Key Words. Approximation by piece-wise linear functions, \((H_n)\)-conditions, fixed-point.

AMS(MOS) subject classifications. 65L05, 65L20, 47J30, 65D05.

1 Introduction↩︎

One of the main ingredients of the direct method of the Calculus of Variations ([1]) to show existence of minimizers for an integral functional of the kind \[I(\boldsymbol{u})=\int_\Omega\psi(\nabla \boldsymbol{u}(\boldsymbol{x}))\,d\boldsymbol{x}\] is its weak lower semicontinuity. Here \(\Omega\subset\mathbb{R}^N\) is a regular (Lipschitz), bounded domain, and feasible mappings \(\boldsymbol{u}:\Omega\to\mathbb{R}^m\) are smooth or Lipschitz, so that \(\nabla \boldsymbol{u}\) is a \(m\times N\)-matrix at each point \(\boldsymbol{x}\in\Omega\). The weak lower semicontinuity property is in turn equivalent to suitable convexity properties of the continuous integrand \(\psi:\mathbf{M}^{m\times N}\to\mathbb{R}\). Morrey ([2], [3]) proved that this weak lower semicontinuity (in \(W^{1, \infty}(\Omega; \mathbb{R}^m)\)) is equivalent to the quasi-convexity of the integrand \(\psi\), namely, \[\psi(\mathbf{F})\le\frac{1}{|D|}\int_D\psi(\mathbf{F}+\nabla \boldsymbol{v}(\boldsymbol{x}))\,d\boldsymbol{x}\] for every \(\mathbf{F}\in\mathbf{M}^{m\times N}\), and every test map \(\boldsymbol{v}\) in \(D\). This concept does not depend on the domain \(D\), and can, equivalently, be formulated in terms of periodic mappings ([4]) so that such a density \(\psi\) is quasiconvex when \[\psi(\mathbf{F})\le\int_Q\psi(\mathbf{F}+\nabla \boldsymbol{v}(\boldsymbol{y}))\,d\boldsymbol{y}\] for all \(\mathbf{F}\in\mathbf{M}^{m\times N}\), and every periodic mapping \(\boldsymbol{v}:Q\to\mathbb{R}^m\). Here \(Q\subset\mathbb{R}^N\) is the unit cube.

Unfortunately, the issue is far from settled by simply saying this, since even Morrey realized that it is not at all easy to decide when a given density \(\psi\) enjoys this property. For the scalar case, when either of the two dimensions \(N\) or \(m\) is unity, quasi-convexity reduces to usual convexity. But for genuine vector situations, it is not so. As a matter of fact, necessary and sufficient conditions for quasi-convexity in the vector case (\(N, m>1\)) were immediately sought, and important new convexity conditions were introduced:

• Rank-one convexity. A continuous integrand \(\psi:\mathbf{M}^{m\times N}\to\mathbb{R}\) is said to be rank-one convex if \[\psi(t_1\mathbf{F}_1+t_2\mathbf{F}_2)\le t_1\psi(\mathbf{F}_1)+t_2\psi(\mathbf{F}_2),\quad t_1+t_2=1, t_1, t_2\ge0,\] whenever the difference \(\mathbf{F}_1-\mathbf{F}_2\) is a rank-one matrix.

• Poly-convexity. Such an integrand \(\psi\) is poly-convex if it can be rewritten in the form \[\psi(\mathbf{F})=g(\mathbb{M}(\mathbf{F}))\] where \(\mathbb{M}(\mathbf{F})\) is the vector of all minors of \(\mathbf{F}\), and \(g\) is a convex (in the usual sense) function of all its arguments.

It was very soon recognized that quasi-convexity implies rank-one convexity (by using a special class of test fields), and that poly-convexity is a sufficient condition for quasi-convexity ([5]) The task suggested itself as trying to prove or disprove the equivalence of these various kinds of convexity. In the scalar case all three coincide with usual convexity, so that we are facing a purely vector phenomenon. It turns out that these three notions of convexity are different, and counterexamples of various sorts have been found over the years. See [6], [7], [8], [9].

If we focus on the equivalence of rank-one convexity and quasi-convexity, Morrey conjectured that they are not equivalent ([2]), though later he simply stated it as an unsolved problem ([3]). The issue remained undecided until the surprising counter-examble by V. Sverak ([4]) after some other additional and very interesting results ([10], [11], [12]). What is quite remarkable is that the original counter-example is only valid when \(m\ge3\), and later attempts to extend it for \(m=2\) failed ([13], [14], [15]). See also [16] for more such examples from a different viewpoint again in cases where \(m>2\). Other counterexamples have not been found. Some efforts by the author were definitely discarded in [17]. References [18], and [19] are also relevant here.

The situation for two-component maps has, therefore, stayed unsolved, though some evidence in favor of the equivalence has been gathered throughout the years. In particular, when additional ingredients or properties are assumed, the equivalence can some times be shown. There is a bunch of very interesting works in this regard; see [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. It is also interesting to point out that for quadratic densities, rank-one convexity and quasi-convexity are equivalent regardless of dimensions. This has been known for a long time ([5], [3]), and it is not difficult to prove it by using Plancherel’s formula. A different point of view is taken in [30]. Another field where the resolution of this equivalence for two components maps would have an important impact is the theory of quasiconformal maps in the plane. There is a large number of references for this topic. See [31] for a rather recent account, or [32] for a more focused article. In particular, if the equivalence between rank-one convexity and quasi-convexity for two component maps turns out to be true, then the norm of the corresponding Beurling-Ahlfors transform equals \(p^*-1\) ([33]). More consequences would likely follow if such equivalence is proved.

Another chapter where many recent efforts have been made, given the intrinsic difficulties of analytical ideas, is concerned with the numerical evidence for a potential counter-example in favor of Morrey’s conjecture. None of them has turned out to be conclusive. See [34], [35], [36], [37].

In this note we prove that indeed for \(m=N=2\), rank-one convexity is equivalent to quasi-convexity. The way in which we are going to think about the problem is by using the dual formulation of this equivalence through Jensen’s inequality. What we will actually show is that, when \(m=N=2\), every homogeneous gradient Young measure is a laminate. See Chapter 9 in [38], and [39]. Equivalently, we will focus on showing that every periodic gradient can be achieved by lamination.

More specifically, take \(l\in\mathbb{N}\), and \(\tau_l\), a regular triangulation of the unit cube \(Q\subset\mathbb{R}^2\) with the three normals \((1, 0)\), \((0, 1)\), and \((1, 1)\). \(l\) is a parameter indicating a certain level of discretization. As \(l\) becomes larger and larger, elements in \(\tau_l\) are finer and finer triangles with the same normals. Suppose we are given a gradient \((\nabla u, \nabla v)\) with two components \[(u, v):Q\subset\mathbb{R}^2\to\mathbb{R}^2,\] where \(Q\) is the unit cube in \(\mathbb{R}^2\), which is \(Q\)-periodic, continuous, piecewise-affine with respect to triangulation \(\tau_l\). By a standard density argument about approximation by continuous, piece-wise affine mappings, it suffices, to reach our goal, to show that the corresponding discrete, homogeneous underlying gradient Young measure is a laminate.

What is essential or special about \(m=2\)? This is a question that one has to understand, as it seems quite central for a final resolution of the problem. The answer turns out to be quite enlightening: for two component maps, one can define an appropriate map going from one component to the other, and show the existence of a fixed point for such a map that translates into a rank-one decomposition for any such two-component gradient. For more than two components, more than one map would be involved, and fixed points for every couple of components may not match. This fixed point result (Kakutani’s) is classical and nothing but a natural generalization of the usual Brower fixed point theorem.

In a more explicit way, the two-component map \((u, v)\) establishes, through triangulation \(\tau_l\), a very clear way of moving from manipulations on the gradient of the first component \(\nabla u\) to the same manipulations on the gradient \(\nabla v\) of the second component by simply replacing \(\mathbf{u}_i\) by the corresponding \(\mathbf{v}_i\) in the same element of the triangulation \(\tau_l\), if the finite support of \((\nabla u, \nabla v)\) is the set of pairs \(\{(\mathbf{u}_i, \mathbf{v}_i)\}_i\). This procedure is incorporated in the definition of our mapping. Such map is in charge of keeping track of decomposition directions as in the definition of laminates and \((H_n)\)-conditions ([40], [39]). We assume readers to be familiar with this material as it is essential to understand our perspective.

Given a probability measure supported in the discrete set of vectors \(\{\mathbf{u}_i\}_i\) of the first gradient \(\nabla u\), that is decomposed in the form of a \((H_n)\)-condition along a set of successive directions and weights, we focus on those decompositions, performed in the same way for the second gradient \(\nabla v\), that preserve the family of decomposition directions coming from the first component, but possibly for a different family of volume fractions or weights. If we define a set-valued operation going from the first vector of weights for the first component to the set of weights of the second component preserving decomposition directions at all levels, intuitively a fixed-point for such a map would respect:

1. decomposition directions for both components (this is ensured by the definition of the map itself); and

2. equal volume fractions for the two components jointly, because the passage from one component to the other through the operation \(\mathbf{u}_i\mapsto\mathbf{v}_i\) respects such volume fractions for a fixed point.

Therefore fixed points for such a map are identified with joint, i.e. simultaneously in the two components, \((H_n)\)-conditions whose decomposition directions are parallel, i.e. with laminates. Our claim, then, reduces to proving the existence of at least one fixed point for such a map.

Most of the technicalities are related to showing that a suitable framework can be set up so that the appropriate assumptions hold for the fixed-point result to be applied. One crucial issue, though, is to understand what is special about a probability measure associated with a gradient \((\nabla u, \nabla v)\), since we know that not every probability measure supported in \(\mathbb{R}^{2\times2}\) should allow the treatment through such fixed point argument.

It will soon be understood why we do not deal with the case \(N>2\). The strategy of the proof hinges on various basic facts whose extension to higher dimension is not clear or does not hold. On the one hand, we need to manipulate in a rather explicit manner periodic, continuous, piece-wise linear functions with respect to triangulations of the unit cube in \(\mathbb{R}^N\). As soon as \(N=3\), this becomes highly technical where enumerative procedures for planar interfaces between elements turn out to be quite tricky to organize as in the practical use of finite elements. Beyond this difficulty, Claim 9 below can hardly admit a parallel result in dimension \(N=3\) or higher. Further insight is therefore required to treat the two-component, higher dimensional case, either to find suitable substitutes for those facts or for finding a counter-example.

Before moving to discuss our main proof, we would like to mention that one of the main applied fields where vector variational problems are relevant is non-linear elasticity ([5]). In particular, polyconvexity has played a major role in existence results. See also [41]. A main hypothesis to be assumed in this area is the rotationally invariance, as well as the behavior for large deformations. See [42] for a discussion on all these notions of convexity under this invariance. Higher-order theories have also been explored, at least from an abstract point of view (Dal?, Maso?, et?, al.?, [43]). More general concepts of quasiconvexity have been introduced in [44]. Recent interesting results about approximation by polynomials are worth mentioning [45]. Explicit examples of rank-one convex functions can be found in various works: [30], [7], [10], among others. See also [46], [47]. The recent book [48] is to be considered.

Finally, it may be instructive to write Theorem 1 in a more transparent form for readers not familiar with the subleties of vector variational problems.

This result is false in the case \(\mathbf{M}^{m\times N}\) for \(m\ge3\) and every \(N\ge2\) ([4]), and it is not known if it is true for \(m=2\) and \(N\ge3\).

2 The overall strategy↩︎

It may be instructive to describe in heuristic but explicit terms what we are trying to accomplish and how we plan to tackle the proof of Theorem 1, so that readers may know in advance where we are heading before getting into precise statements.

Suppose we are given a discrete probability measure \(\mu\) with finite support in \(\mathbb{M}^{2\times 2}\), the space of \(2\times2\)-matrices, and a vanishing barycenter \[\mu=\sum_it_i\delta_{\mathbf{F}_i},\quad t_i>0, \sum_it_i=1,\quad \sum_it_i\mathbf{F}_i=\mathbf{0}.\] How is one to show that \(\mu\) can indeed be reached by lamination? One way or another, we need to find a suitable representation of \(\mu\) in the form \[\label{decomposition} \mu=\sum_j s_j\delta_{\mathbf{A}_j},\quad s_j>0, \sum_js_j=1, \mathbf{A}_j\in\{\mathbf{F}_i\}, \quad t_i=\sum_{j: \mathbf{A}_j=\mathbf{F}_i}s_j,\tag{1}\] in such a way that with the initial set of pairs \[\label{decompositiondos} (s_1, \mathbf{A}_1),\quad (s_2, \mathbf{A}_2), \quad\dots\quad (s_m, \mathbf{A}_m),\tag{2}\] we can build a successive collection of \((H_n)\)-conditions, according to its proper rule, and proceeding recursively up until the final pair \((1, \mathbf{0})\). That basic rule consists in selecting two pairs \(p\), \(q\), from 2 in such a way that \(\mathbf{A}_p-\mathbf{A}_q\) is a rank-one matrix, and passing from 2 to the new collection of pairs replacing those two selected pairs by its associated mixture \[(s_p+s_q, \mathbf{A}_{pq}),\quad \mathbf{A}_{pq}=\frac{s_p}{s_p+s_q}\mathbf{A}_p+\frac{s_q}{s_p+s_q}\mathbf{A}_q,\] and retaining unchanged all the others. In performing such step or mixture we refer to the difference \(\mathbf{A}_p-\mathbf{A}_q\) as the corresponding decomposition direction, and the weights \[\lambda_p=\frac{s_p}{s_p+s_q}, \quad\lambda_q=\frac{s_q}{s_p+s_q}, \quad \lambda_p+\lambda_q=1,\] as the associated relative volume fractions of the mixture. Note that \[\mathbf{A}_p=\mathbf{A}_{pq}+\lambda_q(\mathbf{A}_p-\mathbf{A}_q),\quad \mathbf{A}_q=\mathbf{A}_{pq}-\lambda_p(\mathbf{A}_p-\mathbf{A}_q).\] This process is the basic step of the \((H_n)\)-formalism (check references given in the Introduction), and produces successive mass-points and weights for intermediate probability measures \(\{\mu_k\}\) as well as decomposition directions. The crucial ingredient for laminates is the rank-one condition for all decomposition directions.

If we assume now that \(\mu\) is coming from the discrete gradients of two continuous, scalar functions \((u, v)\) with constant gradients \((\nabla u_i, \nabla v_i)\) with respect to some and the same triangulation \(\tau_l\) (\(l\) is an arbitrary positive integer indicating level of discreteness) \[\label{inicial} \mu=\sum_it_i\delta_{(\nabla u_i, \nabla v_i)},\tag{3}\] where will the decomposition in 1 come from enabling for the \((H_n)\)-formalism to be applied, guaranteeing the rank-one condition for all decomposition directions, and so be sure that \(\mu\) is a laminate? Needless to say, we are seeking a general result to be valid for any such discrete gradient.

Our strategy is the following. Consider the two marginals \[\mu_u=\sum_it_i\delta_{\nabla u_i},\quad \mu_v=\sum_it_i\delta_{\nabla v_i}.\] Take any decomposition like 2 for the first marginal \(\mu_u\) \[\label{paresz} (s_1, \mathbf{u}_1),\quad (s_2, \mathbf{u}_2), \quad\dots\quad (s_m, \mathbf{u}_m),\quad \{\mathbf{u}_j\}=\{\nabla u_i\}.\tag{4}\] We can proceed with arbitrary mixtures for this set of pairs because the support of \(\mu_u\) is contained in \(\mathbb{R}^2\), and therefore the rank-one condition for decomposition directions is not an issue. Because \(\nabla u_i\) and \(\nabla v_i\) are linked to each other being the gradients of \(u\) and \(v\), respectively, in the same element of the triangulation \(\tau_l\), we can consider an identical set of pairs as in 4 replacing \(\mathbf{u}_i\) by \(\mathbf{v}_i\) for all \(i\) if \(\{\mathbf{v}_j\}=\{\nabla v_i\}\). In doing so, we cannot expect in the least that corresponding decomposition directions in successive levels of the form \[\mathbf{U}_j-\mathbf{U}_k,\quad \mathbf{V}_j-\mathbf{V}_k,\] respectively for both sets of pairs, will be parallel to each other to ensure the rank-one condition for the matrix \[\begin{pmatrix}\mathbf{U}_j-\mathbf{U}_k\\\mathbf{V}_j-\mathbf{V}_k\end{pmatrix}.\] Note however that this rank-one condition is guaranteed for the first-level due to the rank-one compatibility across planar interfaces for gradients. We therefore envision the possibility of forcing this parallelism between decomposition directions, at the expense of allowing different volume fractions for the second marginal \(\mu_v\) \[\label{paresy} (S_1, \mathbf{v}_1),\quad (S_2, \mathbf{v}_2), \quad\dots\quad (S_m, \mathbf{v}_m),\quad \{\mathbf{v}_j\}=\{\nabla v_i\}.\tag{5}\] The family of weights \(\{S_j\}\) is to be selected in such a way that corresponding decomposition directions for both components when performing exactly the same mixtures at all levels for both sets of pairs in 4 and 5 turn out to be parallel.

More specifically, imagine we could succeed in deciding on the following elements.

1. One concrete way, skeleton or mold, linked to the underlying triangulation, specifying what triangles are to be mixed at the first stage, and what mixtures of triangles are to be mixed among themselves in successive levels until the final mixture involving the complete mass of the unit cube \(Q\). This mold must be independent of the particular gradients considered and depend only on the elements of \(\tau_l\). It should be selected from the outset. We refer to it as \(\mathscr{S}_l\).

2. A certain collection \(\Theta_l\) of vectors of feasible weights that can be interpreted in a way to produce respective sets of pairs as in 4 , any time a function \(w\) and its gradients \(\{\nabla w_i\}\) with respect to \(\tau_l\), are provided.

Given two arbitrary functions, \(u\) and \(v\), \(Q\)-periodic, continuous, piece-wise affine with respect to \(\tau_l\), we can define a (set-valued) operator \(\mathbf{T}\) from \(\Theta_l\) to itself as follows. Let the vector of weights \(\mathbf{t}\in\Theta_l\) be given, and calculate the associated full set of decomposition directions along \(\mathscr{S}_l\) with the gradients \(\{\nabla u_i\}\). Define the image \(\mathbf{T}(\mathbf{t})\subset\Theta_l\) as the collection of vectors of weights \(\mathbf{r}\in\Theta_l\) that are capable of reproducing the same (parallel) decomposition directions along \(\mathscr{S}_l\) but calculated with the second set of gradients \(\{\nabla v_i\}\). The existence of a fixed point for \(\mathbf{T}\), ensuring the same volume fractions for both components, will definitely be a proof that \(\mu\) in 3 can be reached by lamination by the very way in which the operator \(\mathbf{T}\) has been defined. The arbitrariness of \(l\) leads to our result.

Our job consists in showing that this formal framework with the claimed features is possible. In addition to fixing notation, we need to define \(\Theta_l\), the mold \(\mathscr{S}_l\), how a set of gradients \(\{\nabla w_i\}\) enters into our discussion, and how to define the operator \(\mathbf{T}\) once \((u, v)\) are given and \(\mathscr{S}_l\) selected.

The whole point of our approach is then to use Kakutani’s fixed point theorem to prove the existence of at least one such fixed point for \(\mathbf{T}\). Our main proofs then focus on showing that our analytical framework has been so defined as to allow for the hypotheses of Kakutani’s theorem to be checked. This objective explains why we have to start with care for the description of those main ingredients \(\mathscr{S}_l\) and \(\Theta_l\), and how they relate to each other.

3 \((H_n)\)-conditions↩︎

Since these structures are central to rank-one convexity and laminates, they should play a fundamental role in our strategy. This justifies why we need to be more precise about the ingredients underlying these structures. Two basic sources where \((H_n)\)-conditions are defined and treated are [40], [39] (see also [38]). Though they can be examined in a much more general context, we restrict attention specifically to our situation in this work.

In the bottom-to-top description, pairs in the support of \(\mu_m\) are being mixed successively and at different levels, mixtures of mixtures (of mixtures), to produce the intermediate probability measures \(\mu_k\) until reaching the final trivial measure \(\mu_{1}\). If one is not interested in keeping track of which pairs are mixed at each iteration, couples in the support of each intermediate probability measure \(\mu_k\) can be renumbered so that the first two pairs (\(i_p=1\), \(i_q=2\)) are always the ones to be mixed at every iteration according to the basic rule. However, since it is especially important for us to specify the global way of mixing pairs across the family \(\{\mu_k\}\), we avoid such reorganizations. Indeed, we will explicitly refer to this global way of passing from \(\mu_m\) to \(\mu_{1}\) through the intermediate probabilities \(\mu_k\) as the mold or skeleton of the \((H_n)\)-condition. In our situation here, there is no need to insist in that decomposition directions in ?? should be rank-one, as it is typically demanded, because the \(\mathbf{w}\)’s are taken to be vectors, and hence decomposition directions are always rank-one.

Typically, if initial pairs are labeled through subindex \(i\), \(1\le i\le m\), as in Definition 1, \(\mathscr{S}\) can be determined through the same set of indexes at successive levels using some delimiters, like brackets, to enclose pairs to be mixed as iterations proceed.

Let us stress this issue. Given the initial set of pairs \(\mathbb{H}\), each possible mold \(\mathscr{S}\) for \(\mathbb{H}\) would give rise to a different \((H_n)\)-condition; conversely, given a fixed mold \(\mathscr{S}\) for a certain number \(m\) of pairs, corresponding \((H_n)\)-conditions are generated every time we feed \(\mathscr{S}\) with a real set of pairs weights/vectors \[\mathbb{H}=\{(s_i, \mathbf{w}_i): 1\le i\le m\}.\]

The important point is to realize that given the initial set of pairs \(\mathbb{H}\) and mold \(\mathscr{S}\), the full \((H_n)\)-condition is completely determined by those two elements, and, in particular, all decomposition directions. In this way, if we are given two sets of pairs \(\mathbb{H}_1\) and \(\mathbb{H}_2\) with the same number of ordered elements, the same mold \(\mathscr{S}\) can be used with both, as indicated above, to produce two similar \((H_n)\)-conditions. We can, therefore, compare the full set of decomposition directions across levels produced for both \(\mathbb{H}_1\) and \(\mathbb{H}_2\), and check whether they are parallel or not.

It must be clear by now that there is a great number of possible molds \(\mathscr{S}\) for a given initial set \(\mathbb{H}\) of \(m\) pairs, especially if \(m\) is large. The example above makes us realize that it is not easy to describe such molds in a coherent way in full generality. There is, however, one particular class of molds which admit a more transparent recursive structure. They are given in a dyadic manner.

A simple example will clarify this concept.

The proof is elementary. In the second case of the previous example the permutation is defined through the identification \[[[[1, 6], [4, 7]], [[2, 3], [5, 8]]]\mapsto [[[1, 2], [3, 4]], [[5, 6], [7, 8]]].\]

Because of this proposition, there is no loss of generality in describing \((H_n)\)-conditions organized by a dyadic mold in the following way.

1. Initialization. We identify in the vector \[\mathbf{t}=(t_1, t_2, \dots, t_{2^m})\] our initial set of weights for the \((H_n)\)-condition, and put \[t_k^{(m)}=t_k,\quad \mathbf{w}_k^{(m)}\in\mathbb{R}^2,\] for elements to initialize the \((H_n)\)-condition at the lowest level.

2. Recursion.

   1. Weights for new level, and relative weights. For \[p=m-1, m-2, \dots, 1, 0,\quad 1\le k\le 2^p,\] put \[t_k^{(p)}=t_{2k-1}^{(p+1)}+t_{2k}^{(p+1)},\] and \[\label{pesosz} \lambda_k^{(p)}=\begin{cases} \frac{t_{2k}^{(p+1)}}{t_k^{(p)}},& t_k^{(p)}>0\\ 1/2,& t_k^{(p)}=0\end{cases}.\tag{6}\] In this way \[\label{relabs} t_{2k}^{(p+1)}=t_k^{(p)}\lambda_k^{(p)},\quad t_{2k-1}^{(p+1)}=t_k^{(p)}(1-\lambda_k^{(p)}),\tag{7}\] and \(t_1^{(0)}=1\).

   2. Decomposition direction. For \[p=m-1, m-2, \dots, 1, 0,\quad 1\le k\le 2^p,\] define \[\label{dirdes} \mathbf{W}_k^{(p)}=\mathbf{w}_{2k-1}^{(p+1)}-\mathbf{w}_{2k}^{(p+1)},\tag{8}\] as the corresponding decomposition direction.

   3. Mass points for new level. For \[p=m-1, m-2, \dots, 1, 0,\quad 1\le k\le 2^p,\] set \[\label{equis} \mathbf{w}_k^{(p)}=(1-\lambda_k^{(p)})\mathbf{w}_{2k-1}^{(p+1)}+\lambda_k^{(p)}\mathbf{w}_{2k}^{(p+1)}.\tag{9}\]

This recursive procedure is repeated \(m\) times until one gets one unique final vector \(\mathbf{w}_1^{(0)}\). Decomposition directions \(\mathbf{W}_k^{(p)}\), vectors \(\mathbf{w}_k^{(p)}\), and relative weights \(\lambda_k^{(m-1)}\) as well, depend upon \(\mathbf{t}\), the initial vector of weights, and on \(\{\mathbf{w}^{(m)}_k\}\), the initial set of vectors. But regarding this set of vectors as given, we will make the dependence on \(\mathbf{t}\) explicit by simply putting \[\label{elementos} \mathbf{W}_k^{(p)}(\mathbf{t}), \quad\mathbf{w}_k^{(p)}(\mathbf{t}), \quad\lambda_k^{(m-1)}(\mathbf{t}).\tag{10}\]

This is the bottom-to-top description. Equivalently, we can adopt the top-to-bottom procedure which starting from the trivial probability measure and pair \[\mu_1=\delta_{\mathbf{w}_1^{(0)}},\quad (1, \mathbf{w}_1^{(0)}),\] and given the net of decomposition directions and relative volume fractions \[(\mathbf{W}_k^{(p)}, \lambda_k^{(p)}),\quad p=0, 1, 2, \dots, m-1, 1\le k\le 2^p,\] we can build, in a successive dyadic manner, the intermediate probability measures \(\mu_p\) until the final \(\mu_m\).

There is finally another special class of molds for \((H_n)\)-conditions which will be important for. They build upon dyadic sub-molds.

To illustrate this concept another explicit example may help.

4 \((H_n)\)-conditions for gradients: molds through triangulations↩︎

We will be working with \(Q\)-periodic, continuous, piecewise affine, two-component maps with respect to a specific family of triangulations \(\{\tau_l\}\) of the unit cube \(Q\) of \(\mathbb{R}^2\). In general, the unit cube \(Q\subset\mathbb{R}^N\) can be decomposed in a finite number of simplexes and with a finite number \(d(N)\) of normals to the flat faces of those simplexes. By making small copies of \(Q\) and making use of this decomposition in all copies, we can build a family of exhausting, regular triangulations that provide arbitrary, uniform approximations of Lipschitz functions by piecewise affine maps. This is standard and well-known (see, for instance, [49]). For \(N=2\), three normals suffice, while for dimension \(N=3\), seven are necessary, and so on. Because of the difficulties organizing elements, normals, edges, vertices (much in the same way as it is done in the analysis of finite elements), we restrict attention to \(N=2\). Higher values of \(N\) would require further insight into that way of organizing things. For future reference, we will put \(\mathbb{P}_l\) for the finite-dimensional space of \(Q\)-periodic, \(\tau_l\)-piece-wise linear, continuous functions with gradients that are constant on each element of \(\tau_l\).

The description that follows pretends to setup a way to link triangulations \(\tau_l\) with the \((H_n)\)-hierarchy without reference to specific gradients or to volume fractions. It is just a way to explicitly tag, through the triangles of \(\tau_l\), their planar interfaces, and their unions in a hierarchical form, how gradients and volume fractions are to be mixed once they are provided. Said differently, we would like to determine a mold or skeleton \(\mathscr{S}_l\) associated with triangulation \(\tau_l\) that will be fed with volume fractions (weights) and gradients (vectors) of functions in \(\mathbb{P}_l\).

Before going into the general case for arbitrary values for \(l\), we will deal with a particular case to facilitate the understanding of what we are trying to convey. We will take \(l=2\). Notice that this particular case would correspond to the example in [17].

Suppose we have a triangulation \(\tau_2\) of the unit cube \(Q\) in \(16(=2^2\times2^2)\) small, equal squares, and each one is divided into two triangles along one of the two diagonals (always the same). We will write \[\tau_2=\{T_{s, 2}: s\in\{1, 2, \dots, 32\}\}\] for the full collection of triangle of \(\tau_2\). Since mixtures of gradients across planar interfaces between adjacent elements in \(\tau_l\) must play a fundamental role, our mold will hinge on those in the following way.

For \(192=3\times 2^6\), we want to describe a suitable tri-dyadic mold \(\mathscr{S}_2\) organized through the three subsets of indexes \[\label{tresz} I_1\equiv\{1, 2, \dots, 64\},\quad I_2\equiv\{65, 66, \dots, 128\},\quad I_3\equiv\{129, \dots, 192\},\tag{11}\] which will be identified, in any given way, with the three possible subsets of two normals out of the set of three normals \((1, 0), (0, 1), (1, 1)\) utilized in \(\tau_l\), for instance \[\begin{gather} j=1\mapstonormals (1, 0), (0, 1)\mapsto I_1,\nonumber\\ j=2\mapstonormals (1, 0), (1, 1)\mapsto I_2,\label{normalesz}\\ j=3\mapstonormals (0, 1), (1, 1)\mapsto I_3.\nonumber \end{gather}\tag{12}\] Since planar interfaces in \(\tau_l\) furnish compatibility of gradients across it, our mold will hinge on the full set of those interfaces. Take \(j=1\) to begin with, and focus on \[I_1=\{1, 2, \dots, 64\}, \quad normals (1, 0), (0, 1).\] Each planar interface with either normal \((1, 0)\) or \((0, 1)\) in \(\tau_2\), the normals corresponding to \(j=1\), will be assigned an index \(i\), \(1\le i\le 32\), in an arbitrary manner, but then \(2i\) and \(2i-1\) should be the indexes in \(I_1\) assigned to the adjacent elements of \(\tau_2\) corresponding to interface \(i\). Note that this chosen, specific way of tagging planar interfaces with normals \((1, 0)\) or \((0, 1)\) in \(\tau_2\) with index \(i\) will uniquely determine how subsequent mixtures in later upper levels will be performed as the mixture rule \([2p-1, 2p]\) will always be applied in all levels as it should in a dyadic mold. The arbitrariness of this choice implies that any such manner of labelling planar interfaces is equally valid.

Proceed similarly with \(j=2\), and \(j=3\). This means that for \(j=2\) planar interfaces with normals \((1, 0)\) or \((1, 1)\) will be identified with indexes \(i\), \(33\le i\le 64\), in a manner that \(2i\) and \(2i-1\) in \(I_2\) are the indexes assigned to the two adjacent elements of \(\tau_2\) with the corresponding interface \(i\); and similarly with \(j=3\). Thus all planar interfaces will be assigned two indexes because corresponding normals occur twice in our tag method according to 12 , and triangles in \(\tau_l\) will be given six different indexes, two for each \(j\). Hence, this procedure defines a partition of the set of indexes \(\{1, 2, \dots, 192\}\) in \(32\) subsets \(\mathcal{C}_s\) of six elements \[\label{sumazz} \mathcal{C}_s=\{k_{1s}, k_{2s}, k_{3s}, k_{4s}, k_{5s}, k_{6s}\},\quad k_{is}\in\{1, 2, \dots, 192\},\quad \cup_{1\le s\le 32}\mathcal{C}_s=\{1, 2, \dots, 192\},\tag{13}\] in such a way that the intersection of each \(\mathcal{C}_s\) with the three subsets in 11 is exactly two indexes. Index \(s\) identifies elements in \(\tau_2\), and so 13 implies, as just indicated above, that each triangle in \(\tau_2\) is identified by six different indexes of \(\{1, 2, \dots, 192\}\), two for each of the \(I_j\)’s, and such indexes are compatible with the labelling of planar interfaces as described in the previous paragraph.

Our tridyadic mold \(\mathscr{S}_2\) will proceed simultaneously and independently for all \(j\) in 12 , according to Definition 4, mixing adjacent elements \(2i\) and \(2i-1\) through interface \(i\), in the first level of mixture; and in the associated dyadic manner \([2p-1, 2p]\) inherited by the chosen tag mechanism for interfaces in all upper levels. Once the three independent dyadic processes have been exhausted, fix any form to mix the three final resulting delta measures using again the basic mixing rule of the \((H_n)\)-condition formalism.

The general case for any value of \(l\) is described exactly in the same manner. We explain it in shorter terms.

We fix a family of triangulations \(\{\tau_l\}\), \(l=1, 2, \dots\), in which the two sides of the unit cube \(Q\) of \(\mathbb{R}^2\) parallel to the coordinate axes are divided in \(2^l\) subintervals producing \(2^{2l}\) identical sub-squares of side-length \(2^{-l}\). Each such resulting sub-square is divided into two triangles along the diagonal with normal parallel to the vector \((1, 1)\), for instance. In this way we can write \[\tau_l=\{T_{s, l}: 1\le s\le 2^{2l+1}\}, \quad T_{s, l},triangle of \tau_l with area 2^{-1-2l}.\] After our experience with the case \(l=2\), a partition in \(2^{2l+1}\) pair-wise disjoint subsets \(\mathcal{C}_s\), \(1\le s\le 2^{2l+1}\), as many as triangles \(T_{s, l}\) in \(\tau_l\), of the set of indexes \(\{1, 2, \dots, 3\times2^{2l+2}\}\) is possible complying with the following conditions:

• Each \(\mathcal{C}_s\) has six elements, and it should have two indexes in each one of the subsets \[\begin{gather} I_1=\{1, 2, \dots, 2^{2l+2}\},\quad I_2=\{2^{2l+2}+1, 2^{2l+2}+2, \dots, 2\times2^{2l+2}\},\label{subcon}\\ I_3=\{2^{2l+2}+1, 2^{2l+2}+2, \dots, 3^{2l+2}\}.\nonumber \end{gather}\tag{14}\]

• \(\{\mathcal{C}_s\}\) must be compatible with labels \(i\) for planar interfaces in \(\tau_l\) exactly in the same way as in the case \(l=2\), so that mixtures in a first level are identified with planar interfaces \(i\) and corresponding adjacent triangles \(2i\) and \(2i-1\). The same dyadic rule \([2p-1, 2p]\) is used throughout all levels of mixture.

• Each planar interface in \(\tau_l\) is tagged with two indexes, and each element with six indexes in a compatible manner for the dyadic mold to be applied.

There is, consequently, an underlying tri-dyadic skeleton \(\mathscr{S}_l\) that will be fed with vectors of weights and constant gradients of functions in \(\mathbb{P}_l\). For each \(l\), we fix once and for all the tridyadic mold \(\mathscr{S}_l\) among the many possibilities respecting the conditions just described.

There are no vectors or functions in this discussion. As has been conveyed in Section 3, the chosen tridyadic mold \(\mathscr{S}_l\) will be utilized to produce true \((H_n)\)-conditions and associated decomposition directions, as soon as we feed into \(\mathscr{S}_l\) the gradient \(\nabla w\) of a function \(w\in\mathbb{P}_l\), and a vector of weights coming from a suitable set. Recall that \(\mathbb{P}_l\) is taken to be the finite-dimensional space of \(Q\)-periodic, \(\tau_l\)-piece-wise linear, continuous functions with gradients that are constant on each \(T_{s, l}\). For later reference, we will designate by \(\mathscr{S}_{l, j}\) the sub-mold associated with the independent processes for each \(j=1, 2, 3\). Schematically we can write \[\label{parcial} \{\mathscr{S}_{l, j}: j=1, 2, 3\}\mapsto \mathscr{S}_l\tag{15}\] to stress that each dyadic \(\mathscr{S}_{l, j}\) is a part of the tridyadic mold \(\mathscr{S}_l\), and that the \(\mathscr{S}_{l, j}\)’s proceed independently of each other. In particular, all decomposition directions generated through \(\mathscr{S}_l\) any time weights and gradients are fed, will include those generated by each \(\mathscr{S}_{l, j}\). Recall that \(\mathscr{S}_l\) and \(\mathscr{S}_{l, j}\) are fixed once and for all.

5 The role of weights and functions↩︎

As it was pointed out several times in Section 3, once the tri-dyadic mold has been selected, it should be fed with weights and vectors to produce true \((H_n)\)-conditions.

We start with the description of a suitable set of weights. We again treat first the case \(l=2\) to make the discussion more transparent.

We would like to consider a certain suitable subset \(\Theta_2\) of vectors of weights of the standard simplex \(\Gamma_{192}\) of \(\mathbb{R}^{192}\), associated with the chosen \(\mathscr{S}_2\) and its corresponding partition \(\mathcal{C}_s\) according to Definition 5, namely \[\begin{align} \Theta_2=\{\mathbf{t}=(t_1, t_2, &\dots, t_{192})\in\mathbb{R}^{192}: t_k\ge0, 1\le k\le 192, \tag{16}\\ &t_{2i}+t_{2i-1}=1/96, 1\le i\le 96, \tag{17}\\ &\sum_{k\in\mathcal{C}_s}t_k=1/32, 1\le s\le 32\}.\tag{18} \end{align}\] As in Section 4, bear in mind that:

1. Index \(s\) in 18 identifies triangle \(T_{s, 2}\) in \(\tau_2\). The total mass \(1/32\) of each such triangle is divided into six pieces according to the weights \(t_{k_{is}}\) in 13 .

2. Each index \(i\) in 17 identifies one of the \(48\) planar interface in \(\tau_2\) (recall periodicity). Such identification is made according to the rule in Section 4 concerning flat interfaces in \(\tau_2\) in each of the subsets \(I_j\). Each such interfase occurs twice (corresponding to two different indexes \(i\)), and that is why index \(i\) runs from 1 to 96. Condition 17 furthermore ensures that mixtures coming from planar interfaces of \(\tau_2\) will always be made in a fixed proportion \(1/96\). There is a special reason for this restriction.

The link of the set \(\Theta_l\) with our triangulation \(\tau_l\), mold \(\mathscr{S}_l\), and partition \(\mathcal{C}_s\) should be easy to grasp as it has just been anticipated for the case \(l=2\).

1. Index \(s\) in ?? identifies triangles in \(\tau_l\).

2. Each \(i\) in ?? identifies a planar interface in \(\tau_l\), and each planar interface corresponds to two different values of \(i\), in such a way that triangles sharing a flat interface associated with index \(i\) are to be mixed in proportions \(t_{2i}\) and \(t_{2i-1}\), and they all are assigned a similar weight \(3^{-1}2^{-1-2l}\) every time such interface is utilized in the mold \(\mathscr{S}_l\).

Concerning functions, let \(w\in\mathbb{P}_l\), and consider its associated probability measure \[\label{gradiente} \nu_w=\sum_s2^{-2l-1}\delta_{\left.\nabla w\right|_{T_{s, l}}}\tag{19}\] Recall that \(\tau_l=\{T_{s, l}\}\), and hence each \(\left.\nabla w\right|_{T_{s, l}}\) is a constant vector in \(\mathbb{R}^2\). The \(3\times2^{2l+2}\) vectors to feed, together with weights in \(\Theta_l\), the tri-dyadic mold \(\mathscr{S}_l\) (with partition \(\mathcal{C}_s\)) are \[\mathbf{w}_k=\left.\nabla w\right|_{T_{s, l}} if k\in\mathcal{C}_s,\quad k\in\{1, 2, \dots, 3\times2^{2l+2}\}.\]

According to Definition 4, the \((H_n)\)-conditions corresponding to sub-molds \(\mathscr{S}_{l, j}\) proceed, separately for each \(j\), in a dyadic manner as it has been explicitly described in Section 3. If we write \[\label{tresjotas} \mathbf{t}=(\mathbf{t}_1, \mathbf{t}_2, \mathbf{t}_3)\in\Theta_l,\quad \mathbf{t}_j=(t^{(j)}_k), \quad 1\le k\le 2^{2l+2}, j=1, 2, 3,\tag{20}\] we proceed as in Section 3 in a dyadic manner for each \(\mathbf{t}_j\), though there are a couple of particular issues that need to be explicitly emphasized. We use the same notation utilized in the discussion right after Proposition 3, though, for the sake of simplicity, we ignore (super)-index \(j\), since the process is formally the same for the three dyadic sub-molds.

1. Initialization. As just stressed, we would work separately on each \(I_j\) in 14 , ignoring the index \(j\) for the sake of notational simplicity, and take \[\label{soporte} t_k^{(m)}=t_k,\quad \mathbf{w}_k^{(m)}=\left.\nabla w\right|_{T_{s, l}} if k\in\mathcal{C}_s,\tag{21}\] for \(1\le k\le 2^{2l+2}\).

2. Recursion. The only point to be noticed is that in our particular situation, because the way in which the set \(\Theta_l\) has been defined (recall ?? ), we would have \[\label{pesofijo} t_k^{(m-1)}=2^{1-m}, \quad 1\le k\le 2^{m-1},\tag{22}\] and then \[\label{constante} \lambda_k^{(p)}=1/2,\quad 1\le p\le m-2, 1\le k\le 2^p.\tag{23}\]

All other ingredients of the discussion after Proposition 3 are exactly the same, including the top-to-bottom description, and we will use the same notation \[\label{zsauhoyp} \mathbf{W}_k^{(p)}(\mathbf{t}), \quad\mathbf{w}_k^{(p)}(\mathbf{t}), \quad\lambda_k^{(m-1)}(\mathbf{t})\tag{24}\] for decomposition directions, mass-points of intermediate probability measures, and relative weights. Keep in mind that \(\lambda_k^{(p)}=1/2\) for all \[0\le p\le m-1,\quad 1\le k\le 2^p,\] according to 23 .

Finally, to complete the \((H_n)\)-condition with the tri-dyadic mold \(\mathscr{S}_l\), we need to keep working with the result of the three independent dyadic processes \(\mathscr{S}_{l, j}\), according to what has been decided in \(\mathscr{S}_l\) until we reach the upper, final vanishing barycenter \(\mathbf{0}\in\mathbb{R}^2\). Due to periodicity, we should have \[\label{baricentro} \sum_s2^{-2l-1}\left.\nabla w\right|_{T_{s, l}}=\mathbf{0},\tag{25}\] and we write \[\label{baricentros} \mathbb{W}_j=\frac{1}{1/3}\sum_s\sum_{k \in\mathcal{C}_s\cap I_j} t^{(j)}_k \left.\nabla w\right|_{T_{s, l}},\quad \mathbf{t}_j=(t^{(j)}_k), \quad j=1, 2, 3, \quad \mathbf{t}=(\mathbf{t}_1, \mathbf{t}_2, \mathbf{t}_3)\in\Theta_l,\tag{26}\] for the barycenters of each of the three dyadic sub-molds \(\mathscr{S}_{l, j}\) which are combined in two additional final mixtures in an arbitrary but fixed way. Since \[\frac{1}{3}=\sum_s\sum_{k \in\mathcal{C}_s\cap I_j} t^{(j)}_k\] then \[\label{firstcero} \mathbf{0}=\frac{1}{3}(\mathbb{W}_1+\mathbb{W}_2+\mathbb{W}_3).\tag{27}\] It is convenient to use the same recursive notation for the whole mold \(\mathscr{S}_l\) as has been utilized above for each dyadic \(\mathscr{S}_{l, j}\). In this way, we will indicate \[\begin{gather} t_k^{(m)}=t_k,\quad k\in\{1, 2, \dots, 3\times2^{2l+2}\}, \quad \mathbf{t}=(t_k)\in\Theta_l,\nonumber\\ \mathbf{w}_k^{(m)}=\left.\nabla w\right|_{T_{s, l}},\quad k\in\mathcal{C}_s,\nonumber \end{gather}\] for the initial elements to feed \(\mathscr{S}_l\). The only difference is that now we have \(2l+4\) levels: \(2l+2\) for the three dyadic processes, and two additional ones for the last mixture involving the \(\mathbb{W}_j\)’s in 27 . 10 will also indicate decomposition directions, mass points for intermediate probability measures, and volume fractions for \(m=2l+4\) successive levels in \(\mathscr{S}_l\).

6 The map↩︎

Take an arbitrary value of \(l\). Let \((u, v):Q\to\mathbb{R}^2\) be an arbitrary, two-component vector map of functions \(u\) and \(v\) in \(\mathbb{P}_l\). We turn to our mold \(\mathscr{S}_l\), as has been fixed, to produce \((H_n)\)-conditions when fed with weights from \(\Theta_l\) and values of gradients \(\nabla u\) or \(\nabla v\). As just indicated above, \(m=2l+4\) is the overall number of levels for \(\mathscr{S}_l\). Recall notation in 10 for decomposition directions, mass points, and relative volume fractions.

A joint, simultaneous rank-one description of the joint probability measure \[\label{margi} \nu=\sum_s2^{-2l-1}\delta_{\left.(\nabla u, \nabla v)\right|_{T_{s, l}}},\tag{28}\] demands, as a fundamental ingredient, that all decomposition directions \[\label{dirdescomposicion} \mathbf{U}_k^{(p)}(\mathbf{t}),\quad \mathbf{V}_k^{(p)}(\mathbf{t})\tag{29}\] be proportional to each other for some and the same vector \(\mathbf{t}\) of weights, and for all \[0\le p\le m-1,\quad 1\le k\le 2^p.\] Clearly for a given feasible vector \(\mathbf{t}\in\Theta_l\), those two sets of decomposition directions in 29 generated by the two components \(u\) and \(v\) will, most of the time, not be proportional to each other though it is guaranteed for \(p=m-1\) because decomposition directions at this first lower level are taken to be precisely the three normals occurring in \(\tau_l\) (for both components \(u\) and \(v\)). Yet one may wonder if one could ensure such parallelism for all levels, enabling different vector of weights: once fixed \(u\) and \(v\), for each \(\mathbf{t}\in\Theta_l\), are there feasible vectors of weights \(\mathbf{r}\) such that \[\mathbf{U}_k^{(p)}(\mathbf{t}),\quad\mathbf{V}_k^{(p)}(\mathbf{r})\] are parallel? The definition of our operator is in charge of guaranteeing this property.

Manipulations performed in the preceding section for our \(\mathscr{S}_l\) and for arbitrary \(w\in\mathbb{P}_l\) and \(\mathbf{t}\in\Theta_l\) can be exactly repeated separately for both \(u\) and \(v\), respective vectors of weights \(\mathbf{t}\) and \(\mathbf{r}\) in \(\Theta_l\), corresponding probability measures \[\nu_u=\sum_s2^{-2l-1}\delta_{\left.\nabla u\right|_{T_{s, l}}}, \quad \nu_v=\sum_s2^{-2l-1}\delta_{\left.\nabla v\right|_{T_{s, l}}},\] and initial elements \[\begin{gather} t_k^{(m)}=t_k,\quad r_k^{(m)}=r_k, \quad k\in\{1, 2, \dots, 3\times2^{2l+2}\},\nonumber\\ \mathbf{u}_k^{(m)}=\left.\nabla u\right|_{T_{s, l}},\quad \mathbf{v}_k^{(m)}=\left.\nabla v\right|_{T_{s, l}},\quad k\in\mathcal{C}_s.\nonumber \end{gather}\] The outcome of such a procedure would be the respective sets of decomposition directions, and the sets of mass points for intermediate probability measures \[\label{descomposicion} \mathbf{U}_k^{(p)}(\mathbf{t}), \quad\mathbf{u}_k^{(p)}(\mathbf{t}),\quad \mathbf{V}_k^{(p)}(\mathbf{r}), \quad\mathbf{v}_k^{(p)}(\mathbf{r}),\quad 0\le p\le m-2, 1\le k\le 2^p.\tag{30}\] The families of weights \(\lambda_k^{(m-1)}(\mathbf{t})\), \(\lambda_k^{(m-1)}(\mathbf{r})\) are however independent of \(u\) or of \(v\).

Recall again that the mold \(\mathscr{S}_l\) has already been selected once and for all as described in Section 4, with a global number \(m=2l+4\) of levels.

Note again how \(\mathbf{V}_k^{(m-1)}(\mathbf{r})\), for \(p=m-1\), is always parallel to \(\mathbf{U}_k^{(m-1)}(\mathbf{t})\) precisely because decomposition directions at the level \(p=m-1\) correspond to interfaces between two adjacent elements of the triangulation \(\tau_l\).

The whole point or our concern is the following.

We will be using the following classic result to show the existence of a fixed-point for \(\mathbf{T}\).

This is a classical theorem on fixed-points for set-valued maps, which is but a generalization of the classic Brower’s fixed point theorem. It is well-known and can be found in many places, for instance in [50].

The fundamental properties that the application of this result to our framework require are the non-emptiness, the compactness and the convexity of \(\mathbf{T}(\mathbf{t})\) for each \(\mathbf{t}\in\Theta_l\), in addition to the upper semicontinuity. Once these properties are shown, we will have our main result Theorem 1 as a direct consequence of Proposition 4, Theorem 5, and the standard approximation fact for Lipschitz functions mentioned at the beginning of Section 4.

We proceed to show first those basic requirements for map \(\mathbf{T}\), while we defer the non-emptiness of images to a final independent section.

6.1 Basic properties of the map↩︎

We start with the upper semicontinuity required by Theorem 5. This property is, as a matter of fact, elementary since if \[\mathbf{r}_n\in\mathbf{T}(\mathbf{t}_n),\quad \mathbf{r}_n\to \mathbf{r}, \mathbf{t}_n\to \mathbf{t},\] then, we must necessarily have \(\mathbf{r}\in\mathbf{T}(\mathbf{t})\). This is straightforward because the dependence of elements in 10 on \(\mathbf{t}\) is continuous.

On the other hand, the compactness of each subset \(\mathbf{T}(\mathbf{t})\) is also clear since all these images are closed subsets of the compact set \([0, 1]^q\) for some finite \(q\).

We treat next the convexity of images. Ensuring this convexity property is responsible for the precise definition of the set \(\Theta_l\) we have adopted, and the way in which \((H_n)\)-conditions have been setup in Sections 3 and 5. It is pretty clear after the following statement.

The non-emptiness of images is the true clue of the proof.

7 Non-emptiness of images↩︎

Suppose \(\mathbf{t}\in\Theta_l\) is arbitrary, and consider the family of nested decomposition directions \[\label{direccionesz} \mathbf{U}_k^{(p)}\equiv \mathbf{U}_k^{(p)}(\mathbf{t}),\quad 0\le p\le m-1, 1\le k\le 2^p,\tag{31}\] associated with the first component \(u\) of the vector map \((u, v)\in\mathbb{P}_l\times\mathbb{P}_l\) in a bottom-to-top description. Both \(\mathbf{t}\) and \(u\) are regarded as fixed.

The question we would like to address is:

Given the fixed set of decomposition directions \(\{\mathbf{U}_k^{(p)}\}\) in 31 coming from \(u\in\mathbb{P}_l\) and \(\mathbf{t}\in\Theta_l\) along \(\mathscr{S}_l\), what are the functions \(w\in\mathbb{P}_l\) for which the probability measure corresponding to its gradient \(\nabla w\) as in 19 , admits a decomposition as a \((H_n)\)-condition along \(\mathscr{S}_l\) using such same given set \(\{\mathbf{U}_k^{(p)}\}\) of decomposition directions for whatever weights?

We claim that the answer to this question is the full space \(\mathbb{P}_l\), regardless of what the particular collection of chosen decomposition directions \(\{\mathbf{U}_k^{(p)}\}\), and vector of weights \(\mathbf{t}\in\Theta_l\) are, as long as they truly come from a function \(u\in\mathbb{P}_l\), and a feasible vector of weights \(\mathbf{t}\in\Theta_l\).

It should be clear how we have to resort to the top-to-bottom description of \((H_n)\)-conditions to deal with this issue.

If this claim is correct, the non-emptiness of each set \[\mathbf{T}(\mathbf{t}),\quad \mathbf{t}\in\Theta_l, \quad u, v\in\mathbb{P}_l,\] will hold for any arbitrary such pair of functions \(u, v\).

We now prove Proposition 7.

8 Conclusion↩︎

Once the hypotheses of Theorem 5 have been checked out for our map \(\mathbf{T}\) in Definition 7 (for arbitrary pairs \((u, v)\in\mathbb{P}_l\times\mathbb{P}_l\), mold \(\mathscr{S}_l\), and arbitrary \(l\)) in Subsection 6.1 and Section 7, that theorem lets us conclude the existence of at least one fixed point of it in \(\Theta_l\). Proposition 4 then leads to the desired, fundamental conclusion expressed in Theorem 1: every discrete gradient probability measure supported in \(\mathbb{R}^{2\times2}\) corresponding to a pair in \(\mathbb{P}_l\times\mathbb{P}_l\), for arbitrary \(l\in\mathbb{N}\), is a laminate. As a consequence, rank-one convexity implies quasi convexity for two-dimensional, two-component maps.

References↩︎