Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Planar Geometrically Linearized Hexagonal-to-Rhombic Phase Transformation

The geometrically linearized hexagonal-to-rhombic phase transformation is a very flexible transformation, which allows for numerous exact solutions to the associated three-well problem (4) with different types of boundary data. Here the simplest possible solutions are so-called simple laminates, for which the strain is a one-dimensional function \(e(\nabla u)(x)=f(x\cdot n)\) for some vector \(n \in S^{1}\) and for which i.e., \(e(\nabla u)\) only attains two values. The possible directions of these laminates, as given by the vector \(n \in S^{1}\) are (up to sign reversal) six discrete values which arise as the symmetrized rank-one directions between the energy wells: For each \(i_{1},i_{2} \in \{1,2,3\}\) with \(i_{1} \neq i_{2}\) there exists (up to sign reversal and exchange of the roles of \(a_{i_{1},i_{2}}\) and \(n_{i_{1},i_{2}}\) and renormalization) exactly one pair \((a_{i_{1},i_{2}},n_{i_{1},i_{2}}) \in \mathbb{R}^{2} \setminus \{0\}\times S^{1}\) with the property that The possible vectors are collected in Lemma 16.

In addition to these “simple” constructions, there are further exact solutions to the three-well problem associated with the hexagonal-to-rhombic phase transformation, e.g., there are patterns involving all three variants as depicted in Figs. 24 and 25 in the Appendix.

In the sequel, we study solutions to the hexagonal-to-rhombic phase transformation with affine boundary conditions, i.e., we consider \(u\in W^{1,p}_{\text{loc}}(\mathbb{R}^{2})\) with \(p\in (2,\infty ]\) such that Here we investigate the rigidity/ non-rigidity of the problem by asking whether it has non-affine solutions: Clearly, a necessary condition for this is that \(e(M)\in \operatorname{conv}(K)\). Using the method of convex integration, Müller and Šverák [43] (cf. also the Baire category arguments of [21, 22]) constructed multiple solutions to related differential inclusions, displaying the existence of a variety of solutions to the problem. Noting that these techniques are applicable to our set-up of the three-well problem, ensures that for any \(M\) with \(e(M)\in \operatorname{intconv}(K)\) there exists a non-affine solution to (6). We point out that in the setting of the geometrically linearized hexagonal-to-rhomobic phase transformation, all convex hulls coincide, see Lemma 10, which allows us to work with convex instead of lamination, rank-one or quasiconvex hulls.

In general these convex integration solutions are however very “wild” in the sense that they do not possess very strong regularity properties (cf. [26]). As our inclusion (6) is motivated by a physical problem, a natural question addresses the relevance of this multitude of solutions: Guided by the physical problem at hand and the literature on these problems, natural criteria to consider are surface energy constraints and surface energy regularizations of the minimization problem (1) (cf. for instance [9, 10, 14, 35, 36]). Here the presence of the higher order surface energy contributions gives rise to a length scale which is also reflected in physical experiments (e.g., branching structures are predicted), cf. also our follow-up work [50] for an energetic point of view on convex integration solutions. On the level of our differential inclusion surface energy constraints translate into regularity constraints and lead to the question, whether unphysical convex integration solutions have a natural regularity threshold. Here an immediate regularity property of solutions to (6) is that \(e(\nabla u)\in L^{\infty }( \mathbb{R}^{2})\). With slightly more care, it is also possible to obtain solutions with the property that \(u\in W^{1,\infty }_{\text{loc}}( \mathbb{R}^{2})\) (which in an abstract way follows from the stability arguments from Chap. 3 in [33] but which in this text we will implement by an explicit construction. In particular this allows us to explicitly control the size of the skew part, see Proposition 34, thus justifying the use of the geometrically linearized theory of elasticity). However, prior to this work it was not known whether these solutions can enjoy more regularity, i.e., whether for instance there are convex integration solutions with \(\nabla u \in W^{s,p}(\varOmega )\) for some \(s>0\), \(p\geq 1\). In particular, this shows that there are exact solutions to the (geometrically linearized version of the) minimization problem (1) which enjoy better regularity properties. This makes convex integration solutions potentially also interesting from an energy minimization point of view for energies involving elastic and surface energy contributions (cf. the experimental work of Inamura [29] for first experimental results capturing rather wild microstructures which might be related to convex integration). We however emphasize that our findings do not answer the question (Q2) on the physical relevance of convex integration solutions. They should be viewed as a first attempt at approaching this question.

Let us comment on this result: to the best of our knowledge it represents the first \(W^{s,p}\) higher regularity result for convex integration solutions arising in differential inclusions for shape-memory materials. In addition to providing a regularity result for the displacement gradient \(\nabla u\), we also show higher \(W^{s,p}\) regularity for the characteristic functions of the martensitic phases. Invoking results of Sickel [52], this also implies bounds on the dimension of the singular set of the characteristic functions (cf. Remark 6). This in turn can be interpreted as a measure of the “fractality” of the constructed solutions. In this context, we also point out that our solutions are “piecewise affine” in the sense of Sect. 4 in [33], i.e., at almost every point of the domain, the iterative convex integration solution turns into an exact solution after a finite number of steps (where the number of steps however depends on the respective point). This entails that although our solutions are expected to be “wild”, they are not as wild as solutions to other convex integration schemes, where it might be necessary to iterate infinitely often at each point in the domain.

The given quantitative dependences for \(\theta _{0}\) are certainly not optimal in the specific constants. While it is certainly possible to improve on these numeric values, a more interesting question deals with the qualitative expected dependences: Is it necessary that \(\theta _{0}\) depends on \(\frac{\operatorname{dist}(e(M), \partial \operatorname{conv}(K))}{\operatorname{dist}(e(M),K)}\)?

Since for \(M\in \mathbb{R}^{2\times 2}\) with \(e(M)\in \partial \operatorname{conv}(K)\) there are no non-affine solutions to (6), it is natural to expect that convex integration constructions deteriorate for matrices \(M\) with \(e(M)\) approaching the boundary of \(\operatorname{conv}(K)\). The precise dependence on the behavior towards the boundary however is less intuitive. In this context, it is interesting to note that the regularity threshold \(\theta _{0}>0\) does not depend on the distance to the boundary of \(K\), but rather on the angle which is formed between the initial matrix \(e(M)\) and the boundary of \(\operatorname{conv}(K)\). This is in agreement with the intuition that the larger the angle is, the better the convex integration algorithm becomes, as it moves the values of the iterations which are used to construct the displacement \(u\) further into the interior of \(K\). In the interior of \(K\) it is possible to use larger length scales which increases the regularity of solutions. Whether this dependence is necessary in the value of the product of \(sp\) or whether the product \(sp\) should be independent of this and only the value of the corresponding norm should deteriorate with a smaller angle, is an interesting open question. In a follow-up work, [51], we introduce a different construction which provides a uniform lower bound on the attainable regularity. However, that construction is then not anymore “piecewise affine”.

We remark that in the very special case of additional symmetries it is possible to construct much better solutions. An example is given in the appendix for the case \(M=0\) (cf. also [48] and [11]). For these boundary data and specific domain geometries we show that it is possible to construct a solution \(u\) of the associated differential inclusion such that \(e(\nabla u) \in K\) and \(e(\nabla u)\in BV\). The skew part of the displacement gradient however diverges (and is “unphysical” in this sense). For the hexagonal-to-rhombic phase transformation the boundary data \(M=0\), which correspond to \(e(M)\) lying exactly in the barycenter of the three energy wells, is the only example with such substantially improved regularity properties that we are aware of (cf. [19] for similar examples in the geometrically nonlinear setting). The high symmetry situation with the improved solutions is thus very non-generic in this sense and requires very strong symmetries. It is an important and challenging open question, whether it is possible to exploit further symmetries and thus to construct further solutions with these much better regularity properties.

A fascinating problem in studying solid-solid, diffusionless phase transformations modeling shape-memory materials is the dichotomy between rigidity and non-rigidity. Since the work of Müller and Šverák [43], who adapted the convex integration method of Gromov [27, 28] and Nash-Kuiper [39, 44] to the situation of solid-solid phase transformations, and the work of Dacorogna and Marcellini [21, 22], it is known that under suitable conditions on the convex hulls of the energy wells, there is a very large set of possible minimizers to (1) (cf. also [53] and [33] for a comparison of these two methods). More precisely, the set of minimizers forms a residual set (in the Baire sense) in the associated function spaces. However, in general convex integration solutions are “wild”; they do not enjoy very good regularity properties. This has rigorously been proven for the case of the geometrically nonlinear two-well problem [25, 26], the geometrically nonlinear three-well problem in three dimensions (the “cubic-to-tetragonal phase transformation”) [17, 32] and (under additional assumptions) for the geometrically linear six-well problem (the “cubic-to-orthorhombic phase transformation”) [49]. In these works it has been shown that on the one hand convex integration solutions exist, if the displacement gradient is only assumed to be \(L^{\infty }\) regular. If on the other hand, the displacement gradient is \(BV\) regular (or a replacement of this), then solutions are very rigid and for most constant matrices \(M\) the analogue of (6) does not possess a solution.

Thus, in these specific examples convex integration solutions cannot exist at \(BV\) regularity for the displacement gradient; at this regularity solutions are rigid. At \(L^{\infty }\) regularity they are however flexible and a multitude of solutions exist. Similarly as in it is hence natural to adopt a more quantitative point of view and to ask whether there is a regularity threshold which distinguishes between the rigid and the flexible regime.

It is the purpose of this article to make a first, very modest step into the understanding of this dichotomy by analyzing the \(W^{s,p}\) regularity of a (known) convex integration scheme in an as simple as possible model case.

In our construction of solutions to the differential inclusion (6) we follow the ideas of Müller and Šverák [43] (in the version of [47]) and argue by an iterative convex integration algorithm. For the hexagonal-to-rhombic transformation this is particularly simple, since the laminar convex hull equals the convex hull of the wells and since all matrices in the convex hull are symmetrized rank-one-connected with the wells (cf. Lemma 10). As a consequence it is possible to construct piecewise affine solutions (in the language of [33], Chap. 4). This simplifies the convergence of the iterative construction drastically. It is one of the reasons for studying the hexagonal-to-rhombic phase transformation as a model problem.

Yet, in spite of the (relative) simplicity of obtaining convergence of the iterative construction to a solution of (6) and hence of showing existence, substantially more care is needed in addressing regularity. In this context we argue by an interpolation result (cf. Theorem 2 and Proposition 66): While our approximating displacements \(u_{k}:\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) are such that the \(BV\) norms of the iterations increase (exponentially), the \(L^{1}\) norm of their difference decreases exponentially. If the threshold \(\theta _{0}>0\) is chosen appropriately, the \(W^{s,p}\) norm for \(0< sp< \theta _{0}\) is controlled by an interpolation of the \(BV\) and the \(L^{1}\) norms, which can be balanced to be uniformly bounded. This is based on the interpolation results from [13] which characterizes Besov spaces as interpolation spaces of \(L^{p}\) and \(BV\) spaces and is similar to the strategy from for instance [18] or [53]. To derive the associated bounds, we have to make the iterative algorithm quantitative in several ways which distinguishes it from the “usual” convex integration schemes: For a further discussion of the quantitative aspects of our convex integration scheme and the differences with respect to more standard convex integration algorithms we refer to the discussion in Sect. 3.2 following the presentation of Algorithms 27 and 30.

The remainder of the article is organized as follows: After briefly collecting preliminary results in the next section (interpolation results, results on the convex hull of the hexagonal-to-rhombic phase transition), in Sect. 3 we begin by describing the convex integration scheme which we employ. Here we first recall the main ingredients of the qualitative scheme (Sect. 3.1) and then introduce our more quantitative algorithms in Sects. 3.2–3.3.2. As this algorithm crucially relies on the existence of an appropriate covering, we present an explicit construction of this in Sect. 4. Here we also address quantitative covering estimates for the perimeter and the volume. The ingredients from Sects. 3 and 4 are then combined in Sect. 5, where we prove Theorem 1 for a specific class of domains. In Sect. 6 we explain how this can be generalized to arbitrary Lipschitz domains. Finally, in the Appendix, we recall a very special symmetry based construction for a solution to (6) with \(M=0\) with much better regularity properties for \(e(\nabla u)\) but with unbounded skew part.

Seeking to show higher Sobolev regularity for convex integration solutions, we rely on the characterization of \(W^{s,p}\) Sobolev functions. Here we recall the following two results on an interpolation characterization [13] and on a geometric characterization of the regularity of characteristic functions [52]:

Before proceeding to the proof of Theorem 2, we present an immediate corollary of it: For functions which are “essentially” characteristic functions we obtain the following unified result:

In the sequel, we will mainly rely on Corollary 3, since in our applications (e.g. in Propositions 66, 69) we will mainly deal with functions which are “essentially” characteristic functions.

After this discussion, we come to the proof of Theorem 2:

As an alternative to the interpolation approach, a more geometric criterion for regularity is given by Sickel:

Although this theorem provides good geometric intuition and could have been used as an alternative means of proving Theorem 1, we do not pursue this further in the sequel, but postpone its discussion to future work.

Before discussing our convex integration scheme, we recall some basic notions and properties of the hexagonal-to-rhombic phase transformation, which we will use in the sequel.

We begin by introducing notation for the symmetric and antisymmetric part of two matrices.

In this section we describe the replacement construction that allows us to modify constant gradients by replacing them with an affine construction that preserves the boundary values. Moreover, the resulting new gradients are controlled (cf. Lemma 23).

We will make use of a construction by Otto [47], see also the video at [46], which is a variant of a construction by Conti [15]. We will recall the construction here in detail since it is not publicly available in printed form.

In our applications we make use of a version of this construction with slightly different conventions:

Using this construction as a basic building block, the following lemma allows us to replace a general matrix \(M \in \mathbb{R}^{2\times 2}\) and to restrict the replacement matrices to an \(\epsilon \)-neighborhood of a rank-one line passing through \(M\).

We now show how to apply Lemma 21 to the setting of symmetric matrices in our three-well problem (5):

We emphasize that in our construction in Lemma 23, we have the choice between two different solutions, which differ in the sign of their skew symmetric component and thus in the choice of the corresponding rank-one connection (cf. (28)). This freedom of choice is a central ingredient in the control over the skew symmetric part of the iterated constructions. We summarize this observation in the following corollary.

In this subsection we formulate our convex integration algorithm. It consists of two parts, Algorithms 27 and 30. The first part (Algorithm 27) determines the symmetric part of the iterated displacement vector field, while the second part (Algorithm 30) deals with the choice of the “correct” skew component.

After formulating the algorithms, we prove their well-definedness (i.e. show that it is indeed possible to iterate this construction as claimed).

In the whole section we assume that the domain \(\varOmega \) and the matrix \(M\) in (6) fit together in the sense that \(\varOmega = Q_{ \beta }[0,1]^{2}\), where \(Q_{\beta }\) is the rotation of the Conti construction from Lemma 21 for \(M\) (and the closest energy well \(e^{(i)}\)). We emphasize here the rotation angle \(\beta \), which will become important in our analysis. These “special” domains will play the role of the essential building blocks of the construction of convex integration solutions in general Lipschitz domains (cf. Sect. 6).

We define our convex integration scheme:

While this algorithm prescribes the symmetric part of the iteration, we complement it with an algorithm which defines the choice of the skew part. Here the main objectives are to keep the resulting skew parts uniformly bounded (which is necessary, if we seek to obtain bounded solutions to (6)) and simultaneously to ensure the choice of the “right” rank-one direction (cf. Sect. 5, Lemma 63). Here the rank-one direction has to be chosen “correctly” in the sense that the successive Conti constructions are not rotated too much with respect to one another (which corresponds to the “parallel” case, cf. Definition 29).

In order to make this precise, we introduce two definitions: the first (Definition 28) allows us to introduce an “ordering” on the triangles in \(\{\varOmega _{j,k}\}_{k\in \{1,\dots ,J_{j}\}}\) for different values of \(j\in \mathbb{N}\). With this at hand, we then define the notions of being parallel or rotated (cf. Definition 29).

With this we define the parallel and the rotated cases:

Let us comment on this definition: Intuitively, its objective is to describe whether successive Conti constructions can be chosen as essentially parallel or whether they are necessarily substantially rotated with respect to each other (hence, these notions will also play a crucial role in Sect. 4, where we construct our precise covering). More precisely, let \(SP_{j}\) be as in Algorithm 27 and let \(j\), \(j_{0}\), \(D\), \(\bar{D}\) be as in Definition 29. Then, at the iteration step \(j_{0}\) the triangle \(\bar{D}\) was a subset of one of the Conti rectangles \(R_{j-j_{0},l}^{k}\). Thus, \(u_{j-j_{0}}\) is modified according to the Conti construction with respect to \(\nabla u_{j-j_{0}}|_{\bar{D}}\), \(e_{j-j_{0}}^{(p)}|_{\bar{D}}\) in this domain. In particular, the difference of the matrices \(e(\nabla u_{j-j_{0}}|_{\bar{D}})\), \(e_{j-j_{0}}^{(p)}|_{\bar{D}}\) determines a direction \(e\) in strain space (up to a choice of the skew direction (cf. Corollary 26) this is directly related to the orientation of the Conti rectangle \(R_{j-j_{0},l}^{k}\)). By virtue of Corollary 20 all of the new matrices \(e(\tilde{M}_{0}(\nabla u_{j-j_{0}}|_{\bar{D}})),\dots \), \(e( \tilde{M}_{4}(\nabla u_{j-j_{0}}|_{\bar{D}}))\) essentially lie on the line \(e\) in strain space. Hence the direction which is determined by the difference of \(e^{(p)}_{j-j_{0}+1}|_{D}\) and \(e(\nabla u_{j-j_{0}+1}|_{D})\), is still essentially parallel to the directions \(e\) (in strain space). As by definition (we are now in Step 2(a) of Algorithm 27) the values of \(e^{(p)}_{j-j _{0} + l}|_{D}\) and of \(\nabla u_{j-j_{0} +l}|_{D}\) do not change further until \(l=j_{0}\) is reached, the requirement in (32) implies that the direction \(e\) spanned by \(e(\nabla u_{j-j_{0}}|_{\bar{D}})\), \(e^{(p)}_{j-j_{0}}|_{\bar{D}}\) and the one spanned by \(e(\nabla u_{j}|_{D})\), \(e^{(p)}_{j}|_{D}\) are essentially parallel (cf. Lemma 37 and Remark 38 for the precise statements). If we choose the correct skew directions in Step 2(b) of Algorithm 27, we can hence ensure that the successive Conti constructions are essentially parallel, if (32) is satisfied.

We remark that for this argument to hold and for it to yield new, significant information, it was necessary in Definition 29 to mod out the cases in which Step 2(a) was active, i.e., \(\mathcal{P}_{l}(D) = \{D\}\), as during these there are no changes.

If (33) holds, then the directions of the successive Conti constructions are necessarily substantially rotated with respect to each other (cf. Lemma 39 for the precise bounds). In this case we cannot substantially improve the situation to being more parallel by choosing the skew part appropriately in Corollary 26. Thus, in the sequel, we will exploit these instances as possibilities to control the size of the skew part and to use this, if necessary, to change the sign of the skew direction. The precise formulation of this is the content of Algorithm 30.

Let us comment on these algorithms: Due to the structure of the convex hulls (Lemma 10), our convex integration algorithm produces a (countably) piecewise affine solution (in contrast to the solutions obtained by means of an in-approximation scheme). This is reflected in the fact that the displacement \(u_{j}\) is not further modified in the piecewise polygonal domains in \(\varOmega \setminus \varOmega _{j}\). The preceding algorithm differs from a non-quantitative version of a convex integration scheme in several aspects: These points are central in our higher regularity argument: As we seek to prove higher regularity by means of the interpolation result from Theorem 2 or Corollary 3, we have to control the BV norm of the resulting displacement gradients. However, by a countably infinite (self-similar) covering of the whole domain, this is in general not possible (the total perimeter of the covering triangles is not bounded in general). Hence we only consider finite coverings, which produce a controlled (but growing) BV norm and simultaneously allows us to cover a sufficiently large volume fraction \(v_{0}\) of our domain \(\varOmega _{j}\). That it is possible to satisfy these two competing aims is content of the covering results of the next sections (cf. Propositions 52 and 55). This finite covering of \(\varOmega _{j,k}\) is the cause for the splitting of Step 2 into two parts. Part (a) deals with the triangles which are not covered by Conti constructions and are in this sense “errors” (in the sense that \(u_{j}\) is not modified here), while part (b) deals with the part of the domain that is covered by Conti constructions on which \(u_{j}\) is modified.

The specification of \(\epsilon _{j}\) is of key relevance as well. It distinguishes in a quantitative way whether a new rank-one connection is rotated or not with respect to the corresponding last rank-one connection. In our BV estimate this leads to different bounds (cf. the perimeter estimates in Propositions 52 and 55). In particular we cannot afford substantial rotations, as long as \(\epsilon _{j}\ll \epsilon _{0}\) is very small, since this would yield superexponential growth for the BV norms, which cannot be compensated in our estimates (cf. Fig. 8 and the corresponding explanations for the intuition behind this).

Due to the relation between the size of the scales \(\delta _{j}\) (which itself is directly coupled to the admissible error \(\epsilon _{j}\)) and our regularity estimates, we in general seek to choose the value of \(\epsilon _{j}\) as large as possible without leaving \(\operatorname{intconv}(K)\). By the intercept theorem, it is always possible to choose \(\epsilon _{j}\) to be “relatively large” in the push-out steps (cf. Notation 25). However, for stagnant matrices, this is no longer possible. Here we have to ensure a choice of \(\epsilon _{j}\) which is summable in \(j\in \mathbb{N}\) (in Algorithm 27 we choose it geometrically decaying), in order to avoid leaving \(\operatorname{intconv}(K)\). These considerations lead to the case distinction in the definition of \(\epsilon _{j+1}\) in Step 2(b) of Algorithm 27.

Finally, the quantitative prescription of the skew part is central to deduce the quantitative BV bound of Lemma 63, as we have to take care that, as long as we remain “parallel” in strain space (cf. Definition 29), we approximately preserve the same skew direction. This is necessary to prevent the Conti constructions from being substantially rotated with respect to each other if \(\epsilon _{j}\) is very small and constitutes a crucial ingredient in the derivation of our perimeter and BV estimates in Sects. 4 and 5 (cf. Fig. 8 for the intuition behind this).

The normalization of the initial skew part is convenient (though not necessary).

We now proceed to prove that Algorithms 27 and 30 are well-defined. Here in particular, it is crucial to show that with our choice of the admissible error \(\epsilon _{j}\), we do not leave \(\operatorname{intconv}(K)\) in the iteration except to attain one of the energy wells in \(K\) (cf. Proposition 31). Moreover, we seek to construct solutions to (6) which are Lipschitz regular. These points are the content of the following two Propositions 31 and 34, which deal with the symmetric and anti-symmetric parts respectively. To show these we will rely on several auxiliary observations.

Finally, in this last subsection, we show that Algorithms 27 and 30 can be used to deduce the existence of solutions to our problem (6).

In Sects. 4 and 5 we present a more refined analysis of this construction algorithm. In particular, we give an explicit quantitative construction for the covering procedure from Step 2 in Algorithm 27.

In this section we introduce the central objects of our covering (cf. Definitions 40 and 43) and state our main covering result (Proposition 45).

As a preparation for the main part of this section, we begin by discussing auxiliary results on matrix space geometry. We first estimate the angle formed in strain space between two matrices:

Next we observe the following bounds on the rotation angles:

With these auxiliary results at hand, we proceed to the discussion of our central covering objects. In order to define our set of covering triangles, \(\{\varOmega _{j,k}\}_{k\in \{1,\dots ,J_{j}\}}\), we consider a subclass of triangles with, for our purposes, suitable properties. To this end, we can not ensure that all domains appearing in our covering argument are right angle triangles (due to the presence of the green triangles in the Conti construction in Fig. 10), for which one of the other angles is approximately of size \(\delta _{j}\). However, the following definition provides a family of sets with similar properties. This will allow us to formulate a precise, iterative covering result.

In our construction one prominent reference direction is obtained from Conti’s construction, as detailed in the following definition.

In the sequel, we will give a precise covering result, which shows that in the \(j\)th step of our convex integration Algorithms 27 and 30, we may assume that only very specific triangles are present in the collection \(\{\varOmega _{j,k}\} _{k\in \{1,\dots ,J_{j}\}}\) as (parts of) level sets of \(\nabla u_{j}\). To this purpose we define the following classes of triangles:

Let us comment on this classification: The basic distinction criterion separating the triangles into the different cases is given by checking whether the corresponding triangles are roughly aligned (as in the cases (P1), (P2)) or whether they are substantially rotated (as in the cases (R1), (R2)) with respect to the direction of the relevant Conti construction (the case (R3) is a special “error situation”, which does not entirely fit into this heuristic consideration). Roughly speaking, this determines whether we are in a situation analogous to the first or to the second picture in Fig. 8. This distinction is necessary, as else a control of the arising surface energy is not possible in a, for our purposes, sufficiently strong form.

This distinction (essentially) coincides with our definition of the parallel and the rotated cases (cf. Definition 29): If a triangle \({D_{j}} \in \{\varOmega _{j,k}\} _{k\in \{1,\dots ,J_{j}\}}\) is in the parallel case in step \(j\), then the directions of the Conti construction, which gave rise to \({D_{j}}\), and of the relevant Conti construction at step \(j\) (i.e. the construction by which \({D_{j}}\) is (in part) covered) are essentially parallel (cf. Lemma 37 and Remark 38). Letting \(j_{0}\in \mathbb{N}\) denote the index from Definition 29 and assuming that \(j_{0}=1\), there are three possible scenarios for the relation of \(\delta _{j}|_{D_{j}}\) and \(\delta _{j-1}|_{D_{j}}\): If a triangle \({D_{j}} \in \{\varOmega _{j,k}\}_{k\in \{1,\dots ,J_{j}\}}\) is in the rotated case in step \(j\), then the direction of the Conti construction, which gave rise to \({D_{j}}\), and the direction of the relevant Conti construction at step \(j\) are necessarily substantially rotated with respect to each other (cf. Lemma 39). Again assuming that the index \(j_{0}=1\) (where \(j_{0}\) denotes the index from Definition 29), we now distinguish two cases for the relation between \(\delta _{j}|_{ {D_{j}}}\) and \(\delta _{j-1}|_{D_{j}}\): Here we first note that necessarily (by definition of the rotated case, as occurring only after a push-out step) we have \(\delta _{j}|_{D_{j}} = \delta _{0}\). Then there are two options for \(\delta _{j-1}|_{D_{j}}\): The case (R3) only occurs as an error case as a consequence of our specific covering procedure for the triangles of the types (R1) and (R2).

We relate the different cases to the heuristics given at the beginning of Sect. 4 (cf. Fig. 8). We view the cases (P1) and (R1) as the “model cases” without and with substantial rotation and corresponding to the parallel and orthogonal (triangular) situation depicted in Fig. 8. In both cases (P1) and (R1) the aspect ratio of the given triangle \(D_{j}\) is roughly of order \(\delta _{j}|_{D _{j}}\) (i.e. the quotient of its shortest and of its longest sides are roughly of that order) and we seek to cover it with a Conti construction of comparable ratio \(\delta _{j}|_{D_{j}}\).

The cases (P2) and (R2) are situations in which the underlying triangle \(D_{j}\) is roughly of side ratio \(\delta _{j-1}|_{D_{j}}\) (i.e., the quotient of its shortest and of its longest sides are roughly of that order), where we however seek to cover the triangle with Conti constructions with ratio \(\delta _{0}\). This mismatch is a consequences of our construction of the function \(\delta _{j}|_{D_{j}}\) in Algorithm 27: here we prescribe that the matrices which are pushed out (cf. Notation 25), are allowed to have an error tolerance of \(\delta _{0}\). In particular, it may occur that \(\delta _{j-1}|_{D_{j}}\ll \delta _{j}|_{D_{j}}=\delta _{0}\), which is the situation described in (P2), (R2) either without or with substantial rotation.

The case (R3) is a consequence of how we deal with “remainders” in our covering constructions for the cases (R1), (R2).

Our main result of the present section states that it is possible to find a covering of the level sets which respects Algorithms 27 and 30, such that only the specific triangles from Definition 43 occur. Moreover, we provide bounds for the remaining uncovered “bad” volume and the resulting perimeters.

In the remainder of this section we seek to prove this result and to construct the associated covering. To this end, in Sect. 4.2 we first explain that the “natural covering” of the Conti construction, which is achieved by splitting it into its level sets, satisfies the requirements of Proposition 45. In particular, this implies that the covering, which is obtained in Step 1 of Algorithm 27, satisfies the properties of Proposition 45 (the resulting triangles are of the types (P1), (P2) or (R1), (R2)). Hence, in the remaining part of the section, it suffices to prove that given a triangle of the type (P1)–(R3), we can construct a covering for it which obeys the claims of Proposition 45. To this end, in Sect. 4.3, we first describe a general construction, on which we heavily rely in the sequel. With this construction at hand, in Sect. 4.4 and its subsections we then deal with the cases (P1), (P2), in which there is no substantial rotation involved. Subsequently, we discuss the cases (R1)–(R3) with non-negligible rotations in Sect. 4.5. Finally, in Sect. 4.6 we provide the proof of Proposition 45.

The generalization to more generic domains is detailed in Sect. 6.

We begin with our covering construction by explaining that a Conti construction of ratio \(\delta _{j}|_{D_{j}}\) can be divided into a finite number of triangles which are all of the types (P1), (P2) and (R1)–(R3).

We begin our iterative covering statements by presenting a general building block, which we will frequently use in the sequel. Given a triangle \(D\) we seek to reduce the discussion to that of a rectangle \(R_{2}\), whose long side is aligned with the direction of \(D\) and which is of similar volume as the original triangle. Only in the covering of this rectangle will the situations (P1), (P2) and (R1), (R2) differ. For the case (R3) we argue differently.

We further explain how, given a box \(R\) with some rotation angle with respect to the \(x_{1}\)-axis, we construct a block of the type \(R_{2}\).

In this section we explain how, given a triangle \(D_{j} \in \{\varOmega _{j,k}\}_{k\in \{1,\dots ,J_{j}\}}\) of type (P1) or (P2), we can cover it by a combination of the relevant Conti constructions and some remaining triangles, which are again of the types (P1), (P2) and (R1), (R2). Moreover, we seek to achieve two partially competing objectives: On the one hand, we have to control the volume of \(D_{j}\) which is covered by Conti constructions, from below. On the other hand, we aim at keeping the resulting overall perimeter of the new covering geometry as small as possible. The construction of a covering which balances these two objectives, is the content of Proposition 52, which is the main result of this section.

Motivated by the heuristic considerations at the beginning of Sect. 4 (cf. Fig. 8(a)), we expect that in the cases (P1) and (P2), in which there is no substantial rotation with respect to the relevant Conti construction, the two competing objectives of sufficient volume coverage (Proposition 52(1)) and of a good perimeter bound (Proposition 52(3)), can be satisfied with a surface energy, which is independent of \(\delta _{j}\) and \(\delta _{0}\). Indeed, it is possible to show that in the situation without substantial rotation, in each iteration step the overall perimeter of the covering of a triangle is comparable to the perimeter of the original triangle up to a loss of a controlled universal factor.

In the proof of Proposition 52 we have to be careful in the choice of the covering, in order to keep all the resulting triangles parallel to the direction of \(D_{j}\) or parallel to the relevant Conti construction (cf. Definition 42). This is necessary to ensure a covering such that the sum of the resulting perimeters is comparable to the original perimeter; in particular no factor of \(\delta _{0}\) occurs here. We emphasize that this alignment with the directions of the original triangle or the relevant Conti construction is a central point, since if a (substantial) rotation angle with respect to these directions were to be obtained (e.g. as illustrated in Fig. 17, where the covering gives rise to triangles which are rotated by an angle of \(\frac{\pi }{2}\)), we would inevitably fall into cases similar as the situations described in (R1), (R2), however with a ratio \(\delta _{j}\), which might be substantially smaller than \(\delta _{0}\). As explained at the beginning of Sect. 4, this would entail a growth of the perimeters of the covering by a factor \(\delta _{j}\). As a consequence our BV estimate from Sect. 5 would become a superexponential bound, which could no longer be compensated by the only exponential \(L^{1}\) decay. This would hence destroy all hopes of deducing good higher regularity estimates for the convex integration solutions.

The remainder of this section is organized into three parts: We first discuss the covering constructions for the cases (P1) and (P2) separately in Sects. 4.4.1 and 4.4.2. Then in Sect. 4.4.3 we combine these cases, in order to provide the proof of Proposition 52.

In this section we deal with the covering in the cases (R1)–(R3). As in Sect. 4.4 we seek to simultaneously control the perimeter of the resulting covering and the volume of the domain, which is covered by Conti constructions. Motivated by the discussion from the beginning of Sect. 4, we however expect that it is unavoidable to produce estimates in which the ratio \(\delta _{0}\) appears.

With this expectation, we are less careful in our covering constructions and for instance do not seek to preserve the direction \(n\), in which the corresponding \(\delta _{j}\)-good triangles are oriented. Yet, we still heavily rely on Proposition 48 and only modify the construction within the block \(R_{2}\). This will give rise to certain new “error triangles”, which are of the type (R3). In analogy to Proposition 52 we have:

As in Proposition 52 the proof of this statement is based on separate discussions of the cases (R1), (R2), (R3) and can be deduced by combining the results of Lemmas 56, 58, 59, 46 and Proposition 48. Since this does not involve new ingredients, we restrict our attention to the discussion of the cases (R1)-(R3) and omit the details of the proof of Proposition 52. The analysis of the cases (R1)-(R3) is the content of the following subsections.

We initialize the construction by applying Step 1 in Algorithms 27 and 30. As these initial triangles are obtained as the level sets of a Conti construction with ratio \(\delta _{0}\), they all form \(\delta _{0}\) good triangles (cf. Lemma 46) and hence satisfy the properties of the theorem. It therefore remains to argue that this is preserved in our constructions from Sects. 4.4 and 4.5. Given one of the triangles \(D_{j}\) as in the theorem, the results of Propositions 52 and 55 ensure this, once the rotation angle of the successive Conti constructions is controlled. This however is the achieved by virtue of Remark 38 and Lemma 39.