# Potentials for \(\mathcal {A}\)-quasiconvexity

## Abstract

We show that each constant rank operator \(\mathcal {A}\) admits an exact potential \(\mathbb {B}\) in frequency space. We use this fact to show that the notion of \(\mathcal {A}\)-quasiconvexity can be tested against compactly supported vector fields. We also show that \(\mathcal {A}\)-free Young measures are generated by sequences \(\mathbb {B}u_j\), modulo shifts by the barycentre.

## Mathematics Subject Classification

49J45 35G05## 1 Introduction

*f*. Such subsets \(\mathfrak {C}\) can account for differential constraints and boundary conditions. Modulo terms removed for simplicity of exposition, such functionals could model, for instance, the energy arising from the deformation of a solid body \(\varOmega \), viewed as a sufficiently regular open subset of \(\mathbb {R}^n\), where

*f*is a continuous energy density map characterized by the constitutive properties of the material. In accordance with the Direct Method in the Calculus of Variations, imposing a suitable bound from below on

*f*ensures existence and weak compactness of minimizing sequences \(w_j\). The appropriate continuity property of \(\mathscr {E}\) in this case is that of lower semi-continuity with respect to weak convergence in \({\text {L}}^p\)

*f*satisfying (2) is convex. Of course, convexity of

*f*is sufficient for lower semi-continuity (always understood as weakly sequential throughout this note) in any reasonable class \(\mathfrak {C}\), but it is hardly necessary in general. For instance, if \(\mathfrak {C}\) is the space of weak gradients in \({\text {L}}^2\) and

*f*is a quadratic form, then one can easily show that

*f*being positive on rank-one matrices implies lower semi-continuity. This example, that we will later come back to in more generality, is of particular relevance, as it provides the insight for a second convexity condition, which is necessary for lower semi-continuity with the constraint \(w=\nabla u\): if \(\mathscr {E}\) is lower semi-continuous, then

*f*is convex along rank-one lines. In particular, for integrands

*f*of class \({\text {C}}^2\), this is equivalent to the so-called Legendre–Hadamard ellipticity condition

*f*. Indeed, it was shown by Morrey in [22] that lower semi-continuity of \(\mathscr {E}\) is equivalent with

*quasiconvexity*of

*f*, i.e., the Jensen-type inequality

*u*with compact support in the open cube

*Q*. On one hand, the quasiconvexity assumption is a plausible constitutive relation for energy functionals arising in solid mechanics [5]; on the other hand, it is but a minor improvement of the lower semi-continuity concept, which makes it particularly difficult to check in applications. The counterexample of Šverák [32] rules out the possibility of quasiconvexity being a type of directional convexity (see also [7, Ex. 3.5] for the case of higher order gradients). A tractable sufficient condition for quasiconvexity is

*polyconvexity*, i.e.,

*f*is a convex functions of the minors, also introduced by Morrey in [22] in connection with lower semi-continuity and used by Ball to obtain existence theorems under very mild growth conditions, giving very satisfactory existence results in non-linear elasticity [4]. The fact that quasiconvexity does not imply polyconvexity is much easier to see, at least in higher dimensions, and follows from an old observation of Terpstra concerning quadratic forms [37] (see also [2, 6] and the references therein).

*w*that satisfy a linear, typically under-determined, partial differential constraint, say \(\mathcal {A}w=0\), assumption that we make henceforth. Examples arise in elasticity, plasticity, elasto-plasticity, electromagnetism, and others. The \(\mathcal {A}\)-free framework originates in the pioneering work of Murat and Tartar in compensated compactness [23, 33, 34] and can be correlated with the question of finding energy functionals that are continuous with respect to weak convergence in \(\mathfrak {C}\) [24]. The latter question was also studied in generality by Ball, Currie, and Olver in [7], leading to the generalization of polyconvexity to the case where energy functionals depend on higher order derivatives. In this case, the definition of quasiconvexity extends mutatis mutandis [20]. As to the question of lower semi-continuity, the analysis of the case when

*f*is a quadratic form (see, e.g., [36, Ch. 17] or [35, Thm. 2]) reveals a different necessary condition of directional convexity, namely with respect to the so-called wave cone of \(\mathcal {A}\). It was shown by Dacorogna in [12, Thm. I.2.3] that, in order to have lower semi-continuity, it is sufficient to assume the following generalization of quasiconvexity, namely that

*w*such that \(\int _Qw=0\) and \(\mathcal {A}w=0\). However, it is not clear whether this condition is necessary. More recently, Fonseca and Müller showed in [16] that if one assumes in addition that the fields

*w*are periodic, in which case

*f*is called \(\mathcal {A}\)

*-quasiconvex*, then one indeed obtains a necessary and sufficient condition

^{1}(under suitable growth assumptions on

*f*). Their result holds under the assumption that the symbol map \(\mathcal {A}(\cdot )\) of \(\mathcal {A}\) is a constant rank matrix-valued field away from 0. This condition, introduced in [30, Def. 1.5] to prove coerciveness inequalities for non-elliptic systems, was first used in the context of compensated compactness by Murat and ensures, as noted on [23, p.502], the continuity of the map

In the proof of the main result of [16], considerable difficulty is encountered when proving sufficiency of \(\mathcal {A}\)-quasiconvexity. One reason for this is the absence of potential functions for \(\mathcal {A}\), which, if available, should allow one to test with compactly supported functions in the definition of \(\mathcal {A}\)-quasiconvexity and, perhaps, use more standard methods.

The main result of the present work is to show that the existence of such a potential in Fourier space is equivalent with the constant rank condition.

### Theorem 1

Here \(\mathcal {A}(\cdot )\), \(\mathbb {B}(\cdot )\) denote the (tensor-valued) symbol maps of, respectively, \(\mathcal {A},\mathbb {B}\). We say that \(\mathcal {A}\) has *constant rank* if the map \(0\ne \xi \mapsto {\mathrm{rank}\,}\mathcal {A}(\xi )\) is constant (see Sect. 2 for detailed notation). We will regard \(\mathbb {B}\) as the *potential* and \(\mathcal {A}\) as the *annihilator*, although this terminology is not standard.

*not*, in general, imply for vector fields

*w*that

*w*that allow for usage of the Fourier transform, (5) can be shown to hold (Lemma 2). As a consequence, standard arguments in the Calculus of Variations lead to the fact that a map

*f*is \(\mathcal {A}\)-quasiconvex if and only if

*u*supported in an open cube

*Q*(Corollary 1). It is also the case that, under the constant rank condition, the notions of \(\mathcal {A}\)-quasiconvexity [16, Def. 3.1] and Dacorogna’s \(\mathcal {A}\)-\(\mathbb {B}\)-quasconvexity [12, Eq. (A.12)] coincide. In particular, one can define \(\mathcal {A}\)-quasiconvexity via integration over arbitrary domains (Lemma 3). As a consequence, the lower semi-continuity properties of functionals (1) in the (asymptotically \(\mathcal {A}\)-free) topologies considered in [3, 16], which are natural from the point of view of compensated compactness theory, rely only on the structure of the potential \(\mathbb {B}\).

In fact, we will show that the \(\mathcal {A}\)-quasiconvex relaxation of a continuous integrand can be described in terms of \(\mathbb {B}\) only. From this point of view, it is natural to investigate the Young measures generated by sequences satisfying differential constraints [16, Sec. 4], as they efficiently describe the minimization of energies that are not lower semi-continuous. We recall that the role of parametrized measures for non-convex problems in the Calculus of Variations was first recognized by Young in the pioneering works [39, 40, 41]. See the monographs [26, 27] for a modern, detailed exposition.

Roughly speaking, for \(1<p<\infty \), we consider a sequence \(w_j\) converging weakly in \({\text {L}}^p\) which is asymptotically \(\mathcal {A}\)-free and generates a Young measure \(\varvec{\nu }\). Technically speaking, it suffices to take \(\mathcal {A}w_j\) to be strongly compact in \({\text {W}}^{-k,p}_{{\text {loc}}}\), where *k* is the order of \(\mathcal {A}\). This is (slightly more general than) the topology considered in [16, Rk. 4.2(i)] and is consistent with the topology considered in compensated compactness (see, e.g., [36, Thm. 17.3], which essentially deals with the case of linear Euler–Lagrange equations). In this setting, we will show that the Young measure \(\varvec{\nu }\) is generated by a sequence of smooth maps \(\mathbb {B}u_j\), modulo a shift by the barycentre (Proposition 1).

*strictly*under the constant rank condition). See also [13] and the Appendix of [12].

Since testing with the appropriate quantity is fundamental in the study of partial differential equations, we hope that the observations made in this work will increase the flexibility of analyzing functionals in either class described above. On the other hand, the functional \(\mathscr {F}\) seems better suited for incorporating boundary conditions, which will be pursued elsewhere.

This paper is organized as follows: In Sect. 2 we prove the main Theorem 1, in Sect. 3 we prove that \(\mathcal {A}\)-quasiconvexity can be tested with compactly supported fields \(w=\mathbb {B}u\) (Corollary 1), and in Sect. 4 we prove that \(\mathcal {A}\)-free Young measures are shifts of Young measures generated by sequences \(\mathbb {B}u_j\).

## 2 Proof of Theorem 1

*k*-homogeneous, linear differential operator \(\mathcal {A}\) on \(\mathbb {R}^n\) from

*W*to

*X*we mean

*W*,

*X*. We also define the (Fourier) symbol map

*constant rank*if there exists a natural number

*r*such that

*Moore–Penrose*)

*generalized inverse*, introduced independently in [8, 21, 28], to which we refer plainly as the

*pseudo-inverse*, although the terminology is not standard. For a matrix \(M\in \mathbb {R}^{N\times m}\), its pseudo-inverse \(M^\dagger \) is the unique \({m\times N}\) matrix defined by the relations

*M*. Equivalently, the pseudo-inverse is determined by the geometric property that \(MM^\dagger \) and \(M^\dagger M\) are orthogonal projections onto \(\mathrm {im\,}M\) and \((\ker M)^\perp \) respectively. We refer the reader to the monograph [10] for more detail on generalized inverses.

On the other hand, motivated by a similar construction in [38, Rk. 4.1], one can speculate that \(\mathbb {P}\) and, in fact, \(\mathcal {A}^\dagger (\cdot )\) are rational functions. This is indeed the case, as a consequence of the main result of Decell in [15], building on the fundamental result of Penrose [28, Thm. 2] and the Cayley–Hamilton Theorem.

### Theorem 2

### Proof

(Proof of Theorem 1, sufficiency) Suppose that \(\mathcal {A}\) has constant rank. We put \(M{:}{=}\mathcal {A}(\xi )\) in the above Theorem for \(\xi \in \mathbb {R}^n\setminus \{0\}\), and abbreviate \(\mathcal {H}(\xi ){:}{=}\mathcal {A}(\xi )\mathcal {A}^*(\xi )\). The first, perhaps most crucial, observation is that \(r(\xi )\), as defined by (8), equals the number of non-zero eigen-values of \(MM^*\), which equals the number of singular values of *M*. This is, in turn, equal to \({\mathrm{rank}\,}M\), which is independent of \(\xi \) by the constant rank assumption on \(\mathcal A\).

The necessity of the constant rank condition in Theorem 1 follows from the following Lemma and the Rank–Nullity Theorem.

### Lemma 1

*P*,

*Q*be two matrix-valued polynomials on \(\mathbb {R}^n\). Suppose that there exists

*s*such that

*P*and

*Q*have constant rank in

*S*.

### Proof

*S*. Say \(R_P(S)=\{r_1,r_1+1\ldots ,r_2\}\) for natural numbers \(r_1<r_2\). We also write \(\mathrm {M}_d\) for the map that has input a matrix and returns (a vector of) all its minors of order

*d*. In particular, \(\mathrm {M}_dP\), \(\mathrm {M}_dQ\) are vector-valued polynomials on \(\mathbb {R}^n\). We then have that

^{2}On the other hand,

*S*is Lebesgue-null and we arrive at a contradiction. \(\square \)

It is natural to ask the reversed question, whether a constant rank operator \(\mathbb {B}\) admits an exact annihilator \(\mathcal {A}\). This is indeed the case, as can be shown by a simple modification of the argument above:

### Remark 1

*constant rank*on \(\mathbb {R}^n\) from

*V*to

*W*. Then, we can choose \(M{:}{=}\mathbb {B}(\xi )\) for \(\xi \in \mathbb {R}^n\setminus \{0\}\) in Theorem 2, so that

We conclude the discussion of algebraic properties with two remarks: Firstly, it is quite convenient that the two constructions presented are explicitly computable. On the other hand, performing the computations on simple examples, e.g., involving only \({\text {div}}\), \(\mathrm {grad}\), \({\text {curl}}\), one easily notices that the operators constructed via our formulas are often over complicated. Perhaps more computationally efficient methods, e.g., in the spirit of [38, Sec. 4.2] can be developed.

## 3 \(\mathcal {A}\)-quasiconvexity

The relevance of Theorem 1 for analysis can be seen, for instance, from the fact that periodic \(\mathcal {A}\)-free fields have differential structure:

### Lemma 2

Let \(\mathcal {A}\), \(\mathbb {B}\) be linear, homogeneous, differential operators of constant rank with constant coefficients on \(\mathbb {R}^n\) from *W* to *X*, and from *V* to *W*, respectively. Assume that (4) holds. Then for all \(w\in {\text {C}}^\infty (\mathbb {T}_n,W)\) such that \(\mathcal {A}w=0\) and \(\int _{\mathbb {T}_n}w(x){\text {d}}\!x=0\), there exists \(u\in {\text {C}}^\infty (\mathbb {T}_n,V)\) such that \(w=\mathbb {B}u\). Similarly, for all \(w\in \mathscr {S}(\mathbb {R}^n,W)\) such that \(\mathcal {A}w=0\), there exists \(u\in \mathscr {S}(\mathbb {R}^n,V)\) such that \(w=\mathbb {B}u\).

*n*-dimensional torus, identified in an obvious way with (a quotient of) \([0,1]^n\). The Fourier transform is defined as

### Proof

*l*, then \(\mathbb {B}^\dagger (\cdot )\) is \((-l)\)-homogeneous. We can thus differentiate the sum term by term to obtain

We conclude this Section by showing that one can test with compactly supported smooth maps in the definition of \(\mathcal {A}\)-quasiconvexity.

### Corollary 1

*l*and \(\alpha \in [0,1)\), we have

The proof follows standard arguments; in particular we follow [14, Prop. 5.13] and [17, Thm. 4.2] and include the proof for completeness of the present work.

### Proof

*w*be a periodic field as in the definition of \(Q_\mathcal {A}f(\eta )\). We will construct \(v\in {\text {C}}^\infty _c((0,1)^n,V)\) such that

*l*and define \(u_N(x){:}{=}N^{-l}u(Nx)\) for

*N*sufficiently large. This does not change the value of the integral over the cube. Next, let \(\delta >0\) be sufficiently small and truncate to obtain \(u^\delta _N{:}{=}\rho ^\delta u_N\), where \(\rho ^\delta \in {\text {C}}^\infty _c([0,1]^n)\) is such that \(\rho ^\delta (x)=1\) if \(\mathrm {dist}(x,\partial [0,1]^n)>\delta \) and \(|\nabla ^j\rho ^\delta |\leqslant C\delta ^{-j}\) for \(j=0\ldots l\) and some constant \(C>0\). We impose \(\delta N\ge 1\) and leave \(\delta \) to be determined. It follows, for \(c_1\ge 1\) depending on \(\mathbb {B}\) only, that

*f*is bounded by \(M>0\) on \({\text {B}}(0,|\eta |+c_1C\Vert u\Vert _{{\text {W}}^{\mathbb {B},\infty }})\). Hence, if we choose \(\delta \) such that \(\mathscr {L}^n\left( \{x\in [0,1]^n:\mathrm {dist}(x,\partial [0,1]^n)\leqslant \delta \}\right) \leqslant M^{-1}\varepsilon \), we obtain

*w*such that

*w*such that

*u*with \(u_N(x)=N^{-l}u(Nx)\), where

*u*is extended by periodicity to \(\mathbb {R}^n\), the value of the integral does not change. It suffices to choose

*N*large enough so that \(u_N\) has small \({\text {C}}^{l-1,\alpha }\)-norm. Note that for \(j=0\ldots l-1\) we have

*x*,

*y*lie in the same cube \(z_i+[0,N^{-1}]^n\), we have that

*x*,

*y*lie in different cubes, which we label \(Q_x,Q_y\). Let \(\bar{x}\in \partial Q_x\cap (x,y)\), \(\bar{y}\in \partial Q_y\cap (x,y)\), so that \(|x-y|\ge |x-\bar{x}|+|y-\bar{y}|\), \(|x-\bar{x}|,|y-\bar{y}|\leqslant \sqrt{n}N^{-1}\), and all derivatives of \(u_N\) vanish near \(\bar{x},\bar{y}\). Using these facts and the previous step we get

### Remark 2

Using the argument in Corollary 1, one can show for constant rank operators \(\mathcal {A}\) that \(\mathcal {A}\)-quasiconvexity, as defined by Fonseca and Müller in [16, Def. 3.1], coincides with \(\mathcal {A}\)-\(\mathbb {B}\)-quasiconvexity, as introduced by Dacorogna in [12, 13] (to be precise, in the original definition of \(\mathcal {A}\)-\(\mathbb {B}\)-quasiconvexity, the operator \(\mathbb {B}\) is assumed to be of first order, but this is only a minor technical restriction). In this case, it is not difficult to prove that [13, Thm. 4] is essentially unconditional. A proof of this fact will be given elsewhere.

We also have that \(\mathcal {A}\)-quasiconvexity can be defined by integrals over arbitrary domains, instead of cubes.

### Lemma 3

The proof follows from a simple argument in the Calculus of Variations [14, Prop. 5.11].

### Proof

## 4 \(\mathcal {A}\)-free Young measures

We recall the definition of oscillation Young measures, while also giving a simplified variant of the Fundamental Theorem of Young measures.

### Theorem 3

*parametrized measure*\(\varvec{\nu }=(\nu _x)_{x\in \varOmega }\)) such that for all \(f\in {\text {C}}(\varOmega \times \mathbb {R}^d)\) we have that

*generates the Young measure*\(\varvec{\nu }\) (in symbols, \(z_j\overset{\mathbf {Y}}{\rightarrow }\varvec{\nu }\)). We also recall that a sequence \(z_j\) is said to be uniformly integrable if and only if for all \(\varepsilon >0\), there exists \(\delta >0\) such that for all Borel sets \(E\subset \varOmega \), we have that

*p*

*-uniformly integrable*.

### Lemma 4

*x*, i.e.,

The following is an extension of [16, Lem. 2.15]. The first two steps of the present proof are almost a repetition of their arguments, which we include since the original proof only covers first order annihilators \(\mathcal {A}\).

### Proposition 1

*k*,

*l*, respectively, \(\varOmega \subset \mathbb {R}^n\) be a bounded Lipschitz domain, and \(1<p<\infty \). Let \(w_j,w\in {\text {L}}^p(\varOmega ,W)\) be such that

*p*-uniformly integrable.

A Young measure \(\varvec{\nu }\) satisfying the assumptions of Proposition 1 is said to be an \(\mathcal {A}\)*-free Young measure*.

### Proof

By Lemma 4 and linearity we can assume that \(w=0\). We will identify maps defined on \(\varOmega \) with their extensions by zero to full-space without mention. Uniform integrability considerations strictly refer to sequences defined on \(\varOmega \).

*Step I*. We construct *p*-uniformly integrable \(\tilde{w}_j\in {\text {C}}^\infty _c(\varOmega ,W)\) such that \(\tilde{w}_j\rightharpoonup 0\) in \({\text {L}}^p(\varOmega ,W)\), \(\mathcal {A}\tilde{w}_j\rightarrow 0\) in \({\text {W}}^{-k,q}(\mathbb {R}^n,X)\) for some \(1<q<p\), and \(\tilde{w}_j\) generates \(\varvec{\nu }\).

*p*th moment of \(\varvec{\nu }\). It also follows from Theorem 3 that \((\tau _{\alpha _j}w_j)_j\) is

*p*-uniformly integrable.

We now show that \(\tau _{\alpha _j}w_j\) generates \(\varvec{\nu }\). Since \(w_j\) converges weakly in \({\text {L}}^p(\varOmega ,W)\), it converges weakly in \({\text {L}}^1\), hence is uniformly integrable, so that \(\tau _{\alpha _j}w_j-w_j\rightarrow 0\) in measure. It also follows by elementary manipulations that \(\tau _{\alpha _j}w_j-w_j\rightharpoonup 0\) in \({\text {L}}^p\), so that, indeed, \(\tau _{\alpha _j}w_j\) generates \(\varvec{\nu }\) by Lemma 4.

*p*-uniformly integrable, converges weakly to 0 in \({\text {L}}^p\), and generates \(\varvec{\nu }\).

*Step II*. We project \(\tilde{w}_j\) on the kernel of \(\mathcal {A}\) in \(\mathbb {R}^n\) and show that \(\mathbb {P}\tilde{w}_j\) are

*p*-uniformly integrable in \(\varOmega \), converge weakly to zero in \({\text {L}}^p\), and generate \(\varvec{\nu }\). Here the \({\text {L}}^2\)-orthogonal projection operator \(\mathbb {P}\) is given by the multiplier in (7),

*p*-uniformly integrable, we use the idea in [16, Lem. 2.14.(iv)]. We first note, by boundedness of \(\mathbb {P}\) on \({\text {L}}^p\), that

*p*-uniform integrability of \(\tilde{w}_j\). Note that for each fixed \(\alpha \), \(\mathbb {P}\tau _\alpha \tilde{w}_j\) is bounded in \({\text {L}}^r\) for any \(p<r<\infty \), hence is

*p*-uniformly integrable. Let \(\varepsilon >0\). We choose \(\alpha >0\) such that

*j*. It follows that for all such

*E*,

*j*. The second step is concluded.

*Step III*. Using Lemma 2, we can write \(\mathbb {P}\tilde{w}_j=\mathbb {B}u_j\), where \( \hat{u}_j(\xi ){:}{=}\mathbb {B}^\dagger (\xi )\widehat{\mathbb {P}\tilde{w}_j}(\xi ) \), so that \(u_j\in \mathscr {S}(\mathbb {R}^n,V)\). It remains to cut-off \(u_j\) suitably.

*l*, we first note that

By compactness of the embedding \({\text {W}}^{l,p}(\varOmega )\hookrightarrow {\text {W}}^{l-1,p}(\varOmega )\), we have \(u_j\rightarrow u\) in \({\text {W}}^{l-1,p}(\varOmega ,V)\). Since \(\mathbb {B}u_j\rightharpoonup 0\), we have that \(\mathbb {B}u=0\). On the other hand, \(u=\mathscr {F}^{-1}[\mathbb {B}^\dagger (\cdot )]\star (\mathbb {B}u)=0\), so that \(D^{l-m}u_j\rightarrow 0\) in \({\text {L}}^p(\varOmega )\) for \(m=1,\ldots ,l\).

*p*-uniform integrability of \(\mathbb {B}u_j\) and the choice of \(s_j\). Here \(B_m\) is another collection of bi-linear pairings given by the product rule. It remains to conclude that \(\mathbb {B}(\rho _ju_j)\) converges weakly to zero in \({\text {L}}^p(\varOmega ,W)\), is

*p*-uniformly integrable, and generates \(\varvec{\nu }\). The proof is complete. \(\square \)

## Footnotes

## Notes

### Acknowledgements

The author is grateful to Jan Kristensen for introducing him to the problem and for offering insightful comments and helpful suggestions.

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