Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Planar Geometrically Linearized HexagonaltoRhombic Phase Transformation
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Abstract
In this article we discuss quantitative properties of convex integration solutions arising in problems modeling shapememory materials. For a twodimensional, geometrically linearized model case, the hexagonaltorhombic phase transformation, we prove the existence of convex integration solutions \(u\) with higher Sobolev regularity, i.e., there exists \(\theta _{0}>0\) such that \(\nabla u \in W^{s,p}_{loc}( \mathbb{R}^{2})\cap L^{\infty }(\mathbb{R}^{2})\) for \(s\in (0,1)\), \(p\in (1,\infty )\) with \(0< sp < \theta _{0}\). We also recall a construction which shows that in very specific situations with additional symmetry much better regularity properties hold.
Keywords
Convex integration solutions Elasticity Solidsolid phase transformations Differential inclusion Higher Sobolev regularityMathematics Subject Classification
35B36 35B65 32F321 Introduction
In this article we are concerned with the detailed analysis of certain convex integration solutions which arise in the modeling of solidsolid, diffusionless phase transformations in shapememory materials. We seek to precisely analyze the regularity properties of these constructions in a simple, twodimensional, geometrically linear model case.
 (i)Reduction to the\(m\)well problem. Instead of studying the full variational problem (1), we only focus on exact minimizers. Restricting to the low temperature regime, this implies that we seek solutions to the differential inclusionfor some \(\theta < \theta _{c}\).$$\begin{aligned} \nabla y \in \bigcup_{j=1}^{m} \mathit{SO}(n) U_{j}(\theta ), \end{aligned}$$(2)
 (ii)Small deformation gradient case, geometric linearization. We further modify (2) and assume that \(\nabla y\) is close to the identity. This allows us to linearize the problem around this constant value (cf. Chap. 11 in [6] and also [5] for a comparison of the linearized and the nonlinear theories). Instead of considering (2), we are thus lead to the inclusion problemThe symmetrized gradient \(e(\nabla u)\) represents the infinitesimal displacement strain associated with the displacement\(u\), which is defined as \(u(x):=y(x)x\). The symmetric matrices \(e_{1},\ldots ,e_{m} \in \mathbb{R}^{n\times n}\) are the exactly stressfree strains representing the variants of martensite. After rescaling, we may assume that they are of a size of order one. While this procedure linearizes the geometry of the problem (by replacing the symmetry group \(\mathit{SO}(n)\) by an invariance with respect to the linear space \(\operatorname{Skew}(n)\)), the differential inclusion (3) preserves the inherent physical nonlinearity which arises from the multiwell structure of the problem. In order to ensure the validity of the geometric linearization assumption, in the sequel we will pay particular attention to deriving solutions with bounded displacement gradients of order one, and hence have skew parts of order one (we recall the normalization that our exactly stressfree strains are of order one). For more detailed comments on this, we refer to the discussion on the \(L^{\infty }\) bounds for \(\nabla u\) below (Q2), to (ii) in Sect. 1.3 and to Algorithm 30 in Sect. 3.2 and the discussion following it.$$\begin{aligned} e(\nabla u):=\frac{\nabla u + (\nabla u)^{T}}{2} \in \{e_{1},\dots ,e _{m}\}. \end{aligned}$$(3)
 (iii)Reduction to two dimensions and the hexagonaltorhombic phase transformation. In the sequel studying an as simple as possible model case, we restrict to two dimensions and a specific twodimensional phase transformation, the hexagonaltorhombic phase transformation (this is for instance used in studying materials such as \(\mbox{Mg}_{2}\mbox{Al}_{4} \mbox{Si}_{5}\mbox{O}_{18}\) or MgCd alloys undergoing a (threedimensional) hexagonaltoorthorhombic transformation, cf. [11, 34], and also for closely related materials such as \(\mbox{Pb}_{3}(\mbox{VO}_{4})_{2}\), which undergo a (threedimensional) hexagonaltomonoclinic transformation, cf. [11, 40, 41]). From a microscopic point of view, the hexagonaltorhombic phase transformation occurs, if a hexagonal atomic lattice is transformed into a rhombic atomic lattice. From a continuum point of view, we model it as solutions to the differential inclusionwhere \(\varOmega \subset \mathbb{R}^{2}\) is a bounded Lipschitz domain and$$ \begin{aligned} & u: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, \\ &\frac{1}{2}\bigl(\nabla u + (\nabla u)^{T}\bigr) \in K \quad\mbox{a.e. in }\varOmega , \end{aligned} $$(4)We note that all the matrices in \(K\) are tracefree, which corresponds to the (infinitesimal) volume preservation of the transformation. We note that the set \(K \) is “large” (its convex hull is a twodimensional set in the threedimensional ambient space of twobytwo, symmetric, tracefree matrices, cf. Lemma 10).$$ \begin{aligned} &K:=\bigl\{ e^{(1)}, e^{(2)}, e^{(3)}\bigr\} \quad\mbox{with} \\ &e^{(1)}:= \begin{pmatrix} 1 & 0 \\ 0& 1 \end{pmatrix} ,\quad e^{(2)}:= \frac{1}{2} \begin{pmatrix} 1 & \sqrt{3} \\ \sqrt{3}& 1 \end{pmatrix} ,\quad e^{(3)}:= \frac{1}{2} \begin{pmatrix} 1 & \sqrt{3} \\ \sqrt{3}& 1 \end{pmatrix} . \end{aligned} $$(5)
1.1 Main Result
In addition to these “simple” constructions, there are further exact solutions to the threewell problem associated with the hexagonaltorhombic phase transformation, e.g., there are patterns involving all three variants as depicted in Figs. 24 and 25 in the Appendix.
 (Q1)
Are there (nonaffine) solutions to (6) with \(M \in \mathbb{R}^{2\times 2}\)?
 (Q2)
Are all the convex integration solutions physically relevant? Or are they only mathematical artifacts? Is there a mechanism distinguishing between the “only mathematical” and the “really physical” solutions?
Motivated by these questions, in this article, we study the regularity of a specific convex integration construction and obtain higher Sobolev regularity properties for the resulting solutions:
Theorem 1
Let\(\varOmega \subset \mathbb{R}^{2}\)be a bounded Lipschitz domain. Let\(K\)be as in (5) and let\(M\in \mathbb{R}^{2\times 2}\)be such that\(e(M):= \frac{M+M^{T}}{2} \in \operatorname{intconv}(K)\). Then there exist a value\(\theta _{0}\in (0,1)\), depending only on\(\frac{\operatorname{dist}(e(M), \partial \operatorname{conv}(K))}{ \operatorname{dist}(e(M),K)}\), and a displacement\(u: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\)with\(u\in W^{1,\infty }_{loc}(\mathbb{R} ^{2})\)such that (6) holds and such that\(\nabla u\in W ^{s,p}_{loc}(\mathbb{R}^{2})\cap L^{\infty }(\mathbb{R}^{2})\)for all\(s\in (0,1)\), \(p\in (1,\infty )\)with\(s p< \theta _{0}\).
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 shapememory 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 nonaffine 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 followup 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 hexagonaltorhombic 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 nongeneric 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.
1.2 Literature and Context
A fascinating problem in studying solidsolid, diffusionless phase transformations modeling shapememory materials is the dichotomy between rigidity and nonrigidity. Since the work of Müller and Šverák [43], who adapted the convex integration method of Gromov [27, 28] and NashKuiper [39, 44] to the situation of solidsolid 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 twowell problem [25, 26], the geometrically nonlinear threewell problem in three dimensions (the “cubictotetragonal phase transformation”) [17, 32] and (under additional assumptions) for the geometrically linear sixwell problem (the “cubictoorthorhombic 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.
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.
1.3 Main Ideas
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 hexagonaltorhombic 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 rankoneconnected 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 hexagonaltorhombic phase transformation as a model problem.
 (i)
Tracking the error in strain space. In order to iterate the convex integration construction, it is crucial not to leave the interior of the convex hull of \(K\) in the iterative modification steps. In qualitative convex integration algorithms, it suffices to use errors which become arbitrarily small, and to invoke the openness of \(\operatorname{intconv}(K)\). As the admissible error in strain space is however coupled to the length scales of the convex integration constructions (cf. Lemma 21) and as these in turn are directly reflected in the solutions’ regularity properties, in our quantitative algorithm we have to keep track of the errors in strain space very carefully. Here we seek to maximize the possible length scales (and hence the aspect ratio of the building block constructions) without leaving \(\operatorname{intconv}(K)\) in each iteration step. This leads to the distinction of various possible cases (the “stagnant”, the “pushout”, the “parallel” and the “rotated” case, cf. Notation 25, Definition 29 and Algorithm 27). In these we quantitatively prescribe the admissible error according to the given geometry in strain space.
 (ii)
Controlling the skew part without destroying the structure of (i). Seeking to construct \(W^{1,\infty }\) solutions, we have to control the skew part of our construction. Due to the results of Kirchheim, it is known that this is generically possible (cf. [33], Chap. 3). However, in our quantitative construction, we cannot afford to arbitrarily change the direction of the rankone connection which is chosen in the convex integration algorithms at an arbitrary iteration step. This would entail \(BV\) bounds which could not be compensated by the exponentially decreasing \(L^{1}\) bounds in the interpolation argument. Hence we have to devise a detailed description of controlling the skew part (cf. Algorithm 30).
 (iii)
Precise covering construction. In order to carry out our convex integration scheme we have to prescribe an iterative covering of our domain by constructions which successively modify a given gradient. As our construction in Lemma 21 relies on triangles, we have to ensure that there is a class of triangles which can be used for these purposes (cf. Section 4). Seeking to control both the \(L^{1}\) and the \(BV\) norms of the resulting convex integration solutions, we have to control competing requirements: On the one hand, we have to quantitatively control the overall perimeter (which can be viewed as a measure of the BV norm of \(\nabla u_{k}\)) of the covering at a given iteration step of the convex integration algorithm. This crucially depends on the specific case (“rotated” or “parallel”) in which we are. On the other hand, we have to ensure that a sufficiently large volume fraction of the underlying domain is covered by our building block constructions (which however costs surface energy) in order to obtain good \(L^{1}\) bounds. That it is possible to satisfy both requirements simultaneously is the content of Proposition 45.
1.4 Organization of the Article
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 hexagonaltorhombic 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.
2 Preliminaries
In this section we collect preliminary results which will be relevant in the sequel. We begin by stating the interpolation results of [13] on which our \(W^{s,p}\) bounds rely. Next, in Sect. 2.2 we recall general facts on matrix space geometry and in particular apply this to the hexagonaltorhombic phase transformation and its convex hulls.
2.1 An Interpolation Inequality and Sickel’s Result
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]:
Theorem 2
(Interpolation with BV, [13])
 (i)Let\(p\in [2,\infty )\)and assume that\(\frac{1}{q} = \frac{1 \theta }{p} + \theta \)for some\(\theta \in (0,1)\). Then$$\begin{aligned} \u\_{W^{\theta ,q}(\mathbb{R}^{n})} \leq C \u\_{L^{p}(\mathbb{R} ^{n})}^{1\theta } \u\_{BV(\mathbb{R}^{n})}^{\theta } . \end{aligned}$$(7)
 (ii)Let\(p\in (1,2]\)and let\(\frac{1}{q} = \frac{1\theta }{p} + \theta \)for some\(\theta \in (0,1)\). Let further\((\theta _{1}, q_{1}) \in (0,1)\times (1,\infty )\)be such thatfor some\(\tau \in (0,1)\), where 1− denotes an arbitrary positive number slightly less than 1. Then,$$\begin{aligned} \frac{1}{q_{1}} &= \frac{1\theta _{1}}{2} + \theta _{1}, \\ \bigl(\theta , q^{1}\bigr) &= \tau (0,1) + (1\tau ) \bigl(\theta _{1}, q_{1}^{1}\bigr), \end{aligned}$$with\(1+ := (1)^{1}\).$$\begin{aligned} \ u \_{W^{\theta , q}(\mathbb{R}^{n} )} \leq C \bigl(\u\_{L^{1+}( \mathbb{R}^{n})}^{\frac{\tau }{1\theta }} \u\_{L^{2}(\mathbb{R} ^{n})}^{1\frac{\tau }{1\theta }} \bigr)^{1\theta } \u \_{BV( \mathbb{R}^{n})}^{\theta }, \end{aligned}$$(8)
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:
Corollary 3
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.
Proof of Corollary 3
After this discussion, we come to the proof of Theorem 2:
Proof of Theorem 2

\(\tilde{F}^{s}_{r,2}(\mathbb{R}^{n}) = L^{s,r}(\mathbb{R}^{n})\) for \(s\in \mathbb{R}\), \(1< r<\infty \) and that for this range \(L^{0,r}( \mathbb{R}^{n})=L^{r}(\mathbb{R}^{n})\),

\(\tilde{F}^{s}_{r,r}(\mathbb{R}^{n})= W^{s,r}(\mathbb{R}^{n})\) for \(0< s<\infty \), \(s\notin \mathbb{Z}\), \(1\leq p <\infty \),
As an alternative to the interpolation approach, a more geometric criterion for regularity is given by Sickel:
Theorem 4
(Sickel, [52])
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.
Remark 5
Remark 6
(Fractal Packing Dimension)
2.2 Matrix Space Geometry
Before discussing our convex integration scheme, we recall some basic notions and properties of the hexagonaltorhombic phase transformation, which we will use in the sequel.
We begin by introducing notation for the symmetric and antisymmetric part of two matrices.
Definition 7
(Symmetric and Antisymmetric Parts)
2.2.1 Lamination Convexity Notions
Relying on the notation from Definition 7, in the sequel we discuss the different notions of lamination convexity. Here we distinguish between the usual lamination convex hull (defined by successive rankone iterations) and the symmetrized lamination convex hull (defined by successive symmetrized rankone iterations):
Definition 8
(Lamination Convex Hull, Symmetrized Lamination Convex Hull)
 (i)Let \(U\subset \mathbb{R}^{n\times n}\). Then we setWe refer to \(U^{lc}\) as the laminar convex hull of\(U\) and to \(\mathcal{L}^{k}(U)\) as the laminates of order at most\(k\).$$\begin{aligned} \mathcal{L}^{0}(U) &:= U, \\ \mathcal{L}^{k}(U) &:= \bigl\{ M\in \mathbb{R}^{2\times 2}: M= \lambda A+ (1 \lambda ) B \mbox{ with } AB = a \otimes n, \lambda \in [0,1], \\ & \quad \quad A,B \in \mathcal{L}^{k1}(U)\bigr\} , \quad k\geq 1, \\ U^{lc} &:= \bigcup_{k=0}^{\infty } \mathcal{L}^{k}(U). \end{aligned}$$
 (ii)Let \(U\subset \mathbb{R}^{n \times n}_{sym}\). Then we defineHere \(a\odot b:= \frac{1}{2}(a\otimes b + b\otimes a)\). We refer to \(U^{lc}_{sym}\) as the symmetrized laminar convex hull of\(U\) and to \(\mathcal{L}^{k}_{sym}(U)\) as the symmetrized laminates of order at most\(k\).$$\begin{aligned} \mathcal{L}^{0}_{sym}(U) &:= U, \\ \mathcal{L}^{k}_{sym}(U) &:= \bigl\{ M\in \mathbb{R}^{2\times 2}: M= \lambda A+ (1\lambda ) B \mbox{ with } AB = a \odot n, \lambda \in [0,1], \\ & \quad \quad A,B \in \mathcal{L}^{k1}_{sym}(U)\bigr\} , \quad k \geq 1, \\ U^{lc}_{sym} &:= \bigcup_{k=0}^{\infty } \mathcal{L}^{k}_{sym}(U). \end{aligned}$$
 (iii)
We denote the convex hull of a set \(U\subset \mathbb{R}^{m}\) by \(\operatorname{conv}(U)\).
Remark 9
We note that if \(U \subset \mathbb{R}^{n\times n}\) or \(U\subset \mathbb{R}^{n\times n}_{sym}\) is (relatively) open, then also \(U^{lc}\) or \(U^{lc,sym}\) is (relatively) open.
Lemma 10
(Convex Hull = Laminar Convex Hull)
Proof
The first point follows from an observation of Bhattacharya (cf. [6] and also Lemma 4 in [49]). The second point either follows from a direct calculation or by an application of Lemma 11 below. □
The following lemma establishes a relation between rankone connectedness and symmetrized rankone connectedness. It in particular shows that in two dimensions all symmetric tracefree matrices are pairwise symmetrized rankone connected.
Lemma 11
(RankOne vs Symmetrized RankOne Connectedness)
 (i)There exist vectors \(a\in \mathbb{R}^{n}\setminus \{0\}, n \in S^{n1}\) such that$$\begin{aligned} e_{1}e_{2} = a\odot n. \end{aligned}$$
 (ii)There exist matrices \(M_{1},M_{2} \in \mathbb{R}^{n\times n}\) and vectors \(a\in \mathbb{R}^{n}\setminus \{0\}, n \in S^{n1}\) such that$$\begin{aligned} M_{1}  M_{2} &= a\otimes n, \\ e(M_{1}) &= e_{1}, e(M_{2}) = e_{2}. \end{aligned}$$
 (iii)
\(\operatorname{rank}(e_{1}e_{2})\leq 2\).
Proof
We refer to [49], Lemma 9 for a proof of this statement. □
This lemma allows us to view symmetrized rankone connectedness essentially as equivalent to rankone connectedness.
2.2.2 Skew Parts
We discuss some properties of the associated skew symmetric parts of rankone connections which occur between points in the interior of \(\operatorname{intconv}(K)\). To this end, we introduce the following identification:
Notation 12
(Skew Symmetric Matrices)
We begin by estimating the symmetric and skewsymmetric parts of a symmetrized rankone connection:
Lemma 13
Proof
Using the previous result, we can control the size of the skew part which occurs in rankone connections with \(K\):
Lemma 14
Proof
In the interest of accessibility to a larger audience, the following subsections are phrased in standard matrix space formulation. It would have equally been possible to phrase these results in terms of conformal and anticonformal coordinates, which would allow for a more concise formulation of some statements. For a work using these methods, see for instance [1].
2.2.3 Geometry of the HexagonaltoRhombic Phase Transformation
In this subsection, we discuss the specific matrix space geometry of the hexagonaltorhombic phase transformation. To this end we decompose each matrix of the form $\left(\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right)$ into a component in ${v}_{1}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ and a component in ${v}_{2}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ direction which essentially corresponds to introducing conformal and anticonformal coordinates.
With this notation we make the following observations:
Lemma 15
Proof
Lemma 16
With Lemma 15 at hand, we can also compute the possible (symmetrized) rankone connections which occur between each well and any possible matrix in \(\operatorname{conv}(K)\):
Lemma 17
Proof
This is a direct consequence of Lemma 15 and of the fact that the set \(K\) forms an equilateral triangle in strain space. □
As an immediate consequence of Lemma 16 and Lemma 17, we infer the following result, which is graphically illustrated in Fig. 2(b):
Lemma 18
Proof
3 The Convex Integration Algorithm
In this section we present and analyze our convex integration algorithm (cf. Algorithms 27 and 30). Our discussion of this consists of four parts: First in Sect. 3.1 we introduce a replacement construction in which a displacement gradient can be modified (cf. Lemmas 19–23). Here we follow Otto’s Minneapolis lecture notes [47] and refer to this construction as a version of Conti’s construction (cf. [15, 20] (Appendix) but also [33]).
Next in Sect. 3.2 we explain how the Conti construction can be exploited to formulate the convex integration algorithm (cf. Algorithms 27 and 30). Here we deviate from the more common qualitative algorithms by precisely prescribing error estimates in strain space, by specifying a covering construction and by controlling the skew part quantitatively.
In Sect. 3.3 we analyze our algorithms and show that they are welldefined (cf. Proposition 31). We further provide a control on the skew part of the resulting construction (cf. Proposition 34).
Finally, in Sect. 3.4 we use Algorithms 27 and 30 to deduce the existence of solutions to the inclusion problem (6), cf. Proposition 36.
We remark that our version of the convex integration scheme is based on particular properties of our set of strains: For the hexagonaltorhombic phase transition the laminar convex hull equals the convex hull (cf. Lemma 10). Moreover, we can connect any matrix in \(\operatorname{intconv}(K)\) with the wells \(K\) (cf. Lemma 11). For a general inclusion problem this is no longer possible and hence more sophisticated arguments are necessary. In spite of the restricted applicability of the scheme, we have decided to focus on the hexagonaltorhombic phase transformation, as it yields one of the simplest instances of convex integration and illustrates the difficulties and ingredients which have to be dealt with in proving higher Sobolev regularity in the simplest possible setup.
3.1 The Replacement Construction
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.
Lemma 19
(Variable Conti Construction)
In our applications we make use of a version of this construction with slightly different conventions:
Corollary 20
Proof of Corollary 20
We apply the construction of Lemma 19 with \(1\lambda \) in place of \(\lambda \) and multiply the resulting function by \(\frac{1}{2 \lambda }\). We remark that the matrices of this corollary are the ones given on p. 56 of [47]. □
Proof of Lemma 19
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 rankone line passing through \(M\).
Lemma 21
(Deformed Conti Construction, p. 57 of [47])
Proof
Remark 22
We remark that both the side ratio \(\delta \) as well as the error \(\epsilon \) remain unchanged under rescalings of the form \(\mu u(\frac{x}{ \mu })\) (as this leaves the gradient invariant).
We now show how to apply Lemma 21 to the setting of symmetric matrices in our threewell problem (5):
Lemma 23
(Application to the threewellproblem, p. 60 ff. of [47])
Remark 24
We point out that the strain \(e^{(i)}\) chosen in (26) is not required to be the one closest to \(e(M)\) if the distance to \(e^{(i)}\) is not much larger than the distance to the closest well. This avoids changing the wells constantly, if a matrix has a symmetrized part very close to the middle between two wells. This becomes important in our quantitative analysis in Sects. 4 and 5 since the “rotated” case behaves considerably worse than the “parallel” case.
Proof
Notation 25

motivate us to refer to the matrix \(\tilde{M}_{4}\) as stagnant (with respect to the replaced matrix \(M\)).
 The matrices \(\tilde{M}_{1}\), \(\tilde{M}_{2}\), \(\tilde{M}_{3}\) will also be called pushedout matrices (with the factors \(\frac{4}{3}\) and \(\frac{16}{15}\) respectively), since by constructionand similarly for the other matrices.$$ \frac{4}{3} \bigl\vert e(M)e^{(i)} \bigr\vert \epsilon \leq \bigl\vert e( \tilde{M}_{1})e^{(i)} \bigr\vert \leq \frac{4}{3} \bigl\vert e(M)e^{(i)} \bigr\vert + \epsilon , $$
We refer to the construction of Lemma 23 as the\((\epsilon , \delta )\)Conti construction with respect to\(M\),\(e^{(i)}\). If some of the parameters of this are selfevident from the context, we also occasionally omit them in the sequel.
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 rankone 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.
Corollary 26
3.2 The Convex Integration Algorithm
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 welldefinedness (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:
Algorithm 27
(Quantitative Convex Integration Algorithm, I)
 Step 0:
 State space and data.
 (a)State space. Our state space is given byHere \(j \in \mathbb{N}\) and \(u_{j}: \varOmega \rightarrow \mathbb{R}^{2}\) is a piecewise affine function. The sets$$\begin{aligned} SP_{j}:=\bigl(j,u_{j},\{\varOmega _{j,k}\}_{k\in \{1,\dots ,J_{j}\}}, e_{j} ^{(p)},\epsilon _{j}, \delta _{j}\bigr). \end{aligned}$$(30)are closed triangles, which form a (up to null sets) disjoint, finite partition of the level sets of \(\nabla u_{j}\), for which \(e(\nabla u _{j}) \notin K\). Let \(\varOmega _{j}:= \bigcup_{k=1}^{J_{j}} \varOmega _{j,k}\) denote the set on which \(e(\nabla u_{j})\) is not yet in one of the energy wells.$$\begin{aligned} \varOmega _{j,k}\subset \varOmega \cap \{\nabla u_{j}=\mathrm{const} \}\cap \bigl\{ e(\nabla u_{j})\notin K\bigr\} \end{aligned}$$The functionis constant on each of the sets \(\varOmega _{j,k}\). It essentially keeps track of the well closest to \(e(\nabla u_{j}_{\varOmega _{j,k}})\) for each \(j\), \(k\).$$\begin{aligned} e^{(p)}_{j}:\varOmega \rightarrow K \end{aligned}$$The functionsare constant on each set \(\varOmega _{j,k}\) and vanish in \(\varOmega \setminus \varOmega _{j}\). They correspond to the error and side ratio in the Conti construction which is to be applied in \(\varOmega _{j,k}\). The functions \(\epsilon _{j}\), \(\delta _{j}\) are coupled by the relation$$\begin{aligned} \epsilon _{j}, \delta _{j}: \varOmega \rightarrow \mathbb{R}, \end{aligned}$$Hence, in the following (update) steps, we will mainly focus on \(\epsilon _{j}\) and assume that \(\delta _{j}\) is modified accordingly.$$\begin{aligned} \delta _{j} = \frac{\epsilon _{j}}{10^{2} d_{K}}, \mbox{ where } d_{K}:= \operatorname{dist}\bigl(e(M),K\bigr). \end{aligned}$$
 (b)Data. Let \(M\in \mathbb{R}^{2\times 2}\) with \(e(M)\in \operatorname{intconv}(K)\). Let \(\varOmega = Q_{\beta }[0,1]^{2}\) with \(Q_{\beta }\) denoting the rotation associated with \(M\) (cf. explanations above). Further set$$\begin{aligned} d_{0} &:= \operatorname{dist}\bigl(e(M), \partial \operatorname{conv}(K)\bigr), \\ \epsilon _{0} &:= \min \biggl\{ \frac{d_{0}}{100}, \frac{1}{1600} \biggr\} , \qquad \delta _{0} := \frac{\epsilon _{0}}{10^{2} d_{K}}. \end{aligned}$$
 (a)
 Step 1:
 Initialization, definition of \(SP_{1}\). We consider the data from Step 0 (b) and in addition defineIn the case of nonuniqueness in the above minimization problem, we arbitrarily choose any of the possible options.$$\begin{aligned} u_{0}(x) &=Mx \omega (M)x, \\ e^{(p)}_{0} &=\operatorname{argmin}\operatorname{dist} { \bigl(e(M), K\bigr)}. \end{aligned}$$
Possibly dividing \(\delta _{0}\) by a factor up to 100, we may assume that \(K_{0,0}:=\delta _{0}^{1}\in \mathbb{N}\). We cover \(\varOmega =Q _{\beta }[0,1]^{2}\) by \(K_{0,0}\) many (translated) up to nullsets disjoint \((\epsilon _{0},\delta _{0})\) Conti constructions with respect to \(\nabla u_{0}\) and \(e^{(p)}_{0}\) (cf. Notation 25). We denote these sets by \(R_{0,1}^{1},\dots , R_{0,K_{0,0}}^{1}\). We remark that \(\varOmega = \bigcup_{l=1}^{K_{0,0}} R_{0,l}^{1}\) is possible with (up to null sets) disjoint choices of \(R_{0,l}^{1}\), \(l\in \{1,\dots ,K_{0,0}\}\), as by definition of the domain \(\varOmega \) the sets \(R_{0,l}^{1}\), \(l\in \{1,\dots ,K_{0,0}\}\), are parallel to one of the sides of \(\varOmega \) and as \(\delta _{0}^{1} \in \mathbb{N}\). We apply Step 2(b) on these sets. As a consequence we obtain \(SP_{1}\).
 Step 2:

Update. Let \(SP_{j}\) be given. Let \(M_{j,k}:=\nabla u _{j}_{\varOmega _{j,k}}\) for some \(k\in \{1,\dots , J_{j}\}\). We explain how to update \(u_{j}\) and \(\epsilon _{j}\), \(\delta _{j}\) on \(\varOmega _{j,k}\).
We seek to apply the construction of Lemma 23 with \(\epsilon _{j,k}:=\epsilon _{j}_{\varOmega _{j,k}}\), \(\delta _{j,k}:=\delta _{j}_{\varOmega _{j,k}}\) andin a part of \(\varOmega _{j,k}\). To this end, we cover the domain \(\varOmega _{j,k}\) by a union of finitely many (up to null sets) disjoint triangles and rectangles. The rectangles are chosen as translated and rescaled versions of the domains in the \((\epsilon _{j,k}, \delta _{j,k})\) Conti construction with respect to the matrices from (31). We denote these rectangles by \(R_{j,l}^{k}\), \(l\in \{1,\dots ,K_{j,k}\}\), for some \(K_{j,k}\in \mathbb{N}\) and require that they cover at least a fixed volume fraction \(v_{0}>0\) of the overall volume of \(\varOmega _{j,k}\) (which is always possible, cf. Sect. 4 for our precise covering algorithm).$$\begin{aligned} e^{(p)}_{j,k}, \ M_{j,k} \end{aligned}$$(31)We define new sets \(\tilde{\varOmega }_{j+1,l}^{k}\), \(l\in \{1,\dots , \tilde{K}_{j,k}\}\): These are given by the triangles which are in \(\varOmega _{j,k}\setminus \bigcup_{l=1}^{K_{j,k}}R_{j,l}^{k}\) and by the triangles which form the level sets of the deformed Conti constructions on the rectangles \(R_{l}^{k}\).As a result of Steps 2 (a) and (b) we obtain \(SP_{j+1}\). (a)For \(x\in \varOmega _{j,k}\setminus \bigcup_{l=1}^{K _{j,k}}R_{j,l}^{k}\) we define$$\begin{aligned} u_{j+1}(x) &:=u_{j}(x), \\ \epsilon _{j+1}(x) &:= \epsilon _{j}(x) \quad \bigl(\mbox{and hence } \delta _{j+1}(x):= \delta _{j}(x)\bigr), \\ e^{(p)}_{j+1}(x) &:= e^{(p)}_{j}(x). \end{aligned}$$Further we set \(\varOmega _{j+1,l}^{k}:=\tilde{\varOmega }_{j+1,l}^{k}\). Carrying this out for all \(k\in \{1,\dots ,J_{j}\}\) hence yields a collection of trianglescovering \(\varOmega _{j}\setminus \bigcup_{l=1}^{K_{j,k}}R_{j,l} ^{k}\).$$ \bigl\{ \varOmega _{j+1,l}^{k}\bigr\} _{k\in \{1,\dots ,J_{j}\},l\in \{1,\dots ,K_{j,k} \}} $$
 (b)In the sets \(R_{j,l}^{k}\) we apply the Conti construction with the matrices from (31). In this application we choose the skew part according to Algorithm 30. With \(\tilde{\varOmega }_{j+1,l}^{k} \subset \bigcup_{k=1}^{J_{j}} \bigcup_{l=1}^{K_{j,k}} R_{j,l}^{k}\) as defined in Step 2 (a), we define \(u_{j+1}_{\tilde{\varOmega }_{j+1,l}^{k}}\) as the function from the corresponding Conti construction. More precisely, in each of the rectangles \(R_{j,l}^{k}\) the matrix \(M_{j,k}\) has been replaced by the matriceswith \(e(\tilde{M}_{0}(M_{j,k})) = e^{(p)}_{j,k}\). For each \(x\in \tilde{\varOmega }_{j+1,l}^{k}\) with \(\tilde{\varOmega }_{j+1,l}^{k}\) as above, we define$$\begin{aligned} \tilde{M}_{0}(M_{j,k}), \dots , \tilde{M}_{4}(M_{j,k}), \end{aligned}$$For the definition of \(\delta _{j+1}\) we recall its coupling with \(\epsilon _{j+1}\). We further set$$\begin{aligned} \epsilon _{j+1}(x):= \textstyle\begin{cases} \epsilon _{0} &\mbox{for } \nabla u_{j+1}_{\tilde{\varOmega }_{j+1,k}} \in \{\tilde{M}_{1}(M_{j,k}),\dots ,\tilde{M_{3}}(M_{j,k})\}, \\ \epsilon _{j}(x)/2 &\mbox{for } \nabla u_{j+1}_{\tilde{\varOmega }_{j+1,k}} = \tilde{M}_{4}(M_{j,k}), \\ 0 &\mbox{for } \nabla u_{j+1}_{\tilde{\varOmega }_{j+1,k}} = \tilde{M}_{0}(M_{j,k}). \end{cases}\displaystyle \end{aligned}$$Here we choose an arbitrary possible minimizer if there is nonuniqueness. Finally, we possibly split each of the sets \(\tilde{\varOmega }_{j+1,l}^{k} \in \bigcup_{l=1}^{K_{j,l}}R_{j,l} ^{k}\) into at most four smaller triangles (cf. Sect. 4.2) and add them to the collection \(\{\varOmega _{j+1,l} ^{k}\}_{k\in \{1,\dots , J_{j}\}, l \in \{1,\dots , K_{j,k}\}}\). Upon relabeling this yields a new collection \(\{\varOmega _{j+1,k}\}_{k\in \{1, \dots , J_{j+1}\}}\).$$\begin{aligned} e^{(p)}_{j+1}(x) := \textstyle\begin{cases} \operatorname*{argmin} _{i\in \{1,2,3\}} & \{e(\nabla u_{j+1})_{ \tilde{\varOmega }_{j+1,k}}  e^{(i)}\} \\ &\mbox{for } \nabla u_{j+1}_{\tilde{\varOmega }_{j+1,k}} \in \{ \tilde{M}_{1}(M_{j,k}),\dots ,\tilde{M_{3}}(M_{j,k})\}, \\ e^{(p)}_{j}(x) &\mbox{for } \nabla u_{j+1}_{\tilde{\varOmega }_{j+1,k}} = \tilde{M}_{4}(M_{j,k}), \\ e^{(p)}_{j}(x) &\mbox{for } \nabla u_{j+1}_{\tilde{\varOmega }_{j+1,k}} = \tilde{M}_{0}(M_{j,k}). \end{cases}\displaystyle \end{aligned}$$
 (a)
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” rankone direction (cf. Sect. 5, Lemma 63). Here the rankone 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).
Definition 28
Let \(D\in \{\varOmega _{j,k}\}_{k\in \{1,\dots ,J_{j}\}}\) for \(j\geq 1\). Then a triangle \(\hat{D} \subset D\) is a descendant of\(D\)of order\(l\), if \(\hat{D}\in \{\varOmega _{j+l,k}\}_{k\in \{1,\dots ,J_{j+l} \}}\) is (part of) a level set of \(\nabla u_{j+l}\) and is obtained from \(D\) by an \(l\)fold application of the update step of Algorithm 27 (where we specify the covering to be the one described in Sect. 4). The set of descendants of\(D\)of order\(l\) is denoted by \(\mathcal{D}_{l}(D)\). We define \(\mathcal{D}(D):=\bigcup_{l=1}^{\infty }\mathcal{D}_{l}(D)\).
A triangle \(\bar{D} \in \{\varOmega _{j,k}\}_{k\in \{1,\dots ,J_{j}\}}\) is a predecessor of order\(l\)of\(D\), if \(D\in \mathcal{D}_{l}( \bar{D})\). We then write \(\bar{D}\in \mathcal{P}_{l}(D)\) and also use the notation \(\mathcal{P}(D)\) for the set of all predecessors of \(D\).
With this we define the parallel and the rotated cases:
Definition 29
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 \(R_{jj_{0},l}^{k}\). Thus, \(u_{jj_{0}}\) is modified according to the Conti construction with respect to \(\nabla u_{jj_{0}}_{\bar{D}}\), \(e_{jj_{0}}^{(p)}_{\bar{D}}\) in this domain. In particular, the difference of the matrices \(e(\nabla u_{jj_{0}}_{\bar{D}})\), \(e_{jj_{0}}^{(p)}_{\bar{D}}\) 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 \(R_{jj_{0},l}^{k}\)). By virtue of Corollary 20 all of the new matrices \(e(\tilde{M}_{0}(\nabla u_{jj_{0}}_{\bar{D}})),\dots \), \(e( \tilde{M}_{4}(\nabla u_{jj_{0}}_{\bar{D}}))\) essentially lie on the line \(e\) in strain space. Hence the direction which is determined by the difference of \(e^{(p)}_{jj_{0}+1}_{D}\) and \(e(\nabla u_{jj_{0}+1}_{D})\), 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 \(e^{(p)}_{jj _{0} + l}_{D}\) and of \(\nabla u_{jj_{0} +l}_{D}\) do not change further until \(l=j_{0}\) is reached, the requirement in (32) implies that the direction \(e\) spanned by \(e(\nabla u_{jj_{0}}_{\bar{D}})\), \(e^{(p)}_{jj_{0}}_{\bar{D}}\) 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.
Algorithm 30
(Quantitative Convex Integration Algorithm, II)
 Step 1:
 Step 2:
 Update. Let \(j\in \mathbb{N}, j\geq 1\). Let \(\omega _{j}\) and \(\varOmega _{j,k}\) be given. Suppose that \(\tilde{\varOmega }_{j+1,l} ^{k}\) with \(\tilde{\varOmega }_{j+1,l}^{k}\in \mathcal{D}_{1}(\varOmega _{j,k})\) is constructed from \(\varOmega _{j,k}\) by our covering argument (cf. Step 2 in Algorithm 27). Then we define \(\omega _{j+1}\) as follows:After having carried out the relabeling step, in which we pass from \(\tilde{\varOmega }_{j+1,l}^{k}\) to \(\varOmega _{j+1,l}\), the function \(\omega _{j+1}\) is constant on each of the triangles in \(\varOmega _{j+1,l}\). Together with Algorithm 27 this completes the construction of \(\nabla u_{j+1}\).
 (a)If \(\tilde{\varOmega }_{j+1,l}^{k}\) is not part of one of the Conti constructions in the covering, then we set$$\begin{aligned} \omega _{j+1}_{\tilde{\varOmega }_{j+1,l}^{k}} = \omega _{j}_{\varOmega _{j,k}}. \end{aligned}$$
 (b)If \(\tilde{\varOmega }_{j+1,l}^{k}\) is part of one of the Conti constructions in the covering, then by Algorithm 27 we seek to apply the construction of Lemma 23 with scale \(\epsilon _{j}_{\varOmega _{j,k}}\) and \(e^{(p)}_{j}_{\varOmega _{j,k}}\), \(\nabla u_{j}_{\varOmega _{j,k}}\). Thus, by Corollary 26 we have two possible choices for the skew part of \(\nabla u_{j+1}\). These are determined by their sign. To define the sign, let \(j_{0}\in \mathbb{N}\) be the smallest integer such that \(D:=\mathcal{P}_{j_{0}}(\varOmega _{j,k}) \neq \varOmega _{j,k}\). We then choose the sign of the new skew direction \(\omega _{j+1}_{\tilde{\varOmega }_{j+1,l}^{k}}\) (and hence determine the whole corresponding skew part) according to$$\begin{aligned} &\operatorname{sgn}(\omega _{j+1}_{\tilde{\varOmega }_{j+1,l}^{k}}  \omega _{j}_{\tilde{\varOmega }_{j+1,l}^{k}}) \\ &\quad := \left\{ \textstyle\begin{array}{l@{\quad}l} \operatorname{sgn}(\omega _{j}_{\varOmega _{j,k}} \omega _{jj_{0}}_{ \varOmega _{j,k}}) & \mbox{if } e^{(p)}_{j}_{\varOmega _{j,l}^{k}} = e^{(p)} _{jj_{0}}_{D}, \\ 1 & \mbox{if } e^{(p)}_{j}_{\varOmega _{j,l}^{k}} \neq e^{(p)}_{jj _{0}}_{D} \wedge \omega _{j}_{\varOmega _{j,k}}\geq 0, \\ 1 & \mbox{if } e^{(p)}_{j}_{\varOmega _{j,l}^{k}} \neq e^{(p)}_{jj _{0}}_{D} \wedge \omega _{j}_{\varOmega _{j,k}}< 0. \end{array}\displaystyle \right. \end{aligned}$$
 (a)

We consider finite coverings of \(\varOmega \setminus \varOmega _{j}\) instead of directly covering the whole domain.

We prescribe the choice of \(\epsilon _{j}\) quantitatively.

We prescribe the skew part quantitatively.
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 pushout 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).
3.3 WellDefinedness of Algorithms 27 and 30
We now proceed to prove that Algorithms 27 and 30 are welldefined. 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 antisymmetric parts respectively. To show these we will rely on several auxiliary observations.
3.3.1 Symmetric Part
We begin by discussing the symmetric part and by showing that in our construction it does not leave \(\operatorname{intconv}(K)\), except to reach \(K\).
Proposition 31
(Symmetric Part)
Proof
We prove the statement inductively. For \(j=0\), we note that this holds since \(\epsilon _{j}=\epsilon _{0}\) and \(\nabla u_{0}=M\).
3.3.2 Skew Symmetric Part
In order to deal with the skew part and to show its boundedness, we need several auxiliary results. These are targeted at controlling the maximal number of pushout steps in the parallel case (cf. Lemma 33), where the notions “parallel” and “rotated” are used as in Definition 29. With the control of the maximal number of pushout steps at hand, we can then present a bound on the skew part of the gradients from Algorithms 27 and 30 (cf. Proposition 34). Together with the boundedness of the symmetrized gradient this yields the uniform \(L^{\infty }\) bounds on \(\nabla u_{j}\).
We begin by estimating the distance to the wells.
Lemma 32
The statement of this lemma is very similar to the result of Proposition 31. However, instead of controlling the distance to the boundary, we here estimate the distance to the wells. This can be substantially larger than the distance to the boundary.
Proof
Using Lemma 32 and recalling Definitions 28 and 29, we bound the maximal number of possible pushout steps in the parallel situation:
Lemma 33
Proof
Relying on the previous lemma, we obtain a uniform bound on the skew part:
Proposition 34
(Skew Symmetric Part)
Proof
We prove the claims inductively and note that \(\omega _{0}=0\) satisfies them. We first discuss (35) and show that it remains true for \(\omega _{j}\) with \(j\in \mathbb{N}\). To this end, let \(l\in \mathbb{N}\) and \(D\subset \{\varOmega _{j+l,k}\}_{k\in \{1,\dots ,J _{j+l}\}}\). For abbreviation we set \(M_{j}:= \nabla u_{j}_{D}\), \(\tilde{\omega }_{j}:=\omega _{j}_{D}\) (and recall that \(\omega _{j}_{D} = \omega (\nabla u_{j}_{D})\)) and first assume that \(\tilde{\omega } _{j} \leq 0\) (see Notation 12). We begin by making the following additional assumption:
Assumption 35
We suppose that the skew matrix \(\tilde{\omega }_{j+l}\) is derived from \(\tilde{\omega }_{j}\) by an \(l\)fold application of Algorithms 27 and 30, where in the Conti construction of Corollary 26 we always choose the positive skew direction.
Step 4: Proof of ( 36 ). In order to obtain the estimate (36), we notice that the skew parts associated with values of \(e(\nabla u_{j})\in K\) may on the one hand be strictly larger than the bound given in (35). But on the other hand, they are derived as an \(\tilde{M}_{0}\) matrix in one of the Conti constructions, in which matrices satisfying (35) are modified. This implies that at most a gain of 5 in the modulus of the corresponding skew part is possible, which yields the bound (36). As these domains are not further modified in the convex integration algorithm this bound cannot deteriorate in the course of the application of Algorithms 27 and 30. □
3.4 Existence of Convex Integration Solutions
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).
Proposition 36
(Convex Integration Solutions)
Proof
We apply Algorithm 27 with \(\bar{M}:= M\omega (M)\). By the results of Propositions 31 and 34 this algorithm is welldefined and can be iterated with \(j \rightarrow \infty \). This yields a sequence of functions \(u_{j}:\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) with bounded gradient (with \(\\nabla u_{j}\_{L^{\infty }(\mathbb{R}^{2})}\) depending on \(d_{K}\), cf. Lemma 33). We prove the convergence of this sequence and show that the limiting function \(u_{0}\) solves (5) with boundary data \(\bar{M}\).
Defining \(u(x):=u_{0}(x) + \omega (M)x\) hence concludes the proof of Proposition 36. □
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.
4 Covering Constructions

we seek to cover an as large as possible volume fraction of it, but at least a given fixed volume fraction, \(v_{0}>0\).

We have to control the perimeters of the triangles in the resulting new covering.
We illustrate this in two extreme situations (cf. Fig. 8): Given a rectangle \(R_{1,\delta _{0}}\) with sides of length 1 and \(\delta _{0}\), we seek to cover it with rectangles which have a fixed side ratio \(r\) and whose long sides are either parallel or orthogonal to the long side of the original rectangle \(R_{1,\delta _{0}}\). In order to illustrate the differences between these situations, we for instance assume that \(r=\delta _{0}/2\). In the situation, in which the original rectangle \(R_{1,\delta _{0}}\) is covered by rectangles whose long side is parallel to the long side of \(R_{1,\delta _{0}}\), the covering can be achieved by splitting \(R_{1,\delta _{0}}\) along its central line as illustrated in Fig. 8(a). Thus, the resulting perimeter (we view it as a measure of the \(BV\) energy of the characteristic functions in the Conti covering), which is necessary to cover the volume of \(R_{1,\delta _{0}}\) is bounded by twice the perimeter of \(R_{1,\delta _{0}}\). If the long sides of the covering rectangles of ratio \(\delta _{0}/2\) are however orthogonal to the long side of \(R_{1,\delta _{0}}\), the covering of \(R_{1,\delta _{0}}\) can only be achieved by \(2\delta _{0}^{2}\) small rectangles of side lengths \(\delta _{0}\) and \(\delta _{0}^{2}/2\) (cf. Fig. 8(b)). The necessary perimeter for this covering is thus proportional to \(\delta _{0}^{1} \operatorname{Per}(R _{1,\delta _{0}})\).
For a small value of \(\delta _{0}\) this makes a substantial difference and accounts for the losses in the estimates for the rotated situation.
Although the level sets of the Conti construction consist of triangles and hence our coverings \(\{\varOmega _{j,k}\}_{k\in \{1,\dots ,J_{j}\}}\) will be coverings of triangles by triangles (instead of the previously described rectangular coverings), the heuristics from above still persist.
Motivated by these heuristic considerations, in the sequel we seek to provide covering results and associated \(BV\) bounds, which can be applied in Algorithms 27, 30. We organize the discussion of this as follows: In Sect. 4.1, we introduce some of the fundamental objects (cf. Definitions 40 and 43) and formulate the main covering result (Proposition 45). Here we consider a similar distinction into a parallel and a rotated situation as described in the above heuristics (cf. Definition 43). With the class of triangles from Definition 40 at hand we distinguish several different cases and discuss different covering scenarios. The respective coverings are tailored to the specific situation and are made such that we do not leave our class of triangles during the iteration. Their discussion is the content of Sects. 4.2–4.5. Finally, the various different cases are combined in Sect. 4.6 to provide the proof of Proposition 45.
4.1 Preliminaries
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:
Lemma 37
Remark 38
Proof of Lemma 37
Next we observe the following bounds on the rotation angles:
Lemma 39
Angles
Proof
This is an immediate consequence of Lemma 18. □
Definition 40
 1.
One angle, \(\alpha \), satisfies \(\alpha \in \delta [\frac{1}{10},1000]\),
 2.
The other two angles are contained in \(\frac{\pi }{2} + 2 \delta [1000, 1000]\),
 3.
One of the long sides encloses an angle in \(\delta [1000,1000]\) with \(n\).
If a triangle \(D\) satisfies 1.) and 2.) but not necessarily 3.) we call it \(\delta \)good (which allows for a possible change of orientation).
Remark 41
We note that if \(\alpha \) is small, both long sides could satisfy condition 3.) at the same time. In this case both directions are valid as directions of\(D\).
In our construction one prominent reference direction is obtained from Conti’s construction, as detailed in the following definition.
Definition 42
Let \(D\) be a level set of \(\nabla u_{j}\) and let \(e= e^{(p)}_{j}_{D}\) be the reference well. Let further \(n \in S^{1}\) be the direction of the long side of the Conti rectangle from Step 2 in Algorithm 27 on \(D\). We say that \(n\) is the direction of the relevant Conti construction (at step\(j\)), i.e. the construction by which \({D}\) is (in part) covered. A \(\delta \)good triangle is parallel to Conti’s construction if one of its long sides is parallel to \(n\).
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:
Definition 43
 (P1),
if it is \(\delta _{j}_{D_{j}}\)good with direction \(n \in S^{1}\), where \(n\) denotes the direction of the relevant Conti construction.
 (P2),
if \(\delta _{j}_{D_{j}} = \delta _{0}\) and \(\delta _{j1}_{D _{j}}\neq \delta _{0}\) and if \(D_{j}\) is \(\delta _{j1}_{D_{j}}\)good with direction \(n \in S^{1}\), where \(n\) denotes the direction of the relevant Conti construction.
 (R1),
if \(\delta _{j}_{D_{j}}=\delta _{0}\), the triangle is \(\delta _{0}\)good and if it forms an angle \(\beta \) with \(C\delta _{0} \leq \beta \leq \frac{\pi }{2} C \delta _{0}\) with respect to the direction of the relevant Conti construction (cf. Lemma 39).
 (R2),
if \(\delta _{j}_{D_{j}}=\delta _{0}\), the triangle is \(\delta _{j1}_{D_{j}}\)good with \(\delta _{j1}_{D_{j}} \neq \delta _{0}\) and if it forms an angle \(\beta \) with \(C\delta _{0} \leq \beta \leq \frac{\pi }{2} C \delta _{0}\) with respect to the direction of the relevant Conti construction.
 (R3),if \(\delta _{j}_{D_{j}}=\delta _{0}\) and if the triangle is right angled and such that
 (a)
the other angles are bounded from below and above by \(C \delta _{0}\) and \(\frac{\pi }{2} C \delta _{0}\),
 (b)
one of its sides is parallel to the orientation of the relevant Conti construction.
 (a)
Remark 44
The cases above, as stated, are not distinct since we allow for a factor in our definition of being \(\delta \)good (cf. Definition 40). For instance, there might be triangles which are in both case (P1) and (P2). However, in such situations also the constructions and perimeter estimates are comparable. In situations in which the estimates would differ significantly and where \(\delta _{j1} \ll \delta _{j}=\delta _{0}\), the above definitions yield distinct cases.
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.
 (i)
\(\delta _{j}_{D_{j}} = \delta _{j1}_{D_{j}}/2\). In this case \(\nabla u_{j}_{{D_{j}}}\) was produced as the stagnant matrix (cf. Notation 25) in the iteration step \(j1\). In this case, our covering construction will ensure that \({D_{j}}\) is in case (P1) (not exclusively, cf. Remark 44, but as one option).
 (ii)
\(\delta _{j}_{D_{j}}= \delta _{0}\) but \(\delta _{j1}_{D _{j}}\neq \delta _{0}\). This can for instance occur in a parallel pushout step. In this case, our covering construction will ensure that \({D_{j}}\) is in case (P2) (not exclusively (depending on the value of \(\delta _{j1}\)), cf. Remark 44, but as one option).
 (iii)
\(\delta _{j}_{D_{j}}= \delta _{0}= \delta _{j1}_{{D_{j}}}\). This case can for instance occur in two successive pushout steps. In this case, our covering ensures that \({D_{j}}\) is in the case (P1).
 (i)
\(\delta _{j1}_{D_{j}} = \delta _{0}\). This case can for instance occur in the situation of two successive pushout steps. In this case our covering ensures that \({D_{j}}\) can be taken to be in the case (R1).
 (ii)
\(\delta _{j1}_{D_{j}} \neq \delta _{0}\). This case can for instance occur in the case in which \(\nabla u_{j1}_{D_{j}}\) is produced in a stagnant and \(\nabla u_{j}_{D_{j}}\) in a pushout step. In this situation our covering ensures that \({D_{j}}\) can be taken to
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 \(\delta _{j1}_{D_{j}}\) (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 \(\delta _{j1}_{D_{j}}\ll \delta _{j}_{D_{j}}=\delta _{0}\), 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.
Proposition 45
(Covering)
 (i)
\(\varOmega \setminus \varOmega _{j} \leq (1\frac{7}{8}v_{0})^{j} \varOmega \),
 (ii)
\(\sum_{k=1}^{J_{j+1}}\operatorname{Per}(\varOmega _{j+1,k}) \leq C \delta _{0}^{1} \sum_{k=1}^{J_{j}}\operatorname{Per}(\varOmega _{j,k}) \).
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 nonnegligible 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.
4.2 Covering the Conti Construction by Triangles
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).
Lemma 46
Let\(SP_{j}\)be as in Algorithm27and let\(D_{j} \in \{\varOmega _{j,k}\}_{k\in \{1,\dots ,J_{j}\}}\). Suppose that\(R\subset D_{j}\)is a Conti rectangle of ratio\(\delta _{j}_{D_{j}}\). Let\(M_{1},\dots ,M_{4}\)denote the gradients occurring in the Conti construction with the same convention as in Notation25. Then all level sets in\(R\)on which\(M_{4}\)is attained, can be decomposed into (at most two) triangles which are of the type (P1). The level sets with\(M_{1}\), \(M_{2}\), \(M_{3}\)can be decomposed into triangles of the type (P1), (P2) or (R1), (R2).
Proof
We recall that (after a suitable splitting into in total 16 triangles as depicted in Fig. 10) all except for four triangles in the undeformed Conti construction (cf. Corollary 20) are axisparallel and have aspect ratio approximately \(1:\delta _{j}_{D_{j}}\) (with a factor depending on \(\lambda \); for \(\lambda =\frac{1}{4}\) a factor in the interval \((1/4,4)\) is more than sufficient). After rescaling the \(x_{2}\)axis by \(\delta _{j}_{D_{j}}\) (as in Lemma 21), these aspect ratios are then comparable to \(1:\frac{\delta _{j}_{D_{j}}}{2}\), i.e. \(1: \delta _{j+1}_{R\setminus F}\), where \(F\) denotes the union of the nonaxisparallel level sets in the deformed configuration. Hence, all the axisparallel triangles are of the type (P1) or (R1). The triangles in which \(M_{4}\) is attained, are of type (P1), as the rotation angle of the next Conti construction in the parallel case is controlled by virtue of Lemma 39.
It remains to discuss the remaining triangles contained in \(F\) (green in Fig. 10). These are again \(\delta _{j}_{D_{j}}\)good by a similar estimate on the aspect ratios, and by an estimate on the angle of rotation with respect to the \(x_{1}\)axis. As by our convex integration Algorithm 27, Step 2(b), \(\delta _{j+1}_{F}=\delta _{0}\), they are in general of the type (P2) or (R2), if \(\delta _{j}_{D_{j}} \neq \delta _{0}\), but could also be of the type (P1) or (R1), if \(\delta _{j}_{D_{j}} = \delta _{0}\). □
4.3 A Basic Building Block
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.
Proposition 48
 (i)
\(R_{2} \geq 10^{6}D\).
 (ii)
One corner divides a side of the triangle in the ratio\(\frac{2}{3}:\frac{1}{3}\).
 (iii)
The set\(D \setminus R_{2}\)consists of at most 100 \(\delta \)good triangles which are aligned with the direction of\(D\).
Proof
Let \(P_{1}=(\frac{2}{3},0)\) and let \(g\) be the line of slope \(r\) through \(P_{1}\). Then \(g\) intersects \(\partial D\) in exactly one other point \(P_{2}\). Being aligned along the \(x_{1}\)axis, the rectangle \(R_{2}\) is then uniquely determined by requiring that \(P_{1}\) and \(P_{2}\) are two of its corners. By construction it has aspect ratio \(1:r\).
Adding a vertical line through \(P_{1}\) and a horizontal line through \(P_{1}+(0,\frac{2}{3}\tan (\alpha )) \in \partial D\), we obtain an axis parallel triangle of opening angle \(\alpha \) to the left of \(R_{2}\), another axis parallel triangle of opening angle \(\alpha \) above \(R_{2}\), a foursided box \(B\) to the right of \(R_{2}\) and triangle selfsimilar to \(D\) above the box (cf. Fig. 11).
Remark 49
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 \(R_{2}\).
Lemma 50
 (i)
\(R \subset R_{2}\)and\(R \geq 10^{6}R_{2}\),
 (ii)
The set\(R_{2} \setminus R\)can be decomposed into at most 100 \(\delta \)good triangles, whose direction is either\(n\)or\(e_{1}\).
Proof
We, in particular, note that the lines \(\overline{Q_{1} Q_{2}}\) and \(\overline{Q_{3}Q_{4}}\) are parallel to the \(x_{2}\)axis. Furthermore, the triangles \(Q_{2}P_{1}P_{4}\) and \(P_{4}Q_{1}Q_{2}\) have opening angles \(\arctan (r_{0})\), are parallel to the \(x_{1}\)axis and are either rightangled or have an angle \(\frac{\pi }{2}  \beta \). Hence all of these triangles are \(\delta \)good with direction \(e_{1}\). Similar observations hold for the triangles which are constructed from \(P_{2}\), \(P_{3}\), \(Q_{2}\), \(Q_{3}\).

If \(\beta \in \delta [\frac{1}{10}, 1000]\), we additionally introduce the point \(Q_{9}= (1,0)\) and note that \(P_{1}Q_{9}P_{2}\) is \(\delta \)good and axisparallel, as are \(P_{1}Q_{9}Q_{6}\) and \(Q_{6}Q_{5}P_{1}\) (which both have an opening angle \(\gamma _{2}\)).
 If \(0\leq \beta \leq \delta \frac{1}{10}\), we note that by our restriction on \(\gamma _{2}\),which ensures that \(P_{1}Q_{6}P_{2}\) is \(\delta \)good and parallel to the long side \(n\) of \(R\).$$\begin{aligned} \beta + \gamma _{2} \in \delta \biggl[\frac{1}{10}, 300\biggr], \end{aligned}$$

If \(\beta \in \delta [\frac{1}{10}, 1000]\), we divide the rectangle by a horizontal line through \(P_{1}\), which yields two rectangles of lengths \(1:\tan (\beta )\), \(1:r_{2}\), which are \(\delta \)good.
 If \(\beta \in \delta [0,\frac{1}{10}]\), we note that by the same argument as aboveThus the aspect ratio \(1: (r_{2}+\tan (\beta ))\) results in \(\delta \)good axisparallel triangles.$$\begin{aligned} r_{2} + \tan (\beta ) \in \biggl[\arctan \biggl(\delta \frac{1}{10}\biggr), \arctan (1000 \delta )\biggr]. \end{aligned}$$
Remark 51
4.4 Covering in the Cases (P1), (P2)
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.
Proposition 52
 1.
A volume fraction of at least\(10^{12}D_{j}\)is covered by finitely many rescaled and translated Conti constructions from Lemma21. The Conti constructions can again be covered by finitely many triangles of the types occurring in the cases (P1)–(P2) and (R1)–(R2), where\(j\)is replaced by\(j+1\).
 2.
The complement of the Conti constructions is covered by finitely many triangles occurring in the cases (P1)–(P2), where\(j\)is replaced by\(j+1\).
 3.The overall surface energy of the new triangles\(D_{j+1,l}\in \mathcal{D}_{1}(D_{k})\), is controlled by$$\begin{aligned} \sum_{D_{j+1,l}\in \mathcal{D}_{1}(D_{j})}\operatorname{Per}(D _{j+1,l}) \leq C \operatorname{Per}(D_{j}). \end{aligned}$$
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.
4.4.1 The Case (P1)
We begin by explaining the covering in the case (P1).
Lemma 53
 (i)
\(K_{j}\), with\(K_{j}\in [1,100]\), \(\delta _{j}\)good, up to nullsets disjoint triangles, \(D_{l}\), \(l\in \{1,\ldots ,K_{j}\}\)which are either oriented along the\(x_{1}\)axis or the long side of\(R\),
 (ii)
a rescaled and translated copy\(\tilde{R}\)of\(R\), such that\(\tilde{R} \geq 10^{6}D\).
Proof
We first invoke Lemma 50 with \(R\) and \(\delta = \delta _{j}\). This yields an axisparallel box \(\tilde{R}_{2}\) of side ratio \(r\in [\frac{1}{10},10]\delta _{j}\). This box \(\tilde{R}_{2}\) is admissible in Proposition 48. An application of this proposition with \(D\), \(\tilde{R}_{2}\) and \(\delta =\delta _{j}\) hence yields a covering of \(D\) by \(\delta _{j}\)good triangles which all have \(e_{1}\) as their direction, and a box \(R_{2}\), which is covered as described in Lemma 50. We note that the triangles within \(R_{2}\) are thus also \(\delta _{j}\)good and have as their directions either \(e_{1}\) or the long side of \(R\). The estimate on the perimeter follows, since all the covering triangles have perimeter controlled by \(\operatorname{Per}(D)\) and as \(K_{j} \leq 100\). The estimate on the volume fraction is a consequence of Proposition 48(i) and Lemma 50(i). □
4.4.2 The Case (P2)
As in the case (P1) we have the following main covering result:
Lemma 54
 (i)
\(M_{j}\), with\(M_{j} \in [1,100]\), \(\delta _{j1}\)good triangles\(D_{l}\), \(l\in \{1,\dots ,M_{j}\}\)which are parallel to the\(x_{1}\)axis,
 (ii)\(K_{j}:= \frac{\delta _{0}}{\delta _{j1}}\)translated, disjoint and rescaled copies\(\tilde{R}_{k}\)of\(R\)with the property that$$ \Biggl\bigcup_{k=1}^{K_{j}}\tilde{R}_{k}\Biggr \geq 10^{6}D, $$
 (iii)
\(\tilde{M}_{j}\), with\(\tilde{M}_{j} \in [1,100]\), \(\delta _{0}\)good triangles\(\tilde{D}_{l}\), \(l\in \{1,\dots , \tilde{M}_{j}\}\)which are either parallel to the\(x_{1}\)axis or parallel to the long side of\(\tilde{R}\).
Proof
 each rectangle \(\tilde{R}_{2,k}\) has perimeter bounded by$$\begin{aligned} \operatorname{Per}(\tilde{R}_{2,k})\leq C \frac{\delta _{j1}}{\delta _{0}} \operatorname{Per}(D), \end{aligned}$$

there are \(\frac{\delta _{0}}{\delta _{j1}}\)many axis parallel boxes \(\tilde{R}_{2,k}\).
The main difference of Lemma 54 with respect to Proposition 48 is the step in which we bridge the mismatch in the ratios of the triangle \(D\) (ratio \(\delta _{j1}\)) and the given box \(R\) (ratio \(\delta _{0}\)). Here we pass from a box of ratio approximately \(\delta _{0}\) (which is prescribed for \(R\) and hence for \(\tilde{R}_{2}\)) to a box with ratio approximately \(\delta _{j}\) (for \(\bar{R}_{2}\)) by stacking translates of the boxes \(\tilde{R}_{2,k}\) next to each other.
4.4.3 Proof of Proposition 52
Using the results from Sects. 4.4.1 and 4.4.2 we can now address the proof of Proposition 52.
Proof
The first property of the Proposition follows from Lemma 46 in combination with Lemma 53(ii) (in the case (P1)) or Lemma 54(ii) (in the case (P2)). In particular, by Lemma 46 all the triangles, which are used to cover the Conti constructions, are \(\delta _{j+1}\)good with respect to the relevant Conti construction. The second property is a consequence of Lemma 53(i) combined with Lemma 50 (in the case (P1)) or Lemma 54(i), (iii) (in the case (P2)). We emphasize that all these triangles are either parallel to the original triangle \(D\) or to the relevant Conti construction, implying that both the angles and the orientations are within the admissible margins. Finally, the bound on the perimeters follows from the corresponding claims in Lemmata 53 and 54. □
4.5 Covering in the Cases (R1)–(R3)
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:
Proposition 55
 1.
A volume fraction of\(10^{6}D_{j}\)is covered by finitely many rescaled and translated Conti constructions. The Conticonstructions can again be covered by finitely many triangles of the types occurring in the cases (P1)–(P2) and (R1)–(R3), where\(j\)is replaced by\(j+1\).
 2.
The complement of the Conti constructions is covered by finitely many triangles occurring in the cases (P1), (P2) and (R1)–(R3), where\(j\)is replaced by\(j+1\).
 3.The overall surface energy of the new triangles\(D_{j+1,l}\in \mathcal{D}_{1}(D_{j})\), is controlled by$$\begin{aligned} \sum_{D_{j+1,l}\in \mathcal{D}_{1}(D_{j})}\operatorname{Per}(D _{j+1,l}) \leq C \delta _{0}^{1} \operatorname{Per}(D_{j}). \end{aligned}$$
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.
4.5.1 The Case (R1)
The covering result for the case (R1) is very similar to the one from the case (P1). It only deviates from this by the construction within the rectangle \(R_{2}\):
Lemma 56
 (i)
\(M_{j}\), with\(M_{j} \in [1,100]\), \(\delta _{0}\)good triangles, \(D_{1,k}\)which are aligned with the direction of\(D\),
 (ii)
\(0< K_{j} \leq C \delta _{0}^{2}\)many translated, up to nullsets disjoint and rescaled copies\(R_{2,k}\)of the rectangle\(\bar{R}\), whose union covers a volume of size at least\(10^{6}D\),
 1.
\(0\leq L_{j} \leq C \delta _{0}^{2}\)many triangles\(D_{2,k}\)which are of the type (R3).
Remark 57
By Lemma 39 the angles which occur in our constructions, always satisfy the bound \(\beta \in (C \delta _{0}, \frac{\pi }{2}C \delta _{0})\).
Proof
We note that the error triangles \(D_{2,k}\) are all right angle triangles. Moreover, one of the other angles coincides with the rotation angle \(\beta \). At least one of triangles’ sides is parallel to the orientation of the rectangles \(R_{2,k}\). Thus the triangles \(D_{2,k}\) are of the type (R3).

The number \(K_{j}\) of rectangles \(R_{2,k}\) and the number \(L_{j}\) of triangles \(D_{2,k}\) are bounded by \(C \delta _{0}^{2}\).

The perimeter of each of the rectangles and each of the triangles is controlled: \(\operatorname{Per}(R_{2,k}) + \operatorname{Per}(D_{2,k}) \leq C \delta _{0} \operatorname{Per}(D)\).

There are at most 100 triangles \(D_{1,k}\), each of which has a perimeter controlled by \(\operatorname{Per}(D)\).
4.5.2 The Case (R2)
The case (R2) is the rotated analogue of the case (P2). As we are in a rotated case, we have to be less careful about preserving orientations, and proceed similarly as in the case (R1). Again the main issue is the covering of the rectangle \(R_{2}\). However, in contrast to the case (R1) we now have to deal with a mismatch between the ratio of the triangle \(D\) (with ratio \(\delta _{j} \neq \delta _{0}\)) and the ratio of the Conti construction (with ratio \(\delta _{0}\)). Similarly as in the case (P2) we overcome this issue by a “stacking construction”, which compensates the mismatch.
Lemma 58
 (i)
\(M_{j}\), with\(M_{j} \in [1,100]\), \(\delta _{0}\)good triangles, \(D_{1,k}\)which are aligned with the direction of\(D\),
 (ii)
\(0< K_{j}\leq C \delta _{0}^{1}\delta _{j}^{1}\)many translated and rescaled copies\(R_{2,k}\)of the rectangle\(\bar{R}\), whose union covers a volume of size at least\(10^{6}D\),
 (iii)
\(0< L_{j}\leq C \delta _{0}^{1}\delta _{j}^{1}\)many triangles\(D_{2,k}\)which are of the type (R3).
Proof

Each rectangle \(R_{2,k}\) has perimeter controlled by \(C\delta _{j} \operatorname{Per}(D)\).

There are \(C \delta _{j}^{1}\delta _{0}^{1} \) many such rectangles \(R_{2,k}\).

The perimeters of the error triangles \(D_{2,k}\) are up to a factor controlled by the perimeters of the rectangles \(R_{2,k}\).
4.5.3 The Case (R3)
We deal with the error triangles from the previous step. All of them are right angle triangles in which the other two angles are bounded from below and above by \(C \delta _{0}\) and \(\frac{\pi }{2}C \delta _{0}\). We show that in this situation we can reduce to two model cases, which we discuss below. This allows us to obtain the following result:
Lemma 59
 (i)
\(\bigcup_{k=1}^{K_{j}} R_{k} \geq 10^{6}D\),
 (ii)
\(\sum_{k=1}^{K_{j}} \operatorname{Per}(R_{k}) \leq \frac{C}{\delta _{0}}\operatorname{Per}(D)\).
Our main ingredient in proving this is the following lemma:
Lemma 60
(Covering of Triangles by Rectangles)
Let\(D_{1,m}\)denote a right angle triangle, in which the sides enclosing the right angle are of side lengths 1 and\(m\). Assume that\(m\in (0,50 \delta ^{1})\). Let\(R\)be a rectangle, which has side ratio\(\delta \in (0,1)\). Assume that the longer side of\(R\)is parallel to the side of the triangle\(D_{1,m}\), which is of length\(m\).
 (i)
\(\bigcup_{k=1}^{L}R_{k} \geq 10^{2}D_{1,m}\).
 (ii)The sum of the perimeters of the rectangles \(R_{k}\) satisfies$$\begin{aligned} \sum_{k=1}^{L}\operatorname{Per}(R_{k}) \leq \frac{C(1+m\delta )}{\delta }\leq \frac{C}{\delta }\operatorname{Per}(D_{m,1}). \end{aligned}$$
Remark 61
We remark that for our application, the bound on \(m\) does not impose an additional requirement. Indeed, the triangles of type (R3) only occur as artifacts of the coverings in Lemmas 56 and 58. Here we may estimate \(m\) by \(\tan (\beta )\). For \(\beta =\frac{\pi }{2}\delta \), a Taylor expansion of \(\frac{\sin (\frac{ \pi }{2}\delta )}{\cos (\frac{\pi }{2}\delta )}\) entails the desired estimate.
Before explaining Lemma 60, we show how our main covering result, Lemma 59, can be reduced to the situation of Lemma 60.
Proof of Lemma 59
We first claim that without loss of generality \(D\) can be assumed to be of type (R3) with \(R\) being parallel to one of the short sides of the triangle \(D\). Indeed, if \(R\) is parallel to the long side of the triangle \(D\), then this side is opposite of the right angle of the triangle. In this case, we split the triangle \(D\) into two smaller triangles \(D^{(1)}\), \(D^{(2)}\) by connecting the corner, at which \(D\) has its right angle, by the shortest line to the long side. The resulting triangles \(D^{(1)}\), \(D^{(2)}\) have the same angles as the original triangle (and in particular satisfy the nondegeneracy conditions for the angles, which are required in condition (R3)), but are now such that \(R\) is parallel to one of their short sides.
After this reduction, we seek to apply Lemma 60 with \(\delta =\delta _{0}\) for each of the triangles \(D^{(1)}\), \(D^{(2)}\). To this end we note that as \(\beta _{0} \geq C\delta _{0}\), we have that \(m\leq C \delta _{0}^{1}\). As a consequence, Lemma 60 yields the desired result (by observing that \(\operatorname{Per}(D ^{(1)})+ \operatorname{Per}(D^{(2)}) \leq 2 \operatorname{Per}(D)\)). □
Proof of Lemma 60
4.6 Proof of Proposition 45
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.
5 Quantitative Analysis
After having recalled the qualitative construction of convex integration solutions in Sect. 3, we now focus on controlling the scheme quantitatively. Here we rely on the quantitative covering results from Sect. 4 (cf. Propositions 52 and 55), which allow us to obtain bounds on the \(BV\) norm of the iterates \(u_{k}\) and the corresponding characteristic functions associated with the well \(e^{(i)}\in K\) (Lemma 63). Combined with an \(L^{1}\) estimate and the interpolation inequality from Theorem 2 or from Corollary 3, this then yields the desired \(W^{s,q}\) regularity of the characteristic function of the phases.
As in Sect. 3.2, given a matrix \(M\) with \(e(M)\in \operatorname{intconv}(K)\), we here assume that \(\varOmega := Q_{\beta }[0,1]^{2}\), where \(Q_{\beta }\) is the rotation, which describes how the Conti construction with respect to \(M\) and \(e^{(p)}_{0}\) is rotated with respect to the \(x_{1}\)axis. This special case will play the role of a crucial building block in the situation of more general domains (cf. Sect. 6).
We begin by defining the characteristic functions associated with the corresponding wells:
Definition 62
(Characteristic Functions)
We emphasize that these pointwise limits exists, since for a.e. point \(x\in \varOmega \) there exists an index \(k_{x}\in \mathbb{N}\) such \(x\in \varOmega \setminus \varOmega _{k_{x}}\). By our convex integration algorithm and by Definition 62, the value of \(\chi _{l}(x)\) remains fixed for \(l\geq k_{x}\).
Using the covering results from Sect. 4, we can address the \(BV\) bounds for the characteristic functions \(\chi _{k} ^{(i)}\), \(i\in \{1,2,3\}\):
Lemma 63
(\(BV\) Control)
Proof
 (a)The parallel case. Assume that \(\varOmega _{j,k}\) is of the type (P1) or (P2) (which, by the explanations below Definition 43 holds in the parallel case). Thus, the Conti construction in the \(j\)th and \((j+1)\)th step are nearly aligned in the direction of their degeneracy. In this case, Proposition 52 is applicable and implies thatfor some absolute constant \(C>0\).$$\begin{aligned} \sum_{\varOmega _{j+1,l}\in \mathcal{D}_{1}(\varOmega _{j,k})} \operatorname{Per}( \varOmega _{j+1,k}) \leq C \operatorname{Per}(\varOmega _{j,k}), \end{aligned}$$(45)
 (b)The rotated case. Assume that \(\varOmega _{j,k}\) is of the type (R1)–(R3) (which, by the explanations below Definition 43 holds in the rotated case). Thus, the Conti construction in the \(j\)th and \((j+1)\)th step are not aligned. By Lemma 39 there are even lower bounds on the degree of alignment. In this case, Proposition 55 is applicable and yields that$$\begin{aligned} \sum_{\varOmega _{j+1,l}\in \mathcal{D}_{1}(\varOmega _{j,k})} \operatorname{Per}( \varOmega _{j+1,k}) \leq C \delta _{0}^{1} \operatorname{Per}(\varOmega _{j,k}). \end{aligned}$$(46)
Using the explicit construction of our convex integration scheme, we further estimate the difference of two successive iterates in the \(L^{1}\) norm:
Lemma 64
(\(L^{1}\) Control)
Remark 65
In our realization of the covering argument, which is described in Sect. 4, we have chosen \(v_{0} = 10^{6}\). In particular, it is independent of the boundary condition \(M\) in (6).
Proof
Proposition 66
(Regularity of Convex Integration Solutions)
Proof
Remark 67
(Dependences)

Varying the volume fraction \(\lambda \in (0,1)\) in the Conti construction from Corollary 20.

Choosing a sharper relation between \(\epsilon _{j}\) and \(\delta _{j}\) and modifying the \(j\)dependence of \(\epsilon _{j}\) (for instance by only using summability for the stagnant matrices instead of the geometric decay, which is prescribed in Step 2(b) of Algorithm 27).
A qualitatively different behavior would arise, if in the proof of Lemma 63 only case (a) occurred. Then based on our construction in Proposition 52 and Lemma 63 the \(\delta _{0}\) dependence would improve in Proposition 66: In this case the choice of the product of the exponents\(s\), \(q\) in Proposition 66 would not depend on \(\delta _{0}\), but would be uniform in the whole triangle \(\operatorname{intconv}(K)\). In this case only the value of the \(W^{s,q}\) norm would deteriorate with \(\delta _{0}\).
We however remark that, as a matrix in \(\operatorname{intconv}(K)\) is a convex combination of all three values of \(e^{(1)}\), \(e^{(2)}\), \(e^{(3)}\), the described construction necessarily involves instances of case (b). It is however conceivable that by controlling the number of these steps, it could be possible to improve the dependence of \(s\), \(q\) on \(\delta _{0}\). It is unclear (and maybe rather unlikely), whether it is possible to completely remove it with the described convex integration scheme.
Remark 68
(Fractal Dimension)
We emphasize that in accordance with Remark 6 the \(W^{s,p}\) regularity of \(\chi ^{(i)}\) for \(i\in \{1,2,3\}\) has direct implications on the (packing) dimension of the boundary of the sets \(\{x\in \mathbb{R}^{2}: \chi ^{(i)}(x)=1\}\).
Similarly, we obtain bounds on the displacement and the infinitesimal strain tensor:
Proposition 69
Proof
6 General Domains
In this section we explain how to construct the desired “regular” convex integration solutions in arbitrary Lipschitz domains by using the bounds from the special cases which were discussed in Sect. 5. In this context our main result is the following:
Proposition 70
 (i)\(\bar{\varOmega }_{k}:= \bigcup_{l=1}^{k} \bigcup_{m=1}^{K_{l}} Q_{l}^{m}\), where\(Q_{l}^{m}:= ( [0, \lambda _{l}]^{2} + x_{l,m} )\)are (up to nullsets) disjoint cubes with\(x_{l,m} \in \varOmega \)and\(\lambda _{l}:= 2^{l}\)such that(in the sense of the convergences of their characteristic functions),$$\begin{aligned} \bar{\varOmega }_{k} \nearrow \varOmega \quad\textit{in } L^{1} \bigl(\mathbb{R}^{2}\bigr) \end{aligned}$$
 (ii)
for\(\tilde{\chi }^{(i)}_{k}(x):= \sum_{l=1}^{k} \sum_{m=1}^{K_{l}} \chi _{k}^{(i)}( \frac{xx_{m,l}}{\lambda _{l}})\chi _{\bar{\varOmega }_{k}}(x)\)the estimate (49) remains valid for all\(s\), \(p\)with\(s\in (0,1), p \in (1,\infty ]\)and\(0< sp<\theta _{0}\). In the dependences the constant\(C(s,p)\)however is replaced by\(C(\varOmega ,M,s,p)\)and\(\mu (s,p)\)replaced by\(C(\theta )\mu (s,p)\). Here\(\theta =\theta (s,p)\)is the interpolation exponent associated with\(s,p>0\).
As an immediate consequence we infer the following corollary:
Corollary 71
 (i)the pointwise limit\(\tilde{\chi }^{(i)}\)of the functions\(\tilde{\chi }^{(i)}_{k}\)satisfies$$\begin{aligned} \bigl\ \tilde{\chi }^{(i)}\bigr\ _{W^{s,p}(\varOmega )} \leq C(\varOmega , M, s,p). \end{aligned}$$
 (ii)there exist solutions\(u\)to (6) with$$\begin{aligned} \\nabla u\_{W^{s,p}(\varOmega )} \leq C(\varOmega , M, s,p). \end{aligned}$$
Proof of Corollary 71
We first note that the pointwise limit \(\tilde{\chi }^{(i)}\) exists, since \(\bar{\varOmega }_{k} \rightarrow \varOmega \) and since \(\chi _{k}^{(i)} \rightarrow \chi _{k}\) in a pointwise sense as \(k\rightarrow \infty \). With this at hand, the proof of Corollary 71 follows from Proposition 70 by interpolation in an analogous way as explained in Proposition 66. We therefore omit the details of the proof of the corollary. □
We proceed to the proof of Proposition 70. Here we argue by covering our general domain \(\varOmega \) by the special domains from Sect. 5 (Steps 1 and 2). On each of the special domains, we apply the construction from Sect. 5 (cf. also Algorithms 27 and 30). In order to obtain a sequence with bounded \(W^{s,p}\) norm, we however do not refine to arbitrarily fine scales immediately, but proceed iteratively (cf. Step 3). A central point here is to control the necessary number of cubes at each scale (Claim 72), since this has to be balanced with the corresponding energy contribution (cf. Step 4). To this end, we use a “volume argument”, which by the Lipschitz regularity of the domain allows us to infer information on the number of cubes on each scale (cf. Proof of Claim 72).
Proof of Proposition 70
Step 1: Covering of a general Lipschitz domain. We may assume that \(M=0\) and first consider the case of \(\varOmega \) being a domain which is bounded by the \(x_{2}\)axis, the segment \([0,1]\times \{0\}\), a Lipschitz graph \(f:[0,1]\rightarrow \mathbb{R}\) and the segment \(\{1\}\times [0,f(1)]\). By symmetry we may further assume that \(f(x_{1})\geq 0\) for all \(x_{1}\in [0,1]\). For general domains \(\varOmega \), by the compactness and Lipschitz regularity, we may locally reduce to a similar case, where \(f\) is a Lipschitz curve, but not necessarily a graph. However, all arguments in the following extend to that case as well.
Step 2: Counting cubes. Let \(\tilde{\varOmega }_{l}:= \bigcup_{k=1}^{K_{l}} Q_{l}^{k}\), where \(Q_{l}^{k}\subset \varOmega \) are (up to zero sets) disjoint, grid cubes of an axisparallel grid of grid size \(\lambda _{l}:=2^{l}\). We choose \(K_{l}\in \mathbb{N}\) maximal. Thus, by definition we have that \(\tilde{\varOmega }_{l} \subset \tilde{\varOmega }_{l+1}\subset \varOmega \) for all \(l\in \mathbb{N}\). In the limit \(l\rightarrow \infty \) the sets \(\tilde{\varOmega }_{l}\) eventually cover the whole set \(\varOmega \) (which we assume to be as in Step 1).
We estimate the number of the cubes which are contained in the sets \(\tilde{\varOmega }_{l+1}\setminus \tilde{\varOmega }_{l}\). For these we claim:
Claim 72
The set\(\tilde{\varOmega }_{l+1}\setminus \tilde{\varOmega }_{l}\)contains at most\(C_{f} \lambda _{l+1}^{1}\)of the grid cubes\(Q_{l+1}^{k} \subset \varOmega \).
Proof of Claim 72
Remark 73
The constant \(C_{f}\) from Claim 72 can be controlled by \(C [\nabla f]_{C^{0,1}(\varOmega )}\), for some universal constant \(C>1\).
Notes
Acknowledgements
Open access funding provided by Max Planck Society. At the time of preparation of the article A.R. held a position at the University of Oxford and B.Z. held a position at the University of Bonn. A.R. acknowledges a Junior Research Fellowship at Christ Church. C.Z. and B.Z. acknowledge support from the Deutsche Forschungsgemeinschaft through CRC 1060 “The mathematics of emergent effects”. We would like to thank Sergio Conti for suggesting the problem and for useful discussions. Further, we would like to thank Felix Otto for making his Minneapolis lecture notes available to us and for helpful discussions on the project.
Coflict of Interest
The authors declare that they have no conflict of interest.
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