue as before. Now Lemma 8.4.4 yields a set Wi+1 ∈ Wλi+1 ˜ i as well as (B i+1 )i+1 and u¯i+1 = u¯i such that (8.44)-(8.48) hold with Wi+1 ⊂ W j j=1 for Wi+1 , u ¯i+1 and λi+1 . We now continue with the next iteration step.

˜ i = (Wi \V )∪∂V , where V is the smallest rectangle containing In case b) set W ≥ h∗ ωCmin N (Γ). Observe that |∂V |∗ ≤ |Γ|∗ + C τˆ. Choosing D = D(h∗ ) ≥ 2C h∗ and arguing as in the proof of Lemma 8.3.1 we derive 4ˆ τ

˜ i kω + α(W ˜ i ) ≤ εkWi kω + α(Wi ) + Cεˆ εkW τ − α(V ∩ Wi ) ˆ |H ≤ εkWi kω + α(Wi ). − εh∗ ωmin |∂Wi ∩ N As usual we adjusted the weights as in (8.8). We now may proceed as in case (a) and then continue with the next iteration step. 170

Finally, consider case (c). Let Ψi = Ψj and ψi = ψ j , where Ψj is a set such that (8.35) is violated. As derived above in Section 8.3.2, we nd a boundary component Γm = Γm (Wi ) with |Γm |∞ ≥ τˆ, ω(Γm ) = 1. Moreover, there is a rectangle T ⊂ Qµ with |∂T |H ≤ 4ψi and T ∩ Γ 6= ∅, T ∩ Γm 6= ∅ (cf. Figure 8.8). Let Ai ⊂ (Γl (Wi ))l \{Γ, Γm } be the boundary components with Θl (Wi )∩Ψi 6= i ∅ or, if Γl (Wi ) = Θl (Wi ) ∈ Gi , with M ηl (Γl (Wi )) ∩ Ψi 6= ∅. We now dene an i additional set Bi+1 , where we will `use the energy' to modify Wi . Let   [ [ i Bi+1 = (Ψi ∩ Wi ) ∪ Θl (Wi ) \ Bji , i Bj ∈Bi

Γl (Wi )∈Ai

i

where Bi := {Bji : Bji ∩ Wi ⊂ M ηl (Γl (Wi )) for some Γl (Wi ) ∈ Ai ∩ Gi }. In the i it is essential to subtract the set on the right hand side such denition of Bi+1 that we will be able to assure (8.45)(i). Ψ Γl1

Γl3

Xm

X

Bji2 i

Bj1 

Θl4 Γl2

Figure 8.7: On the left side Ψ (the set surrounded by the dashed grey line) and parts

of Γ, Γm are sketched. Observe that M 21¯τm (Γm ) ∩ Ψ = M 21¯τ (Γ) ∩ Ψ = ∅. Moreover, the picture includes several boundary components with corresponding dodecagonal or decagonal neighborhoods as well as four striped sets Bji1 , . . . , Bji4 . On the right hand i i i )◦ is grey. side the resulting Bi+1 is drawn, where ∂Bi+1 is black and the interior (Bi+1 i i Observe that in general only parts of ∂Bi+1 are contained in Bi+1 . i Note that by (8.45) we have for all Γl (Wi ) ∈ AS i either Θl (Wi ) ⊂ Bi+1 or i Θl (Wi ) ∩ Bi+1 = ∅ depending on whether Θl (Wi ) ∩ B i ∈Bi Bji = ∅ or Θl (Wi ) ⊂ j

i Bji ∈ Bi . Moreover, the components Γl (Wi ) ∈ / Ai clearly satisfy Θl (Wi )∩Bi+1 = ∅. i ˜ Denote by Ai ⊂ Ai the boundary components completely contained in Bi+1 and observe that Gi ∩ Ai ⊂ A˜i . By (8.52) below we obtain |Γl (Wi )|∞ < 19¯ τ for all Γl (Wi ) ∈ Ai which by 2ˆ τl (8.33) for υ suciently small implies N (∂Rl ) ⊂ N 2ˆτ (Γ) ⊂ H(Wi ). Moreover, as |Γl (Wi )|∞ < |Γ|∞ , by the choice of Γ with respect to |·|∞ we obtain |Γl (Wi )|∞ ≤ λi for all Γl (Wi ) ∈ Ai and thus Ai ⊂ Tλi (Wi ). As Tλi (Wi ) ∩ Sλi (Wi ) = ∅, this also yields Ψi ∩Sλi (Wi ) = ∅. This together with (8.46) shows that γ(M 21¯τl (Γl (Wi ))) ≤

171

Dεˆ τl for all Γl (Wi ) ∈ Ai ∩ Gi , where γ(A) := α(A ∩ Wi ∩ Ψi ) + ε|A ∩ (∂Wi ∩ Ψi )|H τl as |Γl (Wi )|∞ ≤ λi for all Γl (Wi ) ∈ Ai . Using for A ⊂ R2 . Observe that ηli = 21¯ the denition of Bi , (8.33), (8.46) and recalling D = D(h∗ ) we nd for υ small enough (with respect to h∗ ) X X i ) ≤ γ(B γ(M 21¯τl (Γl (Wi ))) j Bji ∈Bi Γl (Wi )∈Ai ∩Gi X i ≤ ε|Γl (Wi )|∞ ≤ ε|Bi+1 ∩ ∂Wi |H . ˜ Γl (Wi )∈Ai

In the rst step we used that Bji1 ∩ Bji2 ∩ Wi = ∅ for j1 6= j2 by (8.45)(i) and ∂Wi ∩ Qµ ⊂ Wi . The last step follows from the denition of A˜i . Recall that The i then imply fact that (8.35) is violated and the denition of Bi+1

D(1 − ωmin )−1 εψi < α(Ψi ∩ Wi ) + ε|∂Wi ∩ Ψi |H ≤ α(Ψi ∩ Wi ) + ε|∂Wi ∩ Ψi |H − + ε|∂Wi ∩

X Bji ∈Bi

γ(Bji )

i Bi+1 |H

(8.50)

i i ≤ α(Bi+1 ) + 2ε|∂Wi ∩ Bi+1 |H .

We adjust the weights for components in A˜i : Let Wi∗ = Wi and ω(Γl (Wi∗ )) = ω(Γl (Wi )) − 1−ω2min for Γl (Wi∗ ) = Γl (Wi ) ∈ A˜i and ω(Γl (Wi∗ )) = ω(Γl (Wi )) otherwise. (The set as a subset of R2 is left unchanged, we have only changed the weights of the boundary components.) This implies i kWi∗ kω ≤ kWi kω − 21 h∗ (1 − ωmin )|∂Wi ∩ Bi+1 |H .

(8.51)

∗ We briey note that (8.47), (8.48) are still satised for Wi∗ if we replace ω ˆ i by ω ˆ i+1 , 1−ωmin ∗ ∗ i where ω ˆ i+1 (Γl (Wi )) := 1 − 2 #{j = 0, . . . , i + 1 : Θl (Wi ) ⊂ Bj }. Indeed, as ωi (Γl (Wi )) ≥ ω ˆ i (Γl (Wi )) by (8.48) we nd

ωi (Γl (Wi∗ )) ωi (Γl (Wi )) − (1 − ωmin )/2 ωi (Γl (Wi )) = ≥ ≥1 ∗ ∗ ω ˆ i+1 (Γl (Wi )) ω ˆ i (Γl (Wi )) − (1 − ωmin )/2 ω ˆ i (Γl (Wi )) ∗ for Γl (Wi ) ∈ A˜i . (Observe that ω ˆ i+1 may slightly dier from the desired ω ˆ i+1 as given in (8.47). Below we will see, however, that the properties are still satised for ω ˆ i+1 .) ˜ i = (Wi∗ \ V ) ∪ ∂V , where V is the smallest rectangle containing We set W Γ, Γm and T . As usual we dene ω(∂V ) = 1 and adjust the other weights as in (8.8). We then derive by (8.50) and (8.51)

˜ i kω ≤ kW ∗ kω + |∂T |H ≤ kWi kω − 1 h∗ (1 − ωmin )|∂Wi ∩ B i |H + |∂T |H kW i i+1 2 i 1 ≤ kWi kω + h∗ (1 − ωmin ) 4ε α(Bi+1 ) − 14 h∗ Dψi + |∂T |H ,

172

where for the other boundary components not involved we proceeded as in (8.12). Recall |∂T |H ≤ 4ψi . Now choosing D ≥ h16∗ we conclude

˜ i kω + α(W ˜ i ) ≤ εkWi kω + α(Wi ) + h∗ (1 − ωmin )α(B i ). εkW i+1 Dene λi+1 = |Γ|∞ and u ¯i+1 = u¯i . Observe that Wi∗ ∈ Wλi . In fact, (8.5)(iv) follows from the denition of the weights and (8.52) below. Moreover, (8.5)(ii) is a consequence of (8.48)(ii). As before, following the proof of Lemma 8.4.4, we nd a set Wi+1 = (Wi∗ \ V 0 ) ∪ ∂V 0 ∈ Wλi+1 for a rectangle V 0 ⊃ V with ˜ i and B i+1 = B i \ Sλ (Wi+1 ) for j = 0, . . . , i + 1 such that (8.44), Wi+1 ⊂ W j i+1 j (8.46) hold and (8.47), (8.48) are satised for ω ˆ i+1 . Observe that (8.45) does not i+1 follow from Lemma 8.4.4 as Bi+1 6= ∅. We postpone the proof of (8.45) to Lemma 8.4.5 below. We now continue with the next iteration step. In each iteration step either the number of components satisfying (8.47) increases or the volume of Wi decreases by at least (2C2 s)2 . Consequently, after a nite number of steps, denoted by i∗ , we nd a set U = Wi∗ ∈ WλsU , λU ≥ 0, satisfying (8.47) for all boundary components Γl (U ) with N 2ˆτl (∂Rl (UP )) ⊂ H(U ). Let u¯ = u¯i∗ . Then (8.41) holds for all such boundary components as n (2/3)n < ∞, ω ˆ (Γl (U )) ≥ ωmin for all Γl (U ) by (8.45)(i) and P ∗ since υ∗ can be chosen in dependence of h∗ . Similarly, by (8.45)(i) we nd ij=0 α(Bji ) ≤ 2α(W ) and by (8.48) we get ω(Γl (U )) ≥ ωmin for all Γl (U ). Setting σ = 2(1 − ωmin ), by (8.44) and h∗ ≤ 1 we conclude

εkU k∗ + α(U ) ≤ (1 − 12 σ)−1 εkW k∗ + (1 + σ)α(W ) ≤ (1 + σ)(εkW k∗ + α(W )). As υ is chosen in dependence of h∗ and σ , the constant C2 in (8.43) depends only on h∗ and σ . Finally, the property |W \ U | ≤ ckU k2∞ relies on the isoperimetric inequality and can be derived as in Corollary 8.2.2. It remains to indicate the necessary changes if in some iteration step i the choice of Γ is not unique. If there are several components Γ1 , . . . , Γm with λi+1 := |Γj |∞ > λi for j = 1, . . . , m we choose an order such that Γ1 , . . . , Γm0 , m0 ≤ m, are the components satisfying (8.13), (8.34), (8.35). We now apply Theorem 8.4.1 successively on each Γj , j = 1, . . . , m0 , and replace Tλi+1 (Wi+1 ) in (8.46), S (8.47) by Tλji+1 (Wi+1 ) := Tλi (Wi ) ∪ jk=1 {Γk }. For each Γj , j = m0 + 1, . . . , m, we S 0 proceed as in one of the cases a) - c) and let Tλji+1 (Wi+1 ) := Tλi (Wi ) ∪ m k=1 {Γk } in (8.46), (8.47).  It remains to show (8.45) in case c).

Lemma 8.4.5. If in the i-th iteration step of the above modication procedure case c) is applied, then (8.45) holds for Wi+1 . Proof. We rst show that |Γl (Wi )|∞ < 19¯ τ and dist(Γl (Wi ), Ψi ) ≤ τ¯ 173

for all Γl (Wi ) ∈ Ai .

(8.52)

For sets Γl = Γl (Wi ) intersecting Ψi ⊂ N τ¯ (Γ) this is clear by construction of Ψi and Corollary 8.3.4. (We can assume that property (8.13) holds and Corollary 8.3.4 is applicable as otherwise we would have applied case (a).) Now assume i Γl ∩ N τ¯ (Γ) = ∅ but M ηl (Γl ) ∩ Ψi 6= ∅ for Γl ∈ Gi , which implies dist(Γl , Ψi ) < τ . In particular, this yields Γl ∩ N 22¯τ (Γ) 6= ∅. Recall that ηil ≤ 21υλi ≤ 21¯ Γm ∩ Ψ1 6= ∅ and |Γm |∞ ≥ τˆ ≥ 19 · 22¯ τ for q large enough. Therefore, applying Corollary 8.3.4 for t¯ = 22¯ τ we derive that that |Γl |∞ ≤ 19 · 22¯ τ . Repeating the l τl ≤ υ · 21 · 19 · 22¯ τ < τ2¯ for υ above arguments we obtain dist(Γl , Ψi ) ≤ 21ηi = 21¯ small enough due to the choice of Γ with respect to | · |∞ . This gives the second part of (8.52). Moreover, we have Γl ∩ N τ¯ (Γ) 6= ∅ as M 21¯τl (Γl ) ∩ N τ (Γ) 6= ∅ and τ ≤ τ2¯ by (8.26). This, however, gives a contradiction to the assumption and thus Γl ∩ N τ¯ (Γ) 6= ∅. Then the rst part of (8.52) follows again from Corollary 8.3.4. We now show that (8.45) holds for Wi+1 . Note that by Lemma 8.4.4 we have Wi+1 = (Wi∗ \V 0 )∪∂V 0 for a rectangle V 0 which contains Γ, Γm and T . Moreover, recall that Wi = Wi∗ only dier by the denition of the weights. First of all, to see (8.45)(ii) it suces to show that either Θl (Wi∗ ) ⊂ Bji for S i ∗ some 0 ≤ j ≤ i + 1 or Θl (Wi∗ ) ∩ i+1 j=0 Bj = ∅ and ω(Γl (Wi )) = 1. In fact, we can then follow the argumentation in the proof of Lemma 8.4.4 to obtain the desired property also for the sets Bji+1 = Bji \ Sλi+1 (Wi+1 ), j = 0, . . . , i + 1. i i = ∅ for all (Γl (Wi∗ ))l . Thus, if or Θl (Wi∗ ) ∩ Bi+1 Recall that Θl (Wi∗ ) ⊂ Bi+1 S i Θl (Wi∗ ) 6⊂ Bji for some 0 ≤ j ≤ i + 1 we nd Θl (Wi∗ ) ∩ i+1 j=0 Bj = ∅ by (8.45)(ii) i = ∅. / A˜i as Θl (Wi∗ ) ∩ Bi+1 for iteration step i. This particularly implies Γl (Wi∗ ) ∈ Again by (8.45)(ii) and the construction of the weights in (8.51) we then get ω(Γl (Wi∗ )) = 1, as desired. We concern ourselves with (8.45)(i). First, the assertion is clear for x ∈ Wi+1 \ Wi as Wi+1 \ Wi ⊂ ∂V 0 and ∂V 0 ∈ Sλi+1 (Wi+1 ). For x ∈ / Wi+1 \ Wi it is i+1 i i i+1 for j = 0, . . . , i + 1. As ⊂ B enough to show the property for (B ) since B j j j=1 j Sn i and thus Bi+1 ⊂ Wi , it is elementary to see that it suces to l=1 Θl (Wi ) ⊂ W Sii i conrm Bi+1 ∩ j=0 Bji ⊂ Wi \ Wi+1 . Recall that Γ, Γm ∈ / Ai . By the denition i of Bi+1 we have i Bi+1 ∩

[i j=0


i Bji ⊂ Bi+1 ∩ f (M 21¯τ (Γ)) ∪ f (M 21¯τm (Γm )) ,

where f (A) = A if A ∩ Ψi 6= ∅ and f (A) = ∅ else for A ⊂ R2 . (The possible dierent cases can be seen in Figure 8.5, 8.6, 8.7.) To see this, let A∗ ⊂ {Γ, Γm } such that the boundary component is contained in A∗ if the corresponding neighi borhood intersects Ψi . Observe that if Bji ∩ Bi+1 6= ∅, then by (8.45)(iii) (for Wi ) i i i i ηl we get Bj ∩ Bi+1 ⊂ Bj ∩ Wi ⊂ M (Γl ) for some Γl ∈ Gi \ A˜i = Gi \ Ai . (The last equality follows from Gi ∩Ai ⊂ A˜i .) On the other hand, by (8.45)(ii),(iii) we derive that each Γl (Wi ) ∈ Ai with Θl (Wi ) ⊂ Bji satises Γl (Wi ) ∈ / Gi , Θl (Wi ) ∩ Ψi 6= ∅ ηli and thus Θl (Wi ) ∩ M (Γl ) = ∅ for all Γl ∈ Gi \ (Ai ∪ A∗ ). Likewise, we get 174

i

i

i Ψi ∩ M ηl (Γl ) = ∅ for all Γl ∈ Gi \ (Ai ∪ A∗ ) and thus (Bji ∩ Bi+1 ) ∩ M ηl (Γl ) = ∅ i i for all Γl ∈ Gi \ (Ai ∪ A∗ ). This implies Bji ∩ Bi+1 ⊂ M ηl (Γl ) for some Γl ∈ A∗ . Setting Φ := {x ∈ Qµ : dist(x, Ψi ) ≤ 20¯ τ } and recalling (8.52) we then nd [i
i Bi+1 ∩ Bji ⊂ Φ ∩ f (M 21¯τ (Γ)) ∪ f (M 21¯τm (Γm )) . j=0

We now dier the cases (I) and (II) as considered in Lemma 8.3.6, 8.3.7. In case (I) we get Φ∩Wi+1 = ∅ as by Lemma 8.3.6(i) the rectangle V 0 satises Φ ⊂ V 0 . In (II)(i) the assertion follows as M 21¯τ (Γ), M 21¯τm (Γm ) ∩ Ψi = ∅ . Finally, in (II)(ii) it suces to derive [i i Bi+1 ∩ Bji ⊂ Φ ∩ {x : x1 ≥ −l1 − ψ} ∩ M 21¯τm (Γm ), (8.53) j=0

where without restriction we treat the case Γm ∩ N (Γ) ⊂ N1,− (Γ). Then Lemma 8.3.7(i) gives Φ ∩ {x : x1 ≥ −l1 − ψ} ∩ M 21¯τm (Γm ) ⊂ V 0 which nishes the proof of (8.45)(i). To see (8.53), rst note that f (M 21¯τ (Γ)) = ∅ and Ψi ⊂ {x : x1 ≥ −l1 − ψ}. Consequently, recalling (8.45)(ii),(iii) if the assertion was wrong, there would be some Γl (Wi ) ∈ Ai \ Gi which satises Θl (Wi ) ⊂ M 21¯τm (Γm ) and Θl (Wi ) ∩ {x : x1 < −l1 − ψ} 6= ∅. Again by (8.45)(iii) we then get Θl (Wi ) ⊂ M221¯τm (Γm ) (see Figure 8.6) and therefore Θl (Wi ) ∩ Ψi = ∅. This implies Γl (Wi ) ∈ / Ai and yields a contradiction. i+1 Finally we show (8.45)(iii). It suces to consider Bi+1 as for the other sets the property follows from Lemma 8.4.4. Without restriction we set V 0 = (−v1 , v1 ) × ¯ ¯ = υλi+1 . This is a (−v2 , v2 ). We rst observe that Φ \ V 0 ⊂ N 21λ (∂V 0 ), where λ ¯ = τ¯. We may assume that consequence of the denition of Φ and the fact that λ l1 1 |π1 Γm | ≥ 2 |π2 Γm |. In fact, if l2 ≤ 2 this follows from Corollary 8.3.4, if l2 ≥ l21 then l1 , l2 are comparable and the assumption holds possibly after a rotation of 2 the components by π2 . Recalling |Γm |∞ ≥ τˆ we thus obtain |π1 Γm | ≥ √15 τˆ = √q5hτ¯ . ∗ Recall that |π1 Γm ∩ π1 Γ| ≤ C hτ¯∗ by Lemma 8.3.1(ii). ¯

M221λ (∂V ) B

Ψ ∂T Γm

Γ

∂V ¯

M121λ (∂V )

Figure 8.8: Sketch of the components Γ, Γm , ∂T and in dashed lines the corresponding rectangle V (which in this example coincides with V 0 ). The ball B is chosen large enough such that Φ ⊂ B . (The proportions were adapted for illustration purposes.) 175

We now nd for all x ∈ Φ \ V 0 with x1 , x2 ≥ 0

√ v1 − x1 min{|π1 Γ|, |π1 Γm |} − C τ¯h−1 q 2 τ¯( 5h∗ )−1 − C τ¯h−1 ∗ ∗ ≥ ≥ ≥ qh−1 ∗ x2 − v2 21¯ τ 21¯ τ for q suciently large and may proceed likewise for ±x1 , ±x2 ≥ 0. Thus, upon ¯ i recalling (8.28) and (8.29), we obtain Bi+1 ∩ Wi+1 ⊂ Φ \ V 0 ⊂ M221λ (∂V 0 ). As ¯ , ω(∂V 0 ) = 1 and ∂V 0 ∩ Si+1 B i+1 = 0 (since ∂V 0 ⊂ Sλ (Wi+1 )) we υ|∂V 0 |∞ ≥ λ i+1 j=0 j nally obtain (8.45)(iii). 

Remark 8.4.6.

(i) During the modication process in Theorem 8.4.2 the components Xn+1 (W ), . . . , Xm (W ) at the boundary of Qµ might be changed and the corresponding components of U are given by Xj (U ) = Xj (W ) \ H(U ) for j = n + 1, . . . , m. In particular, we observe |∂Xj (U )|∗ ≤ |∂Xj (W )|∗ arguing as in Lemma 8.1.1. (ii) In general, the components of the set U might not be connected as they can be separated by other components during the modication process. However, by application of Corollary 8.2.2 we obtain a set U 0 ⊂ U with kU 0 k∗ ≤ kU k∗ and |U \ U 0 | ≤ CkU 0 k2∞ ≤ CµkU 0 k∞ such that all components of U 0 are pairwise disjoint and rectangular and thus particularly connected. Moreover, recalling the modication process (cf. Section 8.2) we nd that for each Γ(U ) the corresponding rectangle R(U ) given by (8.5) is contained in a component of U 0 .

8.4.3 Proof of the main theorem We now are in a position to prove our Korn-Poincaré-type inequality. For later purpose in Section 9 we split the proof into three steps and begin with a corollary of Theorem 8.4.2. In what follows, we will frequently employ (8.41) and in doing so we apply the inequalities

|Θ|H ≤ C|Θ|∗ ≤ C|Γ|∞ ≤ C|Γ|H ,

|Θ|∗ ≤ |∂R|∗ ≤ |∂R|H

(8.54)

for a boundary component Θ ⊂ Γ and the corresponding rectangle R given by (8.5). The properties follow from (8.5)(i), (8.6) and Lemma P8.1.1(vi)). Moreover, s we observe that for W ∈ V and a subset A ⊂ Qµ one has Γl (W ) |Γl (W ) ∩ A|H ≤ 2|∂W ∩ A|. Recall the denition of W C2 s in Theorem 8.4.2 as well as (7.3) and (A.1).

Corollary 8.4.7. Let ε, µ, h∗ > 0. Let U ⊂ Qµ = (−µ, µ)2 , U ∈ W C2 s and ˜ = (−˜ uR ∈ H 1 (U ). Assume there is a square Q µ, µ ˜)2 ⊂ Qµ such that (8.41) is satised for all components Θl (U ) having nonempty intersection with Q˜ , where u¯R is the extension of uR dened in (8.37). Then there is a universal constant C such that 176

˜ 2 ≤ (ER (Q)) ˜ 2 ≤ Cµ ˜ |E u¯R |(Q) ˜2 ke(∇uR )k2L2 (U ∩Q) ˜ + CC1 µε|∂U ∩ Q|H |∂U ∩ Qµ |H ,

where C1 is the constant in Theorem 8.4.2. Proof. Recall α(V ) = ke(∇¯uR )k2L2 (V ) for V ⊂ U . Note that by Hölder's inequality we have

2

2

|E u¯R |(V ) ≤ (ER (V )) ≤ C|V |α(V ) + C

ˆ

|[uR ]| dH

1

2

(8.55)

.

V ∩Ju¯R

˜= for V ⊂ U . Moreover, observe that Ju¯R ∩ Q (8.41) and (8.54) ˜ 2 ≤ Cµ ˜ +C (ER (Q)) ˜2 α(Q)

S

l

˜ . We now derive by Θl (U ) ∩ Q 1

X

˜ 2 k[¯ |Θl (U ) ∩ Q| H uR ]kL2 (Θl (U )) ˜ =∅ Θl (U )∩Q6 X ˜ + CC1 ε|∂U ∩ Q| ˜H |Γl (U )|2∞ ≤ Cµ ˜2 α(Q)


2

l

˜ ∩ U ) + CC1 µε|∂U ∩ Q| ˜ H |∂U ∩ Q|H . ≤ Cµ ˜ α(Q 2

In the second and third step we employed Hölder's inequality. In the last step we P ˜ used α(Q\U ) by (8.40) as well as |Γl (U )|∞ ≤ 2µ and l |Γl (U )|∞ ≤ C|∂U ∩Q|H .  We now formulate and prove the main theorem of this chapter rst in terms of sets W ∈ V s . The assertion follows combining Theorem 8.4.2, Corollary 8.4.7 and Theorem B.4.

Theorem 8.4.8. Let ε, µ > 0 and h∗ > 0 suciently small. There is a constant C = C(h∗ ) and a universal constant c¯ > 0 such that for all sets W ⊂ Qµ = (−µ, µ)2 , W ∈ V s with connected boundary components, and all y ∈ H 1 (W ), R ∈ SO(2), the following holds: There is a set U ∈ W C2 s with |U \ W | = 0, |W \ U | ≤ c¯|∂U ∩ Qµ |2∞ and εkU k∗ + ke(∇uR )k2L2 (U ) ≤ (1 + h∗ ) εkW k∗ + ke(∇uR )k2L2 (W )




such that for the square Q˜ = (−˜µ, µ˜)2 with µ˜ = max{µ − 2|∂U ∩ Qµ |H , 0} we have 2 2 2 kuR (x) − (A x + c)k2L2 (U ∩Q) ˜ ≤ Cµ ke(∇uR )kL2 (U ∩Q) ˜ + Cµε|∂U ∩ Q|H 2 for some A ∈ R2×2 skew and c ∈ R .

Proof. Choose σ = σ(h∗ ) ≤ h∗ and apply Theorem 8.4.2 to get a set U ∈ W C2 s

with |U \ W | = 0 satisfying (8.42). We can assume that µ ˜ = µ − 2|∂U ∩ Qµ |H > 0 ˜ and (8.33) it is not as otherwise there is nothing to show. By denition of Q ˜ 6= ∅ fullls hard to see that every boundary component Γl (U ) with Θl (U ) ∩ Q 177

N 2τˆl (∂Rl (U )) ⊂ H(U ) and therefore (8.41) holds. The claim now follows from Theorem B.4 and Corollary 8.4.7.  We can now nally give the proof of Theorem 8.0.1. Proof of Theorem 8.0.1. Let u ∈ SBD2 (Qµ , R2 ) ∩ L2 (Qµ , R2 ) be given. Following the arguments in the proof of Theorem 9.4.1 below (see (9.110)), in particular covering Ju with rectangles, we nd a set W ∈ V s with connected boundary components such that kW k∗ ≤ (1+¯ ch∗ )H1 (Ju ) and a modication u˜ ∈ SBD2 (Qµ , R2 ) u) − e(∇u)k2L2 (Qµ ) ≤ δ for δ > 0 arbitrarily small such with k˜ u − uk2L2 (Qµ ) + ke(∇˜ that u ˜ ∈ H 1 (W ). We now apply Theorem 8.4.8 on u ˜ and W for R = Id and y = id + u˜. Up to a modication by applying Corollary 8.2.2 we can assume that the components of P U are pairwise disjoint rectangles R1 , . . . , Rn with j |Rj |∗ ≤ kU k∗ which yields
P 1 −1 2 |R | ≤ (1 + c ¯ h )kU k ≤ (1 + c ¯ h ) H (J ) + ε ke(∇˜ u )k 2 j ∞ ∗ ∗ ∗ u j L (W ) . Finally, as ˜ we may assume kU k∗ ≤ Cµ (otherwise Q = ∅) we conclude ku(x) − (A x + c)k2L2 (U ∩Q) u(x) − (A x + c)k2L2 (U ∩Q) ˜ ≤ Cδ + k˜ ˜ 2 ≤ Cδ + Cµ2 ke(∇˜ u)k2L2 (U ∩Q) ˜ + CµεkU k∗

≤ C(1 + µ2 )δ + Cµ2 ke(∇u)k2L2 (Qµ ) + Cµ2 εH1 (Ju ). As δ was arbitrary, we obtain the desired estimate.



8.5 Trace estimates for boundary components This section is entirely devoted to the proof of Theorem 8.4.1. We start with some preliminary estimates including an approximation of uR by a piecewise innitesimal rigid motion. Here we also discuss the passage from an estimate in the neighborhood to a trace estimate. Afterwards the proof is performed in several steps. We will rst assume that in a neighborhood of Γ only small boundary components are present (Step 1). Then we suppose that we have a bound on the projection k · kπ (recall denition (8.18)) which will allow us to apply a slicing method in the regions of the domain where too large boundary components exist (Step 2). In this context, we have to be particularly careful at the corners of Γ (Step 3). Finally, we present the general proof taking into account the possible existence of sets Ψ1 , Ψ2 discussed in Section 8.3.2 (Step 4). At this point, the trace theorem we derived in Section 7.2 will play an essential role.

8.5.1 Preliminary estimates Assume h∗ , q, ωmin > 0 have been chosen in the previous section (in this order, see Remark 8.3.2). The parameter υ > 0 considered before is not assumed to 178

be already chosen, but will be specied below. Moreover, let r > 0 such that r(1 − ωmin )3 ≥ υ . This implies υ = υ(h∗ , q, ωmin , r). Moreover, we will show r = r(h∗ , q) and recalling that σ = 2(1 − ωmin ) (see proof of Theorem 8.4.2) as well as using q = q(h∗ ) we will nd υ ∼ C(h∗ )σ 3 (cf. Remark 8.4.3). Let ε > 0, λ ≥ 0 and let y ∈ H 1 (W ) for W = Wi ∈ Wλs . (We drop the subscript i in the following.) Again we drop the subscript R ∈ SO(2) and write u = RT y − id instead of uR . Recall α(U ) = ke(∇u)k2L2 (U ) for U ⊂ W . Let Γ = Γ(W ) with |Γ|∞ ≥ λ and the corresponding neighborhoods N (Γ) = N τ (Γ) ˆ (Γ) = N 2ˆτ (Γ) be given. In addition, we dene N ˜ (Γ) = N (Γ) \ (X 1 ∪ X 2 ), and N 1 1 2 2 where ∂X = Γ , ∂X = Γ are the boundary components satisfying |Γi |∞ ≥ τˆ = q 2 h∗−1 υ|Γ|∞ and Γi ∩ N (Γ) 6= ∅, see Corollary 8.3.4 (note that X1 , X2 = ∅ ˆ and N ˜ if no confusion is possible). As before, for shorthand we will write N , N arises. By Remark 7.1.4(i) it is not restrictive to assume that J = J(Γ) as dened before equation (8.14) consists of (almost) squares. Suppose that (8.13), (8.34), (8.35) and (8.38) for Tτˆ (W ) hold. Assume that N 2ˆτ ⊂ H(W ). ˜ may be strict due to boundary compoNote that the inclusion W ∩ N ⊂ N nents Γl with |Γl |∞ < τˆ having nonempty intersection with N . Observe that by (8.5)(iv),(v) and (8.6) for q large we have |∂Rl |∞ < τˆ for these Γl , where Rl is the corresponding rectangle given by (8.5)(i),(v). Then for υ suciently small we get

N 2ˆτl (∂Rl ) ⊂ N 2ˆτ (Γ) ⊂ H(W )

(8.56)

and thus Γl ∈ Tτˆ (W ). Consequently, we can extend u as an SBV function from ˜ as dened in (8.37). For convenience we denote this extension still N ∩ W to N ˜ ) = α(W ∩ N ). by u. Clearly, one has α(N We now begin with some preliminary estimates. Recall denition (7.3) as well ˜ = N . We apply Theorem as (8.55) and write E instead of ER . First assume N B.4 on each Q ∈ J recalling that the constant is invariant under rescaling of the domain: This yields functions A¯ : N → R2×2 ¯ : N → R2 being constant on skew , c each Q ∈ J such that by (8.34) and (8.38) we obtain ˆ X |u(x) − (A¯ x + c¯)|2 dx ≤ C (E(Q))2 ≤ C(E(N ))2 Q∈J N X
2 1/2 ≤ Cυ|Γ|2∞ α(N ) + C |Θl ∩ N |H k[u]kL2 (Θl ) (8.57) l X
2 2 3 −4 ≤ Cυ |Γ|∞ ε + CC∗ υ ε|∂W ∩ N |H |Θl |∗ l

ˆ |3H ≤ C(1 + C∗ )υ −1 |Γ|3∞ ε ≤ Cυ 2 |Γ|3∞ ε + CC∗ υ −4 ε|∂W ∩ N for some C = C(h∗ , q). In the third step we employed Hölder's P P inequality and 2 |N | ≤ Cυ|Γ|∞ . In the penultimate step we used l |Θl |∗ ≤ C l |Γl |H ≤ C|∂W ∩ ˆ |H by (8.54) and (8.56). The constants used in this section may as usual vary N from line to line but are always independent of the parameters r, ωmin and υ . 179

In the general case, recall the denition of Ψ1 and Ψ2 in Section 8.3.2. By Lemma 8.3.6(ii) and Lemma 8.3.7(ii) it is not restrictive to assume that Ψi1 , Ψi,∗ 2 , i Ψ3 are squares for i = 1, 2. Similarly as in the previous estimate we obtain functions A¯ : N → R2×2 ¯ : N → R2 being constant on each Q ∈ J with skew , c Q ∩ (Ψ1 ∪ Ψ2 ) = ∅ and Ψij , i = 1, 2, j = 1, 2, 3, such that ˆ |u(x) − (A¯ x + c¯)|2 dx ≤ C(1 + C∗ )υ −1 |Γ|3∞ ε, (i) 1 2 ˜ N \(Ψ ∪Ψ ) ˆ (8.58) (ii) |u(x) − (A¯ x + c¯)|2 dx ≤ C(1 + C∗ )υ −3 |Γ|2∞ ψ i ε, Ψij

i To see (ii), we apply Theorem B.4 on the sets Ψi1 , Ψi,∗ 2 , Ψ3 and follow the lines of the previous estimate to obtain that the left hand side is bounded ˆ |2 . We then use α(Ψi ) ≤ by Cυ|Γ|2∞ α(Ψij ) + CC∗ υ −4 ε|∂W ∩ Ψij |H |∂W ∩ N j H −1 i −1 i i Dε(1 − ωmin ) ψ ≤ Dευ ψ and |∂W ∩ Ψ |H ≤ Cυ −1 ψ i by (8.34), (8.35). The goal will be to replace the functions A¯, c¯ in (8.58) by constants A ∈ R2×2 skew and c ∈ R2 such that ˆ (i) |u(x) − (A x + c)|2 dx ≤ C(1 + rC∗ )υ −3 |Γ|3∞ ε, ˜ \(Ψ1 ∪Ψ2 ) N 2 2 ˆ (8.59) 2 −4 2 i (ii) |u(x) − (A x + c)| dx ≤ C(1 + rC∗ )υ |Γ|∞ ψ ε, Ψi2

for i = 1, 2 and for r(1 − ωmin )3 ≥ υ . Then the trace theorem applied on each square (if J or Ψj , j = 1, 2, consist also of rectangles, they can be covered by possibly overlapping squares) implies the assertion: To satisfy the assumptions of Lemma 7.2.1, the jump set has to be the union ˜ to J˜u = S ∂Rl ∩ N ˜ by of rectangle boundaries. Therefore, we extend Ju ∩ N l [u](x) = 0 for x ∈ J˜u \ Ju , where Rl are the corresponding rectangles given in (8.5)(i). We observe that by (8.5)(ii) and (8.54) we get X X X X 2 |Θl |∗ , |∂Rl |−1 |Θ | ≤ C |Θl |∗ . (8.60) |∂Rl |H ≤ C l ∗ H l

l

l

l

Note that by every boundary component Γl is contained in at most C(h∗ ) dierent squares (see Lemma 8.3.1). Then by Lemma 7.2.1, either for µ ∼ υ|Γ|∞ or µ ∼ ψ i , (8.59) and (8.38) we obtain for υ small enough ˆ |u(x) − (A x + c)|2 dH1 (x) Γ X X −4 2 ˜ ) + CC∗ ≤ Cυ|Γ|∞ α(N |∂Rl |H |∂Rl |−1 H ευ |Θl |∗ l l X2 + C(υ|Γ|∞ )−1 ku − (A · +c)k2L2 (N˜ \(Ψ1 ∪Ψ2 ) + C(ψ i )−1 ku − (A · +c)k2L2 (Ψi ) 2

2

i=1

≤ C(1 + rC∗ )υ −2 |Γ|2∞ ε + C(1 + rC∗ )υ −4 |Γ|2∞ ε ≤ C(1 + rC∗ )ευ −4 |Γ|2∗ , 180

2

where for the rst two terms we proceeded similarly as in (8.57), also taking (8.60) into account. Finally, choosing Cˆ = C = C(h∗ , q) and r = r(h∗ , q) small enough (i.e. also υ small enough) such that rCˆ ≤ 12 we get (8.39), as desired. Consequently, it suces to establish (8.59).

8.5.2 Step 1: Small boundary components We rst treat the case that only small components Γl lie in N . For 1 > r ≥ υ > 0 dene T = blogr (υ)c and let

St (Q) = {Γl : Γl ∩ Q 6= ∅, υ 4 r−2t |Γ|∞ < |Γl |∞ ≤ υ 4 r−2t−2 |Γ|∞ } for all t ∈ N and S0 (Q) = {Γl : Γl ∩ Q 6= ∅, |Γl |∞ ≤ υ 4 r−2 |Γ|∞ }.

Lemma 8.5.1. Theorem 8.4.1 holds underSthe additional assumption that there is some T4 + 2 ≤ t ≤ T2 − 1 such that s>t Ss (Q) = ∅ for all Q ∈ J and P −3 2t+3 . Q #St (Q) ≤ υ r Proof. We rst observe that the assumption implies N˜ = N . Let T4 +2 ≤ t ≤ T2 −1 with the above properties be given and write υˆ = υ 2 r− later we note that 7 3√ 3 υ 4 r− 2 ≤ υˆ ≤ υ 2 r.

2t+1 2

for shorthand. For (8.61)

ˆ ˆ υˆ|Γ|∞ (ξ) of length 2ˆ We cover N with squares Q(ξ) =Q υ |Γ|∞ and midpoint ξ . (If the sets in J = J(Γ) constructed in Section 8.3.1 are not perfect squares, the ˆ sets Q(ξ) shall be chosen appropriately. The dierence in the possible shapes, however, does not aect the following estimates by Remark 7.1.4(i).) We will now consider a rectangular path, i.e. a path ξ = (ξ0 , . . . ξn = ξ0 ) of square midpoints intersecting all Q ∈ J such that there are indices i1 , i2 , i3 with ξj − ξj−1 = ±2ˆ υ |Γ|∞ e1 for all 0 ≤ j ≤ i1 , i2 ≤ j ≤ i3 and ξj − ξj−1 = ±2ˆ υ |Γ|∞ e2 else. Observe that the number of squares in a path satises n ≤ C υˆ−1 and that we can nd ∼ υ υˆ−1 disjoint rectangular paths in N . Consequently, by assumption S ˆ and (8.34) we can nd at least one rectangular path P := j Q(ξ j ) such that α(P ) ≤ C υˆε|Γ|∞ ,

X

|Γl |∞ ≤ C υˆ|Γ|∞ , #Sˆt (P ) ≤ C υυˆ υ −3 r2t+3

(8.62)

ˆ ) Γl ∈S(P

ˆ ) = {Γl : Γl ∩ P 6= ∅} for some suciently large constant C = C(h∗ , q), where S(P S S S t ˆ )∩ and Sˆ (P ) = S(P Q St (Q). Here we used that each Γl ∈ Q∈J s≤t Ss (Q) ˆ because |∂Rl |∞ ≤ C υˆ2 r−1 |Γ|∞  intersects at most four adjacent squares Q υˆ|Γ|∞ by (8.6), (8.61) and Γl ⊂ Rl (see (8.5)(i)). Observe that the above path can be chosen in the way that also |P ∩ Q| ≥ C υυˆ |Q| for Q = E±,± , where E±,± 181

denote the squares in the corners of N , i.e. (±l1 , ±l2 ) ∩ E±,± 6= ∅. This implies |P ∩ Q| ≥ C υυˆ |Q| for all Q ∈ J . It is convenient to write the above estimate in the form υ 4 r−2t−1±1 = υˆ2 r±1 , #Sˆt (P )υ 4 r−2t−2 ≤ C υˆr. (8.63) We now apply Lemma 7.1.3(ii) with s = υˆ|Γ|∞ and |V | = |P | ∼ υˆ|Γ|2∞ . Recall p √ 3/2 that we get k[u]kL1 (Θl ) ≤ |Θl |H k[u]kL2 (Θl ) ≤ C C∗ ευ −4 |Θl |∗ by (8.38), (8.54) 2×2 and Hölder's inequality. Arguing similarly as in (8.57) we nd A ∈ Rskew and c ∈ R2 such that by (8.56), (8.62), (8.63) and (8.54) ˆ |u(x) − (A x + c)|2 dx ≤ C υˆ−3 (E(P ))2 P
ε X 3/2 2 ≤ C υˆ−3 υˆ|Γ|2∞ α(P ) + CC∗ υˆ−3 4 (|Θ | ) l ∗ ˆ ) Γl ∈S(P υ X
2 ε (8.64) ≤ C υˆ−1 |Γ|3∞ ε + CC∗ υˆ−1 r−1 |Γ|∞ 4 |Γ | l ∞ t Γl ∈Sˆ (P ) υ
2 ε X |Γ | + CC∗ υˆ−1 r|Γ|∞ 4 l ∞ ˆ )\Sˆt (P ) Γl ∈S(P υ ε ε ≤ C υˆ−1 |Γ|3∞ ε + CC∗ υˆr 4 |Γ|3∞ + CC∗ υˆr 4 |Γ|3∞ . υ υ −1 −4 Observing that υˆ ≤ υˆυ by denition of υˆ, we derive ˆ ε (8.65) |u(x) − (A x + c)|2 dx ≤ C(1 + C∗ r) υˆ 4 |Γ|3∞ =: F. υ P We now pass from an estimate on P to an estimate on N . For later purpose in Section 8.5.3, we consider general subsets V ⊂ N consisting of squares in J . Then repeating (8.64) we obtain by Lemma 7.1.3(iii) ku(x) − (A x + c)k2L2 (P ∩V ) ≤ |P ∩ V ||P |−1 CF. |V | Since |P ∩ Q| ≥ C υυˆ |Q| for all Q ∈ J we nd |N ≥ C |V|P∩P| | ≥ Cυ . Therefore, | ˜ = N ) and υ 2 ≤ υˆr (see (8.61)) we also have by (8.57) (recall that N ˆ |u(x) − (A¯ x + c¯)|2 dx ≤ |V ∩ P ||P |−1 CF ≤ C|V ||N |−1 F. V

¯ c¯ are We apply (7.14) on each Q ⊂ V with B1 = P ∩ Q, B2 = Q noting that A, constant on each square. (Although P ∩ Q is not a rectangle if Q = E±,± , we can still argue as in (7.14) since P ∩Q consists of two rectangles.) As |P ∩Q| ≥ C υυˆ |Q| for all Q ∈ J we obtain ˆ ˆ  υ 2 2 |u(x)−(A¯ x+¯ c)| dx+ |u(x)−(A x+c)|2 dx ku(x)−(A x+c)kL2 (Q) ≤ C υˆ Q P ∩Q and thus summing over all Q ⊂ V we derive ˆ ε |N |− |u(x) − (A x + c)|2 dx ≤ C υˆ−1 υF = C(1 + C∗ r) 3 |Γ|3∞ . υ V Consequently, setting V = N , (8.59)(i) is established, as desired. 182

(8.66)



8.5.3 Step 2: Subset with small projection of components The next step will be the case that k · kπ is not too large. For that purpose, recall (8.18) and the denition of Y (see before (8.15)). Consider some U ∈ Y with |U | ≥ Cυ|Γ|2∞ . Moreover, by Y 0 we denote the set of subsets of N consisting of squares in J . (In contrast to Y the connectedness of the sets is not required.) In this section we show that for all Z ⊂ U , Z ∈ Y 0 , one has ˆ |U |− |u(x) − (AU x + cU )|2 dx ≤ C(1 + rC∗ )υ −3 |Γ|3∞ ε (8.67) Z 2 for AU ∈ R2×2 skew , cU ∈ R . Recall that E±,± denote the squares at the corners of Γ (see construction before (8.14)).

Lemma 8.5.2. Let r ≥ υ > 0. Let U ∈ Y with |U | ≥ Cυ|Γ|2∞ and U ∩ E±,± = ∅ 19 be given and assume that kU kπ ≤ 20 τ . Then there is a subset U 0 ⊂ U , U 0 ∈ Y 0 , with |U \ U 0 | ≤ Cr|U | such that (8.67) holds for all Z ⊂ U 0 , Z ∈ Y 0 . Proof. Let U ∈ Y be given with kU kπ ≤

and assume without restriction U ⊂ N2,+ \ (N1,− ∪ N1,+ ). By the choice of τ in (8.26) we obtain that all Γl having nonempty intersection with U satisfy |Γl |∞ < 19¯ τ . In particular, this ˜ implies U ∩ N = U . Let (∂Rl )l be the rectangles corresponding to (Γl )l as given by (8.5)(i),(v). We rst prove that there is a T4 + 2 ≤ t ≤ T2 − 1 such that P −3 2t+3 as in the assumption of Lemma 8.5.1. If the claim Q⊂U #St (Q) ≤ υ r were false, we would have (assume without restriction that T ∈ 4N)

|∂W ∩ N (1+19C)¯τ |H ≥ C

X T2 −1 t= T4 +2

X Q⊂U

19 τ 20

#St (Q)υ 4 r−2t |Γ|∞ ≥ CT υr3 |Γ|∞

≥ C logr (υ)r3 υ|Γ|∞  υ|Γ|∞

(8.68)

for υ small enough (with respect to r = r(h∗ , q)) giving a contradiction to (8.34). In the rst step we used that |∂Rl |∞ ≤ 19C τ¯ by (8.6) which implies Γl ⊂ N (1+19C)¯τ (Γ) and assures that Γl intersects only a uniformly bounded number of dierent squares Q ⊂ U (independently of r, υ ). As before, we dene 2t+1 υˆ = υ 2 r− 2 for shorthand. As in the previous proof we will select a path in U with certain properties. Recalling (8.18) it is not hard to see that |π2 (Rl ∩U )| ≤ |∂Rl |π . As by assumption 19 kU kπ ≤ 20 τ , we nd a set S ⊂ (l2 , l2 + τ ) being the union of intervals 2k 0 s + τ ˆ = U ∩ (R × S) satises (−s, s), k 0 ∈ Z, with |S| ≥ 20 such that the stripe U ∂W ∩ Uˆ = ∅. We cover U by k horizontal paths P = (Pi )i , i = 1, . . . , k consisting ˆ ˆ υˆ|Γ|∞ (ξ), i.e. k = d(2ˆ of Q(ξ) =Q υ |Γ|∞ )−1 τ e as |π2 U | = τ . We can nd a subset Pˆ1 ⊂ P with #Pˆ1 ≥ c1 k for c1 small enough such that

Γl ∩ P i = ∅

for all |Γl |∞ ≥ C¯ 2 υˆ|Γ|∞ and Pi ∈ Pˆ1 183

(8.69)

S τ and |S ∩ π2 Pi ∈Pˆ1 Pi | ≥ 21 , if C¯ is chosen suciently large. Indeed, for {Γl : |π2 Γl | ≥ C¯ υˆ|Γ|∞ } this follows by an elementary argument. On the other hand, by (8.6) we see that each component in G := {Γl : |Γl |∞ ≥ C¯ 2 υˆ|Γ|∞ , |π2 Γl | ≤ ¯ |Γ|∞ = C¯ dierent Pi ∈ P and thus using (8.34) C¯ υˆ|Γ|∞ } intersects at most ∼ Cυˆυˆ|Γ| ∞ G intersects at most C¯ C¯ 2 υCτ ∼ υˆ|Γ|Cτ∞ C¯  |Γ|τ∞ υˆ dierent Pi ∈ P . ˆ|Γ|∞ Moreover, it is not hard to see that there is a subset Pˆ2 ⊂ Pˆ1 with #Pˆ2 ≥ c2 k for c2 suciently small such that |π2 (Pi ∩ Uˆ )| ≥

1 2ˆ υ |Γ|∞ 22

Recall that we have already found a

for all Pi ∈ Pˆ2 .

(8.70)

−1 such that Q⊂U #St (Q) ≤ S ˆ ˆ υ r . Using (8.69) we can now choose a path P = j Q(ξ j ) ∈ P2 such that (8.62) is satised possibly passing to a larger constant C > 0 depending on C¯ . (The essential dierence to the argument developed in (8.62) is the fact that every boundary component may intersect not only four squares but a number depending on C¯ .) Observe that n ∼ υˆ−1 , where n denotes the number of squares ˆ ) = {Γl : Γl ∩ P 6= ∅} and let in the path P . Recall S(P [ [ Ss (Q)}. (8.71) Sˆ>t (P ) = {Γl : Γl ∩ P 6= ∅, Γl ∈ T 4

+2 ≤ t ≤

T 2

P

−3 2t+3

Q∈J

s>t

ˆ = Q(ξ ˆ j) : Q ˆ ∩ Γl = ∅ for all Γl ∈ Sˆ>t (P )}. By Moreover, dene K = {Q t (8.62) it is elementary to see that #Sˆ> (P ) ≤ C υˆ|Γ|∞ (υ 4 r−2t−2 |Γ|∞ )−1 = C υˆ−1 r. Consequently, as by (8.69) every Γl ∈ Sˆ>t (P ) intersects only a uniformly bounded number of adjacent sets, we nd n − #K ≤ C#Sˆ>t (P ) ≤ C υˆ−1 r.

(8.72)

ˆ ˆ Consider two squares Q(a), Q(b) ∈ K and the path (ξ0 = a, ξ1 , . . . , ξm = b). Sm ˆ ˆ 0 := Q(a) ˆ Dene D = j=0 Q(ξj ). Without restriction we assume Q = µ(−1, 1)2 ˆ m = Q(b) ˆ and Q = µ((2m, 0)+(−1, 1)2 ), where for shorthand we write µ = υˆ|Γ|∞ . We will now derive an estimate of the form (7.18). First of all, Theorem B.4 (see also (7.17)), Theorem B.5 and a rescaling argument show ˆ i) ku − (Ai · +ci )kL1 (∂ Qˆ i ) ≤ CE(Q

(8.73)

i 2 for Ai ∈ R2×2 independent of µ. For shorthand skew , c ∈ R , i = 0, m, and a constant ´ ˆ ˆ let E = E(Q0 ) + E(Qm ) and dene α ˆ (D) = D |e(∇u)|. We claim that

µ2 |a0 − am | + µ|c01 − cm ˆ (D), 1 | ≤ CE + C α 0 m µ|c2 − c2 | ≤ CmE + Cmˆ α(D),

(8.74)

where cij denotes the j -th component of ci , i = 0, m, and a0 , am are dened such   0 ai i that A = . By (8.70) we nd two (measurable) sets B1 , B2 ⊂ µ(−1, 1) −ai 0 184

µ µ such that |Bj | ≥ 44 , dist(B1 , B2 ) ≥ 22 and E := µ(−1, 2m + 1) × B1 ∪ B2 ⊂ W . We apply a slicing argument in the rst coordinate direction and obtain ˆ |u1 (µ(2m − 1), y) − u1 (µ, y)| dy B1 ∪B2

ˆ ≤ B1 ∪B2

ˆ

µ(2m−1)

µ

∂1 u1 (t, y) dt dy ≤ C α ˆ (E).

(8.75)

This together with (8.73) and the triangle inequality yields

k(a0 − am ) · +(c01 − cm ˆ (E). 1 )kL1 (B1 ∪B2 ) ≤ CE + C α Choose f : B1 → R such that id + f : B1 → B2 is piecewise constant and µ µ bijective. Thanks to |B1 | ≥ 44 and f (y) ≥ 22 for y ∈ B1 we derive

µ2 |a0 − am | ≤ Ck(a0 − am ) f (·)kL1 (B1 ) ≤ Ck(a0 − am ) · +(c01 − cm 1 )kL1 (B1 ) 0 m 0 m + Ck(a − a ) (· + f (·)) + (c1 − c1 )kL1 (B1 ) ≤ CE + C α ˆ (E) and likewise µ|c01 − cm ˆ (E). This gives the rst bound in (8.74) since 1 | ≤ CE + C α E ⊂ D. Analogously, we slice in ζ = µ(2m − 2, c) direction for 0 < c < 1. By (8.62) we nd |π2 (∂W ∩ Pi )| ≤ Cµ. Consequently, choosing c small enough and µ 1 recalling (8.70), we nd a set B3 ⊂ µ(−1, 1 − c) with |B3 | ≥ 24 2ˆ υ |Γ|∞ = 12 such ζ ¯ that {µ} × B3 + [0, 1]ζ ⊂ W . Letting ζ = |ζ| we get

ˆ ¯ dy ≤ C α |u(µ, y) · ζ¯ − u((µ, y) + ζ) · ζ| ˆ (E) B3

similarly to (8.75) and thus, using (8.73) and the fact that Am ζ¯ · ζ¯ = 0, we derive ˆ ¯ dy ≤ CE + C α |(A0 − Am ) (µ, y)T · ζ¯ + (c0 − cm ) · ζ| ˆ (E). B3

This together with rst part of (8.74) then leads to m 0 µ|c02 − cm 2 + (a − a )µ|

c ≤ Cα ˆ (E) + CE |ζ|

and implies the second part of (8.74) as E ⊂ D. Summarizing, (8.74) yields

ˆ 0 ) + E(Q ˆ m )) + Cmˆ kc0 − cm + (A0 − Am ) · kL2 (Q(a)) ≤ Cm(E(Q α(D), ˆ

(8.76)

which is an estimate of the form (7.18) with the dierence that ER is replaced by the elastic part of the energy α ˆ in squares not contained in K. We briey note ˆ ˆ that in (8.76) we can replace Q(a) by Q(b) due to (8.74).

185

S ˆ . Recall that the essential point for the derivation of Q Dene P˜ = Q∈K ˆ Lemma 7.1.3 was an estimate of the form (7.18), (7.19). Consequently, arguing similarly as in Lemma 7.1.3(ii) for s = υˆ|Γ|∞ and |V | = |P˜ | ∼ υˆ|Γ|2∞ and derive ku − (A · +c)k2L2 (P˜ ) ≤ C υˆ−3 ((E(P˜ ))2 + (ˆ α(P ))2 )

(8.77)

2 for suitable A ∈ R2×2 skew and c ∈ R . Recall the denition of K (cf. (8.71)) and note that Γl ∩ P˜ = ∅ for all Γl ∈ Sˆ>t . Proceeding as in (8.64) and (8.65), in particular using (8.62) and (8.63), it is not hard to see that

ku − (A · +c)k2L2 (P˜ ) ≤ C(1 + C∗ r) υˆ

ε |Γ|3 . υ4 ∞

(8.78)

Note that the dierence to the estimate in the proof of Lemma 8.5.1 is that due to the above slicing argument it suces to consider the elastic part of the energy in the connected components of P \ P˜ . Now let J 0 ⊂ J be the set S of squares
◦ such0 Q ∈Y that |Q ∩ P˜ | ≥ Cυˆ υ |Γ|2∞ ≥ C υυˆ |Q| for all Q ∈ J 0 . Setting U 0 = Q∈J 0 0 it is not hard to see that |U \ U | ≤ Cr|U | for r small enough as |P \ P˜ | ≤ Cr|P | by (8.72). Let Z ⊂ U 0 , Z ∈ Y 0 . As before in Lemma 8.5.1, applying Lemma 7.1.3(iii) instead of Lemma 7.1.3(ii), (8.78) yields

ε ku − (A · +c)k2L2 (P˜ ∩Z) ≤ C|P˜ ∩ Z||P˜ |−1 (1 + C∗ r) υˆ 4 |Γ|3∞ . υ Then applying (7.14), (8.57) and arguing as in (8.66) we derive ˆ ε 0 |U |− |u(x) − (A x + c)|2 dx ≤ C(1 + C∗ r) 3 |Γ|3∞ . υ Z As |U \ U 0 | ≤ Cr|U |, this gives (8.67), as desired.  0 The next step will be to replace U by U in Lemma 8.5.2. To this end, we will apply the above arguments iteratively.

Lemma 8.5.3. Let r ≥ υ > 0. Let U ∈ Y with |U | ≥ Cυ|Γ|2∞ and U ∩ E±,± = ∅ 19 τ . Then (8.67) holds. be given and assume that kU kπ ≤ 20 Proof. S Dene U
1 = U 0 and J1 = J 0 as given in Lemma 8.5.2. Assume that ◦ ¯

Ui = Q∈Ji Q , Ji ⊂ J , with U1 ⊂ . . . ⊂ Ui is given such that for C > 0 suciently large ¯ i |U | |U \ Ui | ≤ Cr

(8.79)

and for all Z ⊂ Ui , Z ∈ Y 0 , one has ˆ Yi−1
¯ 8j G, |u − (A x + c)|2 dx ≤ |Z||U |−1 C 1 + Cr j=0

Z

186

(8.80)

ε 2 3 where A ∈ R2×2 skew , c ∈ R as given by Lemma 8.5.2 and G := (1 + C∗ r) υ 3 |Γ|∞ . Observe that (8.79), (8.80) hold for i = 1 by Lemma 8.5.2. We now pass from i to i + 1 and suppose i ≤ T + 2. First, it is not restrictive to assume that ¯ i+1 |U | for C¯ > 0 as above since otherwise we may set Ui+1 = Ui . We |U \ Ui | ≥ Cr cover U \ Ui with pairwise disjoint, connected sets Ni1 , . . . , Nim ∈ Y , such that 1 8i r |Nik | 2

i

(8.81)

≤ |Nik \ Ui | ≤ 2r 8 |Nik |

for done in the following way: Let V0 = U = S all k = 1, . . . , m. This can be ˜ 1 let l1 , l2 be the smallest and largest index, ( nj=1 Qj )◦ . First, to construct N i respectively, such that Ql ⊂ U \ Ui and choose l = l1 if l1 < n − l2 and l = l2 otherwise. Then add neighbors Ql−1 , Ql+1 ⊂ U , Ql−2 , Ql+2 ⊂ U , . . . until ˜ 1 | holds. (I.e. the right inequality in (8.81) is satised.) This ˜i1 \ Ui | ≤ 2r 8i |N |N i ¯ i |V0 | ≤ 1 r 8i |V0 | by (8.79) for r is possible due to the fact that |V0 \ Ui | ≤ Cr 2 i ˜ 1 | ≤ |N ˜ 1 \ Ui | holds, in particular the suciently small. Then note that also r 8 |N i i left inequality in (8.81) is fullled. We now dene V1 as the connected component ˜ 1 which is not completely contained in Ui . (If both are contained in Ui of V0 \ N i ˜ j , 1 ≤ j ≤ k, we have nished.) We repeat the procedure on sets Vj to dene N i i satisfying (8.81), where k is the smallest index such that |Vk \ Ui | > 21 r 8 |Vk |. We ˜ j for j < k and Nik := N ˜ik ∪ Vk . now dene Nij = N i i It remains to show that also Nik satises (8.81). Recall |Vk−1 \Ui | ≤ 21 r 8 |Vk−1 |. i ˜ k−1 | ≤ |N ˜ k−1 \ Ui | we have |Vk | ≥ As due to the choice of l and the fact that r 8 |N i i i k−1 k−1 1 1 8i ˜ ˜ ˜ k−1 |, |V | − | N | and |V \ U | = |V \ U | − | N \ U | ≤ r |Vk−1 | − r 8 |N k i k−1 i i i i i 2 k−1 2 i ˜ik implies the desired we nd |Vk \ Ui | ≤ r 8 |Vk |. This together with (8.81) for N property for NSik . k (8.68) we nd some T4 + 2 ≤ t ≤ T2 − 1 Let Ni = m k=1 Ni . Similarly as in P −3 2ti +3 . Again set such that for ti = t + 98 · 2i we have Q⊂Ni #Sti (Q) ≤ υ r 2t+1

υˆ = υ 2 r− 2 . Arguing as in (8.70) we can nd a horizontal path Pi consisting ˆ j) = Q ˆ υˆ|Γ|∞ (ξj ), j = 1, . . . , ni , and lying in Ni such that (8.62), (8.69) and of Q(ξ (8.70) are satised replacing t by ti . By (8.79) and (8.81) we obtain ¯ i+1 υˆ−1 ≤ ni ≤ C Cr ¯ i− 8i υˆ−1 . Cr

(8.82)

Clearly, in general the path Pi is not connected. Dene Sˆ>ti (Pi ) and Ki similarly as in (8.71). By (8.62) it is elementary to see that 9

9

#Sˆ>ti (Pi ) ≤ C υˆ|Γ|∞ (υ 4 r−2ti −2 |Γ|∞ )−1 ≤ C υˆ−1 r 8 i r = C υˆ−1 r 8 i+1 . Therefore, letting P˜i =

S

ˆ Q∈K i 9

ˆ ⊂ Pi we nd by (8.69) (cf. (8.72)) Q

ni − #Ki ≤ C υˆ−1 r 8 i+1

and

9 |Pi | − |P˜i | ≤ Cr 8 i+1 υˆ|Γ|2∞ .

187

(8.83)

We now repeat the slicing arguments above on each Nik and obtain expressions similar to (8.76). As before, applying Lemma 7.1.3(ii) we get (cf. (8.77))

ku − (Ak · +ck )k2L2 (P˜i ∩N k ) ≤ C(nk )3 (E(P˜i ∩ Nik ) + α ˆ (Pi ∩ Nik ))2 i

´ k 2 for suitable Ak ∈ R2×2 ˆ (D) = D |e(∇u)| for skew and c ∈ R , where as before α k D ⊂ N . Here nP denotes the number of squares forming the path Pi ∩ Nik , k particularly ni = m k=1 n . We observe that by (8.62) the estimate in (8.63) can now be replaced by 9

υ 4 r−2ti −1±1 = υˆ2 r− 8 i±1 ,

#Sˆti (P )υ 4 r−2ti −2 ≤ C υˆr.

¯ 78 i υˆ−1 by (8.82), ti = t + 9 · i and following the Consequently, recalling ni ≤ C Cr 8 2 arguments in (8.64), (8.77) and (8.78) we obtain X (nk )−1 ku − (Ak · +ck )k2L2 (N k ∩P˜i ) i k X k 2 k ˜ ≤C (n ) (E(Ni ∩ Pi ) + α ˆ (Nik ∩ Pi ))2 ≤ Cn2i (E(P˜i ) + α ˆ (Pi ))2 k

7 i 9 ≤ C υˆr υˆ−3 (E(P˜i ) + α ˆ (Pi ))2 ≤ C υˆr 4 i r− 8 i F ≤ C υˆr 2 F, 7 i 4

where F was dened in (8.65). Observe that in the calculation the additional 9 9 r− 8 i in front of F occurs as in (8.63) υ 4 r−2t−1±1 was replaced by υ 4 r−2t−1±1 r− 8 i . Moreover, the above estimate can be repeated applying Lemma 7.1.3(iii) instead of Lemma 7.1.3(ii): For Z ⊂ Ni , Z ∈ Y 0 , we obtain ˆ X i k −1 k |u(x) − (Ak x + ck )|2 dx ≤ C υˆr 2 F. (n ) |Ni ∩ P˜i |− k

Nik ∩P˜i ∩Z

Dene Jik ⊂ J such that |Q ∩ (P˜i ∩ Nik )| ≥ Cυ υˆ|Γ|2∞ ≥ C υυˆ |Q| for Q ∈ Jik and ˆ k ∩ Z| ≥ υ 2 |Γ|2 . ˆ k ∩ Z 6= ∅ which implies |N ˆ k = S k Q. Assume N set N ∞ i i i Q∈J i 7

3

i

Observe υˆ ≥ υ 4 r− 2 ≥ υ 2 r− 4 −1 by (8.61) and the fact that i ≤ T + 2. As |Nik |(nk )−1 ≤ C υˆυ|Γ|2∞ , we nd by (8.57) ˆ X k −1 ˆ k (n ) |Ni |− |u(x) − (A¯ x + c¯)|2 dx k k ˆ N ∩Z ˆ i i ≤ C υˆυ −1 |u(x) − (A¯ x + c¯)|2 dx ≤ C υˆυ −1 υ 3 (ˆ υ r)−1 F ≤ C υˆr 4 F. U

Again arguing as in (8.66), in particular applying (7.14), we derive ˆ X i i k −1 ˆ k (n ) |Ni |− |u(x) − (Ak x + ck )|2 dx ≤ C υˆr 4 υˆ−1 υF = Cr 4 υF. (8.84) k

ˆk Z∩N i

ˆ k if |N k \ N ˆ k | ≤ r 8i |N k | and U k = ∅ else for all k = 1, . . . , m. We set Uik = N i i i i i We now estimate the dierence between A, c given in (8.80) and Ak , ck for k = 188

ˆ k . Then |U k | ≥ (1 − r 8i )|N k | and thus 1, . . . , m. Consider Uik such that Uik = N i i i by (8.81) we have ˆ k | ≥ (1 − Cr 8i )|N k | ≥ (1 − Cr 8i )|U k | |Uik ∩ Ui | ≥ |Nik ∩ Ui | − |Nik \ N i i i for r suciently small and some C > 0. Consequently, we are in the position to i apply (7.16) for B2 = Uik , B1 = Uik ∩ Ui and s = τ2 , δ = Cr 8 , where we observe δ ≥ Cs|π1 (Uik )|−1 by (8.81). (Recall the remark in Section 7.1 that B2 does not
Qi−1 ¯ 8j (cf. (8.80)). Using (8.80) and 1 + Cr have to be connected.) Set C¯i = C j=0 ˆ k )k are pairwise disjoint, we nd (8.84), in particular recalling that the sets (N i for Z ⊂ U , Z ∈ Y 0

ku − (A · +c)k2L2 (U k ∩Ui ∩Z) ≤ |Uik ∩ Ui ∩ Z||Uik |−1 H1k , i

ku − (Ak · +ck )k2L2 (U k ∩Z) ≤ |Uik ∩ Z||Uik |−1 H2k , i

i

where H1k = |Uik ||U |−1 C¯i G and H2k = Cnk r 4 υF . Therefore, (7.16) yields i

ku − (A · +c)k2L2 (U k ∩Z) ≤ |Uik ∩ Z||Uik |−1 (1 + Cr 8 )|Uik ||U |−1 C¯i G

(8.85) i i + |Uik ∩ Z||Uik |−1 Cr− 8 nk r 4 υF.
Sm ∗ k ◦ and dene Ui+1 = (Ui ∪U ∗ )◦ . We recall For shorthand we write U = U i k=1 Sm Ni = k=1 Nik as constructed in (8.81). We claim i

(8.86)

|Ni \ U ∗ | ≤ Cri+1 |U |

and postpone the proof of this assertion to the end of the proof. Then (8.82) for C¯ suciently large implies |U ∗ | ≥ |Ni | − Cri+1 |U | ≥ cni υˆ|U |. As for Uik 6= ∅ i |U ∗ | ni we have |Nik | ≤ (1 − r 8 )−1 |Uik |, it is not hard to see that |U k | ≤ C nk and thus i

nk |Uik |−1 ≤ C|U ∗ |−1 ni ≤ C|U |−1 υˆ−1 . Let V ⊂ Ui+1 , V ∈ Y 0 . Now by (8.85), the fact that the sets Uik are pairwise disjoint and F ≤ C υˆυ −1 G we derive X
i i ku − (A · +c)k2L2 (U k ∩V ) ≤ |U ∗ ∩ V ||U |−1 C¯i (1 + Cr 8 )G + Cr 8 G k

i

≤ |U ∗ ∩ V ||U |−1 C¯i+1 G, The last estimate follows for C¯ suciently large. By (8.80) we now conclude for V ⊂ Ui+1 X ku − (A · +c)k2L2 (U k ∩V ) ku − (A · +c)k2L2 (V ) = ku − (A · +c)k2L2 (V \U ∗ ) + k

i

≤ |V \ U ∗ ||U |−1 C¯i G + |U ∗ ∩ V ||U |−1 C¯i+1 G ≤ |V ||U |−1 C¯i+1 G. This yields (8.80). To see (8.79) for i + 1, we apply (8.86) to obtain |U \ Ui+1 | ≤ ¯ i+1 |U |. Here we used that U \ Ui ⊂ Ni . |(U \ Ui ) \ Ni | + |Ni \ U ∗ | ≤ 0 + Cr 189

¯ Finally, we choose i∗ ≤ T + 2 large enough such that |U \ Ui∗ | ≤ Crυ|U | 2 (υ|Γ|∞ ) for r suciently small which implies Ui∗ = U . Consequently, thanks to (8.80), (8.67) holds. ˆik we It remains to show (8.86). First, by (8.83) and the construction of N S k S ˆk P 9 9 ˆ k | ≤ Cr 8 i+1 υ|Γ|2 ≤ Cr 8 i+1 |U |. Therefore, have | k Ni \ k Ni | = k |Nik \ N i ∞ it suces to prove X X ˆ k| ˆ k \ U k | ≤ r− 8i |Nik \ N (8.87) |N i i i k

k

P ˆik \ Uik | ≤ Cri+1 |U |. ˆik | + P |N as then we conclude |Ni \ U ∗ | ≤ k |Nik \ N k ˆ k 6= U k , then |N ˆ k | ≤ |N k | < r− 8i |N k \ N ˆ k |. To see (8.87) we observe that if N i i i i i P ˆk P i P k ˆ k | ≤ r− 8 Consequently, we calculate k |Ni \ Uik | = k:U k =∅ |N |N \ k i i k:Ui =∅ i i P − k k k ˆ |≤r 8 ˆ |, as desired. N |N \ N  i

k

i

i

Remark 8.5.4.

We briey note that the previous proof shows that the assertion of Lemma 8.5.3 holds for U ∈ Y with U ∩ E±,± = ∅ of arbitrary size. In fact, we can choose 0 ≤ i0 ≤ T + 2 such that Cri0 +1 υ|Γ|2∞ < |U | ≤ Cri0 υ|Γ|2∞ and begin the induction in (8.79), (8.80) not for i = 0, but for i = i0 . For the rst step i = i0 we do not apply Lemma 8.5.2, but follow the lines of the proof of Lemma 8.5.3 for one single set Ni10 = U . We now drop the assumption that U ∈ Y does not intersect a corner of Γ.

Corollary 8.5.5. Let r ≥ υ > 0. Let U ∈ Y be given and assume that kU kπ ≤ 19 τ . Then (8.67) holds. 20 Proof. Assume without restriction E+,+ ⊂ U and dene U 0 = U \ E+,+ . Using Lemma 8.5.3 and Remark 8.5.4 we nd ˆ 0 |U |− |u(x) − (A x + c)|2 dx ≤ CG Z

for Z ⊂ U 0 , Z ∈ Y 0 , where G := (1 + C∗ r) υε3 |Γ|3∞ . Let Q ∈ J , Q ⊂ U 0 such that ∂Q ∩ E+,+ 6= ∅. Setting Z = Q in the above inequality and arguing as in (8.57) ˆ ∈ R2 such that we nd Aˆ ∈ R2×2 skew , c ˆ ˆ 2 |u(x) − (A x + c)| dx ≤ CυG, |u(x) − (Aˆ x + cˆ)|2 dx ≤ Cυ 2 r−1 G. Q

Q∪E+,+

Applying (7.14) on B1 = Q and B2 = Q ∪ E+,+ we nd ku − (A · +c)|2L2 (Q∪E+,+ ) ≤ CυG as υ ≤ r. Now it is not hard to see that (8.67) is satised. 

190

8.5.4 Step 3: Neighborhood with small projection of components Recall the covering C of the neighborhood N introduced in (8.15). We now treat the case that kU kπ is small for all U ∈ C . It is essential that adjacent elements of the covering overlap suciently. Therefore, we introduce another covering Cˆ ⊂ C as follows. First assume l2 ≥ l21 . If some U ∈ C intersects only one of the four 1 2 sets Nj,± , j = 1, 2, we let U ∈ Cˆ. Then eight sets U±,± , U±,± remain where i i U±,± ∩ E±,± 6= ∅ and U±,± ⊂ Ni,− ∪ Ni,+ for i = 1, 2. As before E±,± denote the 2 1 to Cˆ. If l2 < l21 we proceed ∪ U±,± sets at the corners of Γ. Add the four sets U±,± 2 1 we add the two sets ∪ U±,± likewise with the only dierence that instead of U±,± 2 1 1 2 Uk,+ ∪ Uk,+ ∪ Uk,− ∪ Uk,− , k = +, −, to Cˆ. (Note that by denition of C we have 1 1 U±,+ = U±,− = N1,± in this case.)

Lemma 8.5.6. Theorem 8.4.1 holds under the additional assumption that kU kπ ≤ 19 τ for all U ∈ C . 20 2 Proof. It suces to show that for all U ∈ Cˆ there are AU ∈ R2×2 skew , cU ∈ R such

that

ˆ |U |− |u(x) − (AU x + cU )|2 dx ≤ C(1 + rC∗ )υ −3 |Γ|3∞ ε

(8.88)

Z

holds for all Z ⊂ U , Z ∈ Y 0 . Indeed, the desired result then follows from the construction of the covering Cˆ and the arguments developed in Section 7.1: Write Cˆ = {U1 , . . . , Un } with Ui−1 ∩ Ui 6= ∅ for all i = 1, . . . , n, where U0 = Un . Now let

D = {Ui \ Ui−1 ∪ Ui+1 : i = 0, . . . , n − 1} ∪ {Ui ∩ Ui+1 : i = 0, . . . , n − 1}, where U−1 = Un−1 . Note that the elements in D are pairwise disjoint. We write D = {V1 , . . . , Vm } such that ∂Vi−1 ∩ ∂Vi 6= ∅ for i = 1, . . . , m, where V0 = Vm . By (8.15) and the denition of the `combined sets' in Cˆ, we nd |Vi | ∼ υ|Γ|2∞ . Clearly, (8.88) also holds for all Vi ∈ D for corresponding innitesimal rigid motions as each set is contained in an element of Cˆ. We can now estimate the dierence of the innitesimal rigid motions of B1 = Vi−1 and B2 = Vi , i = 1, . . . , m, proceeding as in (7.10), (7.12) and (7.14). Here it is essential to observe that assumption (7.8) is satised as B1 ∪ B2 ⊂ U for some U ∈ Cˆ and so (8.88) may be applied. We now obtain (8.59) following the argument in (7.18), (7.19) replacing the squares (Q(ξj ))j by the elements of D and noting that #D is uniformly bounded independently of υ . More general, taking (8.88) and (7.13) into account, we have even shown that ˆ |N |− |u(x) − (A x + c)|2 dx ≤ C(1 + rC∗ )υ −3 |Γ|3∞ ε (8.89) V

191

for all V ⊂ N , V ∈ Y 0 . It remains to establish (8.88) for U ∈ Cˆ. By assumption and Lemma 8.5.3 the assertion is clear if U ∩ E±,± = ∅ as then particularly U ∈ C . Therefore, we rst let l2 ≥ l21 and assume that e.g. U ∩ E+,+ 6= ∅. The necessary changes for the case l2 ≤ l21 are indicated at the end of the proof. P As in (8.68) we nd T4 + 2 ≤ t ≤ T2 − 1 such that Q⊂U #St (Q) ≤ υ −3 r2t+3 . 2t+1

Again let υˆ = υ 2 r− 2 . As before, the main strategy will be to construct a suitable path in U . Let (Γl )l be the boundary components such that the corresponding rectangles (∂Rl )l given by (8.5)(i) and (8.5)(v), respectively, satisfy ∂Rl ∩U 6= ∅ and |∂Rl |π 6= |∂Rl |∞ . Let Vl ⊂ N be the smallest rectangle containing Rl ∩N and (l1 +τ, l2 +τ ). We partition (Vl )l into V1 and V2 depending on whether |π1 Vl | ≤ |π2 Vl | or |π1 Vl | > |π2 Vl |. Recalling (8.18) it is not hard to see that |πj Vl | = |Rl |π for Vl ∈ Vj for j = 1, 2. Let aj = inf{s ∈ R : s ∈ πj Vl for a Vl ∈ Vj } and dene the stripes

A1 = (−∞, a1 ) × (−∞, a2 ) ∩ N1,+ ∩ U,

A2 = (−∞, a1 ) × (−∞, a2 ) ∩ N2,+ ∩ U. ∂Rl3

∂Rl2

A2

∂Rl1 A1 X

N (Γ)

1 2 . The sets A , A are Figure 8.9: Sketch of a part of N (Γ) containing U+,+ ∪ U+,+ 1 2

highlighted in grey.

19 As by assumption kU kπ ≤ 20 τ for all U ∈ C , we nd sets Sj ⊂ (lj , aj ) with τ |Sj | ≥ 20 such that the stripes Aˆ1 = A1 ∩ (S1 × R) ∈ U s and Aˆ2 = A2 ∩ (R × S2 ) ∈ τ U s satisfy ∂W ∩ Aˆj = ∅ for j = 1, 2. Moreover, observe that |aj − lj | ≥ 20 for 1 j = 1, 2. We cover A1 by vertical paths P1 = (Pi )i , i = 1, . . . , k1 , and A2 by ˆ υˆ|Γ|∞ (ξ) = Q(ξ) ˆ , horizontal paths P2 = (Pi2 )i , i = 1, . . . , k2 , consisting of squares Q −1 i.e. kj = d(2ˆ υ |Γ|∞ ) (aj − lj )e. As in (8.70) it is not hard to see that there are subsets Pˆj ⊂ Pj with #Pˆj ≥ ckj ≥ cυ υˆ−1 for c > 0 suciently small such that (8.69) and (8.70) hold for all Pij ∈ Pˆj , j = 1, 2. We can now choose

192

P j ∈ Pˆj , j = 1, 2, such that (8.62) is satised possibly passing to a larger constant. Moreover, this can be done in a way that Q∗ := P 1 ∩ P 2 satises X ˜ −1 υˆ2 |Γ|∞ , |Γk |∞ ≤ Cυ Γk ∩Q∗ 6=∅ (8.90) Q∗ ∩ Γk = ∅ for all Γk : |Γk | ≥ C˜ υˆ2 υ −1 |Γ|∞ , for C˜ > 0 suciently large. To see the latter, note that we have ∼ τ 2 (ˆ υ |Γ|∞ )−2 = υ 2 υˆ−2 possibilities to combine paths in Pˆ1 , Pˆ2 such that (8.62) hold. Moreover, we also have ∼ τ 2 (ˆ υ |Γ|∞ )−2 = υ 2 υˆ−2 possibilities to combine paths in Pˆ1 , Pˆ2 such that Q∗ additionally has empty intersection with all Γk satisfying |Γk |∞ ≥ C˜ υˆ2 υ −1 |Γ|∞ . This follows from (8.69) and the fact that by (8.34) we derive

ˆ : ∃Γk : C˜ υˆ2 υ −1 |Γ|∞ ≤ |Γk |∞ ≤ C υˆ|Γ|∞ , Γk ∩ Q ˆ 6= ∅} ≤ C C˜ −1 υ 2 υˆ−2 . #{Q ˆ Since all other components Γk intersect at most Pfour adjacent squares Q, using ∗ −2 2 again (8.34) we can select Q such that also ˆ Γk ∩Q∗ 6=∅ |Γk |∞ ≤ Cυ|Γ|∞ υ υ holds. Let P = Pˆ 1 ∪ Q∗ ∪ Pˆ 2 , where Pˆ j , j = 1, 2, is the connected component of j P \ Q∗ not completely contained in E+,+ . Denote the midpoints of the squares in P by (ξ1 , . . . , ξn ). Recall the denition of Sˆ>t in (8.71) and let ˆ = Q(ξ ˆ j) : Q ˆ ∩ Γl = ∅ for all Γl ∈ Sˆt (P )} ∪ {Q∗ }. K = {Q > ˆ ˆ Consider two sets Q(a), Q(b) ∈ K and the path (ξ0 = a, . . . , ξm = b). We can repeat the slicing method of the previous proofs and end up with an estimate of the form (cf. (8.76))   ∗ ˆ ˆ ˆ kca − cb + (Aa − Ab ) · kL2 (Q(a)) ≤ Cm E(Q(a)) + E(Q(b)) + E(Q ) + α ˆ (D) ˆ Sm ˆ 2 for suitable Aa , Ab ∈ R2×2 skew , ca , cb ∈ R , where D = j=0 Q(ξj ). In fact, if j ˆ ˆ ˆ Q(a), Q(b) ⊂ P for some j = 1, 2, this follows immediately. Otherwise, we apply ˆ ˆ the arguments leading to (8.74) on each pair Q(a), Q∗ and Q∗ , Q(b) and employ the triangle inequality. S ˆ and arguing as in (8.77), we then obtain Q Dening P˜ = Q∈K ˆ εˆ υ 1 ˆ ∗ ))2 ku−(A ·+c)k2L2 (P˜ ) ≤ C υˆ−3 ((E(P˜ ))2 +(α(P ))2 ) ≤ C(1+C∗ r) 4 |Γ|3∞ +C 3 (E(Q υ υˆ 2 for some A ∈ R2×2 skew and c ∈ R . In the last step we proceeded as in (8.78) (see also (8.64)), observing that the paths P˜ dened here and in the proof of Lemma 8.5.2 dier essentially by the square Q∗ . By (8.61) we get υˆ3 υ −3 ≤ υˆr and using

193

(8.90) as well as (8.38) we derive (cf. (8.64))

X
2 ˆ ∗ ))2 ≤ C υˆ−1 |Γ|3∞ ε + C υˆ−3 C∗ ε υˆ−3 (E(Q (|Θl |∗ )3/2 4 ∗ Γl ∩Q 6=∅ υ
2 ε X −1 3 −1 −1 |Γ | ≤ C υˆ |Γ|∞ ε + C|Γ|∞ υˆ υ C∗ 4 l ∞ Γl ∩Q∗ 6=∅ υ ε ≤ C(1 + C∗ r) υˆ 4 |Γ|3∞ . υ

(8.91)

Consequently, we have re-derived (8.78). Proceeding as in Lemma 8.5.2 we obtain a set U 0 ⊂ U with |U \ U 0 | ≤ Cr|U | such that ˆ ε |U |− |u(x) − (A x + c)|2 dx ≤ C(1 + C∗ r) 3 |Γ|3∞ υ Z for Z ⊂ U 0 , Z ∈ Y 0 . On the other hand, by Corollary 8.5.5 we nd Aj ∈ R2×2 skew , 2 cj ∈ R for j = 1, 2 such that ˆ ε |u(x) − (Aj x + cj )|2 dx ≤ C(1 + C∗ r) 3 |Γ|3∞ |U ∩ Nj,+ |− (8.92) υ Z∩Nj,+ for all Z ⊂ U , Z ∈ Y 0 . Now (8.89) follows by applying (7.13) on B1 = U 0 ∩ Nj,+ and B2 = U ∩ Nj,+ . Finally, the essentially dierence in the treatment of the case l2 ≤ l21 is that in the construction of the path P one has to choose two sets Q∗1 ,Q∗2 where the path changes its direction. Following the above arguments it is not hard to see that these sets can be selected so that the required conditions are satised. Note that in the derivation of (8.92) we then exploit that Corollary 8.5.5 also holds for  sets which are much smaller than υ|Γ|2∞ .

8.5.5 Step 4: General case We are eventually in a position to give the proof of Theorem 8.4.1. We briey remark that the following proof crucially depends on the trace theorem established in Lemma 7.2 and the fact that there are at most two large cracks in the neighborhood of Γ. Proof of Theorem 8.4.1. In the general situation we possibly have N 6= N˜ = N \ (X1 ∪ X2 ). Let Cˆ be the covering considered in Lemma 8.5.6. Let K1 , K2 with dist(K1 , K2 ) ≥ c|Γ|∞ be the sets given by Lemma 8.3.5 and let C˜ be the covering of N \ (K1 ∪ K2 ) consisting of the connected components of the sets U \ (K1 ∪ K2 ), U ∈ Cˆ. To simplify the exposition we prefer to present rst a special case where K1 , K2 have the form K− := K1 = (−τ − l1 , −l1 ) × (−τ, τ ) and K+ := K2 = (l1 , l1 + τ ) × (−τ, τ ). Moreover, we suppose that the sets Ψ± associated to boundary components larger than τˆ  if they exist at all  ± ± ± ± have the form Ψ± = Ψ± 1 ∪ Ψ2 ∪ Ψ3 , where Ψ1 = (±l1 , ±(l1 + τ )) × (ψ , 2τ ), 194

± ± ± ± ± Ψ± 2 = (±l1 , ±(l1 + ψ )) × (−ψ , ψ ) and Ψ3 = (±l1 , ±(l1 + τ )) × (−2τ, −ψ ). ± ± Here ψ denote the corresponding values to Ψ (see Section 8.3.2). + If Ψ− or Ψ+ do not exist, we set Ψ− 2 = K1 , Ψ2 = K2 , respectively, and let ± Ψj , j = 1, 3, be the adjacent squares. In addition, we then dene ψ ± = τ . We will treat both cases simultaneously in the following. This special case already covers the fundamental ideas of the proof as the arguments essentially rely on the property that dist(K1 , K2 ) ≥ c|Γ|∞ and the fact that the shapes of all sets are comparable (through homeomorphisms with constants depending on h∗ ) to squares. We will indicate the necessary adaptions for the general case at the end of the proof. Let N±0 = N ∩ {±x2 ≥ 0} \ (K1 ∪ K2 ). By Lemma 8.3.5 the assumptions of Lemma 8.5.6 are satised on each set N+0 and N−0 . Consequently, there are 2×2 A± ∈ Rskew and c± ∈ R2 such that for all V± ⊂ N±0 , V± ∈ Y , one has ˆ ε 0 (8.93) |N± |− |u(x) − (A± x − c± )|2 dx ≤ C(1 + C∗ r) 3 |Γ|3∞ =: G υ V± − + − by (8.89). We let Ξ+ 0 = (l1 , l1 + ψ ) × {0}, Ξ0 = (−l1 − ψ , −l1 ) × {0} and without restriction (possibly after a small translation in e2 -direction) we can assume H1 (Ξ± 0 ∩ ∂W ) = 0. The goal is to show ˆ ψ± G. (8.94) |(A+ − A− ) x + (c+ − c− )|2 dH1 (x) ≤ C υ|Γ|2∞ Ξ± 0 τ − 3 ˜− We prove this only for Ξ− 0 . As a preparation let Ψ1 = (− 2 − l1 , −l1 ) × (ψ , 2 τ ),
− 3 τ − ˜− ˜− ˜− ˜ − ◦ . We observe Ψ 3 = (− 2 − l1 , −l1 ) × (− 2 τ, −ψ ) and Ψ = Ψ1 ∪ Ψ2 ∪ Ψ3 X |Γl |H ≤ C(1 − ωmin )−1 ψ − (8.95) ˜− Γl ∩Ψ 6=∅

for C = C(h∗ , q). If ψ − ≥ c(1 − ωmin )τ this follows from (8.34) and the fact that ˆ = N 2ˆτ (Γ) for all Γl with Γl ∩ Ψ ˜ − 6= ∅ (recall |Γl |∞ ≤ τˆ by the construction Γl ⊂ N in Section 8.3.2). If ψ − ≤ c(1 − ωmin )τ , by (8.35) we obtain |Ψ− ∩ ∂W |H ≤ D(1 − ωmin )−1 ψ −  τ taking c > 0 suciently small. Thus, we can assume that ˜ − 6= ∅. This implies P ˜ − |Γl |H ≤ C|Ψ− ∩ ∂W |H and gives Γl ⊂ Ψ− if Γl ∩ Ψ Γl ∩Ψi 6=∅ the assertion. Recall that υ ≤ r(1 − ωmin )3 (see beginning of Section 8.5.1). Applying Theorem B.4 we obtain by (8.35), (8.38), (8.95) and the fact that ψ − ≤ υ|Γ|∞ (cf. also (8.57) for a similar estimate) ˆ |u(x) − (Ai x + ci )|2 dx ˜− Ψ i

˜ − |α(Ψ ˜ − ) + CC∗ ευ −4 |∂W ∩ Ψ ˜ −| ≤ C|Ψ i i i ≤ Cυ 2 |Γ|2∞ (1 − ωmin )−1 εψ − + CC∗ (1 − ≤ C(1 + C∗ r)υ −3 |Γ|2∞ ψ − ε 195


2 |Γl |∞ − ˜ Γl ∩Ψi 6=∅ ωmin )−3 ευ −2 |Γ|2∞ ψ −

X

(8.96)

2 for Ai ∈ R2×2 skew and ci ∈ R , i = 1, 3. Likewise using particularly (8.95) and − − 2 |Ψ2 | ≤ C(ψ ) we get ˆ |u(x)−(A2 x + c2 )|2 dx ≤ C(1 + C∗ r)υ −4 |Γ|∞ (ψ − )2 ε (8.97) Ψ− 2

− − 2 for A2 ∈ R2×2 skew and c2 ∈ R . By (8.93) for V± = Ψ1 \ K1 , Ψ3 \ K1 we see that

ku − (A+ x + c+ )k2L2 (Ψ− \K1 ) + ku − (A− x + c− )k2L2 (Ψ− \K1 ) ≤ CυG. 3

1

˜− Applying (7.10) and (7.11) on B1 = Ψ− i \ K1 , B2 = Ψi i = 1, 3, we then derive by (8.96) employing ψ − ≤ υ|Γ|∞ τ 2 |A+ − A1 |2 + |c+ − c1 + (A+ − A1 ) b+ |2 ≤ C(υ|Γ|2∞ )−1 G, τ 2 |A− − A3 |2 + |c− − c3 + (A− − A3 ) b− |2 ≤ C(υ|Γ|2∞ )−1 G,

(8.98)

where b− = (−l1 , −τ )T and b+ = (−l1 , τ )T . Furthermore, Lemma 7.2.1, (8.35), (8.38), (8.60), (8.95) and (8.96) yield ˆ |u(x) − (Ai x + ci )|2 dH1 (x) ˜− ∂Ψ i

˜ −) ≤ C(υ|Γ|∞ )−1 ku − (Ai · +ci )k2L2 (Ψ˜ − ) + Cυ|Γ|∞ α(Ψ i i X X −4 + CC∗ ευ |Θl |∗ |Θl |∗ ˜− ˜− ≤ C(1 +

Γl ∩Ψi 6=∅ −4 rC∗ )υ |Γ|∞ ψ − ε

≤

(8.99)

Γl ∩ Ψ i = 6 ∅ − Cψ (υ|Γ|2∞ )−1 G

for i = 1, 3, where we tacitly assumed that all boundary components are rectangular (cf. discussion after (8.59)). In the penultimate step we again used ψ − ≤ υ|Γ|∞ and υ ≤ r(1 − ωmin )3 . Likewise, we get ˆ |u(x) − (A2 x + c2 )|2 dH1 (x) ≤ Cψ − (υ|Γ|2∞ )−1 G, (8.100) − ∂Ψ− 2 ∪Ξ0

where we replaced (υ|Γ|∞ )−1 by (ψ − )−1 and used (8.97) instead of (8.96). Observe that (8.100) is well dened in the sense of traces since H1 (Ξ− 0 ∩ ∂W ) = 0. Dene ψ− − − − − ˜− Ξ± = (− 2 − l1 , −l1 ) × {±ψ } and note that Ξ+ ⊂ ∂Ψ− 2 ∩ ∂ Ψ1 and Ξ− ⊂ ˜− ∂Ψ− 2 ∩ ∂ Ψ3 . Again up to a small translation in e2 -direction we may suppose H1 (Ξ− ± ∩ ∂W ) = 0. Combining the estimates (8.98), (8.99) for i = 1, 3 we obtain ˆ ˆ 2 1 |u(x) − (A+ x + c+ )| dH + |u(x) − (A− x + c− )|2 dH1 ≤ Cψ − (υ|Γ|2∞ )−1 G. Ξ− +

Ξ− −

Using once more the techniques provided in Section 7.1 we may estimate the dierence of A± , A2 and c± , c2 on the boundaries Ξ− ± (replace the sets B1 , B2 in 196

(7.10), (7.11) by the surfaces Ξ− ± ) and obtain by (8.100) an expression similar to (8.98):

(ψ − )2 |A± − A2 |2 + |c± − c2 + (A± − A2 ) b2 |2 ≤ C(υ|Γ|2∞ )−1 G,

(8.101)

where b2 = (−l1 , 0)T . Together with (8.100) this leads to ˆ |u(x) − (A± x + c± )|2 dH1 (x) ≤ Cψ − (υ|Γ|2∞ )−1 G Ξ− 0

and then by the triangle inequality we derive ˆ |(A+ − A− ) x + (c+ − c− )|2 dH1 (x) ≤ Cψ − (υ|Γ|2∞ )−1 G. Ξ− 0

This gives the desired estimate (8.94). From (8.94) applied on both sets, Ξ− 0 and + Ξ0 , we deduce

−C(l1 υ)2 |(A+ − A− ) e1 |2 + | − (A+ − A− ) l1 e1 + (c+ − c− )|2 ≤ C(υ|Γ|2∞ )−1 G and

−C(l1 υ)2 |(A+ − A− ) e1 |2 + |(A+ − A− ) l1 e1 + (c+ − c− )|2 ≤ C(υ|Γ|2∞ )−1 G. Combining these two estimates we nd for υ suciently small l12 |A+ − A− |2 = 2l12 |(A+ − A− )e1 |2 ≤ C(υ|Γ|2∞ )−1 G and then also |c+ − c− |2 ≤ C(υ|Γ|2∞ )−1 G. (This is the step where we fundamentally use dist(K1 , K2 ) ≥ c|Γ|∞ .) We choose A = A− and c = c− . Recalling the denition of G and |N | ≤ υ|Γ|2∞ we obtain by (8.93) for V± = N±0 ˆ ε |u(x) − (A x − c)|2 dx ≤ C(1 + C∗ r) 3 |Γ|3∞ , 0 ∪N 0 υ N+ − which together with the estimates (8.96) and (8.98) gives (8.59)(i). Finally, (8.101) yields (ψ − )2 |A − A2 |2 ≤ C(υ|Γ|2∞ )−1 G and |c − c2 + (A − A2 ) (−l1 , 0)T |2 ≤ − 2 − C(υ|Γ|2∞ )−1 G. Then by (8.97) and the fact that |Ψ− 2 | ≤ C(ψ ) ≤ Cψ υ|Γ|∞ we conclude ˆ |u(x) − (A x + c)|2 dx ≤ C(1 + rC∗ )υ −3 |Γ|2∞ ψ − ε Ψ− 2

giving (8.59)(ii). The estimate for Ψ+ 2 follows analogously. It remains to briey indicate the necessary adaptions for the general case. The main dierences are (i) the shape of the sets Ψ± i , i = 1, 2, 3 and (ii) the position of the sets K1 , K2 . For (i) we observe that Ψ± i , i = 1, 3, are C(h∗ )-Lipschitz equivalent to a square by Lemma 8.3.6(ii) and Lemma 8.3.7(ii) whereby (8.96) can still be derived (cf. Remark 7.1.4(i)). (Note that the sets are even related 197

by ane mappings.) Likewise, an estimate of the form (8.97) can be derived ± ∗ for sets (Ψ± 2 ) ⊃ Ψ2 which have been constructed in Section 8.3.2. Moreover, although not stated explicitly in Section 7.2, the trace estimate used in (8.99), (8.100) can also be applied for sets being an ane transformation of a square. The rest of the arguments concerning the dierence of innitesimal rigid motions (see (8.98), (8.101)) remains unchanged. For (ii) we observe that in the derivation of |A+ − A− |2 ≤ Cl1−2 (υ|Γ|2∞ )−1 G we fundamentally used that dist(K1 , K2 ) ∼ l1 , but the exact position of the sets K1 , K2 was not essential.  We briey explain Remark 8.4.3. At the beginning of Section 8.5.1 we have already observed that υ ∼ C(h∗ )σ 3 . Now the property for C2 follows immediately ˆ ∗ ) (see end of Section (see (8.43)). For C1 we use (8.47) and the fact that Cˆ = C(h 8.5.1). We close this section with an estimate for the skew symmetric matrices involved in the above results.

Lemma 8.5.7. Let be given the situation of Theorem 8.4.2 for a function u = ¯ T y−id, where y ∈ H 1 (W ) and R ¯ ∈ SO(2). Let V ⊂ Qµ be a rectangle and uR¯ = R let F(V ) be the boundary components (Γl )l = (Γl (U ))l satisfying N τˆl (∂Rl ) ⊂ V and (8.41). Then there is a C3 = C3 (σ, h∗ ) such that p

X Γl ∈F (V )

−1 2 −1 ¯ pp ε|∂U ∩ V |H |Xl |2∞ |Al |p ≤ C3 k∇y − Rk L (V ∩W ) + (εs )




for p = 2, 4, where Xl ⊂ Qµ , Al ∈ R2×2 skew is given in (8.40).

Remark 8.5.8.

Similarly as in Remark 8.4.3 we note that the constant C3 = C3 (σ, h∗ ) has polynomial growth in σ , i.e. C3 (σ, h∗ ) ≤ C(h∗ )σ −z for some z ∈ N.

Proof. Let p = 2, 4. Consider a component Γ = Γl (U ) with corresponding rect-

¯ pp + angle R and X with ∂X = Γ. It suces to show |R|2∞ |A|p ≤ C3 k∇y − Rk ˜) L (N
S −1 p2 −1 τˆ τˆl ˜ (εs ) ε|Γ|H for this component, where N = N (∂R) \ Γl ∈I(Γ) N (∂Rl ) and I(Γ) = {Γl : |Γl |∞ ≤ |Γ|∞ }. Then the assertion follows by summation over all components and the fact that |X|∞ ≤ |R|∞ . As Γ satises (8.41), we observe that we applied Theorem 8.4.1 on ∂R in some iteration step, in particular (8.59) is satised. Choose U ∈ C with U ⊂ N 2,+ as considered in Lemma 8.5.3. By assumption we nd a S set S ⊂ (l2 , l2 + τ ) with 1 τ |S| ≥ 2 20 such that for T = (R × S) ∩ U we have T ∩ Γl ∈I(Γ) N τˆl (∂Rl ) = ∅ by (8.33). It is not restrictive to assume that S is connected as otherwise we follow the subsequent arguments for every connected component of S . Recall |Γ|∞ ≤ |∂R|∞ ≤ C|Γ|∞ by (8.6). The Poincaré inequality and a rescaling argument imply ˆ 2 ¯ 2p |u(x) − cˆ|2 dx ≤ C|T |1− p |Γ|2 k∇y − Rk ∞

T

198

L (T )

for a constant cˆ ∈ R2 and p = 2, 4. This together with (8.59) yields

ˆ 2

|A x + c − cˆ| dx ≤

2 C(υ|Γ|2∞ )1− p |Γ|2∞ k∇y

¯ 2Lp (T ) + C Cˆ − Rk

T

∞   X 2 n n=0

3

υ −3 |Γ|3∞ ε.

For the constant in the latter part see below (8.47). Arguing as in (7.10) we nd ˆ 2 2 |T ||Γ|∞ |A| ≤ C |A x + c − cˆ|2 dx. T

Thus, by |T | ≥ Cυ|Γ|2∞ and an elementary calculation we derive

¯ 4 4 + Cυ −8 ε2 . |R|2∞ |A|4 ≤ C|Γ|2∞ |A|4 ≤ Cυ −1 k∇y − Rk L (T ) As |∂R|∗ ≥ s, we obtain |Γ|H ≥S C|∂R|∗ ≥ Cs by (8.5)(ii). Choose C3 large enough and recall T ⊂ N τ¯ (Γ) \ Γl ∈I(Γ) N τ¯l (∂Rl ) ⊂ V as well as the fact that υ ≥ C(h∗ )σ 3 . This yields

¯ 4 4 ˜ + C3 εs−1 ε|Γ|H . |R|2∞ |A|4 ≤ C|Γ|2∞ |A|4 ≤ C3 k∇y − Rk L (N ) giving the claim for p = 4. Likewise, for p = 2 we deduce

¯ 2 2 + Cυ −4 ε|Γ|∞ |R|2∞ |A|2 ≤ C|Γ|2∞ |A|2 ≤ Cυ −1 k∇y − Rk L (T ) 2 ¯ ≤ C3 k∇y − Rk 2 ˜ + C3 ε|Γ|H . L (N )



199

Chapter 9 Quantitative SBD-rigidity This section is devoted to the proof of Theorem 6.1.1. We will rst establish a local rigidity result measuring the distance of the deformation from a piecewise rigid motion in the H 1 -norm. At this point the Korn-Poincaré-type inequality established in the last chapter is essential. Afterwards, we show that in regions where only small boundary components are present such a local estimate can be used to replace the original deformation by an H 1 -function. Observing that in the modication process the least length of the boundary components increases, we can then apply this estimate iteratively to obtain the rigidity result up to a small set. Finally, by constructing a suitable extension we nd that the result holds on the whole domain.

9.1 Preparations Before we start with a local rigidity estimate, we recall some denitions and introduce further notions. Given a Lipschitz domain Ω ⊂ R2 choose µ0 so large that Ω ⊂ Qµ0 = (−µ0 , µ0 )2 . Recall the point set I s = s(1, 1) + 2sZ2 , s > 0, introduced in Section 7.1 and the denitions of U s , V s in (7.1), (8.1) with respect to the square Qµ0 . We dene additional partitions. Set z1 = (0, 0), z2 = (1, 0), z3 = (0, 1), z4 = (1, 1) and let Iis = szi + 2sZ2 as well as Qsi (p) = p + s(−1, 1)2 for p ∈ Iis , i = 1, . . . , 4. Moreover, for U ⊂ Ω let

Iis (U ) = {p ∈ Iis : Qsi (p) ⊂ U } for i = 1, . . . , 4. For shorthand we also write I s = I4s and Qs = Qs4 . c¯k k We let Ωk be the largest ∂Ω) ≥ c¯k} √ set in V satisfying Ω ⊂ {x ∈k Ω : dist(x, s for k ≥ 0 for some c¯ ≥ 2 large enough. For sets W ⊂ Ω , W ∈ V , we assume that one component in denition (8.1) is given by X = Qµ0 \ Ωk . In particular, all other components X1 , . . . , Xn satisfy ∂Xi ⊂ Qµ0 as Ω ⊂ Qµ0 . We choose an

201

(arbitrary) order of (Γj )j=1,...,n and similarly to (8.2) we dene [ Θi = Θi (W ) = Γi \ Γj j 0 and W ∈ Vts . ˜ ⊂ W , |W \ W ˜ | = 0 and kW ˜ k∗ ≤ kW k∗ ˜ ∈ V s with W (i) Then there is a set W t such that ˜ ) ∩ Γj2 (W ˜)=∅ Γj1 (W

˜ )|∞ ≤ k for i = 1, 2. if |Γji (W

(9.2)

s (ii) Then there is a set U ∈ Vt+k with U ⊂ W , |W \ U | = 0 and kU k∗ ≤ kW k∗ such that

Γ(U ) ∩ Γj (U ) = ∅

for all Γj (U ) 6= Γ(U )

(9.3)

for all Γ(U ) with |Γ(U )|∞ ≤ k. Proof. (i) The strategy is to combine iteratively dierent boundary components.

Clearly, if |Γji (W )|∞ ≤ k for i = 1, 2 with Γj1 (W ) ∩ Γj2 (W ) 6= ∅ we may replace W by W 0 = W \ (Xj1 ∪ Xj2 )◦ and note that W 0 ∈ Vts as well as |W \ W 0 | = 0 and kW 0 k∗ ≤ kW k∗ similarly as in Lemma 8.1.1. (Recall that ∂Xji = Γj1 (W ) for ˜ ∈ Vts with |W \ W ˜|=0 i = 1, 2.) We proceed in this way until we obtain a set W ˜ and kW k∗ ≤ kW k∗ such that (9.2) holds. ˜ ), Γj2 (W ˜) (ii) We apply (i) and then proceed to combine two components Γj1 (W ˜ ) ∩ Γj2 (W ˜ ) 6= ∅ and min{|Γj1 (W ˜ )|∞ , |Γj2 (W ˜ )|∞ } ≤ k . Arguing as before if Γj1 (W 202

we end up with a set U satisfying |W \U | = 0, kU k∗ ≤ kW k∗ and (9.3). It remains s to show that U ∈ Vt+k . Consider some Γ(U ) = ∂X with |Γ(U )|∞ > k and observe ˜ ) = ∂X 0 with |Γ(W ˜ )|∞ > k and Γj (W ˜ ) = ∂Xj , i = 1, . . . , m, that there are Γ(W i i ˜ ˜ ˜ ˜ ) ∩ Γ(W ˜ ) 6= ∅ with |Γji (W )|∞ ≤ k , Γji1 (W ) ∩ Γji2 (W ) = ∅ for i1 6= i2 and Γji (W S ˜ such that X = X 0 ∪ m i=1 Xji . But this implies |πi Γ(U )| ≤ 2k+|πi Γ(W )| ≤ 2k+2t for i = 1, 2, as desired.  In what follows we often modify sets by subtracting rectangular neighborhoods of boundary components. In this context it is particularly important that the components remain connected and do not become too large. By 4 we denote the symmetric dierence of two sets.

Lemma 9.1.2. Let k, t, t0 > 0 with t, t0 ≤ Ck and ν ≥ 0. Let V ⊂ Ωk with s V ∈ Vcon . (i) Assume that for each component Xj = Xj (V ), j = 1, . . . , n, there is a rectangle Zj ∈ U s with Xj ⊂ Zj , |πi ∂Zj | ≤ |πi ∂Xj | + ν|∂Xj |∞ for i = 1, 2 and maxi=1,2 |πi ∂Zj | ≤ 2t0 for all j = 1, . . . , n. Moreover, assume that Zj1 \ Zj2 or Zj2 \ Zj1 is connected for all 1 ≤ j1 < j2 ≤ n. Then there is a set U ∈ Vts0 , S S U ⊂ Ωk , with nj=1 Xj (U ) = nj=1 Zj ∩ Ωk and kU k∗ ≤ (1 + cν)kV k∗ for a universal constant c > 0. ˆ = V 0 \ Sn Zj . Then (ii) In addition let V 0 ∈ Vts be given and dene W j=1 s/2 0 ˆ ˆ there is a set W ∈ Vt+2t0 with |W \ W | = 0, |W \ W | ≤ ctkV k∗ and kW k∗ ≤ (1 + cν)kV k∗ + kV 0 k∗ . Proof. (i) Let V ⊂ Ωk with components (Xj )nj=1 and rectangles (Zj )nj=1 be given.

It suces to show the following: There are connected, pairwise disjoint (Xj0 )nj=1 S S with Xj0 ⊂ Zj , nj=1 Xj0 = nj=1 Zj and

[n

j=1

Xn ∂Xj0 H ≤

j=1

|Θj (V )|H + cν

Xn j=1

|Γj |H .

(9.4)

Moreover, we have Xj0 = Rj \(A1j ∪ A2j ). Here Rj ∈ U s is a rectangle and Aij ∈ U s , i = 1, 2, are (if nonempty) unions of rectangles whose closure intersect the corner cij ∈ ∂Rj , where c1j , c2j are adjacent corners of Rj . Sn 0 Then the claim of the lemma follows forPU = Ωk \ P j=1 Xj . Indeed, to 0 see P kU k∗ ≤ (1 + cν)kV k∗ we rst observe j |∂Xj |∞ ≤ j |∂Zj |∞ ≤ (1 + cν) j |∂Xj |∞ . Moreover, by (9.4) we get

kU kH ≤ |

[n j=1

∂Xj0 |H ≤ (1 + cν)kV kH = (1 + cν)|

[n j=1

∂Xj |H .

(9.5)

In the rst inequality we also used |∂Xj0 |∞ ≤ |∂Zj |∞ ≤ Ck and Ωk ∈ V c¯k for c¯  1. (Arguments of this form will be used frequently in the following and from now on we will omit the details.) Finally, we conclude U ∈ Vts0 as maxi=1,2 |πi ∂Zj | ≤ 2t0 for j = 1, . . . , n. 203

We prove the above assertion by induction. Clearly, the claim holds for n = 1 for X10 = Z1 . Now assume the assertion holds for sets with at most n − 1 components and consider V ⊂ Ωk with components (Xj )nj=1 and corresponding (Zj )nj=1 . Without restriction we assume that maxx∈Zn x2 = maxx∈Snj=1 Zj x2 . By hypothesis we obtain pairwise disjoint, connected sets Xj00 , j = 1, . . . , n − 1, S Sn−1 00 fullling the above properties, in particular n−1 j=1 Xj = j=1 Zj . 1 2 1 2 1 2 1 2 ˜ . . , n−1 Given Zn = (z1 , z1 )×(z2 , z2 ) we set Zn = (z1 , z1 )×(z2 , z2 ]. For j = 1, .S s 0 0 let Zj,i ∈ U be the largest rectangle in Zn satisfying Zj ∩ Zn ⊂ Zj,i ⊂ n−1 j=1 Zj i 0 0 0 0 with z1 ∈ Zj,i for i = 1, 2. If Zj,i 6= ∅ for some i, we let Zj = Zj,i , otherwise we 0 0 0 0 set Zj0 = Zj ∩ Zn . (Note that Zj,1 = Zj,2 if Zj,1 , Zj,2 6= ∅. ) Let J0 ⊂ {1, . . . , n − 1} such that Zj ∩ Zn = ∅ for j ∈ J0 . Let J1 ⊂ {1, . . . , n − 1} \ J0 such that (Zj0 \ Zn ) ∩ {z11 , z12 } = ∅ for j ∈ J1 and J2 ⊂ {1, . . . , n − 1} \ J0 such that Z˜n \ Zj0 is a rectangle for j ∈ J2 . (Observe that J1 ∩ J2 = ∅.) Let S J3 = {1, . . . , n − 1} \ (J0 ∪ J1 ∪ J2 ). Dene Xn0 = Zn \ j∈J2 ∪J3 Zj0 . Moreover, we let Xj0 = Xj00 for j ∈ J0 ∪ J2 ∪ J3 and Xj0 = Xj00 \ Xn0 for j ∈ J1 . Clearly, by S S construction the sets are pairwise disjoint and fulll nj=1 Xj0 = nj=1 Zj . Moreover, we observe that the sets are connected and have the special shape given above. In fact, for j ∈ J0 ∪ J2 ∪ J3 this is clear. For Xn0 we rst note that ˙ 32 , where Zj0 intersects the lower right and the lower left corner of Zn J3 = J31 ∪J for j ∈ J31 and j ∈ J32 , respectively. (It cannot happen that Zj0 intersects only the other corners due to the S choice of Zn .) We observe Xn0 = Rn \ (A1n ∪ A2n ) is S connected, where Rn = Zn \ j∈J2 Zj0 and Ain = j∈J i Zj0 for i = 1, 2. 3

Finally, to see the properties for j ∈ J1 we rst observe that Sj := Zj \ Xn0 is a rectangle. In fact, otherwise due to the special shape of Xn0 it is elementary to see that (Zj0 \ Zn ) ∩ {z11 , z12 } = 6 ∅ and thus j ∈ / J1 . We get Xj0 = Xj00 ∩ Sj =
ˆ j \ (Aˆ1 ∪ Aˆ2 ), where R ˆ j = Sj and Rj ∩ Sj \ (A1j ∪ A2j ) is connected and Xj0 = R j j Aˆij = Aij ∩ Sj for i = 1, 2. It remains to conrm (9.4). We rst observe that Xn Xn [n
|Θj (V )|H = 21 |Γj |H + 21 ∂ Xj H . (9.6) j=1

j=1

j=1

(Recall that dierent boundary components may have nonempty intersection.) Similarly, for the components (Xj0 )j we nd Xn [n [n
∂Xj0 H = 21 |∂Xj0 |H + 12 ∂ Xj0 H . j=1

j=1

j=1

We now treat the two terms on the right separately. By Tj ∈ U s we denote the smallest rectangle containing Xj and observe that |∂Tj |∞ = |Γj |∞ , |∂Tj |H ≤ |Γj |H . Recall |∂Zj |H ≤ |∂Tj |H + cν|∂Tj |∞ ≤ (1 + cν)|Γj |H for j = 1, . . . , n. Due to the special shape of the components Xj0 we nd |∂Xj0 |H ≤ |∂Zj |H and thus Xn Xn |∂Xj0 |H ≤ (1 + cν) |Γj |H . (9.7) j=1

j=1

204

˜ j ⊃ Xj Moreover, it is elementary to see that we can nd a connected set X s ˜ j := ∂ X ˜ j satises |Γ ˜ j |H ≤ (1 + cν)|Γj |H and Zj ∈ U is the smallest such that Γ ˜ rectangle containing Xj . By a projection argument it is then not hard to see that [n ∂

[n [n


˜j Xj0 H = ∂ Zj H ≤ ∂ X H j=1 j=1 j=1 X [n
Xj H + cν |Γj |H . ≤ ∂ j=1

Consequently, we derive by (9.6) and (9.7) Xn Xn [n [n
|Γj |H + 21 ∂ Xj H + cν ∂Xj0 H ≤ 21 |Γj |H j=1 j=1 j=1 j=1 Xn Xn = |Θj (V )|H + cν |Γj |H , j=1

j=1

as desired. 0

(ii) Let (Yj )nj=1 be the components of V 0 and let Tj ∈ U s be the smallest rectangle containing Yj . It is elementary to see that Tj1 \ Tj2 is connected for 1 ≤ j1 , j2 ≤ n0 . Thus, by (i) we obtain pairwise disjoint, connected sets (Yj0 )j with S 0 S Sn0 0 00 k j Yj = j Tj and dene V = Ω \ j=1 Yj . By (i) for ν = 0 we then also obtain 00 0 kV k∗ ≤ kV k∗ . √ Moreover, the isoperimetric inequality yields |V 0 \ V 00 | ≤ ctkV 0 k∗ since |∂Tj |∞ ≤ 2 2t for all j = 1, . . . , n0 . S 0 Let (Xj0 )nj=1 and U ∈ Vts0 as given in (i). We dene W 0 = (U \ nj=1 Yj0 ) ∪ Sn0 0 0 0 0 0 ˆ ˆ j=1 ∂Yj . Clearly, we have |W \ W | = 0, |W \ W | ≤ ctkV k∗ and kW k∗ ≤ 0 (1 + cν)kV k∗ + kV k∗ arguing similarly as in Lemma 8.1.1. Observe that possibly s as the components (Xj0 )nj=1 of U may have become disconnected. Thus, W0 ∈ / Vcon s/2

we now construct a set W ∈ Vcon with |W 0 4W | = 0. By Rj ∈ U s we denote the smallest rectangle such that Xj0 ⊂ Rj for j = S S 1, . . . , n and observe j Rj = j Xj0 . To simplify the exposition we assume that each of the components (Xj0 )j has become disconnected as otherwise we do not have to alter the boundary component in the modication procedure described below. Moreover, we can suppose that for each pair Yj01 , Xj02 , 1 ≤ j1 ≤ n0 , 1 ≤ j2 ≤ n, with Rj2 \Yj01 is not disconnected we have Xj02 \Yj01 is not disconnected. In fact, otherwise we can pass to some Yj∗1 ⊂ Yj01 with |∂Yj∗1 |∗ ≤ |∂Yj01 |∗ such that S S S S Xj02 \ Yj∗1 is not disconnected and j Yj0 ∪ j Xj0 = j Yj∗ ∪ j Xj0 . We now proceed by induction. Let W0 = V 00 and Tj0 = Yj0 for j = 1, . . . , n0 . s Assume there are pairwise disjoint, connected sets Tjl−1 ∈ U 2 , j = 1, . . . , n0 such that [n0 [n0 [l−1 (i) Tjl−1 = Yj0 ∪ Xj0 , (ii) Tjl−1 ∩ Xj02 = Tj01 ∩ Xj02 (9.8) 1 j=1

j=1

j=1

for all 1 ≤ j1 ≤ n0 , l ≤ j2 ≤ n. Moreover, assume that the set Wl−1 := Ωk \ 205

S

j

Tjl−1

satises kWl−1 k∞ ≤

kWl−1 kH ≤ |

[ j

0 j |∂Tj |∞ +

P

∂Tj0 |H

+|

[l−1 i=1

Pl−1

|∂Xi0 |∞ and

∂Xi0

[

i=1

l−1

\

j

Tj0 |H

[ 1X |∂Xi0 ∩ Tj0 |H . (9.9) + j 2 i=1

We now construct Wl . Let J l ⊂ {1, . . . , n0 } such that Tjl−1 ∩ Xl0 6= ∅ with ˙ 2l , where j ∈ J2l if and only if Rl \ Tjl−1 is disconnected. J l = J1l ∪J ˆ 0 , where X ˆ 0 ∈ U 2s is the largest set with If j ∈ J1l , we dene Tjl = Tjl−1 \ X l l l−1 0 l 0 ˆ ⊂ X . It is not hard to see that |∂T |∞ ≤ |∂T |∞ for all j ∈ J l and |∂T l |H ≤ X j

l

l

1

j

j

|∂Tjl−1 \ Xl0 |H + 21 |∂Tjl−1 ∩ Xl0 |H + 21 |∂Xl0 ∩ Tjl−1 |H . As each x ∈ R2 is contained in S P at most two dierent ∂Tjl−1 , we nd j∈J l 21 |∂Tjl−1 ∩ Xl0 |H ≤ | j∈J l ∂Tjl−1 ∩ Xl0 |H . 1 1 Therefore, taking the union over all components we derive [ [ [ 0 [ l−1 l−1 l 0 1 ∂T ∂T ∪ ∂T ≤ + ∩ T (9.10) ∂X l j j H. j j 2 l l l H H j ∈J / 1

j∈J1

j

j∈J1

Here we used (9.8)(ii) and the fact that the sets (Tjl−1 )j are pairwise disjoint. Observe that the above construction together with (9.8)(ii) and the special shape of Tj0 (see proof of (i)) implies that the sets Tjl , j ∈ J1l , are connected. Moreover, (9.8)(ii) holds for j1 ∈ J1l . ˜ 0 = X 0 \ S l T l ∈ U 2s . Due to the fact that X ˆ 0 6= ∅ we observe We dene X l l l j∈J1 j S 0 ˜0 \ that the number of connected components of the sets Xl \ j∈J l Tjl−1 and X l 2 S l−1 coincide. Therefore, letting A1 , . . . , Am be the connected components j∈J2l Tj S l−1 ˜0 \ of X it is elementary to see that m = #J l + 1. l T l

j∈J2

2

j

Up to a rotation by π2 we can assume that each Ai intersects the upper and lower boundary of Rl and that A1 intersects the left boundary. For convenience we denote the components (Tjl−1 )j∈J2l by (Tjl−1 )m−1 i=1 . Suppose Rl = (0, l1 ) × (0, l2 ). i Let ai = inf x∈Ai x1 and di = ai+1 − ai , where am+1 = l1 . Dene Tjl1 = (Tjl−1 ∪ 1

(A1 ∪ A2 ))◦ and Tjli = (Tjl−1 ∪ Ai+1 )◦ for i = 2, . . . , m − 1. Observe that the i sets are pairwise disjoint, connected and that (9.8)(ii) holds for ji ∈ J2l . It is elementary to see that |Tjl1 |∞ ≤ |Tjl−1 |∞ + d1 + d2 and |Tjli |∞ ≤ |Tjl−1 |∞ + di+1 for 1 i i = 2, . . . , m − 1. Thus, we have Xm−1 i=1

|Tjli |∞ ≤

Xm−1 i=1

|Tjl−1 |∞ + |Xl0 |∞ . i

(9.11)

For j ∈ / J l we dene Tjl = Tjl−1 and observe that (9.8)(i) holds by construction and the assumptions before (9.8). Together with (9.10) and (9.8)(ii) we then also get

[l−1 [ [ [ [
Tj0 H + 21 ∂Xl0 ∩ Tj0 H . ∂Tjl H ≤ ∂Tjl−1 H + ∂Xl0 \ ∂Xi0 ∪ j

j

i=1

206

j

j

This in conjunction with (9.9) for Wl−1 implies that P (9.9) holdsPfor Wl . Moreover, by (9.11) it is elementary to see that kWl k∞ ≤ j |∂Tj0 |∞ + li=1 |∂Xi0 |∞ . Finally, we dene W = Wn and observe that W has the desired properties. In ˆ \W | ≤ ctkV 0 k∗ . Moreover, we fact, by (9.8)(i) we have |W 4W 0 | = 0 and thus |W clearly get kW k∞ ≤ kU k∞ + kV 00 k∞ ≤ (1 + cν)kV k∞ + kV 0 k∞ . As each x ∈ R2 is contained in at most two dierent ∂Xl0 , we nd by (9.9) [ [ [n [n ∂Xi0 \ ∂Xi0 ∩ kW kH ≤ kV 00 kH + Tj0 H + Tj0 H i=1

00

i=1

j

j

0

= kV kH + kU kH ≤ kV kH + (1 + cν)kV kH , as desired. Finally, similarly as in Lemma 9.1.1(ii) we obtain |πi Xj (W )| ≤ 2t+4t0 s/2 for i = 1, 2 for all j and thus W ∈ Vt+2t0 .  λ Recall the denition H(·) in (8.32). In addition, for λ > 0 we dene H (W ) ⊃ W as the `variant of W without holes of size smaller than λ': We arrange the sets (Γj )j=1,...,n in the way that |Γj |∞ ≤ λ for j ≥ lλ and |Γj |∞ > λ for j < lλ . Dene [n Xj . (9.12) H λ (W ) = W ∪ j=lλ

In the following, constants which are much smaller than 1 will frequently appear. For the sake of convenience we introduce one universal parameter. For given l ≥ 1 and 0 < s, , m ≤ 1 we let 2 −1 ϑ = l 9 Cm s ,

(9.13)

where Cm = C1 (m, h∗ ) + C3 (m, h∗ ) + m−4 C2−2 (m, h∗ ) with the constants of Theorem 8.4.2 and Lemma 8.5.7 (for xed h∗ ). By Remark 8.4.3, 8.5.8 we nd some z ∈ N such that Cm ≤ C(h∗ )m−z . Moreover, for later let m ˆ = C2 (m, h∗ ) and ¯ recall that by (8.43) we can assume m ˆ  m as well as C m ˆ ≤ m for constants ¯ ¯ C = C(h∗ ). Using only one universal parameter the estimates we establish are often not sharp. However, this will not aect our analysis.

Remark 9.1.3.

All the constants C in the following may depend on h∗ unless they are universal constants indicated as Cu . However, to avoid further notation we drop the dependence here. Only at the end of the proof in Section 9.5, when we pass to the limit h∗ → 0, we will take the h∗ dependence of the constants into account.

9.2 A local rigidity estimate We now establish a local rigidity estimate on a ne partition of the Lipschitz domain Ω. In the following,  will represent the stored elastic energy. We rst construct piecewise constant SO(2)-valued mappings approximating the deformation gradient. Afterwards, we employ Theorem 8.4.8 to nd a piecewise rigid motion being a good approximation of the deformation. 207

9.2.1 Estimates for the derivatives We divide our investigation into two regimes, the `superatomistic' k ≥  and the `subatomisic' k ≤ . Here, we call the -regime the `atomistic regime' as in discrete fracture models  is of the same order as the typical interatomic distance (compare e.g. (6.2) with (1.27)). We begin with the superatomistic regime.

Lemma 9.2.1. Let k > s ≥  > 0 with 1  l := ks . Let m−1 ∈ N and assume ∈ N. Then for a constant C > 0 we have the following: that km s For all U ∈ Vks with U ⊂ Ωk and for all y ∈ H 1 (U ) with ∆y = 0 in U ◦ and γ := k dist(∇y, SO(2))k2L2 (U ) ,

(9.14)

sm with W ⊂ Ω3k , |W \U | = 0, |(U \W )∩Ω3k | ≤ Cu kkW k∗ there is a set W ∈ V(s,3k) and

kW k∗ ≤ (1 + Cu m)kU k∗ + C−1 γ.

(9.15)

Moreover, there are mappings Rˆ i : W ◦ → SO(2), i = 1, . . . , 4, which are constant on the connected components of Qki (p) ∩ W ◦ , p ∈ Iik (Ωk ), such that 4 ˆ i k2 2 (i) k∇y − R L (W ) ≤ Cl γ, ˆ i k4 4 (ii) k∇y − R L (W ) ≤ Cϑγ.

(9.16)

Proof. We rst construct the set W . Let J ⊂ I k (Ωk ) such that k dist(∇y, SO(2))k2L2 (Qk (p)∩U ) > k

(9.17)

for all p ∈ J . Dene

 [ ˆ W = U\

k

p∈J



Q (p) ∪

[ p∈J

∂Qk (p)

s ˆ ∈ V s . In particular, the property W ˆ ∈ Vcon and note that W holds since k max{|π1 Γt (U )|, |π2 Γt (U )|} ≤ 2k . The fact that we add the union of the boundary on the right hand side assures that we do not `combine' components. P boundary −1 k ˆ Moreover, we derive kW k∗ ≤ kU k∗ + C γ . Indeed, p∈J |∂Qp |∗ ≤ 8k · #J ≤ γ ˆ ) we nd a corresponding Γt (U ) (without 8k k by (9.14). For all other Γt (W ˆ ) = Θt (U ) \ S Qk (p) and restriction we use the same index) such that Θt (W p∈J ˆ )|∗ ≤ |Θt (U )|∗ . (Arguments of this form will be used frequently in thus |Θt (W the following and from now on we will omit the details.) Furthermore, we easily ˆ | ≤ Cu kkW ˆ k∗ . deduce |U \ W sm ˆ k∗ , |U \ W | ≤ Then we can nd a set W ∈ V2k with kW k∗ ≤ (1 + Cu m)kW ˆ : dist∞ (x, ∂ W ˆ ) ≤ 2sm}, where dist∞ (x, A) := Cu kkW k∗ and W ◦ ⊂ {x ∈ Ω3k ∩ W 2 2 inf y∈A maxi=1,2 |(x − y) · ei | for A ⊂ R , x ∈ R .

208

Indeed, let M (Γj ) ∈ U sm be the smallest rectangle satisfying M (Γj ) ⊃ {x ∈ R : dist∞ (x, Xj ) ≤ 2sm}, where Xj denotes the component corresponding to ˆ ). Clearly, we obtain |πi ∂M (Γj )| ≤ |πi Γj (W ˆ )| + Cu m|Γj (W ˆ )|∞ for i = 1, 2, Γj (W ˆ ∈ V s . Moreover, it is elementary to see that M (Γj1 ) \ M (Γj2 ) j = 1, . . . , n as W is connected for 1 ≤ j1 , j2 ≤ n since sm  s. Then by Lemma 9.1.2(i) with sm which coincides with Zj = M (Γj ) we obtain a set W ∈ V2k [n [n
ˆ \ Ω3k ∩ W M (Γj ) = Ω3k \ M (Γj ) (9.18) 2

j=1

j=1

up to a set of negligible measure. Here we used sm  k . Moreover, we have ˆ k∗ . |(U \ W ) ∩ Ω3k | ≤ Cu kkW k∗ and kW k∗ ≤ (1 + Cu m)kW Boundary components of W are possibly smaller than 2s due to the modication in (9.18). Therefore, we apply Lemma 9.1.1(ii) to get a (not relabeled) sm such that (9.15) still holds and (9.3) is satised. Now the fact that set W ∈ V3k s sm U ∈ V(s,k) and (9.3) imply W ∈ V(s,3k) . Fix i = 1, . . . , 4 and let F ⊂ Qki (p) ∩ W ◦ be a connected component of k Qi (p) ∩ W ◦ . Dene Fˆ ∈ U s as the smallest (connected) set satisfying

Fˆ ⊃ {x : dist∞ (x, F ) < 2sm}. ˆ ◦ ⊂ U . As |Fˆ | ≤ Cu k 2 , Lemma Due to the construction of W we get Fˆ ⊂ W 7.1.3(i) for µ = 2k implies that there is a rotation R ∈ SO(2) such that k∇y − Rk2L2 (Fˆ ) ≤ Ck 4 s−4 k dist(∇y, SO(2))k2L2 (Fˆ ) = Cl4 γ(Fˆ ), where for shorthand we write γ(Fˆ ) = k dist(∇y, SO(2))k2L2 (Fˆ ) . As ∇y − R is harmonic in Fˆ , the mean value property of harmonic functions for r = sm and Jensen's inequality yield ˆ 4 1 4 (∇y(t) − R) dt |∇y(x) − R| ≤ |Br (x)| Br (x) ˆ (9.19)  2 −2 2 8 −4 −4 2 ≤ C (sm) |∇y − R| ≤ Cl m s γ(Fˆ ) Fˆ

for all x ∈ F . Consequently, as Fˆ intersects at most nine squares Qk (p), p ∈ I k (Ωk ) \ J , by (9.17) and l = ks we get k∇y − Rk2L∞ (F ) ≤ Cl4 m−2 s−2 · k ≤ Cl−4 ϑ as well as

k∇y − Rk4L4 (F ) ≤ Cϑl−4 k∇y − Rk2L2 (Fˆ ) ≤ Cϑγ(Fˆ ). Proceeding in this way for every connected component F of all Qki (p), p ∈ Iik (Ωk ), and noting that every Qs (q), q ∈ I s (Ωk ), intersects at most four dierent associated enlarged sets Fˆ (Qs (q) can intersect more than one set if it lies at the ˆ i : W ◦ → SO(2) with the deboundary of some Qki (p)) we obtain a function R sired properties (9.16).  We now concern ourselves with the subatomistic regime. 209

Lemma 9.2.2. Let M ≥ 0,  > 0 and s ≤ k ≤ . Then for a xed constant C = C(M ) > 0 we have the following: For all U ∈ Vks with U ⊂ Ωk and for all y ∈ H 1 (U ) with γ as dened in (9.14) and k∇yk∞ ≤ M there is a set W ∈ Vks with W ⊂ Ω3k , |W \U | = 0, |(U \W )∩Ω3k | ≤ Cu kkW k∗ and kW k∗ ≤ kU k∗ + C−1 γ

(9.20)

as well as mappings Rˆ i : Ω3k → SO(2), i = 1, . . . , 4, which are constant on Qki (p) ∩ W , p ∈ Iik (Ωk ), such that ˆ i k2 2 k∇y − R L (W ) ≤ Cγ + CkU k∗ .

(9.21)

Proof. Similarly as in (9.17) we let J ⊂ I k (Ωk ) such that (9.22) H1 (∂U ∩ Qk (q)) + k dist(∇y, SO(2))k2L2 (Qk (q)∩U ) > c∗ k

S S for all q ∈ J . Dene W = Ω3k ∩ U \ p∈J Qk (q) ∪ p∈J ∂Qk (q) and note that the kW k∗ ≤ kU k∗ + C−1 γ for c∗ = c∗ (h∗ ) > 0 suciently large. Indeed, for the subset J1 ⊂ J , for which (9.17) holds, we argue as in the √ previous proof. −1 Then with J2 = J \ J1 we note kW k∞ ≤ kU k∞ + C γ + 2 2k · #J2 and kW kH ≤ kU kH + C−1 γ + 8k · #J2 − c∗ k · #J2 . This gives the desired result for c∗ large. Moreover, we get W ∈ Vks and |(U \ W ) ∩ Ω3k | ≤ Cu kkW k∗ . ˜ ∩ W to Q ˜ by ˜ := Qk (q), q ∈ I k (Ωk ). We extend y from Q Consider some Q i i ˜ and v¯(x) = x for all x ∈ Q ˜ \ W . Note that v¯ ∈ SBV (Q) ˜ setting v¯ = y on W ∩ Q 1 ˜ ˜ with Jv¯ = ∂W ∩ Q. By Theorem A.1.9 we obtain a function v ∈ H (Q) such that by a rescaling argument 2 −1 ˜ ≤ CM k p2 −1 k 1− p1 β p1 ≤ CM  p1 β p1 k∇¯ v − ∇vkLp (Q) k∇¯ v k∞ H1 (Jv¯ ∩ Q) ˜ ≤ Ck p

˜ . In the second step we used β ≤ Ck by (9.22) for p < 2, where β = H1 (∂W ∩ Q) and applied k ≤  in the last step. Consequently, we obtain k dist(∇v, SO(2))kpLp (Q) v , SO(2))kpLp (Q) ˜ ≤ Ck dist(∇¯ ˜ + Cβ. 2 ˜ := k dist(∇¯ Thus, since γ(Q) v , SO(2))k2L2 (Q) , the ˜ = k dist(∇y, SO(2))kL2 (Q∩W ˜ ) rigidity estimate in Theorem B.1 yields a rotation R ∈ SO(2) such that p ˜ 1− p2 γ(Q) ˜ p2 + Cβ k∇v − RkpLp (Q) ˜ ≤ Ck dist(∇v, SO(2))kLp (Q) ˜ ≤ C|Q| p

˜ 2 −1 γ(Q) ˜ + Cβ ≤ C2−p p−2 γ(Q) ˜ + Cβ ≤ C2−p γ(Q) ˜ + Cβ. ≤ Cγ(Q) In the second step we used Hölder's inequality and we applied (9.22) in the ˜ fourth step. This implies k∇y − RkpLp (W ∩Q) v − RkpLp (Q) ˜ ≤ k∇¯ ˜ ≤ Cγ(Q) + Cβ 210

ˆi : and proceeding in this way for all Qki (q), q ∈ Iik (Ωk ), we obtain a function R 3k Ω → SO(2) such that for a constant C = C(h∗ ) ˆ i kp p k∇y − R L (W ) ≤ Cγ + CkU k∗ , ˆ i is constant on Qki (p) ∩ W , p ∈ Iik (Ωk ). Finally, by k∇yk∞ ≤ M we where R derive √ 2−p ˆ i k2 2 ˆ i kp p k∇y − R 2) k∇y − R L (W ) ≤ (M + L (W ) ≤ Cγ + CkU k∗ , as desired.  We recall (7.2) and similarly as in Section 8.4 we dene for shorthand αRˆ (F ) = ˆ : F → SO(2). Applying the link¯ eRˆ (∇y)k2L2 (F ) for F ⊂ R2 and a function R earization formula

dist(G, SO(2)) = |¯ eR (G)| + O(|G − R|2 ) for R ∈ SO(2) and G ∈ R2×2 we get ˆ ˆ ˆ 2 2 ˆ 4. |¯ eRˆ (∇y)| ≤ Cu dist (∇y, SO(2)) + Cu |∇y − R| αRˆ (F ) = F

F

(9.23)

(9.24)

F

Here we already see that it suces to establish a rigidity estimate of fourth order as in Lemma 9.2.1 in order to bound the symmetric part of the gradient. One of the main ideas in the following will be to choose l = l(s, , m) in (9.16) such that ϑ ≤ 1 which will imply αRˆ (W ) ≤ Cu γ .

9.2.2 Estimates in terms of the H1-norm We now show that not only the distance of the derivative from a rigid motion can be controlled as derived in (9.16) and (9.21), respectively, but also the distance of the function itself. Once we have such estimates we will be in a position to `heal' cracks (see Section 9.3 below). After the modication of the deformation ν = sd will stand for the minimal distance of two dierent cracks, where d represents the corresponding increase factor. It will turn out that the least crack length will be given by λ = νm−1 . Moreover, k = λm−1 will denote the size of the cell on which we apply Theorem 8.4.2. Dene [ 5 k 8 Si := Q i (p) k 3k p∈Ii (Ω

)

S and note that Ω5k ⊂ 4i=1 Si . Recall (A.1), (9.12) and the denition m ˆ = C2 (m, h∗ ) (see below (9.13)). We will proceed in two steps similarly as in the proof of Corollary 8.4.7 and Theorem 8.4.8 (see Section P48.4.3). Forˆ shorthand 2 we will write γ(F ) = k dist(∇y, SO(2))kL2 (F ) , δp (F ) = i=1 k∇y − Ri kpLp (F ) for p = 2, 4 and subsets F ⊂ W . 211

Lemma 9.2.3. Let k > s > 0,  > 0 such that l := ks = dm−2 for m−1 , d ∈ N with m−1 , d  1. Let λ = sdm−1 = km. Then for constants C, c > 0 we have the following: sm with W ⊂ Ω3k and for all y ∈ H 1 (W ) with For all W ∈ V(s,3k) γ := k dist(∇y, SO(2)k2L2 (W ) ,

δ4 :=

X4 i=1

ˆ i k4 4 k∇y − R L (W )

for mappings Rˆ i : W ◦ → SO(2), i = 1, . . . , 4, which are constant on the connected components of Qki (p) ∩ W ◦ , p ∈ Iik (Ω3k ), we obtain: sm ˆ We nd sets U ∈ V70k , UQ ∈ V smˆ with U ⊂ UQ ⊂ Ω5k , |UQ \ W | = 0 and |(W \ U ) ∩ Ω5k | ≤ Cu kkU k∗ such that (9.25)

kU k∗ ≤ (1 + Cu m)kW k∗ + C−1 (γ + δ4 )

as well as |Qλ (p) ∩ UQ | ≥ cmλ2

for all p ∈ J(UQ ),

(9.26)

3k where J(UQ ) := {p ∈ I λ (Ω ) : Qλ (p) ∩ UQ 6= ∅}. S Moreover, letting UJ = p∈J(UQ ) Qλ (p), for i = 1, . . . , 4 we nd extensions y¯i ∈ ˜ := Q3λ (p) ∩ UJ , SBV 2 (UJ ∩ Si , R2 ) with y¯i = y on UQ ∩ Si such that for all Q j ˜ ⊂ Si we have that Ri = R ˆ i | ◦ ˜ is constant on p ∈ Ijλ (Ω3k ), j = 1, . . . , 4, with Q W ∩Q ˜ and W◦ ∩ Q

n ˜ 2 ≤ Ck 2 Cm min k, γ(W ∩ Q2k (|E(RiT y¯i − id)|(Q)) i (q)) + δ4 (W ∩

Q2k i (q))

+ |∂W ∩

Q2k i (q)|H

o (9.27) ,

where q ∈ Iik (Ω3k ) such that Q˜ ⊂ Qki (q). Proof. Similarly as in the previous proof we let J ⊂ I 3k (Ω3k ) such that H1 (Q3k (p) ∩ ∂W ) + k dist(∇y, SO(2))k2L2 (Q3k (p)∩W ) X4 ˆ i k4 4 3k + k∇y − R L (Q (p)∩W ) > c∗ k

(9.28)

i=1


ˆ = W \ S Q3k (p) ∪ S ∂Q3k (p) and note that for all p ∈ J . Dene W p∈J p∈J choosing c∗ suciently large and arguing as in the previous proof ˆ k∗ ≤ kW k∗ + C−1 (γ + δ4 ), kW

(9.29)

ˆ ∈ V sm as well as |(W \ W ˆ ) ∩ Ω5k | ≤ Cu kkW ˆ k∗ . We now subsequently W (s,3k) ˆ1 ⊃ . . . ⊃ Uˆ4 (the inclusions hold up to sets of negligible meaconstruct sets U ˆ (Step (I)). sure) by application of Theorem 8.4.2 on connected components of W 212

Afterwards, since in Theorem 8.4.2 the trace estimate cannot be derived for components near the boundary, we will further modify the sets in a neighborhood of large boundary components (Step (II)). A nal modication procedure will then assure property (9.26) (Step (III)). (I) Begin with i = 1 and x q ∈ I1k (Ω3k ). Consider a connected component F ˆ ◦ . As R ˆ 1 = R is constant on F we obtain αR (F ) ≤ C(γ(F ) + δ4 (F )) of Qk1 (q) ∩ W by (9.24). Dene Qµ := Qk1 (q) and recall (9.12). Passing to the closure of F (not relabeled) we can regard F as an element of V sm with respect to Qµ (recall (8.1)), where one component is given by X = Qµ \ H(F ) ∈ U sm . (Observe, however, ˆ .) We apply Theorem that Qµ \ H(F ) may intersect several components of W ˆ 8.4.2 on F ⊂ Qµ for ε = , σ = m to obtain a set G ∈ W sm with |G \ F | = 0 and

kGk∗ + αR (G) ≤ (1 + Cu m)(kF k∗ + αR (F )).

(9.30)

(Recall that the sum in kF k∗ runs only over the boundary components having empty intersection with ∂Qµ .) Moreover, similarly as before we have

|F \ G| ≤ Cu kkGk∗

(9.31)

and using (8.41), (8.54) for all Γt (G) ∈ T (G) := {Γt : N 2ˆτt (∂Rt ) ⊂ H(G)} ˆ |[¯ y1 ](x))|2 dH1 (x) ≤ CCm |Γt (G)|2∞ , (9.32) Θt (G)

where y¯1 is the extension (cf. (8.40)) ( y y¯1 (x) = R (Id + At ) x + R ct

ˆ, x∈W x ∈ Xt for Γt (G) ∈ T (G).

(9.33)

Here recall that ∂Rt are the rectangles dened in (8.5)(i)(v) as well as τˆt = CC2 (m, h∗ )|∂Rt |∞ = C m|∂R ˆ t |∞ for C = C(h∗ ) (see (8.6), (8.33), (8.43) and Remark 8.3.2). We proceed in this way for every connected component (Fj )j of all Qk1 (q), ˆ \ S Fj ) ∪ S Gj ∈ V smˆ . (Observe that one may q ∈ I1k (Ω3k ) and dene Uˆ1 = (W j j have H(Fj1 ) ⊂ H(Fj2 ). In this case the above arguments can be omitted for Fj1 .) ˆ1 ) which do not coincide By G we denote the set of boundary components Γ(U with some Γt (Gj ). Note that by (9.24) and (9.29)

ˆ k∗ + α ˆ (W ˆ )) kUˆ1 k∗ ≤ kUˆ1 k∗ + αRˆ1 (Uˆ1 ) ≤ (1 + Cu m)(kW R1 ≤ (1 + Cu m)kW k∗ + C(γ + δ4 ).

(9.34)

ˆ1 ) ∈ G there is a Γ(W ˆ)= The second step follows as by construction for each Γ(U ˆ \Uˆ1 | ≤ ∂X such that Γ(Uˆ1 ) ⊂ X (recall Remark 8.4.6(i)). By (9.31) we also get |W 213

Cu kkUˆ1 k∗ . Moreover, by Remark 8.4.6(ii) we can replace the components of sm ˆ Gj ∈ V smˆ by rectangles such that the resulting set G0j lies in Vcon . Recall that 0 0 the (rectangular) components of Gj satisfy maxi=1,2 |πi Γ(Gj )| ≤ 2k . ˆ \ S Fj ) ∪ S G0j ∈ V smˆ . We now apply Lemma ˆ100 := (W Then we dene U j j 9.1.2(ii) for ν = 0, (Zj )j the rectangular components of (G0j )j and V 0 the set whose ˆ 0 ∈ V smˆ boundary components are given by the elements of G . We obtain a set U 1 5k 0 00 0 0 ˆ1 k∗ ≤ kUˆ1 k∗ and |Uˆ \ Uˆ | ≤ Cu kkUˆ k∗ . (Strictly speaking, we need to with kU 1 1 1 ˆ ˆ pass from V sm to V sm/2 , but do not include it in the notation for convenience.) 0 ˆ | = 0 and |W ˆ \ Uˆ10 | ≤ Cu kkUˆ10 k∗ . Additionally, we ˆ Likewise we observe |U1 \ W ˆ10 ∈ V smˆ such that (9.3) apply Lemma 9.1.1(ii) and get a (not relabeled) set U 6k ˆ 0 ∈ V smˆ since and (9.34) hold. As in the proof of Lemma 9.2.1 this implies U 1 (s,6k) ˆ ∈ V sm , i.e. the least length of components is bounded from below by s. W (s,3k) ˆ 0 ), In the following, by a slight abuse of notation, we say that a component Γt (U 1 0 0 which coincides with some ∂Xt = Γt (G ) for some component G , satises (9.32) if all corresponding (Γts (G))s with Γts (G) ⊂ Xt satisfy (9.32). It is not hard to see that (9.32) is satised for all boundary components with (recall (9.12)) Γt (Uˆ10 ) ∩ S1 6= ∅,

|Γt (Uˆ10 )|∞ ≤ k8 ,

k

N∗ (Γt (Uˆ1 )) ⊂ H 8 (Uˆ10 ),

ˆ1 )) = {x : dist(x, Γt (Uˆ10 )) ≤ C¯ m|Γ where N∗ (Γt (U ˆ t (Uˆ10 )|∞ } for some large constant ¯ ∗ ) > 0. Indeed, assume that there is some Γs = Γts (G) ⊂ Qk1 (q) such C¯ = C(h that for the corresponding rectangle Rs one has N 2ˆτs (∂Rs ) 6⊂ H(G) although the corresponding Γt (G0 ) = ∂Xt fullls the above three properties. First, we 3

k

observe Rs ⊂ Xt by Remark 8.4.6(ii) and thus Rs ⊂ Q14 (q). By (8.6) we get |∂Rs |∞ ≤ C|Γs |∞ . Consequently, since 2τˆs  C1 |∂Rs |∞ for m ˆ small enough (recall 7

k

(8.33)) we have N 2ˆτs (∂Rs ) ⊂ Q18 (q). Since by assumption N 2ˆτs (∂Rs ) 6⊂ H(G), this would imply |∂H(G) ∩ Qk1 (q)|∞ > k8 . ˆ 0 ))n = (∂Xt (Uˆ 0 ))n Consequently, there is a chain of components (Γti (U 1 i=1 1 i=1 i ˆ10 )∩∂Qµ 6= ∅, Xtn (Uˆ10 )∩N 2ˆτs (∂Rs ) 6= ∅ and Γt (Uˆ10 )∩Γt (Uˆ10 ) 6= ∅. such that Γt1 (U i−1 i ˆ10 ) with |Γ∗ (Uˆ10 )|∞ > k such that N 2ˆτs (∂Rs ) ∩ Thus, by (9.3) there is one Γ∗ (U 8 X∗ (Uˆ10 ) 6= ∅. Recalling that Rs ⊂ Xt and 2ˆ τs < C¯ m|Γ ˆ t (Uˆ10 )|∞ for C¯ suciently ˆ1 )) ∩ X∗ (U10 ) 6= ∅. This, however, is a contradiction to large we nd N∗ (Γt (U k N∗ (Γt (Uˆ1 )) ⊂ H 8 (Uˆ10 ). 0 ˆi−1 We now iteratively repeat the above construction for i = 2, 3, 4 for U instead 4 0 ˆ and obtain extensions y¯2 , y¯3 , y¯4 as well as (Uˆi )i=1 and sets Uˆ4 ⊂ . . . ⊂ Uˆ10 ⊂ of W ˆ (the inclusions hold up to a set of negligible measure) with Uˆ40 ∈ V smˆ W (s,15k) such ˆ1 by Uˆ4 . We briey that (9.34) holds for a possibly larger constant replacing U sm ˆ note that the sets are elements of V due to (8.43) and the fact that the least length of components is bounded from below by s. Moreover, for i = 1, . . . , 4, ˆ 0 ) with Γt (Uˆ 0 ) ∩ Si 6= ∅, (9.32) is satised for y¯i and all boundary components Γt (U i i k |Γt (Uˆi0 )|∞ ≤ k8 and N∗ (Γt (Uˆi0 )) ⊂ H 8 (Uˆi0 ). 214

For later we also observe that due to the local nature of the modication process and (9.30) we get

ˆ ∩ Q2k |∂ Uˆi ∩ Qki (q)|H ≤ C|∂ W i (q)|H
2k ˆ ˆ ∩ Q2k + C−1 γ(W i (q)) + δ4 (W ∩ Qi (q)) .

(9.35)

ˆ 0 )4 only hold up to segments, we observe that the Although the inclusions for (U i i=1 sets are `nested' concerning small boundary components in the following sense: ˆ ∗ = Uˆ 0 ∩ (H k8 (Uˆ 0 ))◦ we obtain Letting U i i i Uˆ4∗ ⊂ . . . ⊂ Uˆ1∗ .

(9.36)

Indeed, assume e.g. there was a component X(Uˆ1∗ ) and components X1 , . . . , Xn S S n n ∗ ∗ ˆ∗ ˆ with X(Uˆ ) ⊂ of U 2 1 j=1 ∂Xj ∩ X(U1 ) 6= ∅. Then by construction j=1 Xj and ˆ1∗ ) 6= ∅, |X(Uˆ1∗ ) \ Xi | > 0 of the sets we clearly nd some Xi with ∂Xi ∩ X(U k and |∂Xi |∞ ≤ 8 . This, however, together with (9.3) gives a contradiction to S k k X(Uˆ1∗ ) ⊂ nj=1 Xj . In particular, (9.36) implies H 8 (Uˆ40 ) ⊂ . . . ⊂ H 8 (Uˆ10 ) up to sets of negligible measure and thus for i = 1, . . . , 4, (9.32) is satised for y¯i and all boundary components Γt (Uˆi0 ) with Γt (Uˆi0 ) ∩ Si 6= ∅, |Γt (Uˆi0 )|∞ ≤ k8 , k N∗ (Γt (Uˆi0 )) ⊂ H 8 (Uˆ40 ). We want to remove the third condition. For that reason, we subtract neighborhoods of large boundary components as follows. k ˆ40 ) and let Γ1 (U ∗ ), . . . , Γn (U ∗ ) be the boundary com(II) Let U ∗ = H 8 (U ∗ ˆ ponents. For Γj (U ) let M (Γj ) be the smallest rectangle in U sm satisfying ¯ m} M (Γj ) ⊃ {x ∈ R2 : dist∞ (x, Xj ) ≤ Ck ˆ for the constant C¯ > 0 introduced above, where Xj denotes the component corresponding component to Γj (U ∗ ). Clearly, using the fact that C¯ m ˆ ≤ m (see (9.13)) one has |πi ∂M (Γj )| ≤ ∗ ∗ |πi Γj (U )| + Cu m|Γj (U )|∞ ≤ 31k for i = 1, 2. As the components (Xj )j are pairwise disjoint and connected, we obtain Z(Γj1 ) \ Z(Γj2 ) is connected for all 1 ≤ j1 , j2 ≤ n, where Z(Γj ) denotes the smallest rectangle containing Xj . Consequently, since the neighborhoods M (Γj ) \ Z(Γj ) all have the same thickness ¯ m ∼ Ck ˆ , we get that M (Γj1 ) \ M (Γj2 ) is connected for all 1 ≤ j1 , j2 ≤ n. S ˆi0 ) Then by Lemma 9.1.2(ii) applied on V = U ∗ , V 0 = Ω5k \ |Γj (Uˆ 0 )|≤ k Xj (U i 8 ˜i with |(Uˆi0 \ Sn M (Γj )) \ U˜i | ≤ Cu kkV 0 k∗ . In particular, we we obtain sets U j=1 ˜ = U˜4 and observe that U˜ ∈ V smˆ . Moreover, we obtain kU˜ k∗ ≤ (1 + set U 32k Cu m)kV k∗ + kV 0 k∗ . As Uˆ40 satises (9.3), we derive (∂V ∩ ∂V 0 ) ∩ (Ω5k )◦ = ∅ and ˜ k∗ ≤ (1 + Cu m)kUˆ40 k∗ , i.e. (9.34) holds replacing Uˆ1 by U˜ (possibly therefore kU for a larger constant). Applying Lemma 9.1.1(ii) we get (not relabeled) sets sm ˆ ˜4 ⊂ . . . ⊂ U˜1 up to sets of U˜i ∈ V33k satisfying (9.3). For later we note that U negligible measure. This follows from (9.36) and the fact that in Lemma 9.1.2(ii) the components of V 0 are replaced by corresponding rectangles. Arguing as in

215

(9.36) we also nd k U˜4∗ ⊂ . . . ⊂ Uˆ1∗ , where U˜i∗ = U˜i ∩ (H 8 (U˜i ))◦

(9.37)

k ˜4 ) ⊂ . . . ⊂ H k8 (U˜1 ) up to sets of negligible In particular, this also implies H 8 (U measure. We now see that for i = 1, . . . , 4, (9.32) holds for y¯i for all components satisfying

Γt (U˜i ) ∩ Si 6= ∅,

|Γt (U˜i )|∞ ≤ 81 k.

(9.38)

ˆi .) In fact, (Strictly speaking (9.32) holds for the corresponding components of U ˆ for |Γt (Uˆi0 )|∞ ≤ k8 , due to the construction of U˜i since C¯ m|Γ ˆ t (Uˆi0 )|∞ ≤ k8 C¯ m ˆ 0 )|∞ ≤ k and N∗ (Γt (Uˆ 0 )) 6⊂ H k8 (Uˆ 0 ) are `combined' with components with |Γt (U i i 4 8 ˆ 0 which is larger than k . a boundary component of U 4 8 ˜ ∈ V smˆ satisfying We apply Lemma 9.1.1(i) to obtain a (not relabeled) set U 33k ˜ ), t = 1, . . . , n, let N1 (Γt ), N2 (Γt ) be the smallest rectangles (9.2). For each Γt (U ˆ in U sm satisfying N1 (Γt ) ⊃ {x ∈ R2 : dist∞ (x, Xt ) ≤ min{Bm|Γt (U˜ )|∞ , 2λ}}, N2 (Γt ) ⊃ {x ∈ R2 : dist∞ (x, Xt ) ≤ Bm min{|Γt (U˜ )|∞ , λ}} for some B > 0 (independent of h∗ ) and λ = km, where Xt is the component ˜ ). It is not restrictive to assume that corresponding to Γt (U


k H1 N2 (Γt ) ∩ (∂ U˜ \ (Γt (U˜ ) ∪ ∂H 8 (U˜ )) ≤ CBm min{|Γt (U˜ )|∞ , λ}

(9.39)

˜ ) with |Γt (U˜ )|∞ ≤ k . Indeed, otherwise we replace U˜ by U˜ 0 := for all Γt (U 8
k U˜ \N2∗ (Γt ) ∪∂N2∗ (Γt ), where N2∗ (Γt ) = (N2 (Γt )∩H 8 (U˜ ))◦ , and arguing similarly ˜ 0 k∗ ≤ kU˜ k∗ . Let (Xt0 )t0 , Xt0 6= Xt , as in (9.28) and Lemma 8.1.1 we get kU ˜ having nonempty intersection with N ∗ (Γt ). Clearly, be the components of U 2 S k ˜ 0 on a we have |∂Xt0 |∞ ≤ 8 . We dene T = N2∗ (Γt ) ∪ t0 Xt0 and modify U ˜ 00 = (U˜ 0 \ T ) ∪ ∂T . Arguing similarly as in set of measure zero by letting U ˜ 00 ∈ V smˆ and kU˜ 00 k∗ ≤ kU˜ k∗ . Then by the proof of Lemma 9.1.1 we nd U 33k ˜ 00 which additionally satises (9.2). Lemma 9.1.1(i) we nd a (not relabeled) set U We continue with this iterative modication process until (9.39) is satised for all components smaller than k8 . Finally, by Lemma 9.1.1(ii) we obtain a (not ˜ 00 ∈ V smˆ satisfying (9.3). Noting that during the modication relabeled) set U 34k procedure components larger than k8 do not become smaller than k8 we also nd k k H 8 (U˜ 00 ) ⊂ H 8 (U˜ ). For convenience the set will still be denoted by U˜ in the following.

216

(III) We now nally construct the sets UQ and U . For each t = 1, . . . , n dene the rectangle [ Zt = Qλ (p). (9.40) λ p∈I (N1 (Γt ))

We nd Zt ⊂ N1 (Γt ) and for suciently small components one has Zt = ∅. Choosing B suciently large we get Xt ⊂ Zt if |∂Xt |∞ > k8 . Rearrange the components in a way that Zt = ∅ for t > n0 . This implies k Ω5k \ H 8 (U˜ ) ⊂

[n0 t=1

Zt .

(9.41)

ˆ Let Yt ∈ U sm be the smallest rectangle containing Zt ∪ Xt . By the denition of N1 (Γt ) and Zt we obtain

|πi ∂Yt | = |πi ∂(Zt ∪ Xt )| ≤ |πi Γt (U˜ )| + Cu m|Γt (U˜ )|∞ , i = 1, 2

(9.42)

for some Cu = Cu (B) large enough. As (Xt )t are pairwise disjoint and connected, it is elementary to see that Zt1 \ Zt2 or Zt2 \ Zt1 is connected for all 1 ≤ t1 , t2 ≤ n0 . In fact, assume there were t1 6= t2 such that π1 Zt2 ⊂ π1 Zt1 and π2 Zt1 ⊂ π2 Zt2 . Then due to the denition of the neighborhoods we nd π1 Xt2 ⊂ π1 Xt1 and π2 Xt1 ⊂ π2 Xt2 . This, however, implies Xt1 ∩Xt2 6= ∅ and yields a contradiction. A similar argument yields that Yt1 \Yt2 or Yt2 \Yt1 is connected for all 1 ≤ t1 , t2 ≤ n0 . ˜ \ Sn0 Zj and let Jˆ ⊂ I λ (Ω3k ) such that (cf. also (9.28)) Dene UQ0 = U j=1

H1 (Qλ (p) ∩ ∂UQ0 ) > c∗ λ

(9.43)

S for all p ∈ Jˆ. Then let UQ = (Ω5k ∩ UQ0 ) \ p∈Jˆ Qλ (p). Observe that possibly sm ˆ UQ ∈ / Vcon . Therefore, we now dene a set U ⊂ UQ with connected boundary components. S 0 S By Lemma 9.1.2(ii) for V = Ω5k \ nt=1 Xt , V 0 = Ω5k \ nt=n0 +1 Xt we obtain ˜ \ Sn0 Yt ) \ U 0 | ≤ Cu kkV 0 k∗ such that U 0 ∈ V smˆ . Moreover, a set U 0 with |(U con t=1 sm ˆ for m recalling (9.42) as well as |∂Xt |∞ ≤ k8 for t > n0 , we get U 0 ∈ V69k ˜ satises (9.3) we have kU 0 k∗ ≤ suciently small. Using (9.42) and the fact that U (1 + Cu m)kV k∗ + kV 0 k∗ ≤ (1 + Cu m)kU˜ k∗ . Finally, again using Lemma 9.1.2(ii) sm ˆ we nd a set U ∈ V70k with  [n0 [n [
 5k λ Yt ∪ X ∪ Q (p) \ U (9.44) Ω \ ≤ Cu kkU k∗ t 0 ˆ t=1

t=n +1

p∈J

˜ k∗ . Arguing similarly as in (9.22), (9.28) we nd kU k∗ ≤ kU 0 k∗ ≤ (1 + Cu m)kU 5k ˜ satises (9.34). Moreover, we derive |(W \ U ) ∩ Ω | ≤ This implies (9.25) since U Cu kkU k∗ . ˜i ) = ∂Xt Dene UJ as in the assertion of Lemma 9.2.3. We see that all Γt (U k k ◦ ˜i ) ∩ U 6= ∅ satisfy |Γt (U˜i )|∞ ≤ . In fact, if |Γt (U˜i )|∞ > , we would with Γt (U J 8 8 217

k ˜i ))◦ and thus Xt ⊂ Ω5k \ (H k8 (U˜ ))◦ , where we used have Xt ⊂ Ω5k \ (H 8 (U k k k k H 8 (U˜ 00 ) ⊂ H 8 (U˜ ) ⊂ H 8 (U˜4 ) ⊂ H 8 (U˜i ) up to a set of negligible measure (see ˜ 00 given by the modication described below (9.39) (9.37)). (Recall that the set U ˜ for convenience.) Therefore, by (9.41) we get Γt (U˜i ) ⊂ Xt ⊂ is also denoted by U Sn0 ◦ ˜ j=1 Zj and thus Γt (Ui ) ∩ UJ = ∅ giving a contradiction. Consequently, by (9.38)

˜i ) with Γt (U˜i ) ∩ UJ◦ ∩ Si 6= ∅. (9.32) holds for y¯i for all Γt (U

(9.45)

For later we recall that the corresponding components (Γts (G))s with Γts (G) ⊂ ˜ := Xt (U˜i ) (which satisfy (9.32)) also satisfy (8.5) since G ∈ W smˆ . Consider Q 3λ ˜ Qj (p)∩UJ ⊂ Si . We observe that Q consists of a bounded number of squares and k ˆ◦ ˜ that Q∩U Q is contained in a connected component F of Qi (q)∩ W . Indeed, this follows from the fact that due to the construction of UQ , in particular (9.40), two connected components F1 6= F2 , Ft ∩ Si 6= ∅ for t = 1, 2, for which H(Ft ) is not completely contained in another component H(Ft0 ), fulll dist(F1 ∩UJ , F2 ∩UJ ) ≥ ˜ ◦ is connected, i.e. each Q ⊂ Q ˜ shares 2λ. This observation also implies that Q ˜ . Consequently, Corollary 8.4.7 together with at least one face with the rest of Q (9.32) yield

˜ + CkCm |∂ Uˆi ∩ Qk (q)|2 , ˜ 2 ≤ Cλ2 αR (UQ ∩ Q) (|E(RiT y¯i − id)|(Q)) i H i ˆ i |F . Then (9.35) and (9.28) imply where Ri is the value of the constant function R |∂ Uˆi ∩ Qki (q)|H ≤ Ck which together with (9.24) yields (9.27). For later we note that Corollary 8.4.7 also yields ˜ 2 ≤ Cλ2 αR (UQ ∩ Q) ˜ + CkCm |∂ Uˆi ∩ Qk (q)|2 . (|Dj (¯ yi − Ri id)|(Q)) i H i

(9.46)

ˆ = Qλ (p) with Q ˆ ∩ UQ 6= ∅ and show that It remains to show (9.26). Consider Q ˆ ∩ UQ | ≥ cmλ2 . First note that Q ˆ ∩ UQ = Q ˆ ∩ U˜ . Let Γ = Γ(U˜ ) = ∂X be the |Q ˆ ∞ . If |Γ|∞ ≥ k we get a contradiction for boundary component maximizing |X ∩Q| 8 ˆ ∩ UJ = ∅. Assume |X ∩ Q| ˆ ∞  λ. Then (9.43) and the B large enough as then Q ˆ Q | ≤ Cu P |Xt (U˜ )∩ Q| ˆ 2∞  Cu λ P |Xt (U˜ )∩ isoperimetric inequality imply |Q\U t t ˆ ∞ ≤ Cu λ2 and thus |Q ˆ ∩ UQ | ≥ cmλ2 for m small enough. Therefore, we may Q| assume that 1 k 8

ˆ ∞ ≥ c¯λ = 18 m−1 λ ≥ |Γ|∞ ≥ |X ∩ Q|

(9.47)

ˆ ≥ CBm¯ for c¯ > 0 small enough. It is not hard to see that |(N2 (Γ) \ X) ∩ Q| c2 λ2 . ˆ ≥ CBm¯ Indeed, an elementary argument yields |N2 (Γ) ∩ Q| c2 λ2 . Moreover, if we 2 2 ˆ \ X|  Bm¯ ˆ ⊂ N1 (Γ) and thus Q ˆ ∩ UQ = ∅ by the had |Q c λ , we would get Q k ˜ ) = ∅ since otherwise construction of UQ . We can assume that N2 (Γ) ∩ ∂H 8 (U k ˆ and we derive Q ˆ ∩ UJ = ∅ as before. a component larger than 8 intersects Q ˜ ) with Xj (U˜ ) ∩ N2 (Γ) 6= ∅ By (9.2) this also implies that all components Xj (U 218

˜ ) ∩ Γ = ∅. Thus by the isoperimetric inequality and by (9.39) we get satisfy Xj (U ˆ ∩ U˜ | ≥ |(N2 (Γ) \ X) ∩ Q| ˆ − C(Bλm)2 . This implies |N2 (Γ) ∩ Q ˆ ∩ UQ | = |Q ˆ ∩ U˜ | ≥ |Q ˆ ∩ U˜ ∩ N2 (Γ)| ≥ |(N2 (Γ) \ X) ∩ Q| ˆ − C(Bλm)2 |Q ≥ −CB 2 λ2 m2 + C¯ c2 Bmλ2 ≥ cmλ2 for m suciently small.

Remark 9.2.4.



λ with (i) For later we observe that there is a set U H ∈ V35k

(i) kU H k∗ ≤ (1 + Cu m)kW k∗ + C−1 (γ + δ4 ), (ii) kU H kH ≤ Cu kU H k∗ (9.48) which coincides with the set UJ considered in the previous lemma up to a set of 0 negligible measure. In fact, we apply Lemma 9.1.2(i) on the rectangles (Zt )nt=1 S 0 S 0 0 considered in (9.40) and nd pairwise disjoint (Zt0 )nt=1 with nj=1 Zj = nj=1 Zj0 . We dene  [n0  [ H 5k 0 λ U := Ω \ Zj ∪ Q (p) , ˆ j=1

p∈J

λ where Jˆ as in (9.44). By Lemma 9.1.2(i) we get (i) and U H ∈ V35k since |πi ∂Zt | ≤ 2 · 34k + Cu mk ≤ 70k for i = 1, 2. Moreover, (9.48)(ii) is a consequence of Lemma 8.1.1 and the fact that (Zt )t , (Qλ (p))p∈J are rectangles. Clearly UJ ⊂ U H . Moreover, we see that |U H \ UJ | > 0 can only happen if ˜ ))t of U˜ such that Qλ (p) ⊂ there is a square Qλ (p) ⊂ U H and components (Xt (U S k ˜ ˜ t Xt (U ). Since we can suppose |∂Xt (U )|∞ ≤ 8 (otherwise the components are contained in some rectangle Zt ), this yields a contradiction to (9.2). (ii) For i = 1, . . . , 4 we have

|∂ Uˆi ∩ UJ◦ |H ≤ Cu (kW k∗ + C−1 (γ + δ4 )). ˆi ) with Γt (Uˆi ) ∩ U ◦ 6= ∅ fulll (9.32) In fact, recalling (9.45) we get that all Γt (U J ˆi )|H ≤ Cu |Θt (Uˆi )|∗ and the claim follows from and (8.5). Thus, we obtain |Θt (U ˆ1 by Uˆi . (9.34) replacing U We are now in a position to prove the main result of this section. Recall the denition λ = sdm−1 = km and (9.13).

Lemma 9.2.5. Let k > s,  > 0 such that l := ks = dm−2 for m−1 , d ∈ N with m−1 , d  1. Then for a xed constant C > 0 we have the following: sm For all W ∈ V(s,3k) with W ⊂ Ω3k and for all y ∈ H 1 (W ) with k∇yk∞ ≤ C , γ as dened in (9.14) and δ4 :=

X4 i=1

ˆ i k4 4 , k∇y − R L (W )

δ2 :=

X4 i=1

ˆ i k2 2 k∇y − R L (W )

(9.49)

for mappings Rˆ i : W ◦ → SO(2), i = 1, . . . , 4, which are constant on the connected components of Qki (p) ∩ W ◦ , p ∈ Iik (Ω3k ), we obtain: 219

sm ˆ2 We nd sets V ∈ V71k , UJ ∈ V λ with V ⊂ UJ and V ⊂ Ω6k , |V \ W | = 0, 6k |(W \ V ) ∩ Ω | ≤ Cu kkV k∗ such that

kV k∗ ≤ (1 + Cu m)kW k∗ + C−1 (γ + δ4 )

(9.50)

as well as mappings R¯ j : UJ → SO(2) and c¯j : UJ → R2 , which are constant on Qλj (p), p ∈ Ijλ (Ω3k ), such that 2 2 ¯ j · +¯ (i) ky − (R cj )k2L2 (V ) ≤ CCm λ minp=2,4 (1 + ϑp )(γ + δp + kW k∗ ),
p ¯ j k p ≤ CC 2 δp + ϑp (γ + δ4 + kW k∗ ) , p = 2, 4, (ii) k∇y − R (9.51) m L (V )
p 2 ¯ j1 − R ¯ j2 k p (iii) kR L (UJ ) ≤ CCm δp + ϑp (γ + δ4 + kW k∗ ) , p = 2, 4, ¯ j1 · +¯ ¯ j2 · +¯ (iv) k(R cj 1 ) − ( R cj2 )k2 2 ≤ CC 2 λ2 min (1 + ϑp )(γ + δp + kW k∗ ) L (UJ )

m

p=2,4

for j1 , j2 = 1, . . . , 4, j = 1, . . . , 4, where ϑ4 = ϑ and ϑ2 = 1. Moreover, we have ¯ ¯ j1 · +¯ ¯ j2 · +¯ ¯ j1 − R ¯ j2 k4 ∞ λ−2 k(R cj1 ) − (R cj2 )k2L∞ (UJ ) + kR L (UJ ) ≤ C ϑ

(9.52)

for ϑ¯ = min{ϑ(1 + ϑ), Cm3 } and under the additional assumption that ∆y = 0 in W ◦ we obtain ¯ j · +¯ λ−2 ky − (R cj )k2L∞ (V ) ≤ Cϑ(1 + ϑ).

(9.53)

sm ˆ ˆ Proof. Apply Lemma 9.2.3 to obtain U ∈ V70k , UQ ∈ V sm with |UQ \ W | = 0,

UJ and extensions y¯i : Si ∩ UJ → R2 such that (9.25), (9.26) and (9.27) hold. Consider Q = Qλj (p), p ∈ Ijλ (Ω3k ), j = 1, . . . , 4, with Q ∩ UJ 6= ∅. Moreover, ˜ = Q3λ (p) ∩ UJ . As 6λ < k by m  1, we nd some Qk (q) for some let Q i j 4 5 k ˜ ⊂ Q 8 (q) ⊂ Si and therefore we can apply (9.27). Recall i = 1, . . . , 4 with Q i ˆ ˆ that R := Ri |W ◦ ∩Q˜ is constant due to the construction in Lemma 9.2.3 (see below 2×2 (9.45)). By Theorem B.4 we nd A ∈ Rskew and c ∈ R2 such that ˆ (Id + A) · −R ˆ ck2 2 ˜ = kR ˆ T y¯i − · − (A · +c)k2 2 ˜ k¯ yi − R L (Q) L (Q) ˆ T y¯i − id)|(Q)) ˜ 2 ≤ Ck 2 G, ≤ C(|E(R

(9.54)

where

n o 2k 2k G := Cm min k, γ(W ∩ Q2k (q)) + δ (W ∩ Q (q)) + |∂W ∩ Q (q)| 4 H . i i i ˜ as there are (up to rescaling) only a nite The constant C is independent of Q ˜ ˜ shares at least one number of dierent shapes of Q. (Also recall that each Q ⊂ Q ˜ .) face with the rest of Q

220

˜i ) with Q ˜ ∩ Γt 6= ∅ In the proof of Lemma 9.2.3 we have seen that all Γt = Γt (U k τˆl k satisfy (9.32) for y¯i and |Γt |∞ ≤ 8 as well as N (∂Rt ) ⊂ Qi (q) (cf. (9.45)). Thus, by Lemma 8.5.7 for V = Qki (q) we get X ˆ p p ≤ Ck∇y − Rk ˆ p k∇¯ yi − Rk + C |Xt |2∞ |At |p p ˜ ˜ ˆ L (Q) L (Q∩W ) Γt ∈F (Qki (q)) (9.55) k −1 p2 −1 k ˆ ˆ ≤ CCm δp (Qi (q) ∩ W ) + CCm (s ) |∂ Ui ∩ Qi (q)|H ˆ , Uˆi as dened in the previous proof and Xt , At as in (9.33). for p = 2, 4, where W Recall that the factor s−1 appearing in the estimate is related to the fact that the ˆi is s. Thus, recalling that Uˆi fullls least length of boundary components of U (9.35) we obtain by the denition of G p

ˆ p p ≤ CCm δp (Qki (q) ∩ W ˆ ) + C(s−1 ) 2 −1 G =: Hp . k∇¯ yi − Rk ˜ L (Q)

(9.56)

We repeat the estimate (9.54) with Theorem B.3 instead of Theorem B.4 and obtain by (9.46) and Hölder's inequality

ˆ · −˜ ˆ 2 1 ˜ + C(|Dj (¯ ˆ id)|(Q)) ˜ 2 k¯ yi − R ck2L2 (Q) yi − Rk yi − R ˜ ≤ Ck∇¯ L (Q) 1

2

≤ Cλ4(1− p ) Hpp + Ck 2 G, for c˜ ∈ R2 for p = 2, 4. This together with (9.54) and an argumentation similar 2/p to (7.10) yields λ4 |A|2 ≤ Cλ4−4/p Hp + Ck 2 G and therefore by (9.56)

ˆ ) + Cm−2 G =: H ˆ 2, λ2 |A|2 ≤ CH2 + Cm−2 G ≤ CCm δ2 (Qki (q) ∩ W λ2 |A|4 ≤ CH4 + Cλ−2 m−4 G2 ≤ CH4 + Cλ−1 m−5 Cm G ˆ ) + CϑG =: H ˆ 4. ≤ CCm δ4 (Qki (q) ∩ W

(9.57)

ˆ 4 ≤ C(1 + ϑ)G. By (9.23) there is a rotation R ¯ ∈ SO(2) such Observe that H that ¯ − R(Id ˆ ˆ |R + A)|2 = dist2 (R(Id + A), SO(2)) ˆ ˆ 4 = C|A|4 ≤ Cλ−2 H ˆ 4, ≤ 0 + C|R(Id + A) − R|

(9.58)

ˆ ¯− as e¯Rˆ (R(Id + A)) = 0. Likewise, as |A| ≤ C by k∇yk∞ ≤ C we get |R ˆ ˆ 2 . Consequently, the Poincaré inequality, (9.54) R(Id + A)|2 ≤ C|A|2 ≤ Cλ−2 H and (9.57) yield 2 4 4 2 2 ˆ ¯ · +¯ k¯ yi − (R c)k2L2 (Q) ˜ ≤ Ck G + Cλ |A| ≤ Ck G + Ck minp=2,4 Hp

(9.59)

for some possibly dierent c¯ ∈ R2 . Moreover, we get

ˆ − R| ¯ 4 ≤ Cλ2 |R ¯ − R(Id ˆ λ2 |R + A)|4 + Cλ2 |A|4 ¯ − R(Id ˆ ˆ 4. ≤ Cλ2 |R + A)|2 + Cλ2 |A|4 ≤ C H 221

(9.60)

and likewise

ˆ − R| ¯ 2 ≤ CH ˆ 2. λ2 |R

(9.61)

For xed j = 1, . . . , 4 we proceed in this way on each Qt = Qλj (p), p ∈ Ijλ (Ω3k ), ˜ t = Q3λ (p)∩UJ we obtain constants with Qt ∩UJ 6= ∅ and for the corresponding Q j ˆt, R ¯ t ∈ SO(2) and c¯t ∈ R2 as given in (9.59)-(9.61). Consequently, we nd R ¯ j : UJ → SO(2) and c¯j : UJ → R2 being constant on each Qt , where mappings R ˜ t we choose R ¯j = R ¯ t and c¯j = c¯t . By (9.59) and the observation on each Qt ⊂ Q 2k ˜ t we obtain that every Qi (q) is intersected only by ∼ m−2 squares Q

¯ j · +¯ ky − (R cj )k2L2 (U ) ≤ Ck 2 min (1 + ϑp )m−2 Cm m−2 (γ + δp + kW k∗ ) p=2,4

2

2 ≤ Cλ minp=2,4 (1 + ϑp )m2 Cm (γ + δp + kW k∗ )

(9.62)

where ϑ2 = 1 and ϑ4 = ϑ. Here we used that δ4 ≤ Cδ2 . Likewise, applying (9.49), (9.57), (9.60), (9.61) as well as the triangle inequality we get
¯ j kp p ≤ Cm−2 Cm δp + m−2 ϑp (γ + δ4 + kW k∗ ) k∇y − R L (U ) (9.63)
2 ≤ CmCm δp + ϑp (γ + δ4 + kW k∗ ) for p = 2, 4. We now consider Q1 := Qλj1 (p1 ), Q2 := Qλj2 (p2 ) with Q1 ∩ Q2 6= ∅ ˜ i = Q3λ and Q1 , Q2 ∩ UJ 6= ∅. Moreover, let Q ji (pi ) ∩ UJ be the corresponding enlarged sets. It is not hard to see that there is some Qλ (p), p ∈ J(UQ ), with ˜ 1, Q ˜ 2 and therefore by the denition of UJ , in particular (9.26), we Qλ (p) ⊂ Q ˜1 ∩ Q ˜ 2 ∩ UQ | ≥ cmλ2 . Let R ¯ j ∈ SO(2), c¯j ∈ R2 , i = 1, 2, be the derive |Q i i constants constructed above. We compute −1 ¯ ¯ j1 − R ¯ j2 k p ∞ ¯ p λ2 k R ˜ 1 ∩Q ˜ 2 ∩UQ ) L (Q1 ∩Q2 ) ≤ Cm kRj1 − Rj2 kLp (Q X4 ¯ j kp p ≤ Cm−1 k∇y − R ˜ ∩Q ˜ L (Q j=1

(9.64)

2 ∩UQ )

1

and summing over all squares we get by (9.63) 2 ¯ j1 − R ¯ j2 k p p kR L (UJ ) ≤ CCm δp + ϑp (γ + δ4 + kW k∗ )




(9.65)

for 1 ≤ j1 , j2 ≤ 4 and p = 2, 4. Here we used that each Q3λ j (p) ∩ UJ only appears |π1 (Q1 ∩Q2 )|+|π2 (Q1 ∩Q2 )| −1/2 in a nite number of addends. Note that max and ˜ ∩Q ˜ ∩U )| ≤ Cm |π (Q i=1,2

|Q1 ∩Q2 | ˜ 1 ∩Q ˜ 2 ∩UQ | |Q

i

1

2

Q

≤ Cm−1 . Consequently, arguing similarly as in (7.12) we nd

¯ j1 · +¯ ¯ j2 · +¯ λ2 k(R cj1 ) − (R cj2 )k2L∞ (Q1 ∩Q2 ) 1 ¯ j1 · +¯ ¯ j2 · +¯ ≤ C(m− 2 )2 m−1 k(R cj1 ) − (R cj2 )k2L2 (Q˜ 1 ∩Q˜ 2 ∩UQ ) .

222

(9.66)

(R2×2 skew may be replaced by SO(2), see Remark 7.1.4(iii).) Replacing (9.63) by (9.62) in the above argument we then get X4 ¯ j1 · +¯ ¯ j2 · +¯ ¯ j · +¯ k(R cj1 ) − (R cj2 )k2L2 (UJ ) ≤ Cm−2 ky − (R cj )k2L2 (UQ ) j=1

≤

2 2 CCm λ

minp=2,4 (1 + ϑp )(γ + δp + kW k∗ ). (9.67)

Similarly as in the proof of Lemma 9.2.1 (see the construction in (9.18)) we can sm ˆ2 dene V ∈ V71k with |V \ U | = 0, V ◦ ⊂ {x ∈ U ∩ Ω6k : dist∞ (x, ∂U ) ≥ 2smm} ˆ , 6k kV k∗ ≤ (1 + Cu m)kU k∗ and |(W \ V ) ∩ Ω | ≤ Cu kkV k∗ . By (9.25) this implies (9.50). We note that in this case for components Γj = ∂Xj with Xj ⊂ UJ it suces to consider a corresponding rectangle M (Γj ) with M (Γj ) ⊂ UJ . For later we observe that this construction yields 6k [
Ω \ M (Γj ) 4V = 0. (9.68) V ⊂ UJ , We now see that (9.51) follows directly from (9.62)-(9.67). It remains to show (9.52) and (9.53). By (9.57), (9.60) and (9.64) we nd −2 −2 −1 ¯ ¯ j2 k4 ∞ kRj1 − R L (Q1 ∩Q2 ) ≤ Cλ (1 + ϑ)G + Cλ m G for sets Q1 , Q2 ⊂ UJ as considered above. Recalling the denition of G we then get 2 ¯ j1 − R ¯ j2 k4L∞ (Q ∩Q ) ≤ C(1 + ϑ)λ−2 m−1 Cm k ≤ Cs−1 (1 + ϑ)Cm kR  ≤ C(1 + ϑ)ϑ 1 2

¯ j1 ·+¯ ¯ j2 ·+¯ Likewise, we derive λ−2 k(R cj1 )−(R cj2 )k2L∞ (Q1 ∩Q2 ) ≤ C(1+ϑ)ϑ recalling the denition of G and taking (9.66), (9.59) (for p = 4) and the triangle inequality into account. Similarly, by (9.59) for p = 2 and the observation that δ2 (Qki (q) ∩ ˆ ) ≤ Ck 2 as k∇yk∞ ≤ C we nd using  ≤ k W ¯ j1 · +¯ ¯ j2 · +¯ ˆ 2) λ−2 k(R cj 1 ) − ( R cj2 )k2L∞ (Q1 ∩Q2 ) ≤ Cλ−4 m−2 k 2 (G + H 2 2 3 ≤ Cλ−2 m−4 (m−2 G + Cm k 2 ) ≤ Cλ−2 Cm k ≤ CCm .

This nishes the proof of (9.52). ˜ Finally, to see (9.53), we repeat the argument in (9.19): Let x ∈ Q ∩ V ⊂ Q λ 3λ ˜ = Qj (p) ∩ UJ as considered above and let R ¯ · +¯ for Q = Qj (p) , Q c be the corresponding rigid motion as given in (9.59). Since y is assumed to be harmonic in U ◦ the mean value property of harmonic function for r ≤ smm ˆ and Jensen's inequality yield ˆ 2 1 2 ¯ ¯ (y(t) − (R t + c¯)) dt |y(x) − (R x + c¯)| ≤ |Br (x)| Br (x)

≤ C|Br (x)|−1 (1 + ϑ)k 2 G ≤ C(1 + ϑ)m−2 m ˆ −2 s−2 k 2 G ≤ C(1 + ϑ)Cm m−4 m ˆ −2 ls−1 λ2 ≤ C(1 + ϑ)ϑλ2 . ˜ for all x ∈ Q ∩ V . Here we used (9.59) and the fact that Br (x) ⊂ U ◦ ∩ Q 223



9.2.3 Local rigidity for an extended function We now state a version of Lemma 9.2.5 for an extension of the function y .

Corollary 9.2.6. Let be given the assumptions of Lemma 9.2.3, Lemma 9.2.5 λ sm ˆ be the sets provided by Lemma 9.2.3, Remark 9.2.4, , U H ∈ V35k and let U ∈ V70k respectively. Moreover, assume that ϑ ≤ 1. Then the estimates (9.51)(iii),(iv) hold on U H for functions R¯ j , c¯j , j = 1, . . . , 4. Moreover, we nd an extension yˆ ∈ SBV 2 (U H , R2 ) with yˆ = y on U and ∇ˆ y ∈ SO(2) on U H \ W a.e. such that for every Q = Qλj (p), p ∈ Ijλ (Ω3k ), with Q ∩ U H 6= ∅ we have 2 ¯ j kp p ≤ CCm ¯ ) + δp (N )), p = 2, 4 (i) k∇ˆ y−R (G(N L (Q) 2 ¯ )}, ¯ j · +¯ min{k, G(N (ii) kˆ y − (R cj )k2 2 ≤ Cλ2 Cm

¯j · (iii) kˆ y − (R

L (Q) +¯ cj )k2L1 (∂Q)

2

≤ Cλ

2 Cm

(9.69)

¯ )}, min{k, G(N

¯ ) = where N = N (Q) = {x ∈ W : dist(x, Q) ≤ Ck} and for shorthand G(N 1 γ(N ) + δ4 (N ) + H (N ∩ ∂W ). Furthermore, we have H1 (Jyˆ) ≤ Cu (kW k∗ + C−1 (γ + δ4 )).

(9.70)

Proof. Recall the denition of U in (9.44) and that UJ and U H coincide up to a set

˜ j )4 , of measure zero by Remark 9.2.4. In Lemma 9.2.3 we have dened sets (U j=1 U˜4∗ ⊂ . . . ⊂ U˜1∗ (see (9.37)) and corresponding extensions y¯i |UJ ∩Si . Moreover, in ˜i ) with Γt (U˜i ) ∩ U ◦ ∩ Si 6= ∅ satisfy (9.32) for y¯i and (9.45) have seen that all Γt (U J |Γt (U˜i )|∞ ≤ k8 . By Lemma 9.2.5 we get that (9.51)(iii),(iv) hold. The goal is to provide one single extension yˆ : U H → R2 and to conrm (9.69). Dene [ 9 k Q 16 (p) ⊂ Si Sˆi := p∈Iik (Ω3k )

i

˜ ˜ ˜i ∩ U ◦ ) ∪ and let Di = (U ˜i )⊂Sˆi Xt (Ui ), where Xt (Ui ) is the component corJ Γt (U S 4 ˜i ). We now show that U ◦ ⊂ responding to Γt (U J i=1 Di . To see this, it suces to prove [4 Si ∩ UJ◦ ⊂ Dn , i = 1, . . . , 4. (9.71) S

n=1

Fix i and assume thatS(9.71) has already be established for j > i. As Si ∩ UJ◦ ⊂ Ω5k ⊂ H(U˜i ) = U˜i ∪ Γt (U˜i ) Xt (U˜i ) by the denition of UJ , we nd (Si ∩ UJ◦ ) \ S Di ⊂ (Si ∩ UJ◦ ) ∩ Γt (U˜i )6⊂Sˆi Xt (U˜i ). To see (9.71) for i, it now suces to show ˜i ) with Γt (U˜i ) ∩ U ◦ ∩ Si 6= ∅ satises U ◦ ∩ Xt (U˜i ) ⊂ S4 Dn . that each Γt (U J J n=1 ˜i )|∞ ≤ k for all such components, we derive Xt (U˜i ) ⊂ Sˆj for some Since |Γt (U 8 j = 1, . . . , 4. If j < i, by the construction of the sets U˜1∗ ⊃ . . . ⊃ U˜4∗ we nd (Xts (U˜j ))s such that [ Xt (U˜i ) = (U˜j ∩ Xt (U˜i )) ∪ Xts (U˜j ). s

224

˜j ) ⊂ Sˆj , this implies Xt (U˜i ) ∩ U ◦ ⊂ Dj . The case j = i is clear. If j > i, As Xts (U J ˜i ) ∩ U ◦ ⊂ Sj ∩ U ◦ ⊂ S4 Dn by (9.71). This yields the claim. we obtain Xt (U J J n=1 Set y¯ = y¯4 on D4 ∩ UJ , y¯ = y¯j on (Dj \ Dj+1 ) ∩ UJ for j = 3, 2, 1. It is not hard to see that y¯ is dened on U H (as |U H \ UJ◦ | = 0) and y¯ = y on U . Moreover, by ˆi )i construction there is a set of components (Xt )t consisting of components of (U such that [ [4 [ Jy¯ ⊂ ∂Xt ⊂ Γt (Uˆi ). t

i=1

t

By (9.33) we have y¯(x) = y¯it (x) = Rt (Id + At ) x + Rt ct for x ∈ Xt , where 2 Rt ∈ SO(2), At ∈ R2×2 skew , ct ∈ R and 1 ≤ it ≤ 4 appropriately. Note that due the the denition of the extensions in (9.33) the components Xt are associated ˆi )i , not to (U˜i )i . By Remark 9.2.4(ii) this yields (9.70) for y¯. to the sets (U ˜ = Q3λ (p) ∩ UJ and observe Consider Q = Qλj (p) with Q ∩ UJ 6= ∅. Let Q j 2 ˜ |Q ∩ UJ | ∼ λ . Let I ⊂ {1, . . . , 4} such that for each ι ∈ I we can select some 5 ˜ ⊂ Qι8 k (qι ). Note that #I > 1 is possible. It is not hard to see Qkι (qι ) such that Q that for all Xt with Xt ∩ Q 6= ∅ we get it ∈ I . This follows from the construction ˜ 6⊂ Sι implies Q ˜ ∩ Sˆι = ∅ as λ  k . Following of the sets (Di )i and the fact that Q ˆ ¯ ι ∈ SO(2), c¯ι ∈ R2 the lines of (9.56), (9.59)-(9.61) and using H4 ≤ CG we nd R such that 2 ¯ ι · +¯ k¯ yι − (R cι )k2L2 (Q) ˜ ≤ Ck G,

¯ ι kp p ≤ C H ˆp k∇¯ yι − R ˜ L (Q)

(9.72)

for ι ∈ I . Note that for a special choice of ι ∈ I (for ι = i with i as considered ¯ j x + c¯j which we dened in Lemma in (9.54).) we obtain the rigid motion R 9.2.5. Then arguing as in (9.64) and (9.66), in particular employing the triangle inequality and using (9.72), we derive −2 2 ¯ j · +¯ ¯ ι · +¯ k(R cj ) − ( R cι )k2L2 (Q) ˜ ≤ Cm k G,

¯j − R ¯ ι kp p ≤ Cm−1 H ˆp kR ˜ L (Q)

(9.73)

for ι ∈ I . Likewise we obtain by (9.32) ˆ Xˆ X 2 1 |[¯ y ]| dH ≤ C |[¯ yι ]|2 dH1 ≤ C kCm |∂ Uˆι ∩ Qkι (qι )|H . (9.74) Jy¯ ∩Q

ι∈I

Jy¯ι ∩Q

ι∈I

Here we used that all Xt with Q∩Xt 6= ∅ satisfy |∂Xt |∞ ≤ k8 and thus Xt ⊂ Qkι (q). Now we obtain X ¯ j kp p ≤ ¯ j kp p k∇¯ y−R k∇¯ yι − R L (Q) L (Q) ι∈I X
¯ ι kp p + kR ¯ι − R ¯ j kp p ≤C k∇¯ yι − R L (Q) L (Q) ι∈I

for p = 2, 4. Choosing the constant in the denition of N suciently large and ˆ p (see (9.57)) we obtain by (9.72) and (9.73) recalling the denition of G and H

¯ j kp p ≤ CC 2 (γ(N ) + δp (N ) + |∂W ∩ N |H ). k∇¯ y−R m L (Q) 225

Similarly, recalling λ = mk we derive 2 ¯ j · +¯ k¯ y − (R cj )k2L2 (Q) ≤ Cλ2 Cm min{(γ(N ) + δ4 (N ) + |∂W ∩ N |H ), k}.

Consequently, (9.69)(i),(ii) hold for y¯. For later purposes, it is convenient to have an extension satisfying ∇ˆ y (x) ∈ SO(2) for a.e. x ∈ U H \ W . Arguing as in (9.58) for all components Xt we nd ˜ t ∈ SO(2) such that |R ˜ t − (Rt + Rt At )|2 ≤ C|At |4 . Therefore, by Poincaré's R inequality we nd for some possibly dierent c˜t ∈ R2

˜ t · +˜ kR ct − (Rt (Id + At ) · +Rt ct )k2L2 (Xt ) ≤ C|∂Xt |2∞ |Xt ||At |4

(9.75)

for all Xt and likewise passing to the trace (e.g., argue as in (7.10).) we get

˜ t · +˜ kR ct − (Rt (Id + At ) · +Rt ct )k2L2 (∂Xt ) ≤ C|∂Xt |2∞ |∂Xt |H |At |4 . In particular, note the the constants above do not depend on the shape of Xt as ˜ t x + c˜t for the involved functions are ane. We set yˆ : U H → R2 by yˆ(x) = R 1 x ∈ Xt and yˆ = y else. First, we see that (9.70) holds since H (Jy¯) = H1 (Jyˆ). The denition together with (9.74) yields ˆ ˆ X 2 1 |[ˆ y ]| dH ≤ |[¯ y ]|2 dH1 + C |∂Xt |2∞ |∂Xt |H |At |4 X ∩Q6 = ∅ t Jy¯ ∩Q J ∩Q ˆ y¯ X |[¯ y ]|2 dH1 + Ck |∂Xt |2∞ |At |4 ≤ Xt ∩Q6=∅

Jy¯ ∩Q

≤ CCm k

X4 ι=1

|∂ Uˆι ∩ Qkι (qι )|H .

In the second step we used |∂Xt |H ≤ Ck which follows from (9.35) and (9.28). In the last step we used Lemma 8.5.7 similarly as in P the derivation of (9.55) and ˆι ∩ Qk (qι )|H ≤ Ck , employed s ≥ . Using once more that |Jy¯ ∩ Q|H ≤ 4ι=1 |∂ U ι Hölder's inequality and (9.35) yield ˆ ˆ 2 1 |[ˆ y ]| dH ≤ |Jy¯ ∩ Q|H · |[ˆ y ]|2 dH1 Jy¯ ∩Q

Jy¯ ∩Q

≤ CCm k 2

X4 ι=1

2 2 λ min ≤ CCm



|∂ Uˆι ∩ Qkι (qι )|H .

(9.76)


γ(N ) + δ4 (N ) + |∂W ∩ N |H , k .

˜ t − (Rt + Rt At )|2 ≤ C|At |4 , |At | ≤ C and again using (9.55), (9.35) Recalling |R we obtain X k∇¯ y − ∇ˆ y kpLp (Q) ≤ C |∂Xt |2∞ |At |4 ≤ CCm (γ(N ) + δ4 (N ) + |∂W ∩ N |H ) Xt ∩Q6=∅

226

for p = 2, 4, and analogously by (9.75) we get X
2 2 k¯ y − yˆk2L2 (Q) ≤ C |∂Xt |4∞ |At |4 ≤ CCm λ γ(N ) + δ4 (N ) + |∂W ∩ N |H , Xt ∩Q6=∅

where we employed |∂Xt |∞ ≤ Ck = Cλm−1 . Likewise we derive k¯ y − yˆk2L2 (Q) ≤ 2 2 CCm λ k . Together with the estimates for y¯ this shows (9.69)(i),(ii). It remains to prove (9.69)(iii). By (9.69)(i) for p = 4, (9.23) and the fact that ∇ˆ y (x) ∈ SO(2) H 2 2 for a.e. x ∈ U \W we nd k¯ eR¯j (∇ˆ y )kL2 (Q) ≤ CCm (γ(N )+δ4 (N )+|N ∩∂W |H ). This together with (9.76), |Q| ≤ Cλ2 and Hölder's inequality yields

¯ T yˆ − id)|(Q))2 ≤ CC 2 λ2 (γ(N ) + δ4 (N ) + |∂W ∩ N |H ). (|E(R j m Then Theorem B.5 and a rescaling argument show

¯ j · +¯ ¯ j · +¯ ¯ jT yˆ − id)|(Q))2 kˆ y − (R cj )k2L1 (∂Q) ≤ Cλ−2 kˆ y − (R cj )k2L1 (Q) + C(|E(R 2 ≤ Cλ2 Cm (γ(N ) + δ4 (N ) + |∂W ∩ N |H ).

In the last step we have used Hölder's inequality and (9.69)(ii). Similarly as 2 2 ¯ j · +¯ λ k.  before we also derive kˆ y − (R cj )k2L1 (∂Q) ≤ CCm

9.3 Modication of the deformation The goal of the section is to replace the deformation by an H 1 -function on UJ . In particular, we modify the deformation in such a way that the least crack length is increased. Recall ν = sd = λm.

Lemma 9.3.1. Let k > s,  > 0 such that l := ks = dm−2 for m−1 , d ∈ N with sm m−1 , d  1. Then there is a constant C > 0 such that for all W ∈ V(s,3k) with 3k 1 W ⊂ Ω and for all y ∈ H (W ) with k∇yk∞ ≤ C , γ as dened in (9.14) and δ2 , δ4 as given in (9.49) we have the following: sm ˆ2 ν There are sets U ∈ V71k and U H ∈ V72k with U, U H ⊂ Ω6k , |U \ W | = 0, |U H \ H λ (U )| = 0, |(W \ U ) ∩ Ω6k | + |U \ U H | ≤ Cu kkU k∗ and kU k∗ ≤ (1 + Cu m)kW k∗ + C−1 (γ + δ4 )

(9.77)

as well as a function y˜ ∈ H 1 (U H ) such that 2 (i) k dist(∇˜ y , SO(2))k2L2 (U H ) ≤ C min (1 + ϑ3p )Cm (γ + δp + kW k∗ ),

(ii) (iii) (iv)

p=2,4 2 ¯ + ϑ), ¯ k dist(∇˜ y , SO(2))kL∞ (U H ) ≤ C ϑ(1 2 k∇y − ∇˜ y k2L2 (U ) ≤ CCm (γ + δ2 + kW k∗ ), 2 k˜ y − yk2L2 (U ) ≤ CCm (1 + ϑ)λ2 (γ + δ4 + kW k∗ ),

227

(9.78)

where ϑ¯ = min{ϑ(1 + ϑ), Cm3 } and ϑ2 = 1, ϑ4 = ϑ. Under the additional assumption that ∆y = 0 in W ◦ we get 2 2 δ4 + CCm ϑ(1 + ϑ)2 (γ + δ4 + kW k∗ ). k∇y − ∇˜ y k4L4 (U ) ≤ CCm

(9.79)

sm ˆ2 Proof. Apply Lemma 9.2.5 to obtain sets V ∈ V71k , UJ ∈ V λ satisfying (9.50)

¯ j : UJ → SO(2) and c¯j : UJ → R2 , j = 1, . . . , 4. and (9.51) for mappings R We rst dene U = V and see that the estimate in (9.77). Moreover, we recall that Ω6k \ U is the union of rectangular components (see (9.68)). For the components Γ1 (H λ (V )), . . . , Γn (H λ (V )) we let N (Γj ) ∈ U ν denote the smallest rectangle with N (Γj ) ⊃ Xj , where as before Xj denotes the component corresponding to Γj (H λ (V )). As λν = m, we nd |πi ∂N (Γj )| ≤ |πi Γj (H λ (V ))| + Cu m|Γj (H λ (V ))|∞ for i = 1, 2. Arguing similarly as in the construction of (9.18) we have that N (Γj1 ) \ N (Γj2 ) is connected for 1 ≤ j1 , j2 ≤ n. We apply Lemma 9.1.2(i) to obtain S S pairwise disjoint, connected sets (Xj0 )nj=1 such that nj=1 N (Γj ) = nj=1 Xj0 and dene [n Xj0 . U H = Ω6k \ j=1

It is not hard to see that U ∈ Moreover, we nd U H ⊂ H λ (U ) up to a set of negligible measure and recalling (9.68) we obtain (U H )◦ ⊂ UJ . For later we also observe that H

ν V72k .

kU H k∗ ≤ (1 + Cu m)kH λ (U )k∗ .

(9.80)

This also implies |U \ U H | ≤ Cu kkU k∗ . 3 S λ Let Tj = p∈I λ (Ω3k ) Qj4 (p) and dene a partition of unity (ηj )4j=1 with ηj ∈ j

C ∞ (UJ , [0, 1]), supp(ηj ) ⊂ Tj and k∇ηj k∞ ≤ y˜(x) =

X4 j=1

C . λ

Dene y˜ : UJ → R2 by

¯ j x + c¯j ) ηj (x)(R

¯ j , c¯j are constant on each Qλj (p), and observe that y˜ ∈ H 1 (UJ ) as the functions R λ p ∈ Ij (UJ ). The derivative reads as ∇˜ y (x) = Since

P4

j=1

X4 j=1


¯ j + (R ¯ j x + c¯j ) ⊗ ∇ηj (x) . ηj (x)R

(9.81)

∇ηj = 0 we nd

¯1 + ∇˜ y (x) = R

X4 j=2


¯j − R ¯ 1 ) + (R ¯ j x + c¯j − (R ¯ 1 x + c¯1 )) ⊗ ∇ηj (x) . ηj (x)(R

228

First, we compute by (9.52) 4  X

¯ 1 k4 4 k∇˜ y−R L (UJ ) ≤ C

j=2 4  X

≤C

j=2

 1 ¯ 4 ¯j − R ¯ 1 k4 4 ¯ 1 · +¯ kR + k R · +¯ c − ( R c )k 4 j j 1 L (UJ ) L (UJ ) λ4  ϑ¯ ¯ 4 2 ¯ ¯ ¯ kRj − R1 kL4 (UJ ) + 2 kRj · +¯ cj − (R1 · +¯ c1 )kL2 (UJ ) , λ

3 ¯ j ) ≤ C|R ¯j − R ¯ 1 |2 and thus where ϑ¯ = min{ϑ(1 + ϑ), Cm }. By (9.23) we nd e¯R¯1 (R

y )k2L2 (UJ ) k¯ eR¯1 (∇˜

≤C

4  X j=2

≤C

4  X

 1 ¯ 2 ¯ j )k2 2 ¯ 1 · +¯ + k¯ eR¯1 (R k R · +¯ c − ( R c )k 2 j j 1 L (UJ ) L (UJ ) λ2 ¯j − R ¯ 1 k4 4 kR L (UJ ) +

j=2

 1 ¯ 2 ¯ k R · +¯ c − ( R · +¯ c )k j j 1 1 L2 (UJ ) . λ2

Again using (9.23) and (9.51)(iii),(iv) we derive 2 k dist(∇˜ y , SO(2))k2L2 (UJ ) ≤ C(1 + ϑ3 )Cm (γ + δ4 + kW k∗ ).

Similarly, we get

¯ 1 k2 2 k∇˜ y−R L (UJ ) ≤ C

4  X

¯j − R ¯ 1 k2 2 kR L (UJ ) +

j=2

 1 ¯ 2 ¯ k R · +¯ c − ( R · +¯ c )k j j 1 1 L2 (UJ ) λ2

and thus we nd by (9.51)(iii),(iv) 2 k dist(∇˜ y , SO(2))k2L2 (UJ ) ≤ CCm (γ + δ2 + kW k∗ ),

where we used that δ4 ≤ Cδ2 . This gives (9.78)(i) as (U H )◦ ⊂ UJ . Likewise, we may replace the L2 , L4 -norms in the above estimates by the L∞ -norm. Conse¯ ¯ ¯ 1 k4 ∞ quently, by (9.52) we obtain k∇˜ y −R eR¯1 (∇˜ y )k2L∞ (UJ ) ≤ L (UJ ) ≤ C ϑ(1+ϑ) and k¯ ¯ ¯ . It remains to show C ϑ¯ which then implies k dist(∇˜ y , SO(2))k2L∞ (UJ ) ≤ C ϑ(1+ ϑ) (9.78)(iii),(iv) and (9.79). By (9.51)(i) and the fact that U = V we obtain X4 2 2 ¯ j · +¯ k˜ y − yk2L2 (U ) ≤ Cky − (R cj )k2L2 (U ) ≤ CCm λ (1 + ϑ)(γ + δ4 + kW k∗ ). j=1

P P By (9.81) and the fact that 4j=1 ηj = 1, 4j=1 ∇ηj = 0 we derive X4
¯ j ) + (y(x) − (R ¯ j x + c¯j )) ⊗ ∇ηj (x) . ∇y(x) − ∇˜ y (x) = ηj (x)(∇y(x) − R j=1

Therefore, by (9.51)(i)(ii) for p = 2 we get  X4  1 2 2 2 ¯ ¯ k∇˜ y − ∇ykL2 (U ) ≤ C k∇y − Rj kL2 (U ) + 2 ky − (Rj · +¯ cj )kL2 (U ) j=1 λ 2 ≤ CCm (γ + δ2 + kW k∗ ), 229

where we used that δ4 ≤ Cδ2 . Finally, in the case that ∆y = 0 in W ◦ we obtain by (9.51)(i)(ii) for p = 4 and (9.53)

k∇˜ y−

∇yk4L4 (U )

 ϑ(1 + ϑ) 4 2 ¯ ¯ k∇y − Rj kL4 (U ) + ≤C ky − (Rj · +¯ cj )kL2 (U ) j=1 λ2 2 2 ≤ CCm δ4 + CCm ϑ(1 + ϑ)2 (γ + δ4 + kW k∗ ). X4





9.4 SBD-rigidity up to small sets In this section we prove a slightly weaker version of the rigidity estimate given in Theorem 6.1.1 and postpone the proof of the general version to the next section. Recall denition (6.1).

Theorem 9.4.1. Let Ω ⊂ R2 open, bounded with Lipschitz boundary. Let M > 0 and 0 < η, ρ, h∗  1. Let q ∈ N suciently large. Then there are constants C1 = C1 (Ω, M, η), C2 = C2 (Ω, M, η, ρ, h∗ , q) and a universal constant c > 0 such that the following holds for ε > 0 small enough:´ For each y ∈ SBVM (Ω) with H1 (Jy ) ≤ M and Ω dist2 (∇y, SO(2)) ≤ M ε, there is a set Ωy ∈ Vcρsˆ q−1 , sˆ > 0, with Ωy ⊂ Ω, |Ω \ Ωy | ≤ C1 ρ, a modication y˜ ∈ H 1 (Ωy )∩SBVcM (Ωy ) with ky− y˜k2L2 (Ωy ) +k∇y−∇˜ y k2L2 (Ωy ) ≤ C1 ερ, a partition (Pi )i of Ωy and for each Pi a corresponding rigid motion Ri x + ci , Ri ∈ SO(2) and ci ∈ R2 , such that the function u : Ω → R2 dened by ( y˜(x) − (Ri x + ci ) u(x) := 0

for x ∈ Pi else

(9.82)

satises (i) kΩy k∗ ≤ (1 + C1 h∗ )H1 (Jy ) + C1 ρ, (ii) kuk2L2 (Ωy ) ≤ C2 ε, X (iii) ke(RiT ∇u)k2L2 (Pi ) ≤ C2 ε, (iv) k∇uk2L2 (Ωy ) ≤ C2 ε1−η .

(9.83)

i

We divide the proof into three steps. We begin with a version where the least crack length is almost of macroscopic size. Afterwards, we assume that the jump set consists only of a nite number of cracks of arbitrary size. Finally, we treat the general case applying a suitable approximation argument. In what follows, constants indicated by C1 only depend on M, η, Ω. Generic constants C may additionally depend on h∗ . All constants do not depend on ρ and q unless stated otherwise. As we will eventually let h∗ ∼ ρ in Section 9.5, it is essential that the constant in (9.83)(i) does not depend on h∗ .

230

9.4.1 Step 1: Deformations with least crack length We rst treat the case that the least crack length is almost of macroscopic size.

Theorem 9.4.2. Theorem 9.4.1 holds under theη additional assumption that there is an Ω˜ y ⊂ Ωs , Ω˜ y ∈ Vρsq−1 for some s ≥ ρq−1 ε 8 such that y ∈ H 1 (Ω˜ y ), kΩ˜ y k∗ ≤ ˜ y | ≤ C1 ρ for a constant C1 = C1 (Ω, M, η). (1 + C1 h∗ )H1 (Jy ) + C1 ρ and |Ω \ Ω Proof. Let y ∈ H 1 (Ω˜ y ) be given. Let ρ and dene % = ρq for some q ∈ N,

q ≥ 2 large enough to be specied in the proof of Theorem 6.1.1 (see Section 9.5). Assume without restriction ρ−1 ∈ N large. We apply Theorem B.2 and consider the harmonic part w of y satisfying k∇y − ∇wk2L2 (Ω˜ y ) ≤ Ck dist(∇y, SO(2))k2L2 (Ω˜ y ) ≤ Cε, k∇y − ∇wk4L4 (Ω˜ y ) ≤ Ck dist(∇y, SO(2))k4L4 (Ω˜ y ) ≤ Cε.

(9.84)

In the last inequality we used k∇yk∞ ≤ M . Let k = %ρ−1 = ρq−1 . Apply ˜ y ∩ Ωk for the function w and  = cˆρ−1 ε, m = ρ, where Lemma 9.2.1 on Ω cˆ > 0 is suciently large. (Possibly passing to a smaller s we can assume that η sm such that kε 8 ≤ s  k = ρq−1 .) We nd a set W ⊂ Ω3k , W ∈ V(s,3k)

˜ y k∗ + C−1 ε ≤ (1 + C1 ρ)kΩ ˜ y k∗ + ρ kW k∗ ≤ (1 + C1 ρ)kΩ

(9.85)

˜ y \W )∩Ω3k | ≤ C1 k ≤ C1 ρ. (Here and in the following we choose by (9.15) and |(Ω the constant cˆ = cˆ(h∗ ) always larger then the constant C .) Moreover, there are ˆ i : W ◦ → SO(2), i = 1, . . . , 4, which are constant on the connected mappings R components of Qki (p) ∩ W ◦ , p ∈ Iik (Ω), such that by (9.16)(i) for i = 1, . . . , 4 4 1−η ˆ i k2 2 ˆ 2 k∇y − R , L (W ) ≤ Cε + Ck∇w − Ri kL2 (W ) ≤ Cl ε ≤ Cε η

(9.86) 5

2 −1 where l = ks−1 ≤ Cε− 8 . Moreover, as ϑ = l9 Cm s ε ≤ C(ρ)s−10 ε ≤ C(ρ)ε1− 4 η ≤ 1 for η, ε small enough (recall (9.13)) we also get

ˆ i k4 4 ˆ 4 k∇y − R L (W ) ≤ Cε + Ck∇w − Ri kL4 (W ) ≤ Cε

(9.87)

by (9.16)(ii). Now we apply Corollary 9.2.6 on W ⊂ Ω3k for k = ρq−1 , λ = 3%, sm ˆ ˜ y| = 0 m = 3ρ and  = cˆρ−1 ε. We obtain a set Ωy ∈ V9k with Ωy ⊂ Ω5k , |Ωy \ Ω such that by (9.25), (9.85) and (9.87) we nd

kΩy k∗ ≤ (1 + C1 ρ)kW k∗ + C−1 ε ≤ (1 + C1 h∗ )H1 (Jy ) + C1 ρ

(9.88)

˜ y \ Ωy ) ∩ Ω5k | ≤ C1 k . This together with the assumption |Ω \ Ω ˜ y | ≤ C1 ρ and |(Ω and the fact that |Ω \ Ω5k | ≤ C(Ω)k yields |Ω \ Ωy | ≤ C1 ρ. Moreover, there is a λ λ H H 2 ¯ set ΩH ¯j : ΩH y ∈ V with H (Ωy ) ⊂ Ωy and mappings Rj : Ωy → SO(2), c y → R 231

3% 3k 2 being constant on Q3% ˆ ∈ SBVM (ΩH y ,R ) j (p), p ∈ Ij (Ω ), and an extension y such that by (9.69)(ii) we derive ¯ j · +¯ kˆ y − (R cj )k2 2 H ≤ C%2 ρ−2 C 4 (ε + kW k∗ ) ≤ Cρ2q−3 C 4 ε (9.89) ρ

L (Ωy )

ρ

where Cρ = C m3 is the constant dened in (9.13). Here we used that each x ∈ W is contained in at most ∼ ρ−2 dierent neighborhoods N (Q) considered in Corollary 9.2.6. Moreover, the constant cˆ was absorbed in C . Similarly, recalling ϑ ≤ 1 we get by (9.51)(iii),(iv), (9.69)(i) and (9.86), (9.87) ¯j − R ¯ j k2 2 H ≤ Cρ−3 C 2 ε1−η , ¯ j k2 2 H + kR k∇ˆ y−R L (Ωy )

¯ j k4 4 H R L (Ωy )

1

2

ρ

L (Ωy )

¯ j2 k4 4 H R L (Ωy )

¯ j1 − k∇ˆ y− + kR ¯ j1 · +¯ ¯ j2 · +¯ k(R cj 1 ) − ( R cj2 )k2 2

L (ΩH y )

−3

≤ Cρ Cρ2 ε, 2q−3

≤ Cρ

(9.90)

Cρ2 ε,

for j = 1, . . . , 4 and 1 ≤ j1 , j2 ≤ 4. ◦ 3% Denote the connected components of (ΩH by (PiH )i and dene Pi = y ) ∈ U PiH ∩ Ωy . Let Ji ⊂ I % (Ω) be the index set such that Q% (p) ⊂ PiH for all p ∈ Ji . We now estimate the variation of the rigid motions dened on these squares. Let ¯ 4 |Qt Q1 = Q% (p1 ), Q2 = Q% (p2 ) for p1 , p2 ∈ Ji such that Q1 ∩ Q2 6= ∅. Let Rt = R ¯ j is j = 1, . . . , 4 such that R and ct = c¯4 |Qt for t = 1, 2. Then we nd some P p 2 p ¯ j − Rt k p constant on Q1 ∪ Q2 and thus % |R1 − R2 | ≤ C t=1,2 kR L (Q1 ∪Q2 ) for p = 2, 4. Using the arguments in (7.10), (7.12), and Remark 7.1.4(iii) we get

%4 |R1 − R2 |2 +k(R1 − R2 ) · +c1 − c2 k2L2 (Q1 ∪Q2 ) X ¯ j · +¯ k(R cj ) − (Rt · +ct )k2L2 (Q1 ∪Q2 ) . ≤C

(9.91)

t=1,2

Consequently, considering chains as in (7.6) and (7.18), respectively, following the arguments in the proof of Lemma 7.1.3(i),(ii) and recalling Remark 7.1.4(iv), we obtain Ri ∈ SO(2), ci ∈ R2 such that ¯ 4 · +¯ kˆ y − (Ri · +ci )k2L2 (P H ) ≤ Ckˆ y − (R c4 )k2L2 (P H ) i i X −8 ¯ j1 · +¯ ¯ j2 · +¯ + C% k(R cj 1 ) − ( R cj2 )k2L2 (P H ) , 1≤j1 ,j2 ≤4

k∇ˆ y−

Ri kpLp (P H ) i

i

¯ 4 kp p H ≤ Ck∇ˆ y−R L (Pi ) X ¯ j1 − R ¯ j2 k p p H , + C%−2p kR L (P ) 1≤j1 ,j2 ≤4

p = 2, 4.

i

In the rst estimate we used Hölder's inequality (cf. (7.19)). Summing over all connected components, (9.89) and (9.90) implies X kˆ y − (Ri · +ci )k2L2 (P H ) ≤ C(ρ, q)ε, i i X X (9.92) 4 k∇ˆ y − Ri kL4 (P H ) ≤ C(ρ, q)ε, k∇ˆ y − Ri k2L2 (P H ) ≤ C(ρ, q)ε1−η i

i

j

i

for C(ρ, q) large enough. Dening u as in (9.82) (for y˜ = y ) and taking also (9.88) into account, we immediately get (9.83)(i)(ii),(iv). Finally, (9.83)(iii) is a consequence of the linearization formula (9.24) and (9.92).  232

9.4.2 Step 2: Deformations with a nite number of cracks We now prove a version where the crack set consists of a nite number of components. We rst assume that each crack is at least of atomistic size. The strategy will be to establish an estimate of the form (9.86) and (9.87) by iterative modication of y according to Lemma 9.3.1. First, we introduce some notation and derive preliminary estimates. Let ρ > 0, set % = ρq and assume without restriction ρ−1 ∈ N large. As before we assume k dist(∇y, SO(2))k2L2 (Ω) ≤ Cε. Choose t−1 ∈ N such that t ≤ ρ and set tj = tj+1 . By Remark 8.4.3, 8.5.8 we can assume that T := tz+18 ≤ Ct−2 t18 for z ∈ N suciently large (recall (9.13) for the denition of Ct ). Moreover, set Tj = T j+1 . ˜ y ⊂ Ωs for some s > 0 be given. Let Let Ω
˜ y k∗ + C∗ ρ · Xj−1 ti · Πj−1 (1 + C∗ ti+1 ) Bj = kΩ (9.93) i=0 i=0 and B = limj→∞ Bj for a constant C∗ = C∗ (M, η, Ω) ≥ 1 to be specied below. Furthermore, let P = cˆ2 (1 + ρ−1 B) for cˆ = cˆ(h∗ ) suciently large. Set s0 = κε for κ suciently large, let 0 = cˆ2 ρ−1 ε and subsequently dene j+1 = P Tj−1 j . η 1 We set r = 18 , ω = 36 for notational convenience and for j ≥ 0 we dene j n s r ok j −ω ,ε dj = min , (9.94) j j−1 di . In accordance with Sections 9.2, 9.3 we also dene where sj = s0 Πi=0

lj = dj t−2 j ,

λj = sj dj t−1 j ,

kj = sj lj .

(9.95)

As noted before, dj describes the increase of the minimal distance of dierent cracks and P Tj−1 will be the factor of energy increase. Below we will show that indeed dj  1 for all 0 ≤ j ≤ J ∗ , where

J ∗ = dlog1+r (logT εω )) + ω1 e. One of the main reasons why the iterative application of Lemma 9.3.1 works is the d fact that dj increases much faster than P Tj−1 . We dene the quotient qj := P Tj−1 j

r and observe q0 = d0PT0 = T P −1 (s0 −1 0 ) for ε suciently small. Recalling (9.93) 2 −1 and the denition s0 = κε, 0 = cˆ ρ ε we can rst choose T = T (ρ, h∗ ) so small and then κ = κ(T, ρ, h∗ , z¯) so large that

q0 T 1/r ≥ T −¯z ≥ T −1 ≥ cˆ4 P 2 > 1

(9.96)

for z¯ ∈ N to be specied below. For the third inequality we used the fact that P ≤ C for some C = C(C∗ , ρ, h∗ , M ) independent of T . We nd j

qj = T −1/r (q0 T 1/r )(1+r) 233

(9.97)

η s for j ≤ Jˆ, where Jˆ ∈ N is the largest index such that jj ≤ ε− 2 for all j ≤ Jˆ. Indeed, we rst note that the formula is trivial for j = 0. Assume (9.97) holds for j ≤ Jˆ − 1, then we compute

qj+1

Tj+1  sj+1 r Tj+1  sj dj r qjr Tj+1  sj r qjr dj Tj T = = = T qj1+r = = −1 P j+1 P P Tj j P j P

which gives (9.97) for j + 1, as desired. In particular, taking (9.96) into account, (9.97) implies qj > 1 and thus dj = qj P Tj−1  1 for all j ≤ Jˆ. For Jˆ < j ≤ J ∗ we get dj = ε−ω . In fact, using (9.96) and 0 ≤ cˆ2 t−1 ε we observe for C suciently large 1

1

2

j−1 −(i+1) − 2 j = 0 Πi=0 (P Ti−1 ) ≤ cˆ−2 0 Πj−1 T ) ≤ cˆ−2 0 T − 2 (j+1) i=0 (T ω 2 −ω
≤ εT −C−[log1+r (logT ε )] = εo T −logT −1 ε = ε · o(ε−ω ) sj ≥ j d j sj P Tj−1 j

for ε → 0 for all 1 ≤ j ≤ J ∗ . Consequently, if sj+1 j+1

P Tj−1 = o(ε−ω ) (see (9.98)) and thus

=

(9.98)

η

ω

ε− r = ε− 2 , then dj = ε−ω , ω

≥ ε− r . This then implies

dj = ε−ω for all Jˆ < j ≤ J ∗ . −2 9 2 We introduce ϑj = s−1 j j lj Ctj (recall denition (9.13) and lj = dj tj ) and close the preparations by showing that 0 Tj 2 cˆ j+1

ϑj ≤

for 0 ≤ j ≤ J ∗ .

(9.99)

This particularly implies ϑj ≤ 1 for all j as j ≥ 0 for all j . By (9.94)-(9.97) we obtain η

sj ≥ j ε− 2

or

1/r

sj = j dj

1/r

≥ j qj

z ¯

j

2

≥ j T − r (1+r) ≥ j T −9(j+1) .

(9.100)

for all 0 ≤ j ≤ J ∗ . The last step holds for z¯ ∈ N suciently large as limj→∞ 1r (1+ 2 r)j (9(j + 1)2 )−1 = ∞. Similarly as in (9.98) we see that T −9(j+1) = o(ε−ω ) for η 2 j ≤ J ∗ as ε → 0. Since ε−ω = o(ε− 2 ), we nd sj ≥ j T −9(j+1) for all 0 ≤ j ≤ J ∗ . 1 Therefore, we derive by (9.94), (9.96), the rst line of (9.98) and r = 18 −1

3

2

9 −18 2 ϑj j+1 = s−1 Ctj P Tj−1 j ≤ sj 2 j2 cˆ−2 Tj−3 ≤ cˆ−2 0 T 4(j+1) Tj−3 ≤ cˆ−2 0 Tj j j dj tj

for all 0 ≤ j ≤ J ∗ , as desired. In the second step we used Ct2j t−18 ≤ Tj−1 and j P ≤ Tj−1 . Recall the denition of κ and k0 above (see (9.95) and (9.96)).

Theorem 9.4.3. Theorem 9.4.1 holds under the additional assumption that there is an Ω˜ y ⊂ Ωs , Ω˜ y ∈ Vks0 for some s ≥ κε, such that y ∈ H 1 (Ω˜ y ), kΩ˜ y k∗ ≤ ˜ y | ≤ C1 ρ for a constant C1 = C1 (Ω, M, η). (1 + C1 h∗ )H1 (Jy ) + C1 ρ and |Ω \ Ω 234

Proof. Let y ∈ H 1 (Ω˜ y ) be ηgiven. If s ≥ ε 8 we can apply Theorem 9.4.2, so η

it suces to consider s ≤ ε 8 . Recall s0 = κε for some κ = κ(T, ρ, h∗ , z¯)  1 and assume s ≥ s0 . The strategy is to apply Lemma 9.3.1 iteratively. Set H ˜ y ∈ V s and y0 = y . Recall 0 = cˆ2 ρ−1 ε and dene = W0H = Ω W0 = W−1 k0

γ0 := k dist(∇y0 , SO(2))k2L2 (Ω˜ y ) ≤ C

ρ0 ρ0 , α0 := k dist(∇y0 , SO(2))k4L4 (Ω˜ y ) ≤ C 2 . 2 cˆ cˆ

In the last inequality we used k∇yk∞ ≤ M . Recall (9.95). Set sˆj = sj tˆ2j for j ≥ 0 sˆ s and sˆ−1 = s, where tˆj = C2 (tj , h∗ ) (see (9.13)). Assume Wj ∈ Vkjj−1 , WjH ∈ Vkjj H ˜ y \ Wj | ≤ C1 Pj−1 ki , are given with Wj , WjH ⊂ Ω6kj−1 , |Wj \ Wj−1 | = 0 and |Ω i=0 where we set k−1 = s. Recall that |Wj \WjH | ≤ C1 kj−1 and |WjH \H λj−1 (Wj )| = 0, where λ−1 = 0. Set βj = kH λj−1 (Wj )k∗ and βjd = kWj k∗ − kH λj−1 (Wj )k∗ . Moreover, suppose there is a function yj ∈ H 1 (WjH ) with

γj := k dist(∇yj , SO(2))k2L2 (W H ) , αj := k dist(∇yj , SO(2))k4L4 (W H ) j

j

such that for j ≥ 1

(i) βj + βjd ≤ (1 + C1 tj−1 )βj−1 + C−1 j−1 γj−1 ≤ Bj , −1 (ii) γj ≤ CTj−1 tj−1 (γj−1 + j−1 βj−1 ) ≤ cˆ−1 tj−1 ρj ,

(iii) αj ≤ Cϑj−1 γj ≤ CεTj−1 , (iv) k dist(∇yj , SO(2))k2L∞ (W H ) ≤ Cϑj−1 ,

(9.101)

j

(v) k∇yj − ∇yj−1 k4L4 (Wj ) ≤ CεTj−1 , −1 4 4 (vi) k∇yj − ∇yj−1 k2L2 (Wj ) ≤ CTj−1 (lj−1 γj−1 + j−1 βj−1 ) ≤ Clj−1 j .

Setting ϑ−1 = 1 and t−1 = 1, we note that, provided cˆ is suciently large, in the case j = 0 (iii),(iv) are clearly satised for y0 = y and (i),(ii) hold neglecting sˆj the second terms. We now construct yj+1 , Wj+1 ∈ Vkj+1 with Wj+1 ⊂ Ω6kj , ˜ y \ Wj+1 | ≤ C1 Pj ki as well as W H ∈ V sj+1 . |Wj+1 \ W H | = 0 and |Ω j

j+1

i=0

kj+1

First we apply Theorem B.2 and let wj ∈ H 1 (WjH ) be the harmonic part of yj such that similarly as in (9.84)

k∇yj − ∇wj k2L2 (W H ) ≤ Cγj , j

(9.102)

k∇yj − ∇wj k4L4 (W H ) ≤ Cαj j

s

and so in particular k dist(∇wj , SO(2))k2L2 (W H ) ≤ Cγj . Recall WjH ∈ Vkjj , Wj ⊂ j

Ω6kj−1 and note Ωkj ⊂ Ω6kj−1 . Then apply Lemma 9.2.1 with s = sj , k = kj = ˜ jH ∈ V sj tj sj lj , m = tj = tj+1 ,  = j , U = WjH ∩Ωkj , y = wj and obtain a set W (sj ,3kj ) such that X4 X4 ˆ i k4 4 ˜ H ≤ Cϑj γj , δ2 := ˆ i k2 2 ˜ H ≤ Clj4 γj δ4 := k∇wj − R k∇wj − R L (W ) L (W ) i=1

i=1

j

235

j

ˆ i : (W ˜ H )◦ → SO(2), i = 1, . . . , 4, which are constant on the for mappings R j ˜ H )◦ , p ∈ I k (Ωk ). We now use Lemma connected components of Qki (p) ∩ (W j i ˜ H , y = wj and show 9.3.1 with m = tj , s = sj ,  = j , d = dj , W = W j sˆj sˆj 6kj (9.101) for j + 1. First, we obtain Wj+1 ∈ V71k ⊂ V , kj+1 , with Wj+1 ⊂ Ω j sj+1 sj+1 6kj H H H |Wj+1 \ Wj | = 0, |(Wj \ Wj+1 ) ∩ Ω | ≤ Ckj kWj+1 k∗ and Wj+1 ∈ V72kj ⊂ Vkj+1 H H | ≤ C1 kj . Recall kWjH k∗ ≤ \ H λj (Wj+1 )| = 0 and |Wj+1 \ Wj+1 with |Wj+1 (1 + C1 tj )βj by (9.80). Thus, we have −1 kWj+1 k∗ ≤ (1 + C1 tj )kWjH k∗ + C−1 j (γj + ϑj γj ) ≤ (1 + C1 tj )βj + Cj γj (9.103)

by (9.15), (9.77) and the fact that ϑj ≤ 1 (see (9.99)). Moreover, we get a H ) with (see (9.78), (9.79)) function yj+1 ∈ H 1 (Wj+1

(i) k dist(∇yj+1 , SO(2))k2L2 (W H

j+1 )

≤ CCt2j (γj + j βj ),

(ii) k∇wj − ∇yj+1 k2L2 (Wj+1 ) ≤ CCt2j (γj + lj4 γj + j βj ), (iii) k∇wj − ∇yj+1 k4L4 (Wj+1 ) ≤ CCt2j ϑj (γj + j βj ), (iv) k dist(∇yj+1 , SO(2))k2L∞ (W H

j+1 )

(9.104)

≤ Cϑj ,

where we again used that ϑj ≤ 1. The rst inequality in (9.101)(ii) follows directly noting that Tj−1 tj ≥ Ct2j and for the second inequality we use (9.101)(i),(ii) for iteration step j as well as (9.93) to see

CTj−1 (γj + j βj ) ≤ CTj−1 ρj (1 + ρ−1 Bj ) ≤ ρˆ c−1 P Tj−1 j = cˆ−1 ρj+1 ,

(9.105)

where we choose cˆ suciently large. Likewise, (9.101)(i) follows by (9.103), the d fact that kWj+1 k∗ = βj+1 + βj+1 and d βj+1 + βj+1 ≤ (1 + C1 tj )Bj + ρtj−1
j ˜ y k∗ + C∗ ρ · Xj−1 ti · πt=0 ≤ kΩ (1 + C∗ ti+1 ) + ρtj i=0
Xj i j ˜ y k∗ + C∗ ρ · ≤ kΩ t · πt=0 (1 + C∗ ti+1 ) = Bj+1 . i=0

Here we have again chosen cˆ and C∗ large enough (with respect to C and C1 , respectively). ThisP also implies |(Wj \ Wj+1 )P ∩ Ω6kj | ≤ Ckj by (9.101)(i) and thus j 6kj ˜ y \ Wj+1 )| ≤ C |(Ω | ≤ C ji=0 ki . i=0 ki + |Ω \ Ω Estimate (9.101)(iv) follows from (9.104)(iv). The rst inequality in (9.101)(iii) is a consequence of (9.101)(iv), the second inequality is implied by the fact that ε = cˆ−2 ρ0 , (9.101)(ii) and (9.99). Moreover, (9.101)(v) follows from (9.101)(iii), (9.102), (9.104)(iii) and the fact that ϑj Ct2j (γj +j βj ) ≤ ϑj ρj+1 ≤ CεTj by (9.99) and (9.105). Similarly, (9.101)(vi) follows from (9.104)(ii), (9.102) and (9.105). We now choose j ∗ ∈ N such that

ε3ω ≥ sj ∗ ≥ ε4ω ,

j ∗ ≤ Cε1−ω Tj2∗ 236

(9.106)

holds for ε suciently small. The rst inequality is possible by (9.94) and we obtain j ∗ ≤ J ∗ = dlog1+r (logT εω )) + ω1 e. Indeed, by (9.100) and the fact that η ω z¯ ≥ 1 we get sj ≥ ε− r j = ε− 2 j for j > dlog1+r (logT εω ))e and therefore Jˆ ≤ dlog1+r (logT εω ))e. The second inequality can be derived arguing as in (9.98). −ω Similarly, proceeding as in (9.98) we have t−2 ) for ε → 0 and thus j∗ = o(ε −2 ω 6k % ∗ j ∗ ∗ ∗ kj = sj dj tj ∗ = o(ε ). This implies Ω ⊃ Ω for ε small enough. We let

W∗H = WjH∗ ∩ Ω% ,

y∗ = yj ∗ ,

W∗ =

\j ∗ i=0

Wi ∩ Ω% .

˜ y \ W∗ | ≤ C1 Pj ∗ ki ≤ C%. As sˆj = sj tˆ2 is increasing It is not hard to see that |Ω j i=0 sˆ0 . in j (note that dj ≥ tˆ−2 for all j , see e.g. (9.100)), we nd W ∈ V ∗ j The strategy is now to establish an estimate of the form (9.86) and (9.87). η Observe that sj ∗ ≥ ε 8 , i.e. for the function y∗ ∈ H 1 (W∗H ) we may proceed as in Theorem 9.4.2 (replacing s by sj ∗ ). Similarly as in (9.84), we apply Theorem B.2 and let w∗ be the harmonic part of y∗ with η

k∇w∗ − ∇y∗ k2L2 (W∗H ) ≤ Cε1− 2 ,

∗

k∇w∗ − ∇y∗ k4L4 (W∗H ) ≤ CεT j .

(9.107)

by (9.101), (9.106) and ω ≤ η2 . Apply Lemma 9.2.1 on W∗H ⊂ Ω% for the function η w∗ and k = ρq−1 = %ρ−1 , s = ε4ω ,  = cˆρ−1 ε1− 2 , m = ρ. (Without restriction we s ∗m can assume s−1 ∈ N.) We nd a set W H ⊂ Ω3k , W H ∈ V3kj such that η

η

kW H k∗ ≤ (1 + C1 ρ)kW∗H k∗ + Cˆ c−1 ρε 2 −1 ε1− 2 ≤ kW∗H k∗ + C1 ρ

(9.108)

by (9.15) as well as |W∗H \ W H | ≤ |(W∗H \ W H ) ∩ Ω3k | + C1 k ≤ C1 k ≤ C1 ρ. ˆ i : (W H )◦ → SO(2), i = 1, . . . , 4, which are Moreover, there are mappings R constant on the connected components of Qki (p) ∩ (W H )◦ , p ∈ Iik (Ω), such that by (9.16)(i) and (9.107) ∗

η

ˆ i k4 4 H ≤ Ck∇w∗ − R ˆ i k4 4 H + CεT j ≤ Cϑε1− 2 + Cε ≤ Cε, k∇y∗ − R L (W ) L (W ) η where similarly as before equation (9.87) we compute (recall (9.106) and ω = 36 ) η 1− 41 η −10 −40ω 1−ω 36 2 ϑ ≤ C(ρ, q)s  ≤ C(ρ, q)ε ε = C(ρ, q)ε ≤ ε for ε, η small enough. Likewise, we derive

ˆ i k2 2 H ≤ Ck∇w∗ − R ˆ i k2 2 H + Cε1− η2 ≤ C(1 + l4 )ε1− η2 ≤ Cε1−η k∇y∗ − R L (W ) L (W ) η

as l = ks ≤ Cε−4ω ≤ ε− 8 . sˆ0 We now will construct a set W ∈ V143k which is contained in W H ∩ W∗ ∩ Ω3k ∈ V sˆ0 , where the two sets coincide up to a set of measure smaller than C1 ρ. (Similarly as before the dierence of the sets is related to the denition of the boundary components.) Before we give the exact denition of W and establish ˜ y \ W | ≤ C1 ρ arguing as before an estimate of the form (9.85), we rst observe |Ω and derive estimates similar to (9.86) and (9.87). 237

We iteratively apply (9.101)(v) and derive for i = 1, . . . , 4  Xj ∗  1 4 ˆ i k4 4 ˆ i k4 4 4 k∇y − R ≤ C (εT ) + Ck∇y∗ − R ι−1 L (W ) L (W ) ≤ Cε. ι=1

(9.109)

4 Likewise, observe that by (9.94), (9.95) and (9.106) we have lj−1 j ≤ lj4 j = d4j t−8(j+1) j ≤ ε−4ω ε1−ω Tj ≤ ε1−η Tj . We derive by (9.101)(vi) 1−η ˆ i k2 2 k∇y − R L (W ) ≤ Cε

 Xj ∗

1

ι=1

Tι2

2

1−η ˆ i k2 2 + Ck∇y∗ − R L (W ) ≤ Cε

for i = 1, . . . , 4. sˆ0 and to establish kW k∗ ≤ It remains to give the exact denition of W ∈ V143k 1 ˜ (1 + Ch∗ )H (Jy ) + Cρ. Recall W0 = Ωy and dene Wj ∗ +1 := W H for notational convenience. We now dene W inductively. Let Y0 = Y00 = Y000 = W0 . Assume Yj ∈ V sˆ0 and Yj0 ∈ Vksˆj0 , Yj00 ∈ V sˆ0 are given with |Yj0 \ Yj | + |Yj0 4Yj00 | = 0, |Yj \ Yj0 | ≤ C1 kj−1 and

max{kYj0 k∗ , kYj00 k∗ } ≤ kYj k∗ ≤ kWj k∗ +

Xj−1 i=1

βid ,

where Yj00 has the property that all components not intersecting ∂H λj−1 (Wj )
coincide with components of Yj0 and the set Xt (H λj−1 (Wj )) t of components of H λj−1 (Wj ) is a subset of the components of Yj00 . Moreover, suppose that P T T |Yj0 \ ji=0 Wi | = 0 and | ji=0 Wi \ Yj0 | ≤ j−1 i=0 ki . We now pass to step j +1. Let X1 (Wj+1 ), . . . , Xnj+1 (Wj+1 ) be the components of Wj+1 and dene [nj+1
[nj+1 Yj+1 = Yj00 \ Xt (Wj+1 ) ∪ ∂Xt (Wj+1 ) ∈ V sˆ0 . t=1

t=1

T T W \ Yj+1 | ≤ Wi | = 0 and | j+1 First observe that Yj+1 satises |Yj+1 \ j+1 i=0 S i=0 Hi Pj−1 Snj+1 H t=1 Xt (Wj+1 ) ⊃ t Xt (Wj ) and then i=0 ki . As |Wj+1 \ Wj | = 0, we obtain Snj+1 S λj−1 H (Wj )| = 0 we get t=1 Xt (Wj+1 ) ⊃ t Xt (H λj−1 (Wj )). by the fact that |Wj \H As by hypothesis the components of H λj−1 (Wj ) are also components of Yj00 , we derive recalling βid = kWi k∗ − kH λj−1 (Wj )k∗ and β0d = 0 kYj+1 k∗ ≤ kYj00 k∗ + kWj+1 k∗ − kH λj−1 (Wj )k∗ = kWj+1 k∗ +

Xj i=1

βid .

sˆ0 Observe that possibly Yj+1 ∈ / Vcon . However, by Lemma 9.1.2(ii) we nd a set 0 sˆ0 0 0 Yj+1 ∈ V with |Yj+1 \ Yj+1 | ≤ C1 kj and kYj+1 k∗ ≤ kYj+1 k∗ . Here we essentially used the rectangular shape of the boundary components given by (9.68) and sˆ0 0 0 (9.18), respectively. Then it is elementary to see that Yj+1 ∈ V143k ⊂ Vksˆj+1 j Tj+1 P j 0 00 and | i=0 Wi \ Yj+1 | ≤ Moreover, if j + 1 ≤ j ∗ , we let Yj+1 = i=0 ki . 0 λj λj 00 (Yj+1 ∩ H (Wj+1 )) ∪ ∂H (Wj+1 ) and observe that Yj+1 has the desired proper00 ties. In fact, kYj+1 k∗ ≤ kYj+1 k∗ follows as before. Components not intersecting

238

0 ∂H λj (Wj+1 ) are clearly components of Yj+1 . Finally, by denition components λj 00 of H (Wj+1 ) are also components of Yj+1 . sˆ0 . By (9.93) and (9.101)(i),(ii) we We nally dene W = Yj0∗ +1 ∩ Ω3k ∈ V143k have i i−1 βid ≤ βi−1 − βi + C1 ti βi−1 + C−1 i−1 γi−1 ≤ βi−1 − βi + C1 t B + ρt

˜ y k∗ , kW∗H k∗ ≤ (1+C1 tj ∗ )βj ∗ and using (9.93), for i = 1, . . . , j ∗ . Recalling β0 = kΩ (9.108) as well as t ≤ ρ we conclude Xj ∗ kW k∗ ≤ kYj0∗ +1 k∗ ≤ kW H k∗ + (βi−1 − βi + C1 ti B + ρti−1 ) i=1

˜ y k∗ + C1 ρB + C1 ρ ≤ kW k∗ − βj ∗ + β0 + C1 ρB + C1 ρ ≤ C1 ρ + kΩ ˜ y k∗ + C1 ρ ≤ (1 + C1 h∗ )H1 (Jy ) + C1 ρ, ≤ (1 + C1 ρ)kΩ H

as derided. We now proceed as in the proof of Theorem 9.4.2 after equation (9.87) with the η only dierence that we take sˆ0 instead of s ∼ ε 8 in the application of Corollary sˆ0 m ˆ 9.2.6. However, this does not change the analysis. This leads to a set Ωy ∈ Vck with Ωy ⊂ Ω5k and |Ω \ Ωy | ≤ C1 ρ for k = ρq−1 , m = 3ρ for which (9.83) can be established.  We now additionally treat the subatomistic regime by dropping the assumption s ≥ κε.

Theorem 9.4.4. Theorem 9.4.1 holds under the additional assumption that there is an Ω˜ y ⊂ Ωs , Ω˜ y ∈ Vεs for some 0 < s  ε such that y ∈ H 1 (Ω˜ y ), kΩ˜ y k∗ ≤ ˜ y | ≤ C1 ρ for a constant C1 = C1 (Ω, M, η). (1 + C1 h∗ )H1 (Jy ) + C1 ρ and |Ω \ Ω Proof. Let again ρ−1 ∈ N, s0 = κε and recall k dist(∇y, SO(2))k2L2 (Ω) ≤ Cε. As

κ  1 was chosen in dependence of T and T = T (ρ, h∗ ) (see (9.96)), we can suppose κ = κ(ρ, h∗ ). Applying Lemma 9.2.2 for s, k = ρ−2 κε, m = ρ and  = ˜ y ∩ Ωk there is a set W ⊂ Ω3k with W ∈ V s , |Ω ˜ y \ W | ≤ C1 k ≤ C1 ρ ρ−2 κε, U = Ω k for ε small enough and ˜ y k∗ + C−1 ε ≤ kΩ ˜ y k∗ + ρ. kW k∗ ≤ kΩ The last inequality holds by choosing κ larger than C . Moreover, there are ˆ i : Ω3k → SO(2), i = 1, . . . , 4, which are constant on Qk (q) ∩ W , mappings R i q ∈ Iik (Ωk ), such that −2 −2 ˜ ˆ i k2 2 k∇y − R L (W ) ≤ Cε + Cερ κkΩy k∗ ≤ Cρ κε. −2 ˆ i k4 4 Clearly, we also get k∇y − R L (W ) ≤ Cρ κε as k∇yk∞ ≤ M . Then we apply 2

sm ˆ Lemma 9.3.1 for k = ρ−2 s0 , ν = s0 , m = ρ and  = cˆρ−3 κε to get sets U ∈ V71k ν ν and U H ∈ V72k with U, U H ⊂ Ω6k , |U \ W | = 0, |U H \ H m (U )| = 0 and

˜ y k∗ + C1 ρ kU k∗ ≤ (1 + C1 ρ)kW k∗ + C−1 ρ−2 κε ≤ kΩ 239

as well as |W \ U | ≤ C1 k ≤ C1 ρ for ε small enough. Moreover, we nd a function yˆ ∈ H 1 (U H ) such that by (9.78)

(i) k dist(∇ˆ y , SO(2))k2L2 (U H ) ≤ CCρ2 (ρ−2 κε + ρ−3 κεkW k∗ ) ≤ CCρ2 ρ−3 κε, (ii) k dist(∇ˆ y , SO(2))k2L∞ (U H ) ≤ CCρ6 , (iii) k∇y − ∇ˆ y k2L2 (U 0 ) ≤ CCρ2 ρ−3 κε,

k∇y − ∇ˆ y k4L4 (U 0 ) ≤ CCρ8 ρ−3 κε,

where the second part of (iii) follows from (ii). Note that this also implies k dist(∇ˆ y , SO(2))k4L4 (U H ) ≤ CCρ8 ρ−3 κε. Setting W1 = U , W1H = U H , y1 = yˆ we can now follow the proof of Theorem 9.4.3 beginning with (9.101) with the essential dierence that we have to replace ε by CCρ8 ρ−3 κε. We then obtain the desired result for a possibly larger constant C2 in (9.83). 

9.4.3 Step 3: General case We are now in a position to prove the general version of Theorem 9.4.1. Proof of Theorem 9.4.1. Let y ∈ SBVM (Ω) be given and let ρ > 0. It suces to ˜ with k˜ ˜ ∈ V s , s > 0, and a function y˜ ∈ H 1 (Ω) y kL∞ (Ω) y kL∞ (Ω) nd a set Ω ˜ + k∇˜ ˜ ≤ ε cM for a universal constant c > 0 such that ˜ ≤ C1 ρ, kΩk ˜ ∗ ≤ (1 + C1 h∗ )H1 (Jy ) + C1 ρ, |Ω \ Ω| (9.110) ky − y˜k2L2 (Ω) y k2L2 (Ω) ˜ + k∇y − ∇˜ ˜ ≤ C1 ερ. Then the result follows from Theorem 9.4.4 applied on the function y˜. (Accordingly, replace M by cM in all estimates.) Note that we cannot just apply density results for SBV functions (see Theorem A.1.6) since in general such approximations do not preserve an L∞ bound for the derivative. The problem may be bypassed by construction of a dierent approximation at the cost of a non exact approximation of the jump set which, however, suces for our purposes. Let µ = ερ. Recall that Jy is rectiable (see [6, Section S 2.9] ), i.e. there is a countable union of C 1 curves (Γi )i∈N such that H1 (Jy \ i Γi ) = 0. Covering Jy with small balls and applying Besicovitch's covering theorem (see [40, Corollary 1, p. 35]) we nd nitely many closed, pairwise S disjoint balls Bj = Brj (xj ), j = 1, . . . , n with rj ≤ µ such that H1 (Jy \ nj=1 Bj ) ≤ µ. Moreover, we get H1 (Jy ∩ Bj ) ≥ 2(1 − µ)rj and for each Bj we nd a C 1 curve Γij such that µ Γij ∩ Bj is connected and H1 ((Γij 4Jy ) ∩ Bj ) ≤ 2µrj ≤ 1−µ H1 (Jy ∩ Bj ). For a detailed proof we refer to [26, Theorem 2]. √ We choose rectangles Rj with |∂Rj |∞ ≤ 2 2rj such that H1 (Γij ∩(Bj \Rj )) = 0 and |∂Rj |∞ ≤ H1 (Γij ∩ Bj ). We then obtain X X |∂Rj |∞ ≤ H1 (Γij ∩ Bj ) j j  µ X 1 ≤ 1+ H (Jy ∩ Bj ) ≤ (1 + C1 µ)H1 (Jy ) j 1−µ 240

P and likewise j |∂Rj |H ≤ C1 H1 (Jy ). Choose rectangles Qj with Rj ⊂⊂ Qj such that |∂Qj |∗ ≤ (1 + µ)|∂Rj |∗ and [  1 H ∂Qj ∩ Jy = 0. (9.111) j

As before it is not hard to see that Rj1 \ Rj2 is connected for 1 ≤ j1 , j2 ≤ n. The rectangles (Qj )j can be chosen in a way such that they fulll the same property. Possibly replacing the rectangles by innitesimally larger rectangles we can assume that there is some s > 0 such that Rj , Qj ∈ U s for S j = 1, . . . , n. s ρ Then by Lemma 9.1.2(i) we nd sets W, V ∈ Vε with |V 4(Ω \ j Rj )| = 0 and S |W 4(Ωρ \ j Qj )| = 0. Note that W ◦ ⊂⊂ V ◦ and |Ω \ W | ≤ C1 ρ. It is not restrictive to assume that corners of Rj , Qj do not coincide and thus W ◦ , V ◦ are Lipschitz domains. We get (recall Lemma 8.1.1) X kW k∗ ≤ (1 + µ) |∂Rj |∗ ≤ (1 + C1 ρ + C1 h∗ )H1 (Jy ). (9.112) j

Sn

Moreover, as H (Jy \ j=1 Bj ) ≤ µ we get  [n  [n Rj ) ≤ µ + H1 Jy ∩ (Bj \ Rj ) H1 (Jy \ j=1 j=1   [n   [n ≤ µ + H1 Γij ∩ (Bj \ Rj ) + H1 (Γij 4Jy ) ∩ Bj 1

j=1

j=1

µ ≤µ+ H1 (Jy ) ≤ C1 µ, 1−µ

(9.113)

where in the last step we have used H1 (Γij ∩ (Bj \ Rj )) = 0. We now show that y k2L2 (W ) ≤ C1 ερ there is a function yˆ ∈ SBV (W ◦ ) with ky − yˆk2L2 (W ) + k∇y − ∇ˆ such that k∇ˆ y k∞ ≤ cM and Jyˆ is a nite union of closed segments satisfying 1 H (Jyˆ) ≤ C1 µ ≤ C1 ρ. We apply a result by Chambolle obtained in [26] in an SBD-setting and rather cite the result as repeating the arguments. Therefore, we rst obtain a control only over the symmetric part of the gradient. To derive the desired result we repeat the arguments for the function v = (y 2 , y 1 ) instead of y = (y 1 , y 2 ) to control also the skew part. We dene ˆ ◦ E(y, W ) = V (e(∇y)) + H1 (Jy ∩ W ◦ ) ◦ W ´ 1 ◦ ◦ (ξ T Aξ)2 dξ and Ec (y, W ) = E(y, W ) + cH1 (Jy ∩ W ◦ ), where V (A) := 2π S1 2×2 ◦ 2 ◦ ◦ for A ∈ R . As y ∈ SBVM (W ) ∩ L (W ) with E(y, W ) < +∞ and W ◦ has Lipschitz boundary, by [26, Theorem 1] we nd a sequence yn ∈ SBD(W ◦ ) ∩ L2 (W ◦ ) with kyn −ykL2 (W ◦ ) → 0 such that Jyn is a nite union of closed segments and

lim sup E(yn , W ◦ ) ≤ Ec (y, W ◦ ) ≤ E(y, W ◦ ) + C1 µ n→∞ ˆ ≤ V (e(∇y)) + C1 µ. W◦

241

(9.114)

In the second and third step we used (9.113). The proof is based on a discretization argument. Consequently, as a preparation an extension y 0 to some set W 0 ⊃⊃ W ◦ with E(y 0 , W 0 ) ≤ E(y, W ◦ ) + δ for arbitrary δ > 0 had to be constructed (see [26, Lemma 3.2]). In our framework we can choose y 0 = y due to W ◦ ⊂⊂ V ◦ and (9.111). Moreover, kyn k∞ ≤ kyk∞ holds. Although not stated explicitly in the theorem, the approximations satisfy k∇yn kL∞ (W ◦ ) ≤ ck∇y 0 kL∞ (W 0 ) ≤ ck∇ykL∞ (V ) ≤ cM . (For a precise argument see the proof of [25, Theorem 3.1], where a similar construction is used.) Strictly speaking, the theorem only states that Jyn is essentially closed and contained in a nite union of closed segments. However, the proof shows that up to an innitesimal perturbation of yn (do not set yn = 0 on a `jump square', but yn = c˜ for c˜ ≈ 0) the desired property can be achieved. By [26, Lemma 5.1] we obtain weak convergence e(∇yn ) * e(∇y) in L2 (W ◦ ) up Together with the lower semicontinuity results ´ to a not relabeled subsequence. ´ V (e(∇y)) ≤ lim inf V (e(∇yn )) and H1 (Jy ) ≤ lim inf n→∞ H1 (Jyn ) n→∞ W ◦ W◦ (see [26, Lemma 5.1]) we nd by (9.114) ˆ ˆ ˆ V (e(∇y)) + C1 µ. V (e(∇yn )) ≤ V (e(∇y)) ≤ lim sup W◦

n→∞

W◦

W◦

Consequently, by weak convergence we obtain ˆ 2 lim sup ke(∇yn ) − e(∇y)kL2 (W ◦ ) ≤ c lim sup n→∞

n→∞

≤ c lim sup n→∞

◦ W ˆ



V (e(∇yn − ∇y)) ˆ V (e(∇yn )) −

W◦

 V (e(∇y))

W◦

≤ C1 µ = C1 ερ. Then by (9.114) we also get lim supn→∞ H1 (Jyn ) ≤ C1 µ ≤ C1 ρ. We now repeat the argument for v = (y 2 , y 1 ) instead of y and observe that by construction the approximations can be chosen as vn = (yn2 , yn1 ). We nd that yn ∈ SBV (W ◦ ) and lim supn→∞ k∇yn − ∇yk2L2 (W ◦ ) ≤ C1 ερ. Now choose n large enough such that yˆ := yn satises

ky − yˆk2L2 (W ◦ ) + k∇y − ∇ˆ y k2L2 (W ◦ ) ≤ C1 ερ,

H1 (Jyˆ) ≤ C1 ρ

for C1 > 0 large enough. Choose a nite number of closed segments (Si )m i such S S ◦ 1 that Jyˆ ∩ W ⊂ i Si and H ( i Si ) ≤ C1 ρ. For s > 0 small choose Ti ∈ U s as the smallest rectangle with Si ⊂ Ti . Then by Lemma 9.1.2(i) we obtain a set ˜ ∈ Vεs with Ω [m
˜ W\ Ω4 Tm = 0. j=1

˜ ∗ ≤ kW k∗ + C1 ρ and |W \ Ω| ˜ ≤ Observe that for s suciently small we obtain kΩk C1 ρ. This together with (9.112) gives the two rst parts of (9.110). Finally, dene ˜ by y˜ = yˆ| ˜ and observe that y˜ satises (9.110). the function y˜ ∈ H 1 (Ω)  Ω 242

9.5 Proof of the main SBD-rigidity result This Section is devoted to the proof of the main SBD-rigidity result. We start with some preparations and then split up the proof into two steps concerning a suitable construction of the jump set and the denition of an extension. As before constants indicated by C1 only depend on M, η, Ω and all constants do not depend on ρ and q unless stated otherwise. Let y ∈ SBVM (Ω) be given and let ρ > 0, % = ρq for q ∈ N to be specied below. Set k = ρq−1 and m = ρ. Recall the denition Ωρ = {x ∈ Ω : dist(x, ∂Ω) > s for s Cρ}. We apply Theorem 9.4.1 and obtain a set Ωy ⊂ Ωρ with Ωy ∈ Vck suciently small and |Ω \ Ωy | ≤ C1 ρ such that (9.83) holds for a modication y k2L2 (Ωy ) ≤ C1 ερ. Recall y˜ ∈ H 1 (Ωy ) ∩ SBVcM (Ωy ) with ky − y˜k2L2 (Ωy ) + k∇y − ∇˜ 3% from the proof of Theorem 9.4.2 and Corollary 9.2.6 that there is a set ΩH y ∈ Vck 2 with Ω◦y ⊂ ΩH ˆ : ΩH ˜ satisfying (9.70) and estimates y and an extension y y → R of y of the form (9.69). We rst construct a modication of ΩH y and appropriate Jordan curves which separate the connected components. For a (closed) Jordan curve γ we denote by int(γ) the interior of the curve. As connected components may be not S simply connected we further introduce a generalization: We say a curve γ = γ0 ∪ m j=1 γj is a generalized Jordan curve if it consists of pairwise disjoint Jordan curves γ0 , . . . , γm with γjS∈ int(γ0 ) for j = 1, . . . , m. We dene the interior of γ by int(γ) = int(γ0 ) \ m j=1 int(γj ).

Lemma 9.5.1. Let ρ > 0, M > 0 and q ∈ N. There is a constant C1 = C1 (M ) > 3% 1 ˆ ˆ 0 such that for all ΩH y ∈ Vck as given above we nd Ω ⊂ Ωρ with H (∂ Ω) ≤ C1 , ˆ ˆ |ΩH y \ Ω| ≤ C1 ρ and a set S ⊂ Ωρ \ Ω such that (i) H1 (S) ≤ kΩHy k∗ + C1 ρ, (ii) for all Pˆi there is a generalized Jordan curve γ in S ∪ ∂Ωρ such that ˆ = Pˆi , where (Pˆi )i denote the connected components of Ω ˆ, int(γ) ∩ Ω (iii) int(γ) ∩ Ωˆ 6= ∅ for all Jordan curves γ in S ∪ ∂Ωρ , (iv) dist(x, S) ≤ C1 ρq−2 for all x ∈ Ωρ \ Ωˆ , ˆ is connected for all components Xt (Ω) ˆ of Ωρ \ Ω. ˆ (v) (S ∪ ∂Ωρ ) ∩ Xt (Ω)

Proof. In contrast to the previous sections, where it was essential to avoid the

combination of dierent cracks, we now combine boundary components: Choose 3% H H ˆH ˆH ˆH ˆH ˆH a set Ω satisfying Ω y ∈ V y ⊂ Ωy , |Ωy \ Ωy | = 0 and |Γj (Ωy ) ∩ Γl (Ωy )|H = 0 1 H ˆH for j 6= l. Clearly, by (9.88) and (9.48) we have H1 (Ω y ) ≤ H (Ωy ) ≤ C1 . ˆ H satisfying |∂Yj |∞ ≤ Letting Y1 , . . . , Ym be the connected components of Ω y S ˜ = Ω ˆ H \ m Yj . As |∂Yj |∞ ≤ ρq−2 for j = ρq−2 for j = 1, . . . , m we dene Ω y j=1 243

S q−2 ˆ H 1, . . . , m, the isoperimetric inequality implies | m kΩy kH ≤ C1 ρ j=1 Yj | ≤ C1 ρ H ˜ and thus |Ωy \ Ω| ≤ C1 ρ. ˜ be the largest set in U ρq−2 such that dist∞ (x, ∂ Ω ˜ \ ∂Ωρ ) ≥ ρq−2 Let Z ⊂ Ωρ \ Ω ˆ =Ω ˜ ∪ Z. (Observe that Z is typically not connected.) for all x ∈ Z and dene Ω It is not hard to see that ˆ \ ∂Ωρ ) ≤ C1 ρq−2 dist(x, ∂ Ω

ˆ for all x ∈ Ωρ \ Ω.

(9.115)

1 ˆ ˆ Moreover, we get |ΩH y \ Ω| ≤ C1 ρ and H (Ω) ≤ C1 . In fact, for each connected i component Z i of Z we nd boundary components (Xji = Xji (ΩH y ))j and (Yj )j such S S that ∂Z i ⊂ j Xji ∪ j Yji and thus by |∂Xji |∞ ≤ cρq−1 , |∂Yji |∞ ≤ ρq−2 we obtain P P |∂Z i |H ≤ C1 ( j |∂Xji |H + j |∂Yji |H ). We recall H1 (ΩH y ) ≤ C1 and observe that S i1 S i1 S S i2 i1 for dierent components Z , Z one has ( j Xj ∪ j Yj )∩( j Xji2 ∪ j Yji2 ) = ∅. ˆ and dene F(Pˆi ) = {Xj = Let Pˆ1 , . . . , Pˆn be the connected components of Ω H ˆ Xj (ΩH y ) : Xj ∩ Pi 6= ∅}. (Here it is essential that we take the components of Ωy .) By Zj ∈ U 3% we denote the smallest rectangle containing Xj . (I) As a preparation we consider the special case that there is only one conˆ nected component Pˆ1 . Moreover, we rst suppose that S Ωρ \ Ω consists of one connected component only. We can choose a set S in Zj ∈F (Pˆ1 ) Zj consisting of ˆ is connected, segments such that S ∪ (∂Ωρ \ Ω)

H1 (S) ≤ (1 + C1 ρ)

X Xj ∈F (Pˆ1 )

|Γj |∞ ≤ (1 + C1 ρ)

X Xj ∈F (Pˆ1 )

|Γj |∗

(9.116)

and dist(x, S) ≤ C1 ρq−2 for all x ∈ ∂ Pˆ1 \ ∂Ωρ for a suciently large constant. Indeed, a set with the desired properties can be constructed in the following way. By the denition of | · |∞ we rst see that we can choose a piecewise ane Jordan S curve γ in Xj ∈F (Pˆ1 ) Zj ∪ ∂Ωρ such that Pˆ1 ⊂ int(γ) and S0 := γ ∩ Ω◦ρ satises P H1 (S0 ) ≤ Xj ∈G(S0 ) |Γj |∞ , where G(S0 ) = {Xj : Xj ∩ S0 6= ∅}. (If γ ∩ Ω◦ρ = ∅, we ˆ .) Assume a connected set Sl consisting let S0 = {p0 } for some point p0 ∈ Ωρ \ Ω of segments has been constructed such that X H1 (Sl ) ≤ |Γj |∞ + C1 lρq−1 . (9.117) Xj ∈G(Sl )

If dist(x, Sl ) ≤ C1 ρq−2 for all x ∈ ∂ Pˆ1 \∂Ωρ , we stop. Otherwise, there is some y ∈ ∂ Pˆ1 \∂Ωρ such that dist(y, Sl ) > C1 ρq−2 . By the denition of |·|∞ it is elementary to see that we can nd a piecewise ane, continuous curvePTl+1 with Tl+1 ∩Sl 6= ∅, y ∈ Tl+1 , #(G(Tl+1 ) ∩ G(Sl )) = 1 such that H1 (Tl+1 ) ≤ Xj ∈G(Tl+1 ) |Γj |∞ . Then √ q−1 using that |Γ(ΩH and #(G(Tl+1 ) ∩ G(Sl )) = 1 we nd y )|∞ ≤ 2 2 · ck ≤ C1 ρ that (9.117) is satised for Sl+1 := Sl ∪ Tl+1 . After a nite number of iterations n ∈ N we nd that dist(y, Sn ) ≤ C1 ρq−2 for all y ∈ ∂ Pˆ1 \ ∂Ωρ and set S∗ = Sn . Indeed, this follows from the fact that in 244

each iteration G(Sl ) increases and the assertion clearly holds if Sl intersects all H q−1 boundary components since maxP . As H1 (Tl ) > C1 ρq−2 , it is j |Γj (Ωy )|∞ ≤ C1 ρ not hard to see that n ≤ C1 ρ2−q Xj ∈F (Pˆ1 ) |Γj |∞ and thus (9.116) holds replacing S by S∗ . ˆ is not connected. Therefore, we choose Observe that possibly S∗ ∪ (∂Ωρ \ Ω) ˆ (which may be several if some point y in each connected component of ∂Ωρ \ Ω Ωρ is not simply connected) and repeat the construction below (9.117) for each ˆ is connected. y . We obtain a set S such that (9.116) still holds and S ∪ (∂Ωρ \ Ω) ˆ consists of several connected components Xt (Ω) ˆ , we repeat the arguIf Ωρ \ Ω ments on each component separately possibly starting with S0 = {p0 } for some ˆ . p0 ∈ Xt (Ω) We see that (i),(v) are satised, (ii) holds with γ and (iii) follows from the fact that in the construction of the sets Tl above we do not obtain additional `loops'. ˆ satises dist(x, ∂ Pˆ1 \ Moreover, (iv) follows from the fact that each x ∈ Ωρ \ Ω q−2 ∂Ωρ ) ≤ C1 ρ by (9.115). (II) We now consider an arbitrary number of connected components. Choose S ˆ and H1 (γ i ∩Ω◦ ) ≤ Jordan curves γ i in Xj ∈F (Pˆi ) Zj ∪∂Ωρ such that Pˆi ⊂ int(γ i )∩Ω ρ P i i ˆ ˆ We rst assume that Pi = int(γ ) ∩ Ω, i.e. int(γ ) does not i |Γj |∞ . Xj ∈G(γ )

ˆ , and treat the general case in (III). As the sets contain other components of Ω n ˆ (F(Pi ))i=1 might be not disjoint, S we have to combine the dierent curves in a suitable way. Dene Gi = Xj ∈G(γ i ) Zj . It is not restrictive to assume that S 1≤i≤n Gi is connected as otherwise we apply the following arguments on each component separately. For B ⊂ R2 we dene Int(B) = {x ∈ R2 : ∃ Jordan curve γ i in B : x ∈ int(γ i )}. It is not hard to see S that we can order the sets (Pˆi )iSin a way such that for all 1 ≤ l ≤ n we have 1≤i≤l Gi is connected and Int( 1≤i≤l Gi ) ∩ Pˆj = ∅ for all j > l. In fact, to see the second property, assume the rst l sets G1 , . . . , Gl have already been chosen. Select some other component Pˆk , k > l, with corresponding Gk . If the desired property is satised, we reorder and set Gl+1 = Gk , otherwise S we nd some Pˆk0 , k 0 > l, k 0 6= k , with corresponding Gk0 such that Pˆk0 ⊂ Int( 1≤i≤l Gi ∪ Gk ). Possibly repeating this procedure we nally nd a set S Gl+1 such that Int( 1≤i≤l+1 Gi ) ∩ Pˆj = ∅ for all j > l + 1. We now proceed iteratively. Set S0 = ∅ and assume a connected set Sl has been constructed with X (a) H1 (Sl ∩ Ωρ ) ≤ (1 + C1 ρ) |Γj |∗ + C1 (l − 1)ρq−1 , S i Xj ∈

1≤i≤l

G(int(γ ))

ˆ = Pˆi , (b) for all 1 ≤ i ≤ l there is a Jordan curve γ in Sl such that int(γ) ∩ Ω [l (c) dist(x, Sl ) ≤ C1 ρq−2 for all x ∈ ∂ Pˆi \ ∂Ωρ . (9.118) i=1

245

S Let Tl+1 be the (unique) connected component of γ l+1 \ 1≤i≤l Gi such that Pˆl+1 ⊂ S j j Int( 1≤i≤l Gi ∪ Tl+1 ). Now choose two segments Tl+1 , j = 1, 2, with H1 (Tl+1 )≤ S j j j q−1 ˆ C1 ρ , Tl+1 ∩ Sl 6= ∅, Tl+1 ∩ Tl+1 6= ∅ such that Sl+1 := Sl ∪ Tl+1 ∪ j=1,2 Tl+1 satises Pˆl+1 ⊂ Int(Sˆl+1 ) and X H1 (Sˆl+1 ∩ Ωρ ) ≤ (1 + C1 ρ) |Γj |∗ + C1 lρq−1 . S i Xj ∈

1≤i≤l

G(int(γ ))∪G(Tl+1 )

By the order of the sets (Pˆi )i it is not hard to see that there is a Jordan curve ˆ = Pˆl+1 . Observe that dist(x, γ l+1 ) ≤ C1 ρq−2 for all γ in Sˆl+1 with int(γ) ∩ Ω ˆ x ∈ ∂ Pl+1 \ ∂Ωρ might not hold. Therefore, following the lines of (I) we choose a (possibly not connected) set Rl+1 ⊂ int(γ l+1 ) such that such that Sl+1 := ˆ , dist(x, Sl+1 ) ≤ C1 ρq−2 for Sˆl+1 ∪ Rl+1 is connected in each component of Ωρ \ Ω all x ∈ ∂ Pˆl+1 \ ∂Ωρ and X H1 (Rl+1 ) ≤ (1 + C1 ρ) |Γj |∗ . l+1 ˆ Xj ∈G(int(γ

))\G(Sl+1 )

Now it is not hard to see that (a)-(c) are satised for Sl+1 . After the last iteration step we dene S∗ = Sn ∩Ωρ . Observe that by construction (see before (9.115)) each Pˆi satises |∂ Pˆi |∞ ≥ ρq−2 . Thus n ≤ C1 ρ2−q and q−1 then we obtain H1 (S∗ ) ≤ kΩH ≤ C1 ρ. Similarly as before, y k∗ + C1 ρ since nρ ˆ . Consequently, we S∗ ∪ ∂Ωρ might not be connected in the components of Ωρ \ Ω proceed as in (I) (see construction below (9.117)) to nd a set S ⊃ S∗ such that ˆ . This gives (i) still holds and S ∪ ∂Ωρ is connected in the components of Ωρ \ Ω (v). Moreover, (b) implies (ii) and similarly as in (I) also (iii) holds. (Here we do not have to consider generalized Jordan curves.) Finally, to see (iv) we use (c) ˆ satises dist(x, ∂ Ω ˆ \ ∂Ωρ ) ≤ C1 ρq−2 by (9.115). and the fact that each x ∈ Ωρ \ Ω (III) We now nally treat the case that the components (Pˆi )ni=0 may also ˆ . To simplify the exposition we assume that there contain other components of Ω ˆ . The general case is exactly one component, say Pˆ0 , such that Pˆ0 6= int(γ 0 ) ∩ Ω follows by inductive application of the following arguments. ˆ ) and construct a set We proceed as in (II) (assuming we had Pˆ0 = int(γ 0 ) ∩ Ω 0 S particularly satisfying (i),(iii),(v). We have to verify (ii) for Pˆ0 and nd a set S ⊃ S 0 such that (iv) is satised and (i),(iii),(v) still hold. By (Pˆij )j we denote the components with Pˆij ⊂ int(γ0 ). As (ii) holds for these components we nd S S pairwise disjoint Jordan curves γ1 , . . . , γm with j Pˆij ⊂ m int(γj ) ⊂ int(γ0 ). Sj=1 m Consequently, dening the generalized Jordan curve γ = j=0 γj we nd Pˆ0 = ˆ which gives (ii). int(γ) ∩ Ω ˆ which are completely contained in Let (Yj )j be the components of Ωρ \ Ω S S int(γ0 ). We observe that (iv) may be violated for x ∈ Y ∗ := j Yj \ m j=1 int(γj ). ∗ We now proceed similarly as in (I) to obtain a set R ⊂ Y such that S := S 0 ∪R is ˆ and dist(x, S) ≤ C1 ρq−2 for all connected in the connected components of Ωρ \ Ω 246

x ∈ ∂ Pˆ1 ∩ Y ∗ . This implies (iii),(v) and together with (9.115) also (iv). Arguing similarly as in (II) we nd that (i) is still satised since the sum in (9.118)(a) does not run over the components contained in Y ∗ .  We nally can give the proof of Theorem 6.1.1 by constructing an extension yˆ of the function y˜. We briey note that the function yˆ has to be dened as an extension of the approximation and not of the original deformation y as only in this case we obtain the correct surface energy due to the higher regularity of the jump set of y˜ and the available trace estimates. Recall the denition of Eερ (y, U ) in (6.3), in particular fερ (x) = min{ √xερ , 1}. Proof of Theorem 6.1.1. Let Ωy ⊂ Ωρ with Ωy ∈ V s and ΩHy ∈ V 3% with Ω◦y ⊂ ΩHy be given. Recall that |Ω \ Ωy | ≤ C1 ρ. Let y˜ ∈ H 1 (Ωy ) be the approximation of y ∈ SBVM (Ω) with ky − y˜k2L2 (Ωy ) + k∇y − ∇˜ y k2L2 (Ωy ) ≤ C1 ερ and let yˆ ∈ 2 ˆ be the set SBVcM (ΩH ˜ given by Corollary 9.2.6. Let Ω y , R ) be the extension of y H ˆ ◦ . By constructed in Lemma 9.5.1. We rst consider the jumps of yˆ in (Ωy ∩ Ω) (9.76) and Hölder's inequality we nd ˆ ˆ 2  X 2 1 1 |[ˆ y ]| dH ≤ |[ˆ y ]| dH H Qt ⊂Ωy

◦ Jyˆ ∩(ΩH y )

≤

X Qt

Jyˆ ∩Qt

|Jyˆ ∩ Qt |H ·

≤ CH1 (Jyˆ) ·

X Qt

X Qt

|Jyˆ ∩

Qt |−1 H

ˆ

|[ˆ y ]| dH1

2

Jyˆ ∩Qt

CCρ2 %2 (γ(Nt ) + δ4 (Nt ) + |∂W ∩ Nt |H ),

where W as dened in (9.85), Nt := N (Qt ) = {x P ∈ W : dist(x, Qt ) ≤ Cρq−1 } ˆ i k4 4 y−R and γ(Nt ) = k∇ dist(∇ˆ y , SO(2))k2L2 (W ) , δ4 (Nt ) = 4i=1 k∇ˆ L (W ) (recall −2 (9.87)). As each x ∈ Ω is contained in at most ∼ ρ dierent Nt we nd by (9.84), (9.85), (9.87), (9.70) and the fact that  = cˆρ−1 ε ˆ 2 |[ˆ y ]| dH1 ≤ Cρ−2 · CCρ2 %2  ≤ C%2 ρ−3 Cρ2 ε. ˆ ◦ Jyˆ ∩(ΩH y ∩Ω)

ˆ i were dened (Note that in the general case the set W and the rigid motions R dierently (see e.g. (9.109)), but here and in the following we prefer to refer to the proof of Theorem 9.4.2 for the sake of simplicity.) By Remark 8.4.3, 8.5.8 we get for q = q(h∗ ) suciently large ˆ √ √ 3√ 3 |[ˆ y ]| dH1 ≤ CCρ ρq− 2 ε = Cρq−( 2 +z) ε ≤ ρ2 ε. ˆ ◦ Jyˆ ∩(ΩH y ∩Ω)

Recalling that fερ (x) ≤ ρ−1 √xε for x ≥ 0 we get ˆ ˆ ρ 1 −1/2 −1 fε (|[ˆ y ]|) dH ≤ ε ρ ˆ ◦ Jyˆ ∩(ΩH y ∩Ω)

ˆ ◦ Jyˆ ∩(ΩH y ∩Ω)

247

|[ˆ y ]| dH1 ≤ ρ.

(9.119)

ˆ . Let Yt be a connected We now concern ourselves with the components of ∂ Ω ˆ ∪ S), where S is the set constructed in Lemma 9.5.1. Set component of Ωρ \ (Ω ˆ . We observe that by Lemma 9.5.1(ii),(iii) Γt is a St = S ∩ Yt and Γt = Yt ∩ ∂ Ω Jordan curve if Yt ∩ ∂Ωρ = ∅. ˆ and for Γt we choose J(Γt ) ⊂ J such that Q% (p) ∩ Γt 6= ∅ for Dene J = I % (Ω) S all p ∈ J(Γt ). We set M (Γt ) = p∈J(Γt ) Q% (p). For later purpose, for components with |Γt |∞ > 2ρq−2 we introduce a ner of M (Γt ): Dene J(Γt ) = S partition % ˙ n and the connected sets Bi = p∈I Q (p) such that ρ−2 ≤ #Ii ≤ Cρ−2 , I1 ∪˙ . . . ∪I i i = 1, . . . , n, for a constant C  1. For |Γt |∞ ≤ 2ρq−2 we let I1 = J(Γt ). It is elementary to see that n ≤ max{C|Γt |H ρ2−q , 1} ≤ C|Γt |H ρ−q , where we used |Γt |H ≥ Cρq . ¯ j : ΩH → SO(2) and c¯j : ΩH → R2 , j = 1, . . . , 4, as given in (9.90). Consider R y y H ˜ ˆ Recall the denition Ω = Ω \ Z ⊂ Ωy before (9.115). We extend the function yˆ ˆ by setting yˆ = id on Ω ˆ \Ω ˜ and likewise let R ¯ j = Id, c¯j = 0 on Ω ˆ \Ω ˜ . (If to Ω Z ∩ ΩH y 6= ∅, we redene the function on this set.) Applying Corollary 9.2.6 on 3% ˆ each Q3% j (p) ⊂ Ω with Qj (p) ∩ M (Γt ) 6= ∅, we get ¯ j · +¯ kˆ y − (R cj )k2L2 (Bi ) ≤ C%2 Cρ2 · ρ−2 ρq−1  · #Ii = Cρ3q−6 Cρ2 ε, ¯ j · +¯ kˆ y − (R cj )k2L1 (∂Bi ) ≤ Cρ3q−6 Cρ2 ε,

(9.120)

for j = 1, . . . , 4 and i = 1, . . . , n. Here we used k = ρq−1 ,  = cˆερ−1 and the fact −2 that each N (Q3% = ρ−2 dierent Q3% (p) ⊂ ΩH y . The triangle j (p)) contains ∼ m inequality then yields

¯ j1 · +¯ ¯ j2 · +¯ k(R cj1 ) − (R cj2 )k2L2 (Bi ) ≤ Cρ3q−6 Cρ2 ε for 1 ≤ j1 , j2 ≤ 4 and dierent rigid motions. Remark 7.1.4(iii) to get

i = 1, . . . , n. The strategy will be to cover Yt with n We argue as in (9.91)f. and Lemma 7.1.3(ii) recalling ˆ Ri ∈ SO(2), cˆi ∈ R2 such that

ˆ i · +ˆ kˆ y − (R ci )k2L2 (Bi ) ≤ C(#Ii )4 ρ3q−6 Cρ2 ε ≤ Cρ3q−14 Cρ2 ε. Here we used Hölder's inequality (cf. (7.19)). A similar argument shows that we even nd X ˆ i+j · +ˆ kˆ y − (R ci+j )k2L2 (Bi ) ≤ Cρ3q−14 Cρ2 ε (9.121) j=−1,0,1

ˆ n+1 = R ˆ1, for i = 1, . . . , n, where (in the case that Γt is a Jordan curve) we set R ˆ ˆ cˆn+1 = cˆ1 and R0 = Rn , cˆ0 = cˆn . Without restriction recalling Remark 7.1.4(v) ˆ i ∈ imR¯ (M (Γt )) ⊂ SO(2), where imR¯ denotes the image we can assume that R 4 4 ¯ 4 . For shorthand let R ¯=R ¯ 4 and c¯ = c¯4 . By (9.120) and (9.121) of the function R we get X ˆ i+j · +ˆ ¯ · +¯ k(R ci+j ) − (R c)k2L2 (Bi ) ≤ Cρ3q−14 Cρ2 ε. (9.122) j=−1,0,1

248

Using Hölder's inequality and passing to the trace (argue as in (7.10). on each Q3% (p)) we obtain for all i = 1, . . . , n X ˆ i+j · + cˆi+j ) − (R ¯ · +¯ k(R c)k2L1 (Bi ∩Γt ) j=−1,0,1 X ˆ i+j · +ˆ ¯ · +¯ ≤C |Bi ∩ Γt |H k(R ci+j ) − (R c)k2L2 (Bi ∩Γt ) j=−1,0,1 −2 −1 3q−14

≤ C%ρ

·% ρ

Cρ2 ε ≤ Cρ3q−16 Cρ2 ε.

Together with (9.120) this implies X ˆ i+j · +ˆ kˆ y − (R ci+j )k2L1 (Bi ∩Γt ) ≤ Cρ3q−16 Cρ2 ε. j=−1,0,1

This and the fact that n ≤ C|Γt |H ρ−q yield X X √ q ˆ i+j · +ˆ H1 := kˆ y − (R ci+j )kL1 (Bi ∩Γt ) ≤ C|Γt |H ρ 2 −8 Cρ ε. (9.123) i

j=−1,0,1

For the dierence of the rigid motions we get by the triangle inequality and (9.121) X ˆ i+j1 · +ˆ ˆ i+j2 · +ˆ k(R ci+j1 ) − (R ci+j2 )k2L2 (Bi ) ≤ Cρ3q−14 Cρ2 ε. j1 ,j2 =−1,0,1

˜i = {x ∈ Ω : dist(x, Bi ) ≤ Cρ ¯ q−2 }. Arguing similarly as in (7.14) it is not Let B hard to see that X ˆ i+j1 · +ˆ ˆ i+j2 · +ˆ k(R ci+j1 ) − (R ci+j2 )k2L2 (B˜i ) j1 ,j2 =−1,0,1 (9.124) ≤ C(ρ−2 )2 · ρ−4 · ρ3q−14 Cρ2 ε ≤ Cρ3q−22 Cρ2 ε ˜ i |∞ ˜i | |∂ B |B −4 ˜i ). Again using Hölder's ≤ Cρ and ≤ Cρ−2 . Dene I˜i = I % (B as |B | |∂B i i |∞ inequality, passing from the traces to a bulk integral and recalling n ≤ C|Γt |H ρ−q , ˆ i+j1 · +ˆ ˆ i+j2 · +ˆ #I˜i ≤ Cρ−4 we derive (let · = (R ci+j1 ) − (R ci+j2 ) for shorthand) X X X H2 := k · kL1 (∂Q% (p)) i p∈I˜i j1 ,j2 =−1,0,1 X X X %1/2 k · kL2 (∂Q% (p)) ≤C i p∈I˜i j1 ,j2 =−1,0,1 X 1/2 (9.125) X X 1 2 ˜ 2 %k · k ≤C (#Ii ) 2 (∂Q% (p)) L p∈I˜i j1 ,j2 =−1,0,1 i X 1/2 X √ q ≤C ρ−2 k · k2L2 (B˜i ) ≤ C|Γt |H ρ 2 −13 Cρ ε. i

j1 ,j2 =−1,0,1

ˆ By (Tj )j we denote the connected components of Q% (p)\(Ω∪S) for all Q% (p) with Q% (p) ∩ Yt 6= ∅. We now choose suitable rigid motions: Observe that dist(Γt ∪ ∂Ωρ , x) ≤ C1 ρq−2 for all x ∈ Yt by Lemma 9.5.1(iv) and the fact that Yt is a 249

ˆ connected component of Ωρ \(Ω∪S) . Therefore, for every Tj with dist(Tj , ∂Ωρ )  q−2 ρ we nd some (possibly non unique) Bij with dist(Tj , Bij ) ≤ Cρq−2 . In ˜i choosing C¯ in the denition of B ˜i large enough. We particular, we get Tj ⊂ B j dene ˆ i x + cˆi yˆ(x) = R j j

for x ∈ Tj ∩ Yt ∩ Ω2ρ

(9.126)

for all j and note that we have found an extension yˆ to Yt ∩ Ω2ρ . (If Yt ∩ ΩH y 6= ∅, we redene the function on this set.) Taking Lemma 9.5.1(v) into account the choice of Bij can be done in a way that for neighboring sets T1 , T2 with T 1 ∩T 2 6= ∅ one has i1 − i2 ∈ {−1, 0, 1} and that H1 (Jyˆ ∩ Yt ) ≤ C1 H1 (Γt ). Now by (9.123) and (9.125) it is not hard to see that ˆ √ q |[ˆ y ]| dH1 ≤ CH1 + CH2 ≤ C|Γt |H ρ 2 −13 Cρ ε. (Jyˆ ∩Yt )\S

Repeating the arguments for all components Yt we obtain a conguration yˆ ∈ ˜ , where by Lemma 9.5.1 we have SBVcM (Ωρ , R2 ) with yˆ = y˜ on Ω∗y := Ωy ∩ Ω ∗ |Ω \ Ωy | ≤ C1 ρ. (With a slight abuse of notation we replace Ω∗y by Ωy in the ˆ ≤ C1 assertion of Theorem 6.1.1.) Summing over all Yt and recalling that H1 (∂ Ω) by Lemma 9.5.1 we get X ˆ q fερ (|[ˆ y ]|) dH1 ≤ Cρ 2 −13 Cρ ≤ ρ t

(Jyˆ ∩Yt )\S

for q = q(h∗ ) suciently large. Together with (9.119), Lemma 9.5.1(i) and (9.83)(i) this implies ˆ ˆ ρ 1 fε (|[ˆ fερ (|[ˆ y ]|) dH ≤ y ]|) dH1 + H1 (S) ≤ (1 + C1 h∗ )H1 (Jy ) + C1 ρ. Jyˆ

Jyˆ \S

Choosing h∗ = ρ we nally get ˆ fερ (|[ˆ y ]|) dH1 ≤ H1 (Jy ) + C1 ρ.

(9.127)

Jyˆ

We observe ∇ˆ y ∈ SO(2) on Ωρ \ Ωy (see construction in Corollary 9.2.6, (9.126) ˆ \Ω ˜ ). As k˜ and recall yˆ = id in Ω y − yk2L2 (Ωy ) + k∇˜ y − ∇yk2L2 (Ωy ) ≤ C1 ερ we obtain Eερ (ˆ y , Ωρ ) ≤ Eε (y) + C1 ρ which gives (6.4). Here we used k∇˜ y k∞ + k∇yk∞ ≤ cM and the regularity of the stored energy density W . Let (Pj )j be the connected components of Ωρ \ S . By Lemma 9.5.1(ii),(iii) it is not hard to see that for every index j there is a (unique) connected component ˆ such that Pˆj ⊂ Pj . Then there is either a connected component P H Pˆj of Ω j H H ˆ ˆ of Ωy such that Pj = Pj (see proof of Theorem 9.4.2) or yˆ = id on Pj (see construction before (9.120)). We now dene (6.5) by u(x) = yˆ(x) − (Rj x + cj ) 250

for x ∈ Pj , where Rj x + cj is either the rigid motion on PjH given in Theorem 9.4.1 or Rj = Id, cj = 0, respectively. For later purpose, we note that for (9.127) we can also write ˆ X 1 P (Pj , Ωρ ) + fερ (|[ˆ y ]|) dH1 ≤ H1 (Jy ) + C1 ρ, (9.128) 2 j

Jyˆ \∂P

where ∂P = j ∂ Pj and P (Pj , Ω) denotes the perimeter of Pj in Ωρ . ◦ It remains to conrm (6.6). First, (i) follows by H1 (Jyˆ ∩ (ΩH y ) ) ≤ C1 (see ˆ ≤ C1 (see Lemma 9.5.1) and the fact that the H1 (9.70) and (9.88)), H1 (∂ Ω) measure of the jump set added in the construction of yˆ (see (9.126)) is controlled ˆ and H1 (S). In view of (9.83)(ii)-(iv) (see also (9.92)) the properties by H1 (∂ Ω) ˆ for a suciently large constant C(ρ, q) = C(ρ). (ii)-(iv) already hold on the set Ω (Recall q = q(h∗ ) and the S denition h∗ = ρ. See also Remark 9.1.3.) ˆ Recall that Ωρ \ Ω ⊂ t Yt . Repeating the arguments leading to (9.92) we nd by (9.122), (9.124) and (9.126) X kˆ y − (Rj · +cj )k2L2 (Pj \Ω) ˆ ≤ C(ρ)ε.

S

∗

j

ˆ we have ∇ˆ This gives (ii). Moreover, as on each Q% (p) ⊂ Pj \ Ω y = R for some ˆ R ∈ imR¯4 (Ω) (see construction before (9.122)) we get k∇ˆ y − Rj kpLp (P

ˆ

j \Ω)

¯ 4 − Rj kp ≤ C(ρ)kR ˆ Lp (Pj ∩Ω)  ¯ 4 kp ≤ C(ρ) k∇ˆ y−R y − Rj kpLp (P ˆ + k∇ˆ Lp (P ∩Ω) j

 ˆ

j ∩Ω)

for p = 2, 4. By (9.90) and (9.92) this yields X X 1−η k∇ˆ y − Rj k4L4 (Pj \Ω) ≤ C(ρ)ε, k∇ˆ y − Rj k2L2 (Pj \Ω) . ˆ ˆ ≤ C(ρ)ε j

j

This together with (9.23) gives (iii),(iv).  Having completed the main rigidity result, we can now prove the linearized version. We may essentially follow the proof of Theorem 6.1.1 with some minor changes. The proof, however, is considerably simpler as a lot of estimates and lemmas can be skipped. Proof of Theorem 6.1.3. We only give a short sketch of the proof. Dene y = id + u. As the approximation argument presented in the proof of Theorem 9.4.1 also holds in the SBD-setting, it again suces to prove the result under the ˜ u ∈ Vεs such that u| ˜ ∈ H 1 (Ω ˜ u ). We skip assumption that there is some Ω Ωu ˆ i = Id for i = 1, . . . , 4. Similarly as in Lemma Section 9.2.1 and always set R 3% q−1 9.2.5 we nd sets Ωu , ΩH , % = ρq , as well as mappings u ∈ V9k for k = ρ 3% 2×2 H H 2 3k ¯ Aj : Ωu → Rskew and c¯j : Ωu → R , which are constant on Q3% j (p), p ∈ Ij (Ω ), such that (i) ku − (A¯j · +¯ cj )k2 2 ≤ CC 2 %2 (α + kW k∗ ), ρ

L (Ωu )

(iii) k(A¯j1 · +¯ cj1 ) − (A¯j2 · +¯ cj2 )k2L2 (ΩHu ) ≤ CCρ2 %2 (α + kW k∗ ) 251

for j1 , j2 = 1, . . . , 4, j = 1, . . . , 4, where α = ke(∇u)k2L2 (Ω˜ ) and  = cˆρ−1 ε. u This can be established following the lines of the proof of Lemma 9.2.5 with the ¯, dierence that in (9.58) we do not replace Id + A by a dierent rigid motion R H but proceed with Id + A. Analogously, we nd an extension Ωu as constructed in Corollary 9.2.6 and then we obtain the result up to a small set following the lines of Theorem 9.4.2. Finally, the jump set and the extension to Ωρ may be constructed as in Section 9.5. 

252

Chapter 10 Compactness and Γ-convergence 10.1 Compactness of rescaled congurations This section is devoted to the proof of the main compactness result. For the compactness theorem in GSBD (see Theorem A.1.3) it is necessary that the integral for some integrand ψ with limx→∞ ψ(x) = ∞ is uniformly bounded. We rst give a simple criterion for the existence of such a function which is, loosely speaking, based on the condition that the functions coincide in a certain sense on the bulk part of the domain.

Lemma 10.1.1. For every increasing sequence (bi )i ⊂ (0, ∞) with bi → ∞ there is an increasing concave function ψ : [0, ∞) → [0, ∞) with limx→∞ ψ(x) = ∞ and ψ(bi ) ≤ 2i for all i ∈ N. Proof. Let f : [0, ∞) → [0, ∞) be the function with f (0) = 0, f (bi ) = 2i

which is ane on each segment [bi , bi+1 ]. Clearly, f is increasing and satises f (x) → ∞ for x → ∞, but is possibly not concave. We now construct ψ and rst let ψ = f on [0, b1 ]. Assume ψ has been dened on [0, bi ] and satises ψ(bi ) = f (bi ) = 2i . If f 0 (bi −) ≥ f 0 (bi +) we set ψ = f on [bi , bi+1 ]. Here, f 0 (x±) denote the one-sided limits of the derivative at point x. Otherwise, we let ψ(x) = f (bi ) + f 0 (bi −)(x − bi ) for x ∈ [bi , x ¯], where x¯ is the smallest value larger than bi such that f (¯ x) = f (bi ) + f 0 (bi −)(¯ x − bi ). If x¯ does not exist we are done . If x ¯ exists we assume x¯ ∈ (bj−1 , bj ] and dene ψ = f on [¯ x, bj ] noting that 0 0 ψ (¯ x−) ≥ ψ (¯ x+). We end up with an increasing concave function ψ with ψ ≤ f and ψ(x) → ∞ for x → ∞, as desired.

Lemma Let Ω ⊂ R2 and let (yl )l ⊂ L1 (Ω) be a sequence satisfying S 10.1.2. T |Ω \ n∈N l≥n {|y n − y l | ≤ 1}| = 0. Then there is a not relabeled subsequence such that ˆ ψ(|y l |) ≤ C Ω

for a constant independent of l, where ψ is an increasing continuous function with limx→∞ ψ(x) = +∞. 253

Proof. Dene Cl := max1≤i≤l kyi kL1 (Ω) for all l ∈ N. Let An =

−y l | ≤ 1} and set B1 = A1 as well as P Bn = An \ m=1 Bm for all n ∈ N. The sets (Bn )n are pairwise disjoint with n |Bn | = |Ω|. We choose 0 = n1 < n2 < . . . such that P Sni+1 |Bn | −i i i −i 1≤n≤ni |Ω| ≥ 1 − 4 . We let B = n=ni +1 Bn and observe |B | ≤ 4 |Ω|. We pass to the subsequence of (ni )i ⊂ N and choose E i ⊃ B i such that Cn i Cni+1 |E i | = 4−i |Ω|. Let bi = |Ei+1 + 2 for i ∈ N and note that (bi )i is i| + 2 = 4 |Ω| increasing with bi → ∞. By Lemma 10.1.1 we get an increasing concave function ψ : [0, ∞) → [0, ∞) with limx→∞ ψ(x) = ∞ and ψ(bi ) ≤ 2i for all i ∈ N. Clearly, ψ is also continuous.S ˆ i := Ω \ ni Bn we have |B ˆ i | ≤ 4−i |Ω| and choose Eˆ i ⊃ B ˆ i with For B n=1 C ni C ≤ bi . Now let l = ni . Using Jensen's |Eˆ i | = 4−i |Ω|. We then obtain |Eˆnii| = 4i |Ω| i inequality, the denition of the sets B , kyl kL1 (Ω) ≤ Cl and the monotonicity of ψ we compute ˆ ˆ ˆ X l l ψ(|y |) = ψ(|y |) + ψ(|y l |) 1≤j≤i−1 j ˆi Ω B ˆ ˆB X nj+1 ψ(|y l |) ψ(|y | + 2) + = 1≤j≤i−1 B j ˆi B (10.1) ˆ  ˆ  X j nj+1 i l ˆ |E |ψ − |y | + 2 + |E |ψ − |y | ≤ 1≤j≤i−1 ˆi Ej E X X 2−j . 4−j |Ω|2j + 4−i |Ω|2i ≤ |Ω| ≤ T

l≥n {|y

n

Sn−1

1≤j≤i−1

´

j∈N

As the estimate is independent of l ∈ (ni )i , this yields Ω ψ(|y l |) ≤ C uniformly in l, as desired.  Now we are in a position to give the proof of the main compactness result. In the rst part we show that (6.10), (6.11) and (6.12) hold. Proof of Theorem 6.2.1, part 1. Let (εk )k be an arbitrary null sequence. Let yk ∈ SBVM (Ω) with Eεk (yk ) ≤ C be given. The fact that W (G) ≥ c dist2 (G, SO(2)) for all G ∈ R2×2 implies k dist(∇yk , SO(2))k2L2 (Ω) ≤ Cεk for a constant independent of εk . Moreover, we have H1 (Jyk ) ≤ C for all k ∈ N. Choose ρ0 > 0 and let ρl = 2−3l ρ0 for all l ∈ N. By Theorem 6.1.1 we nd modications ykl ∈ SBVcM (Ω, R2 ) with Eερkl (ykl , Ωρl ) ≤ Eεk (yk ) + Cρl and

kykl − yk k2L2 (Ωl ) + k∇ykl − ∇yk k2L2 (Ωl ) ≤ Cεk ρl , k

k

(10.2)

where Ωlk := Ωykl with |Ω \ Ωlk | ≤ Cρl . We further get Caccioppoli partitions P (Pjk,l )j of Ωρl with j P (Pjk,l , Ωρl ) ≤ C and corresponding piecewise rigid motions P l 2 Tkl (x) := j (Rjk,l x + ck,l j )χP k,l (x) such that the functions vk : Ω → R dened by j

( vkl (x)

=

k,l √1 (Rj )T εk

ykl (x) − (Rjk,l x + ck,l j )




for x ∈ Pjk,l , j ∈ N, else,

0 254

(10.3)

satisfy by (6.6)

H1 (Jvkl ) ≤ C,

kvkl kL2 (Ω) + ke(∇vkl )kL2 (Ω) ≤ Cˆl ,

(10.4) k∇vkl k2L2 (Ω) ≤ Cˆl ε−η k

ˆ l ) > 0 and η > 0 small. Observe that |ck,l | ≤ cM for a universal for some Cˆl = C(ρ j constant as kykl k∞ ≤ cM for all k ∈ N. Clearly, each partition may be extended P k,l to Ω by adding the element Ω \ Ωρl and j P (Pj , Ω) ≤ C is still satised as H1 (∂Ωρl ) ≤ CH1 (∂Ω). Using a diagonal argument we get a (not relabeled) subsequence of (εk )k such that by Theorem A.1.3 for every l ∈ N we nd a function v l ∈ GSBD2 (Ω) with vkl → v l a.e. in Ω

and

2×2 e(∇vkl ) * e(∇v l ) weakly in L2 (Ω, Rsym )

(10.5)

for k → ∞. Moreover, by the compactness result for piecewise constant funcl tions (see Theorem A.2.3) we obtain an (ordered) partition P P (Pl j )j of l Ω with l l j P (Pj , Ω) ≤ C and a piecewise rigid motion T (x) := j (Rj x + cj )χPjl (x) such that for all l ∈ N letting k → ∞ we obtain (again up to a subsequence) l Rjk,l χP k,l → Rjl χPjl and ck,l j χPjk,l → cj χPjl in measure for all j ∈ N. This also j implies X |Pjk,l 4Pjl | + kTkl − T l kL2 (Ω) + k∇Tkl − ∇T l kL2 (Ω) → 0 (10.6) j

for k → ∞, where 4 again denotes the symmetric dierence of two sets. We now show that

kv l kL1 (Ω) ≤ Ckv l kL2 (Ω) ≤ Cˆl ,

H1 (Jvl ) ≤ C,

ke(∇v l )k2L2 (Ω) ≤ C.

(10.7)

The rst two claims follow directly from (10.4) and (A.5). To see the third estimate we let χlk (x) := χ[0,ε−1/8 ] (|∇vkl (x)|) and recall that by the linearization fork

mula (9.23) we have dist2 (G, SO(2)) = |¯ eR (G)|2 +ω(G−R) with sup{|G|−3 ω(G) : |G| ≤ 1} ≤ C and ω(RG) = ω(G) for G ∈ R2×2 , R ∈ SO(2). We compute ˆ C ρl l C ≥ Eεk (yk , Ωρl ) ≥ dist2 (∇ykl , SO(2)) εk Ωρl ˆ   CX ≥ χlk |¯ eRk,l (∇ykl )|2 + ω(∇ykl − Rjk,l ) (10.8) j j P k,l εk j ˆ   1 √ =C χlk |e(∇vkl )|2 + ω( εk ∇vkl ) . εk Ω The second term of the integral can be estimated by ˆ ˆ √ l 1 √ 1 √ l l l l 3 ω( εk ∇vk ) 8 χk (x) ω( εk ∇vk ) = χk (x) εk |∇vk | √ ≤ Cε → 0. k l εk | εk ∇vk |3 Ω Ω 255

(10.9)

As e(∇vkl ) * e(∇v l ) weakly in L2 and χlk → 1 boundedly in measure on Ω by (10.4) for η suciently small, it follows χlk e(∇vkl ) * e(∇v l ) weakly in L2 (Ω). By lower semicontinuity we obtain ke(∇v l )k2L2 (Ω) ≤ C for a constant independent of ρl which concludes (10.7). We now want to pass to the limit l → ∞. Similarly as in the argumentation leading to (10.6), by the compactness result for piecewise constant functions (see Theorem P A.2.3) we nd a partition (Pj )j of Ω and a piecewise rigid motion T (x) := j (Rj x + cj )χPj (x) such that for a suitable (not relabeled) subsequence Rjl χPjl → Rj χPj , clj χPjl → cj χPj in measure for all j ∈ N and thus X |Pjl 4Pj | + kT l − T kL2 (Ω) + k∇T l − ∇T kL2 (Ω) → 0 (10.10) j

for l → ∞. Recalling (10.6) and using a diagonal argument we can choose a (not relabeled) subsequence of (ρl )l and afterwards of (εk )k such that for all l we have X X (10.11) |Pjk,l 4Pjl | ≤ 2−l for all k ≥ l. |Pjl 4Pj | ≤ 2−l , j

j

We see that the compactness result in GSBD cannot be applied directly on the sequence (v l )l as the L2 bound in (10.7) depends on ρl . We now show that by choosing the rigid motions on the elements of the partitions appropriately (see (10.3)) we can construct the sequence (v l )l such that we obtain [ \ n m (10.12) Ω \ {|v − v | ≤ 1} =0 n∈N

m≥n

and thus Lemma 10.1.2 is applicable. We x k ∈ N and describe an iterative procedure to redene Rjk,l , ck,l j for all k,l k,l ˆ , cˆ have been chosen for all l, j ∈ N. Let vk1 as dened in (10.3) and assume R j j k,l k,l j ∈ N (which possibly dier from Rj , cj ) such that (10.4) still holds possibly passing to a larger constant Cˆl . Fix some Pjk,l+1 , j ∈ N, and recall that |Pjk,l+1 | ≥ C(ρl+1 ) as Pjk,l+1 contains squares of size ∼ ρql+1 (see the proof of Theorem 6.1.1.) Dene Dl+1 = Pjk,l+1 ∩ Ωkl+1 and let Dil = Pjk,l ∩ Ωlk be the components with i Pjk,l ∩ Pjk,l+1 6= ∅ for i = 1, . . . , n. Without restriction assume that Pjk,l ∩ Pjk,l+1 1 i has largest Lebesgue measure. If |Pjk,l ∩ Pjk,l+1 | > 2|Pjk,l+1 \ (Ωlk ∩ Ωl+1 k )|, we dene 1

ˆ k,l+1 = R ˆ k,l , R j j1

cˆk,l+1 = cˆk,l j j1

on

Pjk,l+1 .

ˆ k,l+1 = Rk,l+1 and cˆk,l+1 = ck,l+1 . In the rst case we then Otherwise we set R j j j j k,l k,l+1 1 l l+1 obtain |D1 ∩ D | ≥ 2 |Pj1 ∩ Pj | ≥ C(ρl+1 ) and thus for p = 2, 4 we get by l+1 l k∇yk k∞ , k∇yk k∞ ≤ cM  ˆ k,l+1 − Rk,l+1 |p ≤ C(ρl+1 ) k∇y l − R ˆ k,l kp p l + k∇y l − ∇y l+1 k2 2 l l+1 |R k k j j j1 L (D ) k L (D1 ∩D ) 1  + k∇ykl+1 − Rjk,l+1 kpLp (Dl+1 ) . 256

The calculation may be repeated to estimate the dierence of the rigid motions. Summing over all components and recalling (10.2), (10.4) (for l) as well as the estimates for the original rigid motions (for l + 1, see (9.92)) we nd that (10.4) still holds possibly passing to larger constants. T We dene Ak,l = n≤m≤l {|vkm − vkn | ≤ 12 } for all n ∈ N and n ≤ l ≤ k . If we show (10.13)

|Ω \ Ak,l | ≤ C2−n ,

then (10.12) follows. Indeed, for given l ≥ n we can choose K = K(l) ≥ l so m large that |{|vK − v m | > 41 }| ≤ 2−m for all n ≤ m ≤ l since vkm → v m in measure for k → ∞. This implies \ X m Ω \ {|v m − v n | ≤ 1} ≤ |Ω \ AK,l | + |{|vK − v m | > 1 }| ≤ C2−n . n≤m≤l

n≤m≤l

4

Passing to the limit l → ∞ we nd |Ω \ m≥n {|v m − v n | ≤ 1}| ≤ C2−n and taking the union over all n ∈ N we derive (10.12). To show (10.13) we proceed in two steps. Employing the redenition of the piecewise rigid motions we rst show that the set where Tkm , n ≤ m ≤ l, dier is small. Afterwards we use (10.2) to nd that the set where ykm , n ≤ m ≤ l, dier T is small. We dene Bk,l = n≤m≤l {Tkm = Tkn } and prove that

T

|Ω \ Bk,l | ≤ C2−n

(10.14)

for all k ≥ l ≥ n. To this end, consider {Tkm = Tkm+1 } for n ≤ m ≤ l − 1 P k,m+1 4Pjk,m | ≤ 3 · 2−m . Dene and rst note that by (10.11) we have j |Pj m+1 )| for all j ∈ J1 and J1 ⊂ N such that |Pjk,m ∩ Pjk,m+1 | ≤ 2|Pjk,m+1 \ (Ωm k ∩ Ωk k,m+1 k,m k,m+1 1 let J2 ⊂ N \ J1 such that |Pj ∩ Pj | > 2 |Pj | for all j ∈ J2 . Observe k,m+1 k,m+1 k,m that |Pj | ≤ 2|Pj \ Pj | for j ∈ J3 := N \ (J1 ∪ J2 ). Due to the above S construction of the rigid motions we obtain {Tkm = Tkm+1 } ⊃ j∈J2 (Pjk,m+1 ∩Pjk,m ) m+1 and therefore recalling |Ω \ (Ωm )| ≤ C2−3m k ∩ Ωk X X |Ω \ {Tkm = Tkm+1 }| ≤ |Pjk,m+1 \ Pjk,m | + |Pjk,m+1 | j∈J2 j∈J1 ∪J3 X X ≤ |Pjk,m+1 \ Pjk,m | + 2|Pjk,m+1 \ Pjk,m | j∈J2 j∈J1 ∪J3 X m+1 + 2|Pjk,m+1 \ (Ωm )| ≤ C2−m . k ∩ Ωk j∈J1

Summing over n ≤ m ≤ l − 1 we establish (10.14). Now recalling (10.2), (10.14), |Ω \ Ωlk | ≤ Cρl and the fact that (ρl )l ⊂ (2−3l ρ0 )l we nd X √ |Ω \ Ak,l | ≤ |Ω \ Bk,l | + |{|ykm+1 − ykm | > 2−m−1 εk }| ≤ C2−n n≤m≤l−1

for all k ≥ l ≥ n, as desired. 257

By (10.7) and (10.12) we can apply Lemma 10.1.2 on the sequence (v l )l . We employ Theorem A.1.3 and obtain a function v ∈ GSBD(Ω) and a further not relabeled subsequence with v l → v a.e in Ω and e(∇v l ) * e(∇v) weakly in L2 (Ω, R2×2 sym ). We now select a suitable diagonal sequence such that (6.11) and (6.12) hold. Observe that by (10.8), (10.9) the functions vˆkl := χlk vkl fulll ke(∇ˆ vkl )kL2 (Ω) ≤ C −1/8 and k∇ˆ vkl k∞ ≤ εk for a constant independent of k, l ∈ N. As weak convergence 2 in L is metrizable on bounded sets and convergence in measure is metrizable ´ (take (f, g) 7→ Ω min{|f − g|, 1}) we can apply a diagonal sequence argument and nd a not relabeled subsequence (yn )n and a corresponding diagonal sequence (wn )n∈N ⊂ (ˆ vkl )k,l with corresponding partitions (Pjn )j and piecewise rigid motions (Tn )n such that by (10.5), (10.6) and (10.10)

wn → v in measure on Ω,

2×2 e(∇wn ) * e(∇v) weakly in L2 (Ω, Rsym ),

Tn → T in L2 (Ω), ∇Tn → ∇T in L2 (Ω), χPjn → χPj in measure on Ω for all j ∈ N, for n → ∞. Up to a further subsequence we can assume wn → v a.e. and ∇Tn → ∇T a.e. Finally, dene un = ∇Tn wn for all n ∈ N and u = ∇T v and observe that (6.11), (6.12) hold. Moreover, by (10.2), (10.3), the fact that −1/8 k∇un k∞ ≤ εn , k∇yn k∞ ≤ cM and the regularity of W it is not hard to see that ˆ ˆ √ 1 1 T W (Id + εn ∇Tn ∇un ) ≤ W (∇yn ) + o(1). εn Ω εn Ω This gives (6.10)(ii). As χlk → 1 boundedly in measure on Ω and |Ω \ Ωlk | → 0 for k, l → ∞ we also get (6.10)(i) recalling (10.2), (10.3) and possibly passing to a further subsequence.  It remains to show (6.13). Proof of Theorem 6.2.1, part 2. The sets Jvc := {x ∈ Jv : [v](x) = c} for c ∈ B1 (0) \ {0} are pairwise disjoint with H1 -σ nite union, i.e. H1 (Jvc ) = 0 up to at most countable values of c. Consequently, we can choose a sequence (cj ) with P c −c 0 ≤ |cj | < 21 such that H1 (Jv i j ) = 0 for i 6= j . Replacing v by v˜ = v + j cj χPj S we thus obtain H1 (∂P \ Jv˜) = 0 (recall that ∂P = j ∂ ∗ Pj , where ∂ ∗ denotes the essential boundary.) We rst show that it suces to prove X lim inf H1 (Jyk ) ≥ H1 (Jv˜). (10.15) k→∞

j

P To see this we have to show H1 (Jv˜) = H1 (Ju \ ∂P ) + 21 j P (Pj , Ω). By (A.9) S P we obtain 2H1 (Jv˜ ∩ ∂P ) = 2H1 ( j ∂ ∗ Pj ∩ Ω) = j P (Pj , Ω) and thus H1 (Jv˜) = P P H1 (Jv˜ \ ∂P ) + 21 j P (Pj , Ω) = H1 (Ju \ ∂P ) + 21 j P (Pj , Ω), as desired. We now show (10.15) in two steps rstP passing to the limit k → ∞ P and then l l l l l l letting l → ∞. We replace vk by v˜k = vk + j cj χP k,l and v by v˜ = v + j cj χPjl j

258

noting that v˜kl → v˜l for k → ∞ and v˜l → v˜ for l → ∞ in the sense of (A.5). In the following we write Jkl = Jv˜kl ∩ Ωρl for shorthand. Recalling (10.3) we obtain by (9.128) ˆ 1X 1 vkl ]|) dH1 + gρl (|[˜ H (Jyk ) ≥ P (Pjk,l , Ωρl ) − Cρl j l 2 k,l J \∂P ˆk (10.16) ≥ gρl (|[˜ vkl ]|) dH1 − Cρl Jkl

S where ∂P k,l = j ∂ ∗ Pjk,l and gρl (x) = min{ ρxl , 1}. Here we note that the passage from vkl to v˜kl does not aect the estimate. We cannot directly apply lower semicontinuity results for GSBD functions due to the involved function gρl . We therefore pass to the limit k → ∞ on one-dimensional sections. Let σ > 0 and µ ˆσ,ν be the measure given in (A.6) inserting the concave vl function gσ for θ and let µ ˆ0,ν be the measure when setting θ ≡ 1. By Lemma vl A.1.4 we have µ ˆσ,ν (U ) ≤ lim inf µ ˆσ,ν (U ) v˜l v˜kl k→∞ ´ for all σ ≥ 0, ν ∈ S 1 and for every open set U ⊂ Ω. Let κ1 = S 1 |ξ · ν| dH1 (ν) for some ξ ∈ S 1 which clearly does not depend on the particular choice of ξ . Using Fatou's lemma and (A.6) we compute for l suciently large ˆ 1 lim inf H (Jyk ) + Cρl ≥ lim inf gσ (|[˜ vkl ]|) dH1 k→∞

ˆ ≥ κ−1 1

k→∞

Jl

ˆk

gσ (|[˜ vkl ](x)|)|ξv˜kl (x) · ν| dH1 (x) dH1 (ν) Jkl S1 ˆ ˆ σ,ν 1 −1 −1 (Ωρl ) dH1 (ν). µ ˆσ,ν lim inf µ ˆv˜l (Ωρl ) dH (ν) ≥ κ1 ≥ κ1 v˜l lim inf k→∞

S1

k→∞

k

S1

We pass to the limit l → ∞ (i.e. ρl → 0) and obtain by the dominated convergence theorem ˆ 1 −1 1 lim inf H (Jyk ) ≥ κ1 µ ˆσ,ν v˜ (Ω) dH (ν). k→∞

S1

Recall that gσ → 1 pointwise for σ → 0. Now letting σ → 0 we obtain by the dominated convergence theorem ˆ 1 −1 1 lim inf H (Jyk ) ≥ κ1 µ ˆ0,ν v˜ (Ω) dH (ν) k→∞ 1 ˆS ˆ = κ−1 |ξv˜(x) · ν| dH1 (x) dH1 (ν) = H1 (Jv˜). 1 S1

Jv˜

This gives (10.15) and completes the proof. The proof of Corollary 6.1.2 is now straightforward. 259



Proof of Corollary 6.1.2 . Let y ∈ SBV (Ω) with H1 (Jy ) < ∞ as well as ´

dist2 (∇y, SO(2)) = 0 be given. Dene an arbitrary null sequence (εk )k and the sequence yk = y for all k ∈ N. Applying Theorem 6.2.1 we obtain a subsequence and congurations uk converging almost everywhere by (6.12). Moreover, we obtain piecewise rigid motions T, Tk such that Tk → T , ∇Tk → ∇T in L2 (Ω) by (6.11) and yk − Tk → 0 a.e. for k → ∞ by (6.10)(i). This implies y = T is a piecewise rigid motion, as desired.  Ω

10.2 Admissible & coarsest partitions and limiting congurations We now prove Theorem 6.2.4 and begin with some preliminary observations. In the following let (yk )k be a (sub-)sequence as considered in Theorem 6.2.1. Recall Denition 6.2.3. For notational convenience we will drop the dependence of (yk )k in the sets ZP , Zu , ZT . Moreover, recall the denition of the set of piecewise innitesimal rigid motions A((Pj )j ) in (6.9). We introduce a partial order on := (Pj2 )j in the admissible partitions ZP : Given two partitions P 1 := (Pj1 )j , P 2 S S ∗ 1 ∗ 2 ZP we say P 1 ≥ P 2 if P 1 is subordinated to P 2 , i.e. j ∂ Pj ⊂ j ∂ Pj up 1 1 2 2 1 to an H -negligible set. We observe that if P ≥ P and P ≥ P , abbreviated by P 1 = P 2 hereafter, then the Caccioppoli partitions coincide: After a possible reordering of the sets we nd |Pj1 4Pj2 | = 0 for all j ∈ N. We begin with the observation that the piecewise rigid motion is uniquely determined in the limit.

Lemma 10.2.1. Let (yk )k be a (sub-)sequence as considered in Theorem 6.2.1. Then there is a unique T ∈ ZT (P ) for all P ∈ ZP . ˆ . Let Proof. Assume there are P , Pˆ ∈ ZP and T ∈ ZT (P), Tˆ ∈ ZT (P)

(uk , P k , Tk ), (ˆ uk , Pˆ k , Tˆk ) ∈ D for k ∈ N be the triples according to Denition 6.2.3(ii). As uk − u ˆk − ( √1εk (Tk − Tˆk )) → 0 a.e. by (6.10)(i) and uk − uˆk converges pointwise a.e. (and the limits lie in R a.e.) by (6.12) we get Tk − Tˆk → 0 pointwise almost everywhere. This implies T = Tˆ, as desired.  From now on T will always denote the rigid motion given by Lemma 10.2.1. We state a lemma giving an equivalent characterization of the coarsest partition (recall Denition 6.2.3(iv)).

Lemma 10.2.2. Let (yk )k be a (sub-)sequence as considered in Theorem 6.2.1. Then P ∈ ZP is coarsest if and only if it is a maximal element in the partial order (ZP , ≥), i.e. Pˆ ≥ P implies Pˆ = P . Proof. (1) Assume P = (Pj )j was not coarsest. According to Denition 6.2.3

let be (uk , P k , Tk ) ∈ D and u be given such that (u, P, T ) ∈ D∞ and (6.10)(6.13) hold. Without restriction possibly passing to a subsequence we assume 260

that


|R1k − R2k | + |ck1 − ck2 | ≤ C for all k ∈ N (cf. (6.14)). By (9.23) we k k k k k T k obtain Ak ∈ R2×2 skew with |A | ≤ C such that R1 − R2 = R1 (Id − (R1 ) R2 ) = √ R1k ( εk Ak + O(εk )). Passing to a (not relabeled) subsequence we then obtain √1 εk


1 S(x) := lim √ (R1k − R2k ) x + ck1 − ck2 k→∞ εk
√ 1 √ = lim √ εk R1k Ak x + ck1 − ck2 + O( εk ) = RA x + c k→∞ εk

(10.17)

2 k ˆ k ˆ ˆ ˆk , uˆ for some A ∈ R2×2 skew , c ∈ R and R = limk→∞ R1 . We now dene P , P , Tk , u k k k k k k as follows. Let Pˆ1 = P1 ∪ P2 , Pˆ2 = ∅, Pˆj = Pj for j ≥ 3 and likewise for the limiting partition Pˆ . Let Tˆk (x) = R1k x + ck1 for x ∈ Pˆ1k and Tˆk (x) = Tk (x) for x ∈ Ω \ Pˆ1k . It is elementary to see that (6.11) holds as |R1k − R2k | + |ck1 − ck2 | → 0 for k → ∞. Furthermore, we let


1 uˆk = uk + √ (R1k − R2k ) x + ck1 − ck2 χP2k εk ˆ and u ˆ = u + (RA x +
c)χP2 (see (10.17)). Then (6.10)(i) clearly holds P as Tk − Tk = 1 k k k k 1 (R1 −R2 ) x+c1 −c2 χP2k . It is not hard so see that H (Ju \∂P )+ 2 j P (Pj , Ω) ≥ S P H1 (Juˆ \ ∂ Pˆ ) + 21 j P (Pˆj , Ω), where Pˆ = j ∂ ∗ Pˆj and thus also (6.13) is satised. It remains to verify (6.12) and (6.10)(ii). First, (6.12)(iii) is obvious and (6.12)(i) follows from (10.17) and the denition of u ˆ. To see (6.12)(ii) we use R1k = √ k k k k R2 + εk R1 A + O(εk ), |A | ≤ C and observe X √ χPjk e((R1k )T ∇uk ) + χP2k e((R1k )T R1k Ak ) + O( εk ) uk ) = χPˆ1k e((R1k )T ∇ˆ j=1,2 X √ χPjk e((Rjk )T ∇uk ) + O( εk ) = j=1,2 X * χPj e(RjT ∇u) = χPˆ1 e(RT ∇ˆ u) j=1,2

weakly in L2 (Ω, R2×2 sym ). Finally, to establish (6.10)(ii) we nd by k∇uk kL∞ (Ω) ≤ −1/8

Cεk

√ 3/8 ∇TˆkT ∇ˆ uk = (R1k )T ∇uk + Ak + O( εk ) = (R2k )T ∇uk + Ak + O(εk )

(10.18)

a.e. in P2k . Observe that W (G) = 21 Q(e(G−Id))+ω(G−Id) with sup{|F |−3 ω(F ) : |F | ≤ 1} ≤ C and ω(RG) = ω(G) for G ∈ R2×2 , R ∈ SO(2) by the assumptions on W , where Q = D2 W (Id). Thus, we obtain by (10.18) ˆ  ˆ  √ 1 1 1 √ T W (Id + εk ∇Tˆk ∇ˆ uk ) = Q(e(∇TˆkT ∇ˆ uk )) + ω( εk ∇TˆkT ∇ˆ uk ) εk P2k εk Pk 2 ˆ 2 √ 3 ω( εk ∇ˆ uk )  1 T = Q(e(∇Tk ∇uk )) + + O(εk4 ) εk P2k 2 261

and likewise ˆ  ˆ  √ 1 1 1 √ T T W (Id + εk ∇Tk ∇uk ) = Q(e(∇Tk ∇uk )) + ω( εk ∇uk ) . εk P2k εk P2k 2 In both estimates the second terms converge to 0 arguing as in (10.9) and using −1/8 k∇uk k∞ , k∇ˆ uk k∞ ≤ Cεk . Consequently, we get ˆ ˆ √ √ 1 1 T ˆ uk ) = W (Id + εk ∇Tk ∇ˆ W (Id + εk ∇TkT ∇uk ) + o(1) (10.19) εk P2k εk P2k for εk → 0. Thus, Pˆ is an admissible partition and thus P is not maximal. (2) Conversely, assume that P = (Pj )j was not maximal, i.e. we nd Pˆ = S S ˆ (Pj )j with Pˆ ≥ P , Pˆ = 6 P , i.e. j ∂ ∗ Pj and j ∂ ∗ Pˆj dier by a set of positive H1 measure. We may assume without restriction that P1 ∩Pˆ1 and P2 ∩Pˆ1 have positive L2 -measure. According to Denition 6.2.3(i) let (uk , P k , Tk ), (ˆ uk , Pˆ k , Tˆk ) ∈ D and ˆ T ) ∈ D∞ and (6.10)-(6.13) hold. As by u, uˆ be given such that (u, P, T ), (ˆ u, P, (6.12) uk and u ˆk convergence pointwise a.e., by (6.10) also √1εk (Tk − Tˆk ) converges ˆk| + pointwise a.e. (and the limits lie in R a.e.). But this implies √1εk |Rjk − R 1
k k |cj − cˆ1 | ≤ C for j = 1, 2 and k ∈ N. Then the triangle inequality shows that (6.14) is violated and thus P is not a coarsest partition.  The alternative characterization now directly implies that there is at most one coarsest partition.

Lemma 10.2.3. Let (yk )k be a (sub-)sequence as considered in Theorem 6.2.1. Then there is at most one maximal element in (ZP , ≥). Proof. Assume there are two maximal elements P 1 , P 2 ∈ ZP with P 1 6= P 2 .

As before, without restriction we may assume that P11 ∩ P12 and P21 ∩ P12 have positive L2 -measure. We proceed as in the proof of Lemma 10.2.2(2) to see that the partition P 1 is not coarsest and thus not a maximal element in (ZP , ≥).  We now analyze the admissible congurations if the partitions are given. Recall that T is uniquely determined by Lemma 10.2.5.

Lemma 10.2.4. Let (yk )k be a (sub-)sequence as considered in Theorem 6.2.1 ˆ . Then Zu (P) = uˆ + ∇T A(P). and P, Pˆ ∈ ZP such that P ≤ Pˆ . Let uˆ ∈ Zu (P) Proof. (1) To see Zu (P) ⊂ uˆ + ∇T A(P) we have to show that u − uˆ ∈ ∇T A(P)

for all u ∈ Zu (P). To this end, consider Pj1 ∈ P , Pˆj2 ∈ Pˆ such that |Pj1 \ Pˆj2 | = 0. Let uk , u ˆk and Tk , Tˆk be given according to Denition 6.2.3. As uk − uˆk and thus 1 ˆk ) converge pointwise a.e. we nd |Rk − R ˆ k | + |ck − cˆk | ≤ C √εk . √ (Tk − T j1 j2 j1 j2 εk Repeating the argument in (10.17) we derive u(x) − u ˆ(x) = limk→∞ uk (x) − uˆk (x) = limk→∞ √1εk (Tk (x) − Tˆk (x)) = ∇T (x)(A x + c) for a.e. x ∈ Pj1 for some 2×2 A ∈ Rskew , c ∈ R2 .

262

(2) Conversely, to see Zu (P) ⊃ u ˆ + ∇T A(P)P we rst consider the special case P = Pˆ = (Ph )h . Let u¯ ∈ Zu (P) and A(x) = h (Ah x + ch )χPh be given. We have to show that u := u ¯ + ∇T A ∈ Zu (P). We rst note that H1 (Ju \ ∂P ) = H1 (Ju¯ \ ∂P ) and thus (6.13) is satised. According to Denition 6.2.3(iii) let (¯ uk , P k , T¯k ) ∈ D be given such that (6.10)¯ k x + c¯k for x ∈ P k . Now (6.13) hold. Assume that T¯k has the form T¯k (x) = R j j j √ k k k ¯ k (Id + √εk Aj ), SO(2)) and ¯ (Id + εk Aj )| = dist(R choose Rj such that |Rj − R j j √ let ckj = c¯kj + εk Rjk cj . Dene X Tk (x) = (Rjk x + ckj )χPjk j

as well as uk = u ¯k + √1εk (Tk − T¯k ). Clearly, this implies (6.10)(i). By (9.23) we 1 ¯ k + √εk Rk Aj + ωj,k with ε− 2 |ωj,k | → 0 for all j ∈ N. Moreover, we have Rjk = R j j k nd X √
√ Tk = T¯k + εk Rjk Aj x + ωj,k x + εk Rjk cj χPjk → T j

in measure for k → ∞. Then it is not hard to see that Tk → T and ∇Tk → ∇T in L2 which gives (6.11). Likewise, we obtain 
X  k 1 1 ¯ Rj (Aj x + cj ) + √ ωj,k x χPjk uk − u¯k = √ Tk − Tk = j εk εk X → ∇T (Aj x + cj )χPj = ∇T A j

pointwise a.e. which implies uk → u ¯ +∇T A and shows (6.12)(i). We observe that there are decreasing sets Vk with |Vk | → 0 for k → ∞ such that k∇AkL∞ (Ω\Vk ) ≤ P −1/8 −1/2 Cεk and k j χPjk εk ωj,k kL∞ (Ω\Vk ) → 0 for k → ∞. Consequently, we obtain

k∇uk − ∇¯ uk kL∞ (Ω\Vk ) ≤ k∇AkL∞ (Ω\Vk ) + k

−1/2

X j

χPjk εk

−1/8

ωj,k k2L∞ (Ω\Vk ) ≤ Cεk

and therefore, replacing uk by χΩ\Vk uk we nd that (6.10)(i) still holds and (6.12)(iii) is fullled. Arguing similarly as in (10.18) and taking k∇¯ uk kL∞ (Ω\Vk ) + −1/8 k∇AkL∞ (Ω\Vk ) ≤ Cεk we nd −1/2

uk (x) + Aj + (Rjk )T εk wj,k (Rjk )T ∇uk (x) = (Rjk )T ∇¯ 1/4 −1/2 ¯ k )T ∇¯ = (R uk (x) + O(ε ) + Aj + (Rk )T ε wj,k j

k

j

k

for a.e. x ∈ Pjk \ Vk . Thus, also (6.12)(ii) follows from the fact that (6.12)(ii) holds for the sequence u ¯k and ˆ X

¯ jk )T ∇¯ uk |2 |e (Rjk )T ∇uk − e (R j

Pjk \Vk

≤ Ck

−1/2

X j

χPjk εk

263

1/2

ωj,k k2L∞ (Ω\Vk ) + Cεk

→ 0.

Finally, the above estimates together with a similar argumentation as in (10.19) yield (6.10)(ii). In the general case we have to show u := u ˆ + ∇T A ∈ Zu (P) for given uˆ ∈ ˆ ˆ Zu (P), P ≥ P and A ∈ A(P). As P ∈ ZP we nd some u¯ ∈ Zu (P) which by (1) ¯ satises u ¯ − uˆ = ∇T A¯ for A¯ ∈ A(P). Consequently, we get u = u¯ + ∇T (A − A) and by the special case in (2) we know that u ∈ Zu (P), as desired.  To guarantee existence of coarsest partitions we show that each totally ordered subset has upper bounds such that afterwards we may apply Zorn's lemma.

Lemma 10.2.5. Let (yk )k be a (sub-)sequence as considered in Theorem 6.2.1. Let I be an arbitrary index set and let {Pi = (Pi,j )j : i ∈ I} ⊂ ZP be a totally ordered subset, i.e. for each i1 , i2 ∈ I we have Pi1 ≤ Pi2 or Pi2 ≤ Pi1 . Then there is a partition P ∈ ZP with Pi ≤ P for all i ∈ I . Proof. To prove the existence of an upper bound we rst show that it suces to

consider a suitable countable subset of {Pi : i ∈ I}. For notational convenience we write i1 ≤ i2 for i1 , i2 ∈ I if Pi1 ≤ Pi2 . Choose an arbitrary i0 ∈ I and note that it suces to S nd an upper S bound for all i ≥ i0 . We observe that for each i ≥ i0 we have j ∂ ∗ Pi,j ⊂ j ∂ ∗ Pi0 ,j (up to an H1 -negligible set). Thus, S k k for each k ∈ N there are (coarsened) partitions Pik = (Pi,j )j with j ∂ ∗ Pi,j =
S S S ∗ ∗ ∗ 1 j≥k ∂ Pi0 ,j \ ∂ ( j≥k Pi0 ,j ) up to H -negliglible sets for all i ≥ i0 . j ∂ Pi,j \ Observe that typically Pik are not elements of {Pi : i ∈ I}, but satisfy [  [ 1 ∗ ∗ k H ∂ Pi,j \ ∂ Pi,j ≤ ω(k) j

j

with ω(k) → 0 for k → ∞. After identifying partitions whose boundaries only dier by H1 -negligible sets we nd that each {Pik : i ≥ i0 } contains only a nite number of dierent elements and therefore contains a maximal element P k = (Pjk )j . Now we can choose i0 ≤ i1 ≤ i2 ≤ . . . such that P k = Pikk for k ∈ N. It now suces to construct an upper bound P = (Pj )j ∈ ZP with P ≥ Pik for all k ∈ N. Indeed, we then obtain for all i ≥ i0 [  [  [ [ 1 ∗ ∗ 1 ∗ ∗ k H ∂ Pj \ ∂ Pi,j ≤ H ∂ Pj \ ∂ Pi,j j j j j [  [ ≤ H1 ∂ ∗ Pj \ ∂ ∗ Pjk j j [  [ 1 ∗ ∗ ≤H ∂ Pj \ ∂ Pik ,j + ω(k) = ω(k) j

j

and as k ∈ N was arbitrary, we derive j ∂ ∗ Pj ⊂ j ∂ ∗ Pi,j , as desired. Now consider the totally ordered sequence of partitions (Pik )k . For notational convenience we will denote the sequence by (Pi )i∈N in the following. By the compactness theorem for Caccioppoli partitions (see Theorem A.2.2) we get an (ordered) Caccioppoli partition P = (Pj )j such that χPi,j → χPj

S

264

S

S in measure for i → ∞ for all j ∈ N. This also implies H1 ( j ∂ ∗ Pj \ E) ≤ S 1 lim inf i→∞ H1 ( j ∂ ∗ Pi,j \ E) for every H1 -measurable S ∗set with SH ∗(E) < ∞ (see e.g. [35, Theorem 2.8]). Consequently, we apply j ∂ Pi,j ⊂ j ∂ Pk,j for i ≥ k S S to derive H1 ( j ∂ ∗ Pj \ j ∂ ∗ Pk,j ) = 0 for all k ∈ N. This implies P ≥ Pk for all k ∈ N and therefore it suces to show that P ∈ ZP . To this end, we will construct partitions P n , rigid motions Tn ∈ R(P n ) and a limiting function u by a diagonal sequence argument. For all i ∈ N, according to Denition 6.2.3(i), we nd (uki , Pik , Tik ) ∈ D and a sequence of admissible limiting congurations ui ∈ Zu (Pi ) such that (6.10)-(6.13) hold as k → ∞. The strategy is to select ui in a suitable way such that we nd a limiting conguration u ∈ GSBD(Ω) with ui → u a.e., e(∇T T ∇ui ) * e(∇T T ∇u), lim inf i→∞ H1 (Jui ) ≥ H1 (Ju ). k(n)

(10.20)

Then we can choose a diagonal sequence (¯ un ) := (un )n converging to the triple (u, P, T ) in the sense of (6.10)-(6.13). Indeed, k(n) can be selected such that k(n) k(n) letting P¯ n = (P¯jn )j = Pn and T¯n = Tn ∈ R(P n ) we nd χP¯jn → χPj in measure for all j ∈ N and T¯n → T , ∇T¯n → ∇T in L2 (Ω) which gives (6.11). Moreover, possibly passing to a further subsequence this can be done in a way −1/2 that u ¯n → u a.e., u¯n − εn (yn − T¯n ) → 0 a.e. and therefore also (6.10), (6.12)(i) hold. Likewise, (6.12)(ii) can be achieved by (10.20) and the fact the weak convergence is metrizable as ke(∇(Tik )T ∇uki )kL2 (Ω) ≤ C for a constant independent of k, i. The last property follows from the construction of vˆkl in the proof of Theorem 6.2.1 (see (10.8), (10.9)). Moreover, (6.12)(iii) and (6.10)(ii) directly follow from the corresponding estimates for the functions uki . Finally, to see (6.13) it suces to prove     1X 1X lim inf i→∞ H1 (Jui \∂Pi )+ P (Pi,j , Ω) ≥ H1 (Ju \∂P )+ P (Pj , Ω) , j j 2 2 S ∗ where ∂Pi = j ∂ Pi,j . This can be derived arguing as in (10.15): We may P consider an innitesimal perturbation of the form u ˜i = ui + j cj χPi,j , u˜ = P u + j cj χPj with cj small such that H1 (∂Pi \ Ju˜i ) = H1 (∂P \ Ju˜ ) = 0 and the convergence in (10.20) still holds after replacing ui , u by u ˜i , u˜, respectively. Then the claim follows from (10.20). Consequently, P ∈ ZP due to Denition 6.2.3(i). It suces to show (10.20). Clearly, we have ke(∇T T ∇ui )k2L2 (Ω) ≤ C and H1 (Jui ) ≤ C for a constant independent of i ∈ N. This follows by a lower semicontinuity argument taking (6.13) and ke(∇(Tnk )T ∇ukn )kL2 (Ω) ≤ C into account. Consequently, in order to apply Theorem A.1.3 we have to nd an increasing continuous function ψ : [0, ∞) → [0, ∞) with limx→∞ ψ(x) = +∞ such that ´ ψ(|u |) ≤ C. i Ω 265

We proceed similarly as in the proof of Theorem 6.2.1 and dene ui iteratively. Choose S ∗ u1 ∈ Z Su (P∗1 ) arbitrarily. Given ui we dene ui+1 as follows. We recall ≤ i2 . Consider some Pi+1,j and choose l1,j < j ∂ Pi2 ,j ⊂ j ∂ Pi1 ,j for i1 S 2 l2,j < . . . such that Pi+1,j = ∞ k=1 Pi,lk,j up to an L - negligible set (observe that the union may also be nite). Without restriction assume that Pi,l1,j has largest Lebesgue measure. Choose some u ˜i+1 ∈ P Zu (Pi+1 ). By Lemma 10.2.4 ˆ for P = Pi , P = Pi+1 we get (ui − u ˜i+1 )χPi+1,j = ∞ k=1 (Alk,j x + clk,j )χPi,lk,j for 2×2 2 Alk,j ∈ Rskew , clk,j ∈ R . Now dene

ui+1 (x) = u˜i+1 (x) + Al1,j x + cl1,j for x ∈ Pi+1,j and observe that ui = ui+1 on Pi,l1,j . Proceeding in this way on all Pi+1,j we nd some A˜i+1 ∈ A(Pi+1 ) such that ui+1 := u ˜i+1 + A˜i+1 ∈ Zu (Pi+1 ) applying Lemma 10.2.4 for P = Pˆ = Pi+1 . Moreover, S there is a corresponding i i i A ∈ A(Pi ) such that ui+1 = ui + A with A = 0 on j Pi,l1,j . P We now show that i∈N |Ai | < +∞ a.e. To see this, we recall that χPi,j → χPj in measure for all j ∈ N. Consequently, as due to the total order of the partitions the sets Pi,j are increasing for xed j ∈ N, the construction of the functions (ui )i 1 i implies P A = i0 on Pi,j for i so large that |Pi,j | > 2 |Pj |. Thus, for a.e. x ∈ Pj the sum i≥1 |A (x)|P is a nite sum and therefore nite. Taking the union over all j ∈ N we obtain i∈N |Ai | < +∞ a.e. Consequently, there is an increasing continuous ψ : [0, ∞) → [0, ∞) P function i with limx→∞ ψ(x) = ∞ such that kψ(|u1 | + i∈N |A |)kL1 (Ω) < ∞. Using the denition P ui+1 = ui + Ai and the monotonicity of ψ we nd kψ(|ui |)kL1 (Ω) ≤ kψ(|u1 | + k∈N |Ak |)kL1 (Ω) < ∞ for all i ∈ N, as desired.  After these preparatory lemmas we are nally in a position to prove Theorem 6.2.4. Proof of Theorem 6.2.4. First, (i) follows from Lemma 10.2.1. The uniqueness of the coarsest partition is a consequence of Lemma 10.2.3 and Lemma 10.2.2. We obtain existence by Zorn's lemma: As (ZP , ≥) is a partial order and every chain has an upper bound by Lemma 10.2.5, there exists a maximal element P¯ ∈ ZP . Lemma 10.2.2 shows that P¯ is a coarsest partition which gives (ii). Finally, ¯ = v + ∇T A(P) ¯ for some v ∈ Zu (P) ¯ , follows from assertion (iii), namely Zu (P) Lemma 10.2.4 for the choice P = Pˆ = P¯ . 

10.3 Derivation of linearized models via Γ-convergence This section is devoted to the proof of Theorem 6.3.1. Proof of Theorem 6.3.1. (i) Thanks to the preparations in the last section the lower bound is almost immediate. Let (u, P, T ) ∈ D∞ be given as well as a 266

sequence (yk )k ⊂ SBVM (Ω) with corresponding (uk , P k , Tk ) ∈ D such that (6.10)(6.13) hold. By (6.13) it suces to show that ˆ ˆ 1 1 lim inf W (∇yk ) ≥ Q(e(∇T T ∇u)). k→∞ εk Ω 2 Ω We proceed as in (10.8): Recall that W (G) = 21 Q(e(G − Id)) + ω(G − Id) with sup{|F |−3 ω(F ) : |F | ≤ 1} ≤ C by the assumptions on W , where Q = D2 W (Id). We compute by (6.10)(ii) ˆ ˆ √ 1 1 W (∇yk ) ≥ W (Id + εk ∇TkT ∇uk ) + o(1) εk Ω ε ˆk Ω  1 1 √ = Q(e(∇TkT ∇uk )) + ω( εk ∇TkT ∇uk ) + o(1) εk Ω 2 as k → ∞. The second term converges to 0 arguing as in (10.9) and us−1/8 ing k∇uk k∞ ≤ Cεk (see (6.12)). As e(∇TkT ∇uk ) * e(∇T T ∇u) weakly in 2 2×2 L (Ω, Rsym ) by (6.12)(ii) and Q is convex we conclude ˆ ˆ 1 1 lim inf W (∇yk ) ≥ Q(e(∇T T ∇u)), k→∞ εk Ω Ω 2 as desired. (ii) By a general density result in the theory of Γ-convergence together with Theorem A.1.6, Theorem A.1.8 and the fact that the limiting functional E(u, P, T ) is continuous in u with respect to the convergence given in Theorem A.1.6 and Theorem A.1.8 it suces to provide recovery sequences for u ∈ W(Ω). Moreover, as in the proof of Theorem 6.2.1 we may assume that H1 (∂P \ Ju ) = 0 up to an innitesimal small perturbation of u (a similar argument was used in the proof of Lemma 10.2.5). Let (u, P, T ) ∈ D∞ and εk → 0 be given. De√ ne yk (x) = T x + εk u(x) for all x ∈ Ω. It is not hard to see that (yk )k ⊂ SBVM (Ω) for εk small enough. Moreover, dene P k = P , Tk (x) = T x and
1 uk = √εk yk − Tk ≡ u for all k ∈ N. Then (6.10),(6.11), (6.13) and the rst two parts of (6.12) hold trivially. To see the (6.12)(iii) it suces to note that −1/8 k∇uk k∞ = k∇uk∞ ≤ C ≤ Cεk . We nallyPconrm limk→∞ Eεk (yk ) = E(u, P, T ). As for all k ∈ ´ N we have 1 1 1 1 H (Juk ) = 2 j P (Pj , Ω)+H (Ju \∂P ), it suces to show limk→∞ εk Ω W (∇yk ) = ´ 1 Q(e(∇T T ∇u)). Using again that W (G) = 21 Q(e(G − Id)) + ω(G − Id) we Ω 2 compute ˆ ˆ 1 1 W (∇yk ) = W (∇TkT ∇yk ) εk Ω εk Ω ˆ   1 1 √ = Q(e(∇TkT ∇uk )) + ω( εk ∇TkT ∇uk ) 2 εk ˆΩ ˆ √ 1 1 T = Q(e(∇T ∇u)) + O( εk ) → Q(e(∇T T ∇u)). Ω 2 Ω 2 267

This nishes the proof.



Remark 10.3.1. Due to the assumptions in the density result of Theorem A.1.8 we have to suppose that u ∈ L2 (Ω) in Theorem 6.3.1(ii). A possible strategy to drop this additional assumption is to show that each limiting conguration u given by Theorem 6.2.1 can be approximated in the sense of (6.10)-(6.13) by a sequence (v l )l ⊂ GSBD(Ω) ∩ L2 (Ω) such that E(u, P, T ) = liml→∞ E(v l , P, T ). A natural candidate seems to be the sequence (v l )l given in the proof of Theorem 6.2.1, but the verication of the convergence of the surface energy appears to be a subtle problem. The proof of Corollary 6.3.2 is now straightforward. Proof of Corollary 6.3.2. To see the liminf-inequality assume without restriction that Eεk (yεk ) ≤ C and yεk → y in L1 for k → ∞. By (6.10)(i), (6.11) we obtain y = T for some T ∈ R(P) for a Caccioppoli partition P . Moreover, Theorem 6.3.1 yields lim inf k→∞ Eεk (yk ) ≥ Eseg (y). A recovery sequence is obviously given by yk = y for all k ∈ N. 

10.4 Application: Cleavage laws We are nally in a position to prove the cleavage law in Theorem 6.3.3. Proof of Theorem 6.3.3. The proof is very similar to the proof of Theorem 1.6.5 and we only indicate the necessery changes. Let (yεk )k be a sequence of almost minimizers. Passing to a suitable subsequence, by Theorem 6.2.1 we obtain a triple (uk , P k , Tk ) ∈ D and a limiting triple (u, P, T ) ∈ D∞ such that (6.10)(6.13) hold and

E(u, P, T ) ≤ lim inf ε→0 inf{Eε (y) : y ∈ A(aε )} by Theorem 6.3.1(i). Due to the boundary conditions it is not hard to see that 2×2 on each component Pj ∈ P we nd Aj ∈ Rskew and cj ∈ R2 such that −1/2

u1 (x) = limk→∞ εk

(e1 · (Id − Rjk ) x − e1 · ckj + aε x1 )

= e1 · Aj x + e1 · cj + ax1

(10.21)

for a.e. x ∈ Ω0 with x1 < 0 or x1 > l and x ∈ Pj . In particular, this implies

u1 (x1 , x2 ) − u1 (ˆ x1 , x2 ) = |x1 − xˆ1 |a

(10.22)

for a.e. x ∈ Ω0 with x ˆ1 < 0, x1 > l and (x1 , x2 ), (ˆ x1 , x2 ) ∈ Pj . We rst derive the limiting minimal energy and postpone the characterization of the sequence of almost minimizers to the end of the proof. The argument in (10.21) shows that ∇T = Id on Pj if |Pj ∩ {x : x1 < 0 or x1 > l}| > 0. As in the 268

proof of Lemma 10.2.5 (cf. also proof of Theorem 6.3.1(ii)) we may assume that H1 (∂P \ Ju1 ) = 0 after a possible innitesimal perturbation. Consequently, it is not restrictive to assume ∇T T ∇u = ∇u a.e. Indeed, we may replace u by ∇T u in a component Pj which does not intersect the boundaries without changing the energy. By (6.16), a slicing argument in GSBD (see Theorem A.1.5 or [34, Section 8,9]) and the fact that inf{Q(F ) : eT1 F e1 = a} = αa2 (see Section 6.3) we obtain ˆ ˆ 1 Q(e(∇u)) + |νu · e1 |dH1 + E(u) E(u, P, T ) ≥ Ju Ω0 2 ˆ 1ˆ l  α T ≥ (e1 ∇u(x)e1 )2 dx1 + S x2 (u) dx2 + E(u), 0 0 2 where S x´2 denotes the number of jumps of u1 on a slice (−η, l + η) × {x2 } and E(u) = Ju (1 − |νu · e1 |)dH1 . If S x2 ≥ 1 the inner integral is bounded from below by 1. By the structure theorem for Caccioppoli partitions (see Theorem A.2.1) we nd that ((−η, 0) ∪ (l, l + η)) × {x2 } ⊂ Pj for some j ∈ N for H1 a.e. x2 with S x2 = 0 and then arguing as in (5.18) we derive that the term is bounded from below by 21 αla2 due to the boundary conditions (10.22). This implies E(u) ≥ min{ 12 αla2 , 1}. Otherwise, it is not hard to see that the congurations yεelk = x + F aεk x for x ∈ Ω0 satisfy Eεk (yεelk ) → 21 αla2 for k → ∞. Likewise, we get Eεk (yεcrk ) = 1 for all k ∈ N, where yεcrk (x) = xχx1 < 1 + (x + (laεk , 0))χx1 > 1 for x ∈ Ω and 2 2 yεcrk = (x1 (1 + aεk ), x2 ) for x ∈ Ω0 \ Ω. This completes (6.17). It remains to characterize the sequences of almost minimizers. Let u be a minimizer of E under the boundary conditions (10.21). Let rst |a| < acrit . Repeating the arguments in the proof of Theorem 1.6.5(i) we nd that u ∈ H 1 (Ω0 ) with u(x) = F a x + A x + c 2×2 for x ∈ Ω and suitable A ∈ Rskew , c ∈ R2 , where the matrix A appears as in contrast to the proof of Theorem 1.6.5 we cannot derive eT1 ∇ue2 = 0, but only eT1 ∇ue2 + eT2 ∇ue1 = 0 (cf. (10.21)). In particular, this implies P consists −1/2 only of P1 = Ω0 and thus by (10.21) we get A = limk→∞ εk (Id − R1k ) and −1/2 −1/2 e1 · c = − limk→∞ εk e1 · ck1 . Letting s = limk→∞ e2 · (εk ck1 + c) (which exists by (6.10)(i), (6.12)(i)), we now conclude by (6.10) (i) for a.e. x ∈ Ω −1/2

lim εk

k→∞

(yεk (x) − x) −1/2

= u(x) + lim εk k→∞

(R1k − Id) x + ck1




(10.23)

= u(x) − A x − c + (0, s) = (0, s) + F a x. If |a| > acrit we argue as in the proof of Theorem 1.6.5(ii) to nd p ∈ (0, l), 269

2 Ai ∈ R2×2 skew , ci ∈ R for i = 1, 2 such that ( A 1 x + c1 u(x) = A 2 x + c2

for x1 < p, for x1 > p,

where we used that E(u) = 0 i νu = ±e1 a.e. Indeed, the linearized rigidity estimate in [25] can also be applied in the GSBD-setting as it relies on a slicing argument and an approximation which is also available in the generalized framework (see [54, Section 3.3]). (The only dierence is that the approximation does not converge in L1 but only pointwise a.e. which does not aect the argument.) Now repeating the calculation in (10.23) for the sets P1 = {x ∈ Ω0 : x1 < p} and P2 = Ω0 \ P1 we nd s, t ∈ R such that for x ∈ Ω a.e. −1/2

lim εk

k→∞

(yεk (x) − x) = u(x) − (A1 x + c1 )χx1

p (x) + (0, s)χx1

p (x).

This nishes the proof.



270

Appendix A Functions of bounded variation and Caccioppoli partitions A.1 (G)SBV and (G)SBD functions In this section we collect the denitions and fundamental properties of the function spaces needed in this thesis. Let Ω ⊂ Rd open, bounded with Lipschitz boundary. Recall that the space SBV (Ω, Rd ), abbreviated as SBV (Ω) hereafter, of special functions of bounded variation consists of functions y ∈ L1 (Ω, Rd ) whose distributional derivative Dy is a nite Radon measure, which splits into an absolutely continuous part with density ∇y with respect to Lebesgue measure and a singular part Dj y whose Cantor part vanishes and thus is of the form

Dj y = [y] ⊗ ξy Hd−1 bJy , where Hd−1 denotes the (d − 1)-dimensional Hausdor measure, Jy (the `crack path') is an Hd−1 -rectiable set in Ω, ξy is a normal of Jy and [y] = y + − y − (the `crack opening') with y ± being the one-sided limits of y at Jy . If in addition ∇y ∈ L2 (Ω) and Hd−1 (Jy ) < ∞, we write y ∈ SBV 2 (Ω). See [6] for the basic properties of this function space. Likewise, we say that a function y ∈ L1 (Ω, Rd ) is a special function of bounded T deformation if the symmetrized distributional derivative Ey := (Dy) 2 +Dy is a d×d nite Rsym -valued Radon measure with vanishing Cantor part. It can be decomposed as

Ey = e(∇y)Ld + E j y = e(∇y)Ld + [y] ξy Hd−1 |Jy ,

(A.1)

where e(∇y) is the absolutely continuous part of Ey with respect to the Lebesgue measure Ld , [y], ξy , Jy as before and a b = 21 (a ⊗ b + b ⊗ a). For basic properties of this function space we refer to [5, 8]. 271

To treat variational problems as considered in Section 6 (see in particular (6.2)) the spaces SBV (Ω) and SBD(Ω) are not adequate due to the lacking L∞ bound being essential in the compactness theorems. To overcome this diculty the space of GSBV (Ω) was introduced consisting of all Ld -measurable functions u : Ω → Rd such that for every φ ∈ C 1 (Rd ) with the support of ∇φ compact, the composition φ ◦ u belongs to SBVloc (Ω) (see [38]). In this new setting one may obtain a more general compactness result (see [6, Theorem 4.36]). Unfortunately, this approach cannot be pursued in the framework of SBD functions as for a function u ∈ SBD(Ω) the composite φ ◦ u typically does not lie in SBD(Ω). In [34], Dal Maso suggested another approach which is based on certain properties of one-dimensional slices. First we have to introduce some notation. For every ν ∈ Rd \ {0}, for every s ∈ Rd and for every B ⊂ Ω we let

B ν,s = {t ∈ R : s + tν ∈ B}.

(A.2)

Furthermore, dene the hyperplane Πν = {x ∈ Rd : x · ν = 0}. Moreover, for every function y : B → Rd we dene the function y ν,s : B ν,s → Rd by

y ν,s (t) = y(s + tν)

(A.3)

and yˆν,s : B ν,s → R by yˆν,s (t) = y(s + tν) · ν . If yˆν,s ∈ SBV (B ν,s , R) and Jyˆν,s denotes the the approximate jump set we dene

Jyˆ1ν,s := {x ∈ Jyˆν,s : |[ˆ y ν,s ](x)| ≥ 1}. The space GSBD(Ω, Rd ) of generalized functions of bounded deformation is the space of all Ld -measurable functions y : Ω → Rd with the following property: There exists a nonnegative bounded Radon measure λ on Ω such that for all ν ∈ S d−1 := {x ∈ Rd : |x| = 1} we have that for Hd−1 -a.e. s ∈ Πν the function yˆν,s = y ν,s · ν belongs to SBVloc (Ων,s ) and ˆ   |Dˆ y ν,s |(B ν,s \ Jyˆ1ν,s ) + H0 (B ν,s ∩ Jyˆ1ν,s ) dHd−1 (s) ≤ λ(B) Πν

for all Borel sets B ⊂ Ω. If in addition e(∇y) ∈ L2 (Ω) and H1 (Jy ) < ∞, we write y ∈ GSBD2 (Ω). We refer to [34] for basic properties of this space. We recall fundamental compactness, slicing and approximation results.

Compactness and lower semicontinuity We state a version of Ambrosio's compactness theorem in SBV adapted for our purposes (see e.g. [6]):

272

Theorem A.1.1. Let (yk )k be a sequence in SBV (Ω, Rd ) such that ˆ

|∇yk (x)|2 dx + Hd−1 (Jyk ) + kyk k∞ ≤ C Ω

for some constant C not depending on k. Then there is a subsequence (not relabeled) and a function y ∈ SBV (Ω, Rd ) such that yk → y in L1 (Ω), and ∇yk * ∇y in L2 (Ω),

Dj yk *∗ Dj y as Radon measures.

(A.4)

An important subset of SBV is given by the indicator functions χW , where W ⊂ Ω is measurable with Hd−1 (∂ ∗ W ) < ∞. Sets of this form are called sets of nite perimeter (cf. [6]). As a direct consequence of Theorem A.1.1 we get the following compactness result.

Theorem A.1.2. Let (Wk )k ⊂ Ω be a sequence of measurable sets satisfying Hd−1 (∂Wk ) ≤ C for some constant C independent of k . Then there is a subsequence (not relabeled) and a measurable set W such that χWk → χW in measure for k → ∞. We now state a compactness result for the generalized spaces. In [34, Theorem 11.3] we nd the following theorem which we slightly adapt for our purposes.

Theorem A.1.3. Let (yk )k be a sequence in GSBD(Ω). Suppose that there exist a constant M > 0 and an increasing continuous functions ψ : [0, ∞) → [0, ∞) with limx→∞ ψ(x) = +∞ such that ˆ

ˆ

|e(∇yk )|2 + H1 (Jyk ) ≤ M

ψ(|yk |) + Ω

Ω

for every k ∈ N. Then there exist a subsequence, still denoted by (yk )k , and a function y ∈ GSBD(Ω) such that yk → y pointwise a.e. in Ω, 2×2 e(∇yk ) * e(∇y) weakly in L2 (Ω, Rsym ), 1

(A.5)

1

lim inf H (Jyk ) ≥ H (Jy ). k→∞

An analogous result holds in GSBV replacing e(∇yk ), e(∇y) by ∇yk , ∇y (see [4, Theorem 2.2]). The lower semicontinuity result for the jump set can be generalized considering one-dimensional slices. For a concave function θ : (0, ∞) → [0, 1] let ˆ ˆ

µ ˆνy (B) :=

θ(|[ˆ y ν,s ](t)|) dH0 (t) dHd−1 (s) Πν

B ν,s ∩Jyˆν,s

for all Borel sets B ⊂ Ω. 273

Lemma A.1.4. Let (yk )k be a sequence in GSBD(Ω) converging to a function y ∈ GSBD(Ω) in the sense of (A.5). Then µ ˆνy (U ) ≤ lim inf µ ˆνyk (U ) k→∞

for every ν ∈ S d−1 and for all open sets U ⊂ Ω. Proof. As yk → y in the sense of (A.5) we may assume that (yν,s )k → yν,s in

GSBV (U ν,s ) for Hd−1 -a.e. s ∈ U ν := {s ∈ Πν : U ν,s 6= ∅}. This is one of the essential steps in the proof of Theorem A.1.3 (cf. [34, Theorem 11.3] or [8, Theorem 1.1] for an elaborated proof in the SBD-setting). The desired claim now follows from the corresponding lower semicontinuity result for GSBV functions (see e.g. [6, Theorem 4.36]) and Fatou's lemma.  We briey note that using the area formula (see e.g. [6, Theorem 2.71, 2.90])) and ne properties of GSBD functions (see [34]), µ ˆνy (B) can be written equivalently as ˆ ν µ ˆy (B) = θ(|[y] · ν|)|ξy · ν| dHd−1 (A.6) Jy ∩B

for all ν ∈ S d−1 and all Borel sets B ⊂ Ω (see also [34, Remark 9.3]).

Slicing We briey state the main slicing properties of SBV functions. For a proof we refer to [6, Section 3.11]. Recall denitions (A.2) and (A.3).

Theorem A.1.5. Let y ∈ SBV (Ω, Rd ). For all ν ∈ S d−1 and Hd−1 -a.e. s in Πν = {x : x · ν = 0} the function y ν,s belongs to SBV (Ων,s , Rd ). Moreover, one has ∇y(s + tν) · ν = (y ν,s )0 (t) for a.e. t ∈ Ων,s , Jyν,s = {t ∈ R : s + tν ∈ Jy }, ˆ ˆ d−1 #Jyν,s dH (s) = |ξy · ν| dHd−1 . Πν

Jy

There is an analogous result in (G)SBD (see [34, Section 8,9]): Replace ∇(s + tν) · ν by ν T e(∇y(s + tν))ν and y ν,s by yˆν,s .

Approximation results Another basic tool we need are density results. We start with a result in the space of SBV2 (see [31]). We dene W(Ω, Rd ) as the space of all SBV 2 functions y such that Jy is a nite, disjoint union of (d-1)-simplices and y ∈ W k,∞ (Ω \ Jy , Rd ) for all k . 274

Theorem A.1.6. The space W(Ω, Rd ) is dense in SBV 2 (Ω, Rd ) in the following sense: For every y ∈ SBV 2 (Ω, Rd ) there exists a sequence yn ∈ W(Ω, Rd ) such that for n → ∞ (i) kyn − ykL1 (Ω) → 0, (ii) k∇yn − ∇ykL2 (Ω) → 0, (iii) H1 (Jyn ) → H1 (Jy ). We now state a density result for SBV functions due to Cortesani and Toader being appropriate for anisotropic surface energies (see [32]). Moreover, a proof very similar to that of Proposition 2.5 in [51] shows that we may also impose ˜⊃ suitable boundary conditions on the approximating sequence. Assume that Ω d Ω is a bounded, open domain in R with Lipschitz boundary dening the Dirichlet ˜ of Ω. Moreover, let ΩD,δ := {x ∈ Ω ˜ : dist(x, ∂D Ω) ≤ boundary ∂D Ω = ∂Ω ∩ Ω ˜ \ Ω) for δ > 0. δ} ∪ (Ω

˜ . For every u ∈ SBV 2 (Ω, ˜ Rd ) ∩ L∞ (Ω, ˜ Rd ) Theorem A.1.7. Let g ∈ W 1,∞ (Ω) ˜ with u = g on Ω \ Ω, there exists a sequence un and a sequence of neighborhoods ˜ of Ω ˜ \ Ω such that un = g on Ω 1 , un ∈ W 1,∞ (Un ) and un |Vn ∈ Un ⊂ Ω D, n W(Vn , R2 ), where Vn ⊂ Ω is some neighborhood of Ω \ Un , such that kun k∞ ≤ kuk∞ and ˜ Rd ), ∇un → ∇u strongly in L2 (Ω, ˜ Rd ), (i) un → u strongly in L1 (Ω, ´

´

(ii) lim supn→∞ Jun φ(νun )dH1 ≤ Ju φ(νu )dH1 for every upper semicontinuous function φ : S 1 → [0, ∞) satisfying φ(ν) = φ(−ν) for every ν ∈ S 1 .

´ ˜ and thus it will be penalized in φ(νu ) dH1 if Recall that u is dened on Ω Ju u does not attain the boundary condition g on the Dirichlet boundary ∂D Ω. The following result proved in [54] together with Theorem A.1.6 provides a density result in GSBD.

Theorem A.1.8. Let u ∈ GSBD2 (Ω, Rd ) ∩ L2 (Ω, Rd ). Then there exists a sequence un ∈ SBV 2 (Ω, Rd ) such that each Jun is contained in the union of a nite number of closed connected pieces of C 1 -surfaces, each un belongs to W 1,∞ (Ω \ Jun , R2 ) and the following properties hold: (i) kun − ukL2 (Ω) → 0, (ii) ke(∇un ) − e(∇u)kL2 (Ω) → 0, (iii) H1 (Jun ) → H1 (Ju ). The last result is an adaption of [26] to the GSBD-setting. In the proof of Theorem 9.4.1 we will draw ideas from [26] to establish a slightly dierent variant of the density result [31] where an L∞ -bound for the derivative is preserved. 275

Distance to weakly dierentiable functions The distance of an SBV function to Sobolev functions can be measured by the distribution curl ∇y (see [25, Proposition 5.1]).

Theorem A.1.9. Let Q = (0, 1)d . Let y ∈ SBV∞ (Q) := {y ∈ SBV (Q, Rd ) : k∇yk∞ < ∞, Hd−1 (Jy ) < ∞}. Then µy := curl ∇y is a measure concentrated on Jy such that |µy | ≤ Ck∇yk∞ Hd−1 |Jy .

d there is a constant C = C(p) > 0 such that for all Moreover, for p < d−1 y ∈ SBV∞ (Q) there is a function u ∈ H 1 (Q, Rd ) such that

k∇u − ∇ykLp (Q) ≤ C|µy |(Q) ≤ Ck∇yk∞ Hd−1 (Jy ).

A.2 Caccioppoli partitions We rst introduce the notions of perimeter and essential boundary. Consider E ⊂ Rd measurable and let ˆ  1 d div(ϕ) : ϕ ∈ Cc (Ω, R ), kϕk∞ ≤ 1 P (E, Ω) = sup (A.7) E

be the perimeter of E in Ω (see [6, Section 3.3]). Moreover, we dene the essential boundary by   [ |E ∩ B% (x)| ∗ d d ∂ E=R \ x ∈ R : lim%↓0 =t . (A.8) t=0,1 |B% (x)| By [6, (3.62)] we have

P (E, Ω) = Hd−1 (Ω ∩ ∂ ∗ E).

P We say a partition P = (Pj )j of Ω is a Caccioppoli partition of Ω if j P (Pj , Ω) < +∞. We say a partition is ordered if |Pi | ≥ |Pj | for i ≤ j . In the whole thesis we will always tacitly assume that partitions are ordered. Given a rectiable set S we say that a Caccioppoli partition is subordinated to S if (up to an Hd−1 -negligible set) the essential boundary ∂ ∗ Pj of Pj is contained in S for every j ∈ N. The local structure of Caccioppoli partitions can be characterized as follows (see [6, Theorem 4.17]).

Theorem A.2.1. Let (Pj )j be a Caccioppoli partition of Ω. Then [ j

(Pj )1 ∪

[ i6=j

contains Hd−1 -almost all of Ω. 276

∂ ∗ Pi ∩ ∂ ∗ Pj

Here (P )1 denote the points where P has density one (see [6, Denition 3.60]). Essentially, the theorem states that Hd−1 -a.e. point of Ω either belongs to exactly one element of the partition or to the intersection of exactly two sets ∂ ∗ Pi , ∂ ∗ Pj . In particular, the structure theorem implies (see [6, (4.24) and Theorem 4.23]) [  X 2Hd−1 ∂ ∗ Pj ∩ Ω = P (Pj , Ω). (A.9) j

j

We now state a compactness result for ordered Caccioppoli partitions (see [6, Theorem 4.19, Remark 4.20]).

Theorem A.2.2. Let Ω ⊂ Rd bounded, open with Lipschitz boundary. Let Pi =P(Pj,i )j , i ∈ N, be a sequence of ordered Caccioppoli partitions of Ω with supi j P (Pj,i , Ω) ≤ C independent of i ∈ N. Then there exists a Caccioppoli partition P = (Pj )j and a not relabeled subsequence such that Pj,i → Pj in measure for all j ∈ N as i → ∞. Caccioppoli partitions are naturally associated to piecewise constant functions. We say y : Ω → Rd is piecewiese constant in Ω if there exists P a Caccioppoli d partition (Pj )j of Ω and a sequence (tj )j ⊂ R such that y = j tj χPj . We close this section with a compactness result for piecewise constant functions (see [6, Theorem 4.25]).

Theorem A.2.3. Let Ω ⊂ Rd bounded, open with Lipschitz boundary. Let (yi )i ⊂ SBV (Ω, Rd ) be a sequence of piecewise constant functions such that supi (kyi k∞ + Hd−1 (Jyi )) ≤ C independent of i ∈ N. Then there exists a not relabeled subsequence converging in measure to a piecewise constant function y.

277

Appendix B Rigidity and Korn-Poincaré's inequality The following geometric rigidity result by Friesecke, James and Müller (see [49]) is fundamental in the thesis.

Theorem B.1. Let Ω ⊂ Rd a (connected) Lipschitz domain and 1 < p < ∞. Then there exists a constant C = C(Ω, p) such that for any y ∈ W 1,p (Ω, Rd ) there is a rotation R ∈ SO(d) such that k∇y − RkLp (Ω) ≤ C kdist(∇y, SO(d))kLp (Ω) .

(B.1)

One ingredient in the proof is the following decomposition into a harmonic and a rest part.

Theorem B.2. Let Ω ⊂ R2 open and 1 < p < ∞. There is a constant C = C(p) such that all y ∈ W 1,p (Ω, R2 ) can be split into y = w + z , where w is a harmonic function and z satises k∇y − ∇wkLp (Ω) = k∇zkLp (Ω) ≤ Ck dist(∇y, SO(2))kLp (Ω) . Note that the constant C is independent of the domain Ω. In higher dimensions one additional needs k∇yk∞ ≤ M for M > 0. Proof. Following the singular-integral estimates in [30, Section 2.4] we nd k∇zkLp (Ω) ≤ ckcof∇y −∇ykLp (Ω) . The assertion follows from the fact that |cofA− A|p ≤ Cp distp (A, SO(2)) for all A ∈ R2×2 (see also (3.11) in [49]).  We state a Poincaré inequality in BV (see [6]).

Theorem B.3. Let Ω ⊂ Rd bounded, connected with Lipschitz boundary. Then there is a constant C > 0, which is invariant under rescaling of the domain, such that for all u ∈ BV (Ω, Rd ) ku − ck

d

L d−1 (Ω)

≤ C|Du|(Ω)

for some c ∈ Rd , where | · | denotes the total variation. 279

Finally, we recall a Korn-Poincaré inequality and a trace theorem in BD (see [21, 62]).

Theorem B.4. Let Ω ⊂ Rd bounded, connected with Lipschitz boundary and let P : L2 (Ω, Rd ) → L2 (Ω, Rd ) be a linear projection onto the space of innitesimal rigid motions. Then there is a constant C > 0, which is invariant under rescaling of the domain, such that for all u ∈ BD(Ω, Rd ) ku − P uk

where Eu =

DuT +Du 2

d

L d−1 (Ω)

≤ C|Eu|(Ω),

is the symmetrized distributional derivative.

Theorem B.5. Let Ω ⊂ R2 bounded, connected with Lipschitz boundary. There exists a constant C > 0 such that the trace mapping γ : BD(Ω, R2 ) → L1 (∂Ω, R2 ) is well dened and satises the estimate
kγukL1 (∂Ω) ≤ C kukL1 (Ω) + |Eu|(Ω)

for each u ∈ BD(Ω, R2 ). For sets which are related through bi-Lipschitzian homeomorphisms with Lipschitz constants of both the homeomorphism itself and its inverse uniformly bounded the constants in Theorem B.1 and Theorem B.4 can be chosen independently of these sets, see e.g. [49].

280

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Logos Verlag Berlin

Effective Theories for Brittle Materials: A Derivation of Cleavage Laws and Linearized Griffith Energies from Atomistic and Continuum Nonlinear Models

The main focus of this book lies on the derivation of effective models for brittle materials in the simultaneous passage from discrete-to-continuum and nonlinear to linearized systems. In the first part the cleavage behavior of brittle crystals is investigated including the identification of critical loads for failure and the analysis of the geometry of crack paths that occur in the fractured regime. In the second part Griffith functionals in the realm of linearized elasticity are derived from nonlinear and frame indifferent energies by means of a quantitative geometric rigidity result for special functions of bounded deformation.

M. Friedrich

A thorough understanding of crack formation in brittle materials is of great interest in both experimental sciences and theoretical studies. Such materials show an elastic response to very small displacements and develop cracks already at moderately large strains. Typically there is no plastic regime in between the restorable elastic deformations and complete failure due to fracture.

ASMPI

ISSN 1611-4256

28

ISBN 978-3-8325-4028-9