## Abstract

The paper ‘Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity’ published in *Proc. R. Soc. A* 2013, 469, 20130075 (doi:10.1098/rspa.2013.0075) is clarified. Notably, much more general boundary conditions are given under which sharp lower bounds on the integrals of certain quadratic functions of the fields can be obtained. More precisely, if the quadratic form is *Q**-convex then any solution of the Euler–Lagrange equations will necessarily minimize the integral. As a consequence, strict *Q**-convexity is found to be an appropriate condition to ensure uniqueness of the solutions of a wide class of linear Euler–Lagrange equations in a given domain *Ω* with appropriate boundary conditions.

## 1. Introduction

Quasi-convexity of a function *f*(∇**U**), as introduced by Morrey [1], generalizes the notion of convexity and is important in the direct method of the calculus of variations for establishing the existence in a space of dimension *d* of minimizers of integrals
**U**(**x**) satisfying suitable boundary conditions. The function *f*(∇**u**) is said to be quasi-convex if and only if
*D* of *d* matrices ** ξ**, and for all smooth functions

**(**

*ϕ***x**) which vanish on the boundary of

*D*. Roughly speaking, if a function

*f*(∇

**U**) is quasi-convex then fine-scale oscillations in

**U**do not lead to lower values of

*I*(

**U**): refer to the book of Dacorogna [2] for a good introduction to quasi-convexity. It can be shown that if (1.2) holds for one

*D*then it holds for all

*D*.

Quasi-convexity has found applications in existence theorems for nonlinear elasticity [3], in the theory of shape memory materials [4,5], in bounding the effective moduli of composite materials (e.g. [6–9]), in bounding the yield surface of polycrystalline materials [10], and in bounding the volume of an inclusion in a body using boundary measurements [11]. The most interesting quasi-convex functions are the extremal ones: these lose their quasi-convexity whenever a convex function is subtracted from them. Recently, the first non-trivial extremal quasi-convex function was found [12], and a connection was discovered between extremal quasi-convex functions and extremal polynomials [13].

The definition of quasi-convexity can be generalized [14,15] to functions *f*(∇**U**,∇∇**U**,…,∇^{k}**U**) in which case the function is said to be quasi-convex if and only if
*D* of *ξ*_{1},*ξ*_{2},…,*ξ*_{k−1},** ξ** and for all smooth functions

**(**

*ϕ***x**) which vanish on the boundary of

*D*.

To establish the existence of a minimizer in the direct method of the calculus of variations one must first show that the functional *I*(**U**) is bounded below. Once this is established it follows that there exist minimizing sequences **U**_{1},**U**_{2},…**U**_{n}… such that *I*(**U**_{n}) approaches its infimum as *n* tends to infinity. The next step is to show that there is a subsequence such that **U**_{n} converges in an appropriate sense to some **U**_{0} in the function space. The final step is to show *I*(**U**) is lower semi-continuous, i.e. for any sequence **U**_{n} converging to some **U**_{0} (and in particular for our subsequence) the liminf of *I*(**U**_{n}) as *n* tends to infinity is not less than *I*(**U**_{0}). Quasi-convexity is used to establish this lower semi-continuity.

For quadratic functions *f* things simplify. Given a solution to the Euler–Lagrange equations, then it suffices to show that the functional *I*(**U**) is bounded below to establish that this Euler–Lagrange equation solution minimizes *I*(**U**). Here, in this addendum to the paper [16], we show that the *Q**-convexity of *f* (defined below) ensures that any Euler–Lagrange equation solution minimizes *I*(**U**). Using this fact, we obtain sharp bounds on *I*(**U**) in terms of boundary values. Such inequalities can be regarded as generalizations of the divergence theorem, which expresses the integral of the null-Lagrangian *f*(∇**U**)=*Tr*(∇**U**) in terms of the boundary values of **U**. We believe that the most useful inequalities are going to come from extremal *Q**-convex functions: these lose their *Q**-convexity whenever a quadratic convex function is subtracted from them. The motivation for our work comes from the fact that sharp lower bounds on integrals of the form (1.1) are of interest for obtaining sharp bounds on the electrical or elastic response of inhomogeneous bodies, and these can be used in an inverse manner to derive, for example, bounds on the volume fraction of an inclusion in a body from boundary value measurements [17,11].

In the paper [16], which generalized a result in [11], sharp lower bounds on the integral
**x** lies in a *d*-dimensional space, were obtained for sharply *Q**-convex quadratic functions *f*(**E**(**x**)), where **E** derives some potential **U**, for certain special boundary conditions on **U** and its derivatives. Specifically, the field **E** with *m* components *E*_{r}(**x**), *r*=1,2,…,*m* derives from a real or complex potential **U**(**x**), with ℓ components *U*_{1}(**x**),…,*U*_{ℓ}(**x**) through the equations
*r*=1,2,…,*m*, where *L*_{rq} is the differential operator
*t* in a space of dimension *d* with real or complex valued constant coefficients *A*_{rq} and *f*(**E**(**x**)) is said to be a *Q**-convex function if it satisfies
**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**) . The angular brackets denote volume averages over the unit cell of periodicity. (By all periodic functions, we also meant for all primitive unit cells of periodicity, including parallelepiped-shaped ones.) The function *f* is sharply *Q**-convex if in addition one has the equality
*Q**-special fields. When the quadratic function *f* only depends on ∇**U** then *Q**-convexity is equivalent to rank-one convexity which in turn is equivalent to quasi-convexity [18,19]. More generally, *Q**-convexity and quasi-convexity are not equivalent since the quasi-convexity condition (1.3) only involves the highest order coefficients *f* is quadratic.

In fact, such inequalities can easily be obtained for many more boundary conditions than considered in [16], even for functions that are *Q**-convex but not sharply *Q**-convex. An obvious example is when **E**=∇*U* where *U* is a scalar, and when *f*(**E**)=**E**⋅**E**. If we find some potential *U*(**x**) satisfying the boundary condition that **x**∈∂*Ω*, where **n** is the outward normal to the surface ∂*Ω* of *Ω*. The inequality is of course sharp when **x** in *Ω*.

Generalizing this, we should define a solution field in *Ω* to be any field **S** is the Hermitian *m*×*m* matrix defining the quadratic form *f*
**L**^{†} is the formal adjoint of **L**,
**L** and **L**^{†} are formal adjoints the quantity
**E**(**x**) deriving from a potential **U**(**x**) that matches the appropriate boundary data of the potential **U**(**x**) and its derivatives when *t*>1), provided certain further supplementary conditions hold.

We remark in passing that *Q**-convexity is only a sufficient condition, not a necessary condition for (1.13) to hold. It has the advantage (thanks to the quadratic nature of *f* and the Fourier space arguments of Tartar and Murat [20–22]) that the condition for *Q**-convexity reduces to an algebraic condition and is thus relatively easy to test [16]. Given that a solution to the Euler–Lagrange equations exists, a necessary and sufficient condition for (1.13) to hold is that the operator **L**^{†}**S****L** be positive semidefinite on the space of functions *δ***U** that vanish (and when *t*>1 have appropriate derivatives vanishing) on the boundary of *Ω*. In the case when the quadratic *f* only depends on ∇**U** the positive semidefiniteness of **L**^{†}**S****L** on this space of functions is equivalent to the quasi-convexity of *f* since vanishing boundary conditions on *δ***U** correspond to setting *D*=*Ω*, ** ξ**=0 and

**=**

*ϕ**δ*

**U**in (1.2). More generally, whether

**L**

^{†}

**S**

**L**is positive semidefinite on this space of functions will depend on the shape and size of

*Ω*, and in contrast to

*Q**-convexity will not be that easy to test.

To establish the inequality in (1.13), we first find a parallelepiped *C* that contains *Ω*. Inside *Ω* define *δ***E**(**x**) and *δ***U**(**x**) outside *Ω* so that it is periodic with *C* as a unit cell. In this cell, but outside *Ω*, we set *δ***E**(**x**)=0 and *δ***U**(**x**)=0. The boundary data on *Ω* for the potential **U**(**x**) are chosen so the equation *δ***E**=**L***δ***U** holds weakly across the boundary of *Ω*. Defined in this way, the relation *δ***E**=**L***δ***U** holds in a weak sense, and so the inequality (1.7) implies
*C*| is the volume of *C*. Since *f* is quadratic,
*δ***E**〉. Given a *m*-dimensional constant vector **J**^{0}, we have
**A** is the matrix with elements *A*_{rq}. (The last equality in (1.18) is established using integration by parts and the fact that *δ***U** is zero in the vicinity of the boundary of *C*.) Since this holds for all **J**^{0}, we deduce that
*f*(〈*δ***E**〉) to be zero is that
**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 (1.17) implies the inequality in (1.13). In summary, we have proved the following theorem:

### Theorem 1.1

*When f is Q***-convex and the supplementary condition* (*1.20*) *holds* (*in which* **S** *is the Hermitian m×m matrix entering the quadratic form f and* **A** *is the zeroth-order term in the differential operator L*)

*any solution*

*to the Euler–Lagrange equations*(

*1.10*)

*necessarily minimizes the integral*(

*1.4*)

*with suitable boundary conditions on the admissible fields. The minimum f*

_{0}

*given by*(

*1.12*)

*can be computed in terms of boundary terms using integration by parts.*

One corollary of this analysis is that strict *Q**-convexity is an appropriate condition to ensure uniqueness of solutions to the Euler–Lagrange equations for given boundary conditions. Strict *Q**-convexity means that if (1.13) is satisfied as an equality, then **E**(**x**)=〈**E**〉. If there were two solution fields *Ω*, and the supplementary condition (1.20) held, then the inequality (1.13) would imply
*Ω*, **E** is replaced by *Q**-convexity implies
*C*∖*Ω*, the right-hand side must be zero also and we conclude that inside *Ω*,
*f* is sharply *Q**-convex, rather than strictly *Q**-convex, uniqueness could still hold. Indeed the field *δ***E**(**x**) must be a *Q**-special field which in addition vanishes in *C*∖*Ω*, and there may be no non-zero such fields.

Examples of the solution fields **u** is a complex *m*-component vector, **k** is a complex *d*-dimensional vector, and *h*(*z*) is an analytic function of *z* such that the singularities of *h*(**x**⋅**c**) lie outside *Ω*. The function *h*(*z*) could be chosen to be *e*^{z} in which case *h*(*z*)=1/(*z*−*z*_{0}). Introduce the elements
*m*×ℓ matrix **k** to be a solution of
**e** to be an associated null-vector of **M**(**k**). More generally, a wide variety of solution fields to the Euler–Lagrange equations could be generated by superpositions of fields of the form (1.24).

Similarly, we can bound integrals of the form
**J**(**x**) with *m* real or complex components *J*_{r}(**x**), *r*=1,2,…,*m*, that satisfy the differential constraints
*g*(**J**) which are *Q**-convex in the sense that the inequality
**J** satisfying **L**^{†}**J**=0.

In this context, we should define the solution fields *Ω* to be those fields for which there exists at least one potential *Ω* such that the Euler–Lagrange equations
**T** is the Hermitian *m*×*m* matrix entering the quadratic form *g*:
*δ***J**(**x**) outside *Ω* so that it is are periodic with *C* as a unit cell. In this cell, but outside *Ω*, *δ***J**(**x**)=0. The boundary data on *Ω* for **J**(**x**) are chosen so the equation **L**^{†}*δ***J**=0 holds weakly across the boundary of *Ω*. Defined in this way, the relation **L**^{†}*δ***J**=0 holds in a weak sense, and analogously to (1.17) we have
*δ***J**〉. Let *m*-dimensional constant vectors **E**^{0} expressible in the form **E**^{0}=**L****U**^{0}(**x**) for some polynomial potential **U**^{0}(**x**). Given such a vector **E**^{0}, we have
*δ***J**=0 in a vicinity of the boundary of *C*. Since this holds for all *g*(〈*δ***J**〉) to be equal to be zero is that the range of **T** be a subset of

### Theorem 1.2

*When g is Q***-convex and the range of T is a subset of*

*where*

**T**is the Hermitian m×m matrix entering the quadratic form g and*denotes the vector space of m-dimensional constant vectors*

**E**^{0}

*expressible in the form*

**E**^{0}=

**LU**^{0}(

**)**

*x**for some polynomial potential*

**U**^{0}(

**))**

*x**any solution*

*to the Euler–Lagrange equations*(

*1.31*)

*necessarily minimizes the integral*(

*1.29*)

*with suitable boundary conditions on the admissible fields. The minimum g*

_{0}

*given by*(

*1.33*)

*can be computed in terms of boundary terms using integration by parts.*

Again strict *Q**-convexity is an appropriate condition to ensure uniqueness. Strict *Q**-convexity means that if (1.30) is satisfied as an equality, then **J**(**x**)=〈**J**〉. If there were two solution fields *Ω*, and the ancillary condition that the range of **T** be a subset of *Ω*.

We remark that for all the results to hold it clearly suffices for *f* or *g* to be *C*-periodic functions with a unit cell *C* containing *Ω*, and not necessarily for all periodic functions.

We finish with an example in the context of three-dimensional linear elasticity, where **J** represents a (symmetric 3×3 matrix valued) stress field, satisfying ∇⋅**J**=0, with **T** defined through its action on **J**:
*Q**-convex). It has an inverse **T**^{−1} given by its action
*ϕ*(**x**) can be anything and the components of *Ψ*(**x**) must be harmonic functions. For example, we could take
** Υ** are harmonic functions. The associated stress is

**t**on the boundary of

*Ω*, one can always find potentials

*ϕ*(

**x**) and

**(**

*Υ***x**) such that

**n**is the normal to the boundary, though certainly given potentials

*ϕ*(

**x**) and

**(**

*Υ***x**) one could choose the traction

**t**to be equal to

**J**satisfying the traction condition that

**J**=0) one has the sharp inequality

## Funding statement

The author thanks the National Science Foundation for support through grant no. DMS-1211359.

- Received November 15, 2014.
- Accepted February 6, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.