## Abstract

Subject to suitable boundary conditions being imposed, sharp inequalities are obtained on integrals over a region *Ω* of certain special quadratic functions *f*(**E**), where **E**(**x**) derives from a potential **U**(**x**). With **E**=∇**U**, it is known that such sharp inequalities can be obtained when *f*(**E**) is a quasi-convex function and when **U** satisfies affine boundary conditions (i.e. for some matrix **D**, **U**=**D****x** on ∂*Ω*). Here, we allow for other boundary conditions and for fields **E** that involve derivatives of a variety orders of **U**. We define a notion of convexity that generalizes quasi-convexity. *Q**-convex quadratic functions are introduced, characterized, and an algorithm is given for generating sharply *Q**-convex functions. We emphasize that this also solves the outstanding problem of finding an algorithm for generating extremal quasi-convex quadratic functions. We also treat integrals over *Ω* of special quadratic functions *g*(**J**), where **J**(**x**) satisfies a differential constraint involving derivatives with, possibly, a variety of orders. The results generalize an example of Kang, and the author in three spatial dimensions where **J**(**x**) is a 3×3 matrix-valued field satisfying ∇⋅**J**=0.

## 1. Introduction

The divergence theorem
1.1of Lagrange, Gauss, Ostrogradsky and Green, is one of the most important theorems in mathematics, particularly in applied mathematics. (Here, **n** is the outward normal to the surface ∂*Ω* of the region *Ω*). In one-dimension, it reduces to the fundamental theorem of calculus, and by setting **U**=*w***V**, one obtains the multidimensional version of integration by parts:
1.2which with **V**=∇*v* yields Green's first identity,
1.3Note that the left-hand side of (1.1) can be re-expressed as
1.4where **E**=∇**U** is the gradient of the potential **U**.

This immediately leads to the question: for what functions *f* can one express
1.5in terms of boundary data, when **E** derives from a potential? Such functions are known as null-Lagrangians. One example in two-dimensions of a null-Lagrangian is the determinant of **E** with **E**=∇**U** and **U** being a two-component vector: one has
1.6and we directly see that the quantity on the right-hand side is a divergence. With
1.7another null-Lagrangian is
1.8which again is a divergence. It has been shown by Ball *et al*. [1] that any *C*^{1} null-Lagrangian can be expressed as a divergence, so that the evaluation of (1.5) reduces to an application of the divergence theorem. Necessary and essentially sufficient algebraic conditions to determine whether a (quadratic or non-quadratic) function is a null Lagrangian have been given by Murat [2–4] (see also [5]). In the important case where **E**=∇**U**, *f*(**E**) is a null Lagrangian if and only if it is a linear combination of the subdeterminants (minors) of any order of **E**. For references, see the paper of Ball, Currie and Olver, who also show that when **E**(**x**)=∇^{k}**U**(**x**) there are no new null Lagrangians *f*(**E**) beyond those obtained by applying the result for *k*=1 to the *d*ℓ^{k−1}-component potential ∇^{k−1}**U**(**x**).

Instead of seeking equalities, we search for sharp inequalities and try to find functions *f* and boundary data for which one can obtain sharp bounds on integrals of the form (1.5). Thus, whereas the divergence theorem is an equality relating to the integral of a linear function of **E** to boundary fields, we derive sharp inequalities relating the integral of certain quadratic functions of **E** to boundary fields, for certain boundary fields. This is the sense in which our sharp inequalities generalize the divergence theorem. If **E**=∇**U** and affine boundary conditions **U**=**D****x** on ∂*Ω* are imposed, then one possible value of **E** in the interior of *Ω* is of course **E**=**D** (we follow the usual convention that ∇*U*_{1},∇*U*_{2},…,∇*U*_{ℓ} are the rows of ∇**U**, so that **E** is an ℓ×*d* matrix), and functions *f* that satisfy the inequality
1.9are called quasi-convex and for such functions the inequality is obviously sharp. (Here, |*Ω*| denotes the volume of *Ω*). For a good introduction to quasi-convexity, the reader is referred to the book of Dacorogna [6] and references therein. As discussed there, examples of quasi-convex functions include convex and polyconvex functions. So far, only the quadratic quasi-convex functions have been completely characterized: see Tartar & Murat [7–9]. In particular, the quadratic function *f*(∇**U**) is quasi-convex if and only if *f*(**H**) is non-negative for all rank one ℓ×*d* matrices **H**. One of the contributions of this paper is to show that for quadratic quasi-convex functions that are sharply quasi-convex, sharp bounds on the integral (1.5) can be obtained for a wide variety of boundary conditions, and not just affine ones. We emphasize (see [10], p. 26 and [11]) that it follows from an example of Terpstra [12], and was shown more simply by Serre [13,14] that there are quadratic quasi-convex functions that are not the sum of a convex quadratic function and a null-Lagrangian. (Terpstra's results imply that such quadratic forms exist only if ℓ≥3 and *d*≥3). This indicates that, in general, the sharp bounds we obtain cannot be obtained using null-Lagrangians.

We allow for more general fields **E**(**x**), namely those with *m* components *E*_{r}(**x**),*r*=1,2,…,*m*, that derive from some real or complex potential **U**(**x**), with ℓ components *U*_{1}(**x**),…,*U*_{ℓ}(**x**) through the equations
1.10for *r*=1,2,…,*m*, where *L*_{rq} is the differential operator
1.11of order *t* in a space of dimension *d* with real- or complex-valued constant coefficients *A*_{rq} and , some of which may be zero.

We will find that sharp inequalities on the integral (1.5) can be obtained for certain boundary conditions when *f* is a sharply *Q**-convex function that is quadratic. A *Q**-convex function is defined as a function that satisfies
1.12for all periodic functions **E**=**L****U** that derive from a potential **U**(**x**) that is the sum of a polynomial **U**^{0}(**x**) and a periodic potential **U**^{1}(**x**). Here, the angular brackets denote volume averages over the unit cell of periodicity. (By all periodic functions, we also mean for all primitive unit cells of periodicity, including parallelepiped-shaped ones). In the degenerate case, as pointed out to me by Marc Briane (2013, private communication), there may be potentials that give rise to periodic fields, but are not expressible as the sum of a periodic part and a polynomial part, e.g. with *L*=∂/∂*x*_{1} the potential is not so expressible. However, such potentials are not of interest to us.

The function *f* is sharply *Q**-convex, if, in addition, one has the equality
1.13for some non-constant periodic function that derives from a potential that is the sum of a polynomial and a periodic potential . We call the fields and its potential *Q**-special fields. When the elements *A*_{rq} vanish, and the coefficients are zero for all *h*≠*t*, then *Q**-convexity may be equivalent to quasi-convexity, but otherwise they are not equivalent: see [1] for a precise definition of quasi-convexity.

Note that there could be a sequence of functions **E**_{1}(**x**),**E**_{2}(**x**),**E**_{3}(**x**),…,**E**_{n}(**x**)… each having the same average value **E**^{0}, but not converging to **E**^{0}, such that
1.14in that case, we would call *f* marginally *Q**-convex, but not sharply *Q**-convex unless there existed a field such that the equality (1.13) held. In this paper, we are not interested in marginally *Q**-convex functions that are not sharply *Q**-convex.

In a nutshell, the main argument presented in this paper can be summarized as follows. When **E** is the *Q**-special field , then we can use integration by parts to show that the integral (1.5) can be evaluated exactly in terms of boundary values, for all compact regions *Ω* and in particular for regions *Ω* within a cell *C* of periodicity. Then, we modify **U** within *Ω* while keeping in *C*\*Ω*, and maintaining the boundary conditions, so that **E**=**L****U** holds weakly, including across the boundary ∂*Ω*. The field **E** is extended outside *C* to be periodic with unit cell *C*. Then, the inequality (1.12) must hold. This inequality (and an additional supplementary condition which ensures that ) shows the integral of *f*(**E**) over *Ω* can increase only when **E** is modified in this way. This gives the desired sharp inequality on the integral.

An associated problem, which we are also interested in, is to obtain sharp bounds on integrals of the form
1.15for fields **J**(**x**) with *m* real or complex components *J*_{r}(**x**),*r*=1,2,…,*m*, that satisfy the differential constraints
1.16for *q*=1,2,…,ℓ, where
1.17in which the bar denotes complex conjugation. Observe that the operators in (1.11) and (1.17) are formal adjoints.

Again, using essentially the same argument, we will find that sharp inequalities on the integral (1.15) can be obtained for certain boundary conditions when *g* is a sharply *Q**-convex function that is quadratic. In this setting, a *Q**-convex function *g* is defined as a function that satisfies
1.18for all periodic functions **J** satisfying **L**^{†}**J**=0. The function is sharply *Q**-convex if, in addition, one has the equality
1.19for some non-constant periodic function satisfying . We will call a *Q**-special field.

Presumably the assumption that *f* and *g* are quadratic is not essential, but if they are not, it becomes a difficult task to find those that are sharply *Q**-convex. In addition, there should be some generalization of the theory presented here to allow for *Q**-special fields and which are not periodic, but we do not investigate this here.

The theory developed here generalizes an example given in [15], reviewed in §2, that itself stemmed from developments in the calculus of variations, the theory of topology optimization and the theory of composites: see the books [16–19]. A key component of this is the Fourier space methods developed by Tartar & Murat [7–9] in their theory of compensated compactness for determining the quasi-convexity of quadratic forms.

## 2. An example

To make the general analysis easier to follow, we first review an example given in [15], which was used to obtain sharp estimates of the volume occupied by an inclusion in a body from electrical impedance tomography measurements made at the surface of the body. This example serves to introduce the central arguments and the notion of *Q**-special fields, and reviews the method of Tartar & Murat [7–9] for establishing the quasi-convexity of quadratic forms.

Let us consider in dimension *d*=3 a 3×3 real-valued matrix-valued field **J**(**x**) satisfying ∇⋅**J**=0 *J*_{ij,i}=0. (Thus, following the usual convention, but opposite to the convention adopted in [15,18] its three rows, not columns, are each divergence-free). Alternatively, **J**(**x**) could be regarded as a nine component vector that corresponds to the case *m*=9 and *t*=1 in (1.16) and (1.17). We want to show that for certain fluxes **q**=**J****n** at the boundary ∂*Ω* one can obtain sharp lower bounds on the integral (1.15) when
2.1This special function, introduced in [18], §25.7, is known to be quasi-convex (which, in this case, is equivalent to *Q** convexity), in the sense that the inequality
2.2holds for all periodic functions **J**(**x**) satisfying ∇⋅**J**=0 (as we are dealing with divergence-free fields rather than with gradients, this is the appropriate definition of quasi-convexity) where the angular brackets denote volume averages over the unit cell of periodicity. Note that because *g*(**J**) is not convex, we cannot use Jensens inequality to establish this. Following the ideas of Tartar & Murat [7–9], the reason can be seen in Fourier space where the inequality (by Parseval's theorem, because the function *g* is real and quadratic) takes the form
2.3which holds if *g*(**H**) is non-negative for all rank two real matrices **H**, and in particular for the matrices and which are at most rank two because ∇⋅**J**=0 implies . To prove that *g*(**H**) is non-negative for all rank two real matrices (i.e. rank 2 convex), one first notes that *g*(**H**) is rotationally invariant in the sense that
2.4holds for all rotations **R**. This implies it suffices to check non-negativity with matrices **H** such that **H**[0,0,1]^{T}=0, i.e. of the form
2.5in which case
2.6is clearly non-negative and zero only when *h*_{11}=*h*_{22}, *h*_{12}=−*h*_{21} and *h*_{31}=*h*_{32}=0. More generally, *g*(**H**) will be zero if and only if the rank two matrix **H** has matrix elements of the form
2.7for some constants *α*_{0} and *β*_{0}, and for some vector **k** which is a vector such that **H****k**=0, where *ϵ*_{ijm} is the completely antisymmetric Levi–Civita tensor taking the value +1 when *ijm* is an even permutation of 123, −1 when it is an odd permutation, and 0 otherwise.

This establishes (2.2) and shows one has the equality
2.8for *Q**-special fields of the form
2.9where the are elements of a constant matrix **J**^{0} and *α*(**x**) and *β*(**x**) are arbitrary periodic functions (in this case, has Fourier components of the form (2.7)). This can be established directly by noting that
2.10where
2.11and so (2.8) follows by integration by parts:
2.12

Now suppose **J**(**x**) defined within *Ω* satisfies the boundary condition **q**=**J****n** at the boundary ∂*Ω* where *q*_{ℓ} has components
2.13for some functions *α*(**x**) and *β*(**x**) defined in the neighbourhood of ∂*Ω*, and for some constants . The key point is that one can extend *α*(**x**) and *β*(**x**) beyond this boundary, so that they are periodic (with a unit cell of periodicity *C* containing *Ω*) and define a field given by (2.9). The extension of **J**(**x**) equals on *C*\*Ω* and is defined periodically over the entire domain with periodic cell *C*. The boundary conditions (2.13) ensure that this periodic function satisfies ∇⋅**J**(**x**) in a weak sense and therefore (2.2) implies
2.14where |*C*| is the volume of *C*. In addition, (2.8) implies
2.15Furthermore, because ∇⋅**J**=0 weakly, integration by parts implies that
2.16and thus it follows that . So subtracting (2.15) from (2.14) gives
2.17Using (2.10) and integrating by parts allows us to evaluate the right-hand side:
2.18where **E**_{0} is given by (2.11).

In summary, for boundary fluxes of the form (2.13), we obtain the inequality
2.19which is sharp, it being satisfied as an equality when within *Ω*. In fact, there is a huge range of fields for which one has equality, because one is free to change *α*(**x**) and *β*(**x**) in the interior of *Ω* so long as the right-hand side of (2.13) remains unchanged at the boundary ∂*Ω*.

## 3. Conditions for *Q**-convexity

Here, we solve the problem of characterizing those quadratic functions *f* or *g* which are *Q**-convex. This is a straightforward extension of the ideas developed by Tartar & Murat [7–9] in their theory of compensated compactness for characterizing quadratic quasi-convex functions. Again, the key step is to study the inequality defining quadratic *Q**-convex functions in the Fourier domain, where the differential constraints on the fields become algebraic constraints, and to use Parseval's theorem.

Because *f* and *g* are quadratic, we can let
3.1where the *S*_{rs} or *T*_{rs} are the real- or complex valued elements of some Hermitian *m*×*m* matrix **S** or **T**.

We first consider periodic functions **E**(**x**) and **J**(**x**) and potentials **U**(**x**) that can be expressed in the form
3.2where **U**^{1}(**x**) is periodic with zero average value and **U**^{0}(**x**) is a polynomial with elements
3.3where the coefficients *B*_{q} and are chosen so that
3.4is a constant for *r*=1,2,…,*m*. We let denote the vector space spanned by all constant fields **E**^{0} with elements expressible in the form (3.4).

We expand **E**(**x**), **U**^{1}(**x**) and **J**(**x**) in a Fourier series
3.5where the sum is over all **k** in the reciprocal lattice space (the reciprocal lattice consists of all **k** where the Fourier transform of the periodic functions **E**(**x**), **U**^{1}(**x**) and **J**(**x**) have their natural support so that their primitive unit cell of periodicity is also a cell of periodicity of the wave e^{ik⋅x}, i.e. e^{ik⋅a}=1 for all primitive lattice vectors **a** of these periodic functions). From these expansions, we see that the differential constraints (1.10) or (1.16) imply that for all **k**≠0 in the reciprocal lattice
3.6where and are the matrix elements
3.7of the *m*×*m* matrix and its adjoint , defined for all . In other words, the differential constraints imply that is in the range of and that is in the null-space of , where again these spaces are defined for all and not just those **k** in the reciprocal lattice. We let *Γ*_{1}(**k**) denote the projection onto and *Γ*_{2}(**k**) denote the projection onto . Because these spaces are orthogonal complements it follows that
3.8where **I** is the *m*×*m* identity matrix.

By substituting the Fourier expansion (3.5) for **E**(**x**) in (1.12), we see (by Parseval's theorem) that (1.12) holds if and only if the expression
3.9is non-negative. A necessary and sufficient condition for this to hold for all possible primitive unit cells is that for all non-zero
3.10or equivalently that
3.11where the inequality holds in the sense of quadratic forms. This is an algebraic condition that can be checked numerically, and in some cases analytically.

If **S** is real (and hence symmetric), then (3.9) reduces to
3.12If, in addition, the coefficients *A*_{rq} are real, and the coefficients are real when *h* is even and purely imaginary when *h* is odd, then is real and if and only if the real and imaginary parts of **H** lie in . Thus, in this case, to guarantee (1.12) it suffices that (3.10) holds for real **H**, for all **k**≠0.

Similarly, we look for Hermitian matrices **T** which are *Q**-convex in the sense that (1.18) holds, and a necessary and sufficient condition for this is that
3.13or equivalently that
3.14where again the inequality hold in the sense of quadratic forms. If **T** is real and the coefficients *A*_{rq} are real, and the coefficients are real when *h* is even and purely imaginary when *h* is odd, then it suffices to check (3.13) holds for real **H**.

## 4. Sharply *Q**-convex quadratic functions and their associated *Q**-special fields

Here, we are interested in sharply *Q**-convex functions *f*(**E**) for which one has the equality (1.13) for some non-constant periodic function that derives from a potential that is the sum of a polynomial and a periodic potential . We will see that associated with is a companion field satisfying . As remarked in §5, this allows us to express the integral of over *Ω* in terms of boundary values. Similarly, we are interested in sharply *Q**-convex functions *g*(**J**) for which one has the equality (1.19) for some non-constant periodic function satisfying . (This is basically the Euler–Lagrange equation associated with the minimization of 〈*f*(**E**)〉 when 〈**E**〉 is held constant.) We will see that associated with is a companion field and a companion potential such that . This will allow us to express the integral of over *Ω* in terms of boundary values.

The *Q**-convexity conditions (3.11) and (3.14) are clearly satisfied when **S** and **T** are positive definite, but in this case, one only has equality in (3.10) and (3.13) when **H**=0. For any **k**≠0, let denote the subspace of all *m* dimensional vectors for which one has equality in (3.10), and let denote the subspace of all ℓ dimensional vectors **G** such that . We call *f* sharply *Q**-convex if, for some **k**≠0, contains at least one non-zero vector **H**. Note that if , then
4.1and because from (3.11) *Γ*_{1}**S***Γ*_{1} is positive semi-definite, we deduce that **H** must in fact be a null-vector of this matrix, which implies that
4.2and hence that
4.3Associated with a sharply *Q**-convex *f* are periodic potentials expressible in the form
4.4where for all **k**≠0. The unit cell of periodicity has to be chosen, so the **k** in the reciprocal lattice includes some **k** such that contains at least one non-zero vector.

Let us introduce the companion *Q**-special fields
4.5in which so that for some polynomial potential and where
4.6With these *Q**-special fields satisfy
4.7and
4.8so that (1.13) is satisfied.

Similarly, for all **k**≠0, let denote the subspace of all *m* dimensional vectors for which one has equality in (3.13). We call *g* sharply *Q**-convex if, for some **k**≠0, contains at least one non-zero vector **H**. Analogous to (4.2), (3.14) implies
4.9Because is the range of , there exists a (possibly non-unique) ℓ dimensional vector **G** such that
4.10Now associated with a sharply *Q**-convex *g* are the *Q**-special fields
4.11where
4.12and so that for some polynomial potential . The unit cell of periodicity has to be chosen, so the **k** in the reciprocal lattice includes some **k** such that contains at least one non-zero vector. We choose the periodic potential
4.13so that its Fourier coefficients satisfy
4.14for **k**≠0. With , these *Q**-special fields satisfy
4.15and
4.16so that (1.19) is satisfied.

## 5. Sharp inequalities on the integrals over *Ω*

Because the operators **L** and **L**^{†} are formal adjoints the quantities
5.1can be computed in terms of boundary terms using integration by parts. Now, we show that
5.2for all fields **E**(**x**) deriving from a potential **U**(**x**) that matches the appropriate boundary data of the potential (generally involving both **U**(**x**) and its derivatives when *t*>1) or for all fields **J**(**x**) that match the appropriate boundary data of , provided certain further supplementary conditions hold. First, note that these inequalities are clearly sharp, being attained when or when .

To establish the first inequality in (5.2), we first find a parallelepiped *C* that contains *Ω* and that is formed from an integer number of the primitive unit cells of . (If *Ω* lies inside a primitive unit cell of , then we can take *C* as this primitive cell, but otherwise we need to join a set of these primitive cells together to obtain a parallelepiped that covers *Ω*). We extend **E**(**x**) outside *Ω* so that it is periodic with *C* as a unit cell. In this cell, but outside *Ω*, **E**(**x**) equals . The boundary data on the potential are chosen, so the equation holds weakly across the boundary of *Ω*. (For example, if *m*=ℓ=*d*=1 and , then this would require continuity of both *U* and ∂*U*/∂*x*_{1} at the interface.) We extend the potential **U**(**x**) outside *C* so that is *C*-periodic: if **x**_{0} is any lattice vector then for **x**∈*C* we set
5.3Defined in this way, the relation **E**=**L****U** holds in a weak sense, and so the inequality (1.12) is satisfied, which we rewrite as
5.4where |*C*| is the volume of *C*. Also (4.8) holds, which we rewrite as
5.5so subtracting these equations gives
5.6This inequality, in general, requires us to know 〈**E**〉 and . Given an *m*-dimensional constant vector **J**^{0}, we have
5.7where **A** is the matrix with elements *A*_{rq}. (In establishing the last equality in (5.7), we have used integration by parts and the fact that **U** and are equal in the vicinity of the boundary of *C*). Because this holds for all **J**^{0}, we deduce that
5.8Therefore, a sufficient condition for *f*(〈**E**〉) to be equal to is that
5.9so that the range of **A** is in the null space of **S** (which if **S** is non-singular requires that **A**=0). When this supplementary condition holds, then clearly (5.6) implies the first inequality in (5.2)

To establish the second inequality in (5.2), we first find a parallelepiped *C* that contains *Ω* and that is formed from an integer number of the primitive unit cells of . Then, we extend **J**(**x**) outside *Ω* so that it is *C*-periodic, and within the unit cell, *C* equals outside *Ω*. The boundary data on the field are chosen, so the equation holds weakly across the boundary of *Ω*. Then, analogous to (5.6), we have
5.10This inequality, in general, requires us to know 〈**J**〉 and . Given an *m*-dimensional constant vector so that **E**^{0}=**L****U**^{0}(**x**) for some polynomial potential **U**^{0}(**x**), we have
5.11where the last equality follows from integration by parts, using the fact that in a vicinity of the boundary of *C*. Because this holds for all , we deduce that a sufficient condition for *g*(〈**E**〉) to be equal to is that the range of **T** be a subset of and if this supplementary condition holds then clearly (5.10) implies the second inequality in (5.2).

## 6. An algorithm for generating sharply *Q**-convex quadratic functions and their associated *Q**-special fields and for generating extremal quasi-convex functions

For applications, one needs a way of generating *Q**-convex quadratic functions and their associated *Q**-special fields. This section shows how to do this, but is fairly technical and so can be skipped readers not interested in the details. However, it is important to emphasize that the approach presented here also solves, for the first time, the problem of generating arbitrary extremal quasi-convex functions.

To begin with let us suppose **S** can be expressed in the form
6.1where **V** is Hermitian and positive definite, and *α* is real and positive. The *Q**-convexity of *f* is equivalent to the inequality
6.2holding for all complex *γ* and for all , for all non-zero , as can be established by taking the maximum of the right-hand side over *γ*, thereby recovering (3.10). Rewriting this inequality as
6.3and taking the maximum of the right-hand side over we see that an equivalent condition is that
6.4where, assuming the ℓ×ℓ matrix is non-singular for all **k**≠0,
6.5In the context of quasi-convexity, the condition (6.4) was first derived in [20], stimulated by a result of Kohn & Lipton [21], who found that inequalities like (6.4) were useful to estimate a non-local term entering the Hashin–Shtrikman variational principle. The simpler derivation presented here was suggested by an anonymous referee of a later paper (see also [18], §24.9). If **V**=**I**, then ** Γ**(

**k**) is equal to

*Γ*_{1}(

**k**), the projection onto the range of . More generally,

**(**

*Γ***k**) is defined to be the matrix such that for all vectors

**v**, 6.6implying that 6.7where the inverse is to be taken on the space . (Thus,

**(**

*Γ***k**) is the psuedo-inverse of

*Γ*_{1}(

**k**)

**V**

*Γ*_{1}(

**k**)). To see that (6.6) implies (6.7), we use the fact that and are orthogonal subspaces. Hence, (6.6) implies 6.8Combining these gives 6.9which implies (6.7), because it holds for all

**v**.

Clearly (6.4) is just satisfied if
6.10and in this case, *f* is marginally *Q**-convex. However, if this supremum is attained for some **k**≠0, then *f* is sharply *Q**-convex.

As an example, consider fields of the form
6.11where *U*(**x**) is a scalar potential. Suppose **V**=**I** and
6.12where **t** is a real *d*-component unit vector. Then, ** Γ**(

**k**) is the projection 6.13in which

*k*

^{2}=

**k**⋅

**k**and so 6.14takes its maximum value

*α*=2 when

**k**=

**t**. (It is clear that the maximum occurs when

**t**and

**k**are parallel and then it is just a matter of simple algebra to show (6.14) is less than or equal to 2.) This example shows that the supremum can be attained, and in some cases may be attained only at one non-zero value of

**k**.

Returning to the general case, let denote the set of those **k**≠0 which attain the supremum in (6.10). Then, for , *α*=**v**⋅** Γ**(

**k**)

**v**and the set consists of those

**G**of the form 6.15where

*a*is an arbitrary complex constant. Thus, 6.16satisfies 6.17where 6.18has the property that 6.19implying . So the potential and the

*Q**-special fields and take the form 6.20where and 6.21The reciprocal lattice needs to be chosen so that it includes some to ensure the

*a*(

**k**) in (6.20) are not all zero.

When **V**=**I**, then ** Δ**(

**k**) is equal to the projection

*Γ*_{2}(

**k**) onto the null-space of . More generally,

**(**

*Δ***k**) is defined to be the matrix such that for all vectors

**w**, 6.22implying 6.23where the inverse is to be taken on the space .

We now use the fact that for an appropriate choice of the scalar polynomial *q*(**k**),
6.24is also polynomial in **k**: to see this it suffices to take
6.25although it is better to take the lowest degree polynomial that works. Then, setting
6.26where the are the Fourier components of some scalar periodic function *c*(**x**), we see that
6.27which also gives the *Q**-special fields and . The function *c*(**x**) has to be chosen, so that is zero if . (We do not have to worry about this latter constraint if it happens that consists of all non-zero vectors in ).

We assumed **V** was positive definite, but, in fact, it is easy to check that the derivation goes through if the quadratic form associated with **V** is *Q**-convex. It could also be marginally *Q**-convex but, in this case, one needs to choose **v** so that
6.28because otherwise the supremum in (6.10) is surely infinite. Thus, by induction, successively setting
6.29for *j*=1,2,…,*n* (where **V**_{0} has an associated *Q**-convex quadratic form, **v**=**v**_{j} has to be chosen so that (6.28) is satisfied, and *α*=*α*_{j} is given by (6.10)) one finally reaches a point such that the quadratic function *f* associated with
6.30is an extremal quadratic *Q**-convex function in the sense that one cannot subtract any non-zero positive semi-definite tensor from **S** and retain its *Q**-convexity. In this case, with **V**=**S** the supremum in (6.10) is infinite for all non-zero . We emphasize that this also solves the outstanding problem of generating extremal quadratic quasi-convex functions (or equivalently ‘extremal translations’): see [20] and [18], chapters 24 and 25 for other insights into this problem.

In the end, one obtains *Q**-special potentials of the form
6.31Here, the *n* scalar periodic potentials *c*_{j}(**x**) have Fourier components satisfying
6.32and, when one has made the substitutions (6.29), the **P**_{j} are polynomials given by (6.24) and the consist of those **k** which attain the supremum in (6.10),

Similarly, supposing **T** (the matrix defined by (3.1) that is associated with the quadratic form *g*) can be expressed in the form
6.33where **V** is positive definite, we see that *g* is *Q**-convex if and only if
6.34If we choose
6.35then *g* will be marginally *Q**-convex and sharply *Q**-convex if this supremum is attained for some **k**≠0. Let denote the set of those **k**≠0 which attain the supremum. Then, for , *β*=**w**⋅** Δ**(

**k**)

**w**, and the set consists of those

**H**of the form 6.36where

*b*is an arbitrary complex constant. In this case, we have 6.37where 6.38So the potential and the

*Q**-special fields and take the form 6.39where and 6.40

We next look for the polynomial *q*(**k**) of lowest degree such that
6.41is also polynomial in **k**. Set
6.42where the are the Fourier components of some scalar periodic function *c*(**x**) such that now is zero if . Then, we see that
6.43which also gives the *Q**-special fields and (in the case where **T** is non-singular).

The derivation still goes through, if the quadratic form associated with **V**^{−1} is *Q**-convex or marginally *Q**-convex. In the latter case, one needs to choose **w** so that
6.44because otherwise, from (6.23), the supremum in (6.35) is surely infinite. Thus, by induction, successively setting
6.45for *j*=1,2,…,*n* (where **W**_{0} has an associated *Q**-convex quadratic form, **w**=**w**_{j} has to be chosen so that (6.28) is satisfied, and *β*=*β*_{j} is given by (6.35)) one finally reaches a point such that the quadratic function *g* associated with
6.46is an extremal quadratic *Q**-convex function in the sense that one cannot subtract any non-zero positive semi-definite tensor from **T** and retain its *Q**-convexity. In this case, with **V**^{−1}=**T** the supremum in (6.35) is infinite for all non-zero .

## 7. A generalization

Following the ideas of Cherkaev & Gibiansky [22] (see also [18], §§24.1 and 30.4), the arguments presented here extend directly to integrals of the form
7.1where *h* is quadratic, and the *m*-component field **J**(**x**) still satisfies **L**^{†}**J**=0, whereas the *n*-component field derives from a *p*-component potential according to the relations
7.2for *r*=1,2,…,*n*, where is the differential operator
7.3of order *v* in a space of dimension *d* with real- or complex-valued constant coefficients and .

The function can be expressed in the form
7.4where the matrix **M** is Hermitian. It is chosen to be sharply *Q**-convex, so that for periodic fields and **J**(**x**) satisfying the differential constraints one has
7.5with equality for some non-constant *Q**-special fields and .

## Acknowledgements

The author thanks Hyeonbae Kang for his collaboration on the paper [15] which included the example reviewed in §2, and thanks him and Marc Briane for helpful comments. In addition, the author is also grateful to Ben Eggleton, CUDOS and the University of Sydney for the provision of office space during his visit there and to the National Science Foundation for support through grant no. DMS-1211359.

- Received February 5, 2013.
- Accepted May 30, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.