## Abstract

We determine an improved lower bound for the conductivity of three-component composite materials. Our bound is strictly larger than the well-known Hashin–Shtrikman bound outside the regime where the latter is known to be optimal. The main ingredient of our result is a new quantitative rigidity estimate for gradient fields in two dimensions.

## 1. Introduction

The theory of homogenization permits the determination of the macroscopic properties of microstructured materials. This corresponds to determining an appropriate weak limit of PDEs with rapidly oscillating coefficients. In the case of periodic homogenization of scalar problems, one considers periodic microstructures which can be described by a position-dependent conductivity *σ*∈*L*^{∞}(*Y*), with *Y*=(0, 1)^{2} the unit square. Throughout the paper, *σ* will be scalar so that we will be concerned with locally isotropic media. We assume the ellipticity condition *σ*≥*σ*_{1}>0 a.e. The homogenized (or macroscopic) conductivity can be determined by(1.1)for every ** ξ**∈

^{2}. Here, is the set of

*u*∈

*W*

^{1,2}(

*Y*) such that is

*Y*-periodic (in the sense of traces) and with zero mean over

*Y*. It is easy to see that

*σ*

_{hom}is positive definite.

The theory of composites is aimed at understanding which are the possible values of *σ*_{hom}, with *σ* varying in some subset of *L*^{∞}(*Y*). This rich theory has been the subject of many monographic texts including Bensoussan *et al*. (1978), Sanchez-Palencia & Zaoui (1987), Dal Maso (1993), Jikov *et al*. (1994), Braides & Defranceschi (1998), Cioranescu & Donato (1999), Cherkaev (2000) and Milton (2002). In this paper, we focus on the case where the macroscopic conductivity *σ*_{hom} is proportional to the identity matrix Id,(1.2)The methods of Hashin & Shtrikman (1962) prove that for every isotropic microstructure *σ* (i.e. for every microstructure obeying (1.2)), one has(1.3)By duality, there is also a corresponding upper bound on *s*_{hom}. Since the techniques for treating the dual problem are identical, we treat only the lower bound in the present paper. It is natural to ask whether this bound can be improved, given some class of possible values of *σ*. The answer depends of course on the chosen class.

In problems coming from shape optimization, one typically deals with mixing a finite number of components, with conductivities *σ*_{1}<*σ*_{2}<⋯<*σ*_{N}, and prescribed volume fractions *m*_{1}, *m*_{2}, …, *m*_{N}∈[0, 1], with (Kohn & Strang 1986; Bendsoe & Kikuchi 1988; Cherkaev 2000; Allaire 2002; Milton 2002). The microscopic conductivity *σ* can vary in the set(1.4)In the case of two materials, the bound (1.3) is optimal, in the sense that for any admissible *σ*_{1}, *σ*_{2}, *m*_{1}, *m*_{2}, one hasIn fact, in this case, the infimum is attained (Tartar 1985; Vigdergauz 1994; Astala & Nesi 2003).

For the rest of this paper, we will focus on the case of three materials (*N*=3). This case turns out to be substantially more complex, and one has to distinguish several parameter ranges. In the case *m*_{1}≥*α*(1−*m*_{2}) with(1.5)optimality of (1.3) was proven by Milton (1981) by a construction based on ‘coated spheres’. Later, Lurie & Cherkaev (1985) found different microstructures saturating the bound exactly in the same regime. A corresponding result for anisotropic composites was also given by Milton & Kohn (1988).

In the broader range(1.6)optimality of (1.3) was proven by Gibiansky & Sigmund (2000) with constructions that were inspired by numerical simulations. A simpler proof, using finite-order laminates, which additionally permits treatment of the case of anisotropic composites, was then obtained by Albin (2006; see also Albin *et al*. 2007). The question of optimality of (1.3) in the case that (1.6) does not hold remained open.

The classical lower bound (1.3) was improved by Nesi (1995), who proved that(1.7)The function *Δ*_{N} is however strictly positive only in a subset of the range of parameters for which (1.6) does not hold.

Albin *et al*. (2007) showed that, if (1.6) is not satisfied, then(1.8)for all *σ*∈Σ with *C*^{1} smooth level sets (again in the case *σ*_{hom}=*s*_{hom}Id). The question of the existence and regularity of *σ*∈*Σ* which minimizes *s*_{hom} in general remains open.

In this paper, we make no smoothness assumption and prove that the Hashin–Shtrikman bound (1.3) can be improved whenever (1.6) does not hold, thereby answering completely the question of optimality of (1.3) for the case of three components.

*There is a function Δ such that for all σ*_{i} *and m*_{i}*, and all σ∈Σ such that σ*_{hom}*=s*_{hom} *Id, it holds that*(1.9)*with Δ>0 whenever* *(1.6)* *is not satisfied.*

One key idea, which permits in particular a simple proof of (1.3), is to ‘vectorialize’. This idea appeared in print for the first time in Tartar (1979) and later in a more detailed version in Lurie & Cherkaev (1984) and Tartar (1985). By considering (1.1) for two vectors ** ξ** and , one obtains(1.10)Here , is the vectorial counterpart of and . Transforming the problem into a vectorial one permits the direct use of rigidity of gradient fields. It also displays in a very direct way the strong connection between bounds on conductivities and vectorial variational problems, including the concept of quasiconvexity. One crucial observation is that, the determinant being a divergence, one has(1.11)This condition permits a straightforward proof of (1.3), see §2; in the language of the calculus of variations, it turns out that the Hashin–Shtrikman bound is the best polyconvex bound.

Gradient fields, however, satisfy many more constraints than (1.11). In particular, the improved bound obtained by Nesi and stated in (1.7) can be obtained from the theorem 1 of Alessandrini & Nesi (2001) which, in our case, shows that for any which solvesfor some *σ*∈Σ, the sign of the determinant is prescribed by the boundary data in the sense that if det *E*>0 then det ∇*u*>0 almost everywhere. A precise statement is given in theorem 3.1. Originally, the bound was based on a weaker result established in Bauman *et al*. (2001).

The key idea in the proof of theorem 1.1 is that for all *σ*∈Σ such that the two sides of (1.3) are close, the gradient field ∇*u* entering (1.10) is close to the set (after scaling, see §2 for details)(1.12)(a matrix is anticonformal if it is traceless and symmetric). Using Alessandrini and Nesi's result, we then show that ∇*u* is actually close to the smaller setThe same sets *Z* and *Z*^{b} were already used in proving (1.8) in Albin *et al*. (2007); one key ingredient of that proof was indeed that ∇*u*∈*Z* a.e. implies ∇*u*∈*Z*^{b} a.e. Here we make the estimate quantitative. Precisely, we obtain the following optimal quantitative version of Alessandrini and Nesi's estimate for this pair of sets.

*For all* *with* det *E>0, and all* *, one has*(1.13)*for some c>0 independent of E and u. The scaling of the estimate is optimal.*

A similar result can be obtained for gradient Young measures supported on the set *Z* (see §3*c* for an illustration of the concept of gradient Young measure).

*Let ν be a W*^{1,p}*-gradient Young measure with average E for some p>1 and with supp ν⊂Z*, det *E>0. Then, ν is actually a W*^{1,∞}*-gradient Young measure, and supp ν⊂Z*^{b}.

The study of rigidity results for gradient fields, such as theorem 1.2, or for Young measures, such as theorem 1.3, has been an important theme in the vectorial calculus of variations over many years. This ranges from classical results such as Korn's inequality or Liouville's rigidity theorem, to recent breakthroughs, including, in particular, the optimal quantitative version of Liouville's rigidity derived by Friesecke *et al*. (2002), corresponding two-well results (Chaudhuri & Müller 2004; De Lellis & Székelyhidi 2006), rigidity for the four-gradient problem by Chlebík & Kirchheim (2002), rigidity of conformal matrices (Faraco & Zhong 2005), and the localization result for gradient Young measures by Faraco & Székelyhidi (2006). In all these cases, one deals with gradient fields supported around a set with some specific structure (such as *Z* here), and proves that the gradient constraint makes only a part of the set efficiently usable (*Z*^{b} here).

The work by Faraco & Székelyhidi (2006) contains some results on sets which are very similar to the set *Z* considered here. In particular, they show that if a gradient field or a *W*^{1,2} gradient Young measure is supported on the setfor some *k*>0, then it is supported either in its bounded or in its unbounded component (by taking the closure, the same extends to the limit *k*→∞). Here, *λ*_{1} and *λ*_{2} denote the singular values of *F*. Although the context is similar, their result and their method of proof differ significantly from the present work.

One common key ingredient in the proof of most of the cited rigidity results is that one separates the gradient field in two components: one which solves some ‘good’ equation, which has to be constructed for the purpose, and one which is small. In Friesecke *et al*. (2002) and generalizations, the equation was Laplace equation, and in Faraco & Székelyhidi (2006), it was a nonlinear Beltrami equation. In this paper, it is an almost degenerate linear elliptic equation, see (3.3) or, equivalently, a linear Beltrami equation; the origin of the exponent 2/3 in (1.13) is related to the degeneracy of the equation.

The proofs of these results are given in the following sections. In §2, we prove theorem 1.1; in §3, we prove theorem 1.2; and in §4, we prove theorem 1.3.

## 2. The improved bound

Before proving the improved bound, we present a proof of the bound (1.3) using the techniques developed in Tartar (1979, 1985) and Lurie & Cherkaev (1984).

(The Hashin–Shtrikman bound). *Let 0<σ*_{1}*<σ*_{2}*<σ*_{3} *and m*_{1}*, m*_{2}*, m*_{3}*≥0 be given with m*_{1}*+m*_{2}*+m*_{3}*=1. Let σ∈Σ (see* *(1.4)**) be given such that σ*_{hom}*=s*_{hom}*Id. Then*(2.1)

We begin with the characterization of *σ*_{hom} given in (1.10), and rewrite the integral using (1.11) asThe integrand on the right-hand side can be seen as a quadratic form in ∇*u*. We denote it more compactly bywhere is the position-dependent linear operator defined as(2.2)It is straightforward to check that this operator is self-adjoint and positive semi-definite almost everywhere. Furthermore, *L* has a non-trivial null-space only for those *x* for which *σ*(*x*)=*σ*_{1} (remark 2.1). This notation gives an alternate form for the characterization of *σ*_{hom}. Namely,(2.3)The bound (2.1) arises by removing the constraint that the field ∇*u* is a gradient(2.4)The inequality follows from the observation that for any , ∇*u* is an admissible *F* in the latter minimization problem.

This minimization problem is straightforward to address. We seek to minimize a quadratic functional subject to a linear constraint. Fix . From the Euler–Lagrange equations of (2.4), it is easy to see that a minimizer *F* of the problem satisfies *LF*=*G* a.e. in *Y* for some . In particular, if we take(2.5)then *F* has the correct average and(2.6)It follows easily that *F* is a minimizer for (2.4). Substituting the definition of *E*, (2.5) and (2.6) into (2.4) gives (2.1). ▪

To understand *L*, it is useful to study the operator independently in the setsand to recall that for all , one hasThe operator *L* has eigenvalues *σ*_{i}+*σ*_{1} and *σ*_{i}−*σ*_{1}, with associated eigenspacesrespectively. In particular, *L* is invertible in *Y*_{i} for *i*≠1. In *Y*_{1}, *L* has the non-trivial nullspace . Its eigenvalues are 2*σ*_{1} and 0 with the same eigenspaces as above. Thus, in *Y*_{1}, (2.6) uniquely determines only the projection of *F* onto .

We now turn our attention to the proof of theorem 1.1. The idea is to get a quantitative estimate on the error in passing from (2.3) to (2.4).

*Step 1. Estimate of the HS defect by a good comparison field*. In order to simplify some computations, we shall rescale the problem so that(2.7)Then the optimality condition (2.6) becomes(2.8)

We fix the volume fractions *m*_{1}, *m*_{2}, *m*_{3} and let *σ*∈Σ be an admissible material layout with *σ*_{hom}=*s*_{hom}Id. We now split any into a solution to (2.8) and an error. Specifically, we define by(2.9)with *Y*_{i}={*x*∈*Y*:*σ*(*x*)=*σ*_{i}}. Thus, *F*_{u} is either a multiple of the identity or of the form *β*Id+*H*, with ; we shall later rescale *F*_{u} to a matrix field *G* taking values in *Z*. The subscript *u* makes explicit the dependence of *F*_{u} on *u*.

Since, by construction, *F*_{u} satisfies (2.8) almost everywhere, we have(2.10)The last equality holds becauseFrom (2.8), (2.9) and (2.10), we findThis, together with the characterization (2.3) of *σ*_{hom}=*s*_{hom}Id, shows thatwhereThe improved bound arises by proving a lower bound on .

In *Y*_{i} for *i*≠1, we have (remark 2.1)In *Y*_{1} instead, we have , and from remark 2.1 we see thatFrom these two observations, we find that(2.11)Thus, we have(2.12)whereIndeed, (2.12) holds if *C* is replaced by the larger term in the above inequality. We have chosen *C* independent of the volume fractions *m*_{i}.

*Step 2. Lower bound for* *using the rigidity estimate.* It is convenient to make another affine change of variables to bring the set *Z* into play. We setwhere(2.13)Then , and(2.14)whereIn other words, *α*_{1}=1, *α*_{3}=0 and *α*_{2}=*α* from (1.5). Thus, almost everywhere, with *Z* as in (1.12). Further, , where . In particular, .

We have to establish a lower bound for if (1.6) fails. Our proof is based on comparing the global relation (corresponding to (1.11))(2.15)with local estimates on det ∇*v*. As a warm-up, we first consider the situation ∇*v*−*G*=0 a.e. Then and by theorem 1.2 det ∇*v*≥0 a.e. Since ∇*v*=*G*=0 in *Y*_{3}, we see thatComparing with (2.15) shows that , which is equivalent to (1.6), concluding the proof in this warm-up case.

We now use the quantitative rigidity estimate to derive a lower bound for if (1.6) fails. We may, without loss of generality, only consider those *v* for which (otherwise , and we may choose in (1.9)). We start from *Y*_{i}, for *i*≠1, and writeSince *G*=α_{i}Id in *Y*_{i}, applying Hölder's inequality to the second term we findwith two terms disappearing for *i*=3 since *α*_{3}=0. Equivalently,(2.16)

For *Y*_{1} we proceed differently. Consider any mapping *H*∈*L*^{∞}(*Y*; *Z*^{b}). Then we estimate as before with *H* in place of *G*, noting that |*H*|≤2 and det *H*≥0 a.e.,

Taking the supremum in *H* of the right-hand side, we obtainAfter applying theorem 1.2, we havewhich translates into(2.17)for a universal constant *c*>0. Combining (2.15)–(2.17), we find(2.18)where(2.19)clearly *κ*>0 when (1.6) is violated. We may take the maximum of the second term above with 0 since the left-hand side of (2.18) is clearly non-negative.

Using straightforward estimates on the left-hand side of (2.18), we findwhere *c*^{*} is a constant that depends on *c* and *σ*_{i}. Thus, we have (see the comments before (2.16)) either is sufficiently large already, or else the above inequality holds, which implies that at least one of the terms on the left-hand side is larger than *κ*/4. Thus, by choosing a constant *C*^{*}>0 sufficiently small and recalling (2.12), we haveand the right-hand side can be taken as *Δ* for the theorem; *C* depends only on *c* and *σ*_{i}, but not on *m*_{i}. ▪

By more carefully estimating (2.11) and (2.18), we obtain the following refinement of the theorem.

*Theorem 1.1* *holds with**where α is defined in* *(1.5)*; *γ is defined in* (2.13)*; κ is defined in* *(2.19)**; and c is universal*.

## 3. Rigidity of the set *Z*

### (a) The underlying PDE

In order to illustrate the main ideas in the proof, we first present a special case. Assume that satisfies the differential inclusion(3.1)whereand assume that det *E*>0. The key property of *Z*_{0} is that for any *F*∈*Z*_{0}, there is a number *σ*_{F}>0 such thatIndeed, it suffices to take *σ*_{F}=3 if *F*=Id/2 and *σ*_{F}=1 else. Performing this construction at each point *x*∈*Y*, we obtain a function such that(3.2)(it suffices to take *σ*(*x*)=3 if ∇*u*(*x*)=1/2Id, and *σ*(*x*)=1 otherwise). Since the cofactor of a gradient field is divergence free, taking the divergence (in a distributional sense) of (3.2) shows that *u* solves the elliptic equation(3.3)Since 1≤*σ*≤3, this equation has a unique solution in , hence it uniquely identifies *u*. This permits us to use the following injectivity result.

(From Alessandrini & Nesi (2001), theorem 1) *Let* *be the weak solution of* div(*σ*∇*u*)=0, *where* , *σ=σ*^{T}*,* 1*/k≤σ≤k a.e., for some k≥*1 *and* *. If* det *E*>0*, then*

By (3.3), we can apply this result to our *u*. Then, (3.1) becomes(3.4)In particular, we have shown that any which solves the differential inclusion ∇*u*∈*Z*_{0}, where *Z*_{0} is the unbounded set defined above, is actually Lipschitz, i.e. that only a bounded part of *Z*_{0} is actually used. The key argument used to derive (3.4) has been to construct an elliptic PDE which has the given *u* as a solution, and then to use properties of that PDE. In §3*b*, we shall consider maps which are only approximate solutions of the differential inclusion, and derive an optimal quantitative rigidity for them. In §4, we shall instead consider maps with weaker integrability, and in particular *W*^{1,p}-gradient Young measures, and show that they exhibit the same rigidity. In §3*d*, we show that our rigidity result cannot follow from polyconvexity alone.

### (b) Quantitative estimate

Let *F*: *Y*→*Z* be a measurable map such that dist(∇*u*, *Z*)=|∇*u*−*F*|. Pick some *δ*∈(0, 1). We defineClearly, *F*_{δ}∈*Z*; if *F*_{δ} is of the form *α*Id, then *δ*≤α≤1. At the same time, |*F*_{δ}−∇*u*_{δ}|=(1−*δ*)|*F*−∇*u*|. We define by settingAs above, this choice ensures that(3.5)Now let(3.6)It is clear that , and that(3.7)Given *f*_{δ}, the elliptic equation (3.7) has a unique solution in , and this is *u*_{δ}. Here and below *E*_{δ}=(1−*δ*)*E*+δId. We remark that if Tr *E*≥0 then det *E*_{δ}>0 for all *δ*, otherwise det *E*_{δ}>0 provided (1−*δ*)Tr *E*+δ>0.

We define as the solution ofBy theorem 3.1, we have det ∇*v*_{δ}>0 a.e. in *Y*, which implies(3.8)To see this, let be such that dist(*G*, *Z*)=|*G*−*H*| for some *H*∈*Z*\*Z*^{b} (otherwise there is nothing to prove). We can, without loss of generality, assume *G*=diag(*λ*_{1}, *λ*_{2}), with *λ*_{2}>2+*λ*_{1}>2 (consider the plane of diagonal matrices, where *Z*\*Z*^{b} is a subset of a straight line). We compute , and .

It remains to estimate the distance between ∇*u*_{δ} and ∇*v*_{δ}. To do this, we consider the difference *w*_{δ}=*u*_{δ}−*v*_{δ}. By the linearity of the equation, *w*_{δ} is the unique solution in ofTesting this equation with *w*_{δ} givesLet . Then , and thus(3.9)In order to estimate the norm of , we combine (3.5) and (3.6) to obtainwhich implies that pointwise(3.10)Therefore, by (3.9) and the fact that by definition 1≤*σ*_{δ}≤2/*δ*, we obtainFinally, recalling (3.8),and, for all admissible *δ*,Careful optimization in *δ* (when Tr *E*<0, choose ) givesOptimality of the scaling follows from lemma 3.1. ▪

### (c) Optimality: gradient Young measures and laminates

Optimality is proven constructing a gradient field such that its *L*^{2} distance from *Z* is of order *ϵ*, and its *L*^{2} distance from *Z*^{b} is of order *ϵ*^{2/3}. Instead of writing an explicit test function, it is much simpler to first perform a construction in matrix space, and then to use general tools, in particular the concept of a laminate, to obtain existence of a test function. We therefore start by briefly sketching the necessary background material.

Given a sequence converging weakly in to the affine function , one says that the sequence (*u*_{k}) generates the *W*^{1,p} (homogeneous) gradient Young measure if(3.11)for all . By simple truncation arguments, the same will automatically hold for all continuous *f* satisfying for some *q*<*p* the growth condition . Taking *f* to be the identity mapping, one sees that the average of *ν* is *E*. It can be shown that every weakly converging sequence generates such a measure. The Young measure gives the *volume distribution* of the values of the gradient (see Pedregal (1997) and Müller (1999) for details).

It is a very difficult question to decide which measures on can be generated this way. There is, however, a large class of measures that can be easily generated, the class of so-called laminates. A laminate of zeroth order with average *F* is a Dirac delta, i.e. it has the form *ν*=δ_{F} and is generated by the (constant) sequence *u*_{k}(*x*)=*Fx*. A laminate of *n*th order is defined inductively from a laminate of order *n*−1 by replacing each of the terms *cδ*_{F} by a sum , where , *λ*∈[0, 1] and rank (*F*_{1}−*F*_{2})≤1. One says that the matrix *F* has been split into *F*_{1} and *F*_{2}. For example, first-order laminates have the form , with *F*_{1}−*F*_{2}=*a*⊗*n*, and are the limits of the gradient distributions of the mapsThe Lipschitz function is defined by *χ*(0)=0, *χ*′(*t*)=1 if *t*∈(*z*, *z*+λ) and *χ*′(*t*)=0 if *t*∈(*z*+λ, *z*+1), for . For large *k*, the gradients ∇*u*_{k} oscillate on a fine scale between the values *F*_{1} and *F*_{2}, with average *F*=λ*F*_{1}+(1−*λ*)*F*_{2}. As *k*→∞, the sequence *u*_{k} converges weakly-^{*} in *W*^{1,∞} to the affine function ; the function *f*(∇*u*_{k}) converges weakly-^{*} in *L*^{∞} to . Refining this argument, one can show that mixtures are always possible between rank-one connected matrices, hence that all laminates as defined above are attainable as weak limits of gradients (for details, see Dacorogna (1989), Müller (1999) and Dolzmann (2003)).

*For any ϵ>*0*, there exists* *with *det *E>*0 *and* *such that**and*

If *ϵ*≥1 one can take ; therefore, we can assume *ϵ*<1. We shall first construct a laminate which obeys the mentioned inequalities, and then a test function *u*. The laminate is supported on diagonal matrices. Let 0<*δ*<1. We consider *F*_{1}=diag(*δ*, 2−*δ*)∈*Z*^{b}; *E*=diag(*δ*/2, 2−*δ*); *F*′=(−*δ*, 2−*δ*). Thenis a laminate with average *E*. We now split *F*′ into the two matrices, *F*_{2}=diag(–*δ*, 2+*δ*) and *F*_{3}=diag(−*δ*, 0), to obtain the laminate (figure 1)Since *F*_{1}, *F*_{2}∈*Z*, we haveHowever, *F*_{2}∉*Z*^{b}. Therefore,Let now *δ*=*ϵ*^{2/3}, and pick a sequence *u*_{k} which generates the laminate *ν*. ThenTherefore, taking *k* sufficiently large the lemma is proven. ▪

### (d) Rigidity does not hold for polyconvex measures

We finally show that our rigidity result uses in a substantial way the fact that we are dealing with a gradient field. In particular, the same cannot be proven just arguing with null Lagrangians, i.e. on the basis of polyconvex bounds. To prove this, we exhibit a polyconvex measure which violates the statement. By polyconvex measure, we mean a measure *ν* on which obeys the equivalent for measures of (1.11), i.e. such that the determinant of the average is the average of the determinant. In particular, let *E*=diag(1/7, 1/3) andIt is clear that supp *ν*⊂*Z*, and that det *E*>0. Easy computations prove thathence *ν* is a polyconvex measure with average *E*. However, it is apparent that supp *ν*⊄*Z*^{b}.

## 4. Gradient Young measures supported on Z

The key point in the proof of theorem 1.3 is that a *W*^{1,p}-gradient Young measure supported on *Z* necessarily has much better integrability, precisely, it is a *W*^{1,q}-gradient Young measure for any *q*≥1. Let us suppose that *ν* is a *W*^{1,q}-gradient Young measure for some *q*>2 and which is generated by a sequence {*u*_{j}}. Taking *f*(*F*)=dist^{2}(*F*, *Z*) in (3.11) gives . Theorem 1.2 implies that as well. Finally, applying (3.11) now for *f*(*F*)=dist^{2}(*F*, *Z*^{b}) shows that supp *ν*⊂*Z*^{b}. Moreover, the boundedness of supp *ν* then implies by a truncation argument that *ν* is in fact a *W*^{1,∞}-gradient Young measure.

The proof of sufficient integrability relies on the following theorem due to Boyarskii (1957) (see also Meyers (1963) for the generalization in *n* dimensions).

(From Boyarskii (1957) and Meyers (1963)) *Let k≥1 be given. There exist constants p*_{c}*(k)<*2*<q*_{c}*(k) such that if p*_{c}*(k)<p<q*_{c}*(k), if* *satisfies*(4.1)*and if* *, then a unique solution to the equation**exists and* *. The exponents p*_{c}*(k) and q*_{c}*(k) obey*

Theorem 1.3 now follows from the following higher integrability result, which for notational simplicity is formulated in *W*^{1,3} (any exponent *p*>2 would do).

*Let ν be a homogeneous W*^{1,p}*-gradient Young measure with supp ν⊂Z and p>1*. *Then ν is also a W*^{1,3}*-gradient Young measure.*

By our initial discussion, we may assume *p*≤2. We begin by choosing *k*>1 such that *p*_{c}(*k*)<*p*<3<*q*_{c}(*k*), where *p*_{c}(*k*) and *q*_{c}(*k*) are defined in theorem 4.1. Let {*u*^{j}}⊂*W*^{1,p} be a generating sequence for *ν*. By a standard localization argument, we may assume(4.2)where is the average of *ν*.

Now we proceed analogously to the proof of theorem 1.2, with more care to the growth. Let *q* be such that *p*_{c}(*k*)<*q*<*p*, and let *δ* be such thatThis ensures that *σ*_{δ}, which we shall construct, will satisfy (4.1). For each *j*, we define , , and as in the proof of theorem 1.2. Then (3.10) givesand since *q*<*p*, (3.11) shows

Now consider the two PDEs(4.3)and(4.4)Since, by construction, *p*_{c}(*k*)<*q*<3<*q*_{c}(*k*), theorem 4.1 guarantees the existence of a unique solution to both equations. Furthermore, since and recalling (4.2), the solutions to (4.3) and (4.4) necessarily coincide. The estimate in theorem 4.1 applied to (4.4) implies that , hence the sequence is bounded in *W*^{1,3} and it generates a *W*^{1,3}-gradient Young measure. Moreover, by theorem 4.1 applied to (4.3), we haveCombining this with (3.11) shows that the gradient Young measure generated by coincides with *ν*. ▪

## Acknowledgments

The work of N.A. was supported by the NSF Postdoctoral Research Fellowship award no. 0603611, S.C. was supported by the DFG through SPP 1253 and V.N. was supported by COFIN 2006 ‘Omogeneizzazione e metodi variazionali in matematica applicata’.

## Footnotes

- Received February 12, 2007.
- Accepted May 10, 2007.

- © 2007 The Royal Society