Direct methods in the calculus of variations for differential forms

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R E S E A R C H

Open Access

Direct methods in the calculus of variations

for differential forms

Tingting Wang

*

_{, Gejun Bao and Guanfeng Li}

*_{Correspondence:}

[email protected]

Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China

Abstract

The purpose of this paper is to establish the general theory of the direct methods to functionalsIdeﬁned on the Grassmann algebra employing the classical approaches. In this paper, various notions of convexity conditions for weak lower semicontinuity of

Iare discussed, and existence theorems for minimizers ofIare obtained. Lastly, we present some examples to illustrate our main results.

MSC: 35A15; 58A10

Keywords: weak lower semicontinuity; diﬀerential forms; variational problem; direct methods; existence theorem

1 Introduction

Differential forms as invaluable tools have been available and applicable to various fields of study, see [–], while the systematic investigation of variational theory for differential forms has been less studied. Direct methods is a classical and fundamental method, used in variational problems, which has been widely applied for solving differential equations that possess variational structure, see [–]. More precisely, direct methods work for the minimization problem for certain energy functionalsI, whose critical points are usually related to solutions of certain differential equations. Various conditions for the existence of minimizers ofIhave been greatly studied, see [–].

This paper is intended to generalize and apply the classical direct methods in variational problems for differential forms with an aim to enlarge the range of applications of the variational principle. Our work is motivated by Iwaniec and Lutoborski [], who first used differential forms to express variational problems.

Letbe a nonempty, bounded, open subset ofRn_{. Suppose that integers }_≤_l

, . . . ,lm≤

nandW:∧l× · · · × ∧lm→ ∧n_{is a continuous function satisfying the growth condition}

Wξ, . . . ,ξm≤Cξp+· · ·+ξmpm (.)

for allξ = (ξ_{, . . . ,ξ}m₎_{∈ ∧}l× · · · × ∧lm_{, where  <}_p

, . . . ,pm<∞. Writep= (p, . . . ,pm).

Giveng= (g_{, . . . ,}_gm₎_∈_W,p_(,_∧l× · · · × ∧lm_{), we consider the integral}

I[F] =

W(g+dF) =

Wg+df, . . . ,gm+dfm (.)

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deﬁned for vectorial diﬀerential forms

F=f, . . . ,fm∈W,p,∧l–× · · · × ∧lm–_,

wherep= (p, . . . ,pm) is imposed by the growth condition (.). Let

M=F|F∈W_,p,∧l–× · · · × ∧lm–_.

We consider the existence of the minimizer ofIinM, that is, we want to prove that

inf

MI[F]

is achieved.

In particular, associated with the variational kernelW(ξ) =|ξ|p_{, (.) is reduced to the}

p-Dirichlet integral

I[u] =

|g+du|p,

and the minimizers of the p-Dirichlet integral are exactly the weak solutions to the p-harmonic equation

d∗|g+du|p–(g+du)= .

2 Notation and preliminary results

This section is devoted to the notation of the exterior calculus and a few necessary pre-liminaries. Readers can refer to [–] for more details.

We denote by∧l₌_∧l₍_Rn_{) the space of}_l_{-covectors in}_Rn_{, and the direct sum}

∧Rn₌ n

l= ∧l_Rn

is a graded algebra with respect to the wedge product∧. We shall make use of the exterior derivative

d:C∞,∧l→C∞,∧l+

and its formal adjoint operator

d∗= (–)nl+∗d∗:C∞,∧l+→C∞,∧l,

known as the Hodge codiﬀerential, where the symbol∗denotes the Hodge star duality operator. Note that each of the operatorsdandd∗applied twice gives a zero.

LetC∞(,∧l_{) be the class of inﬁnitely diﬀerentiable}_l_{-forms on}_⊂_Rn_{. Since}_{is a}

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Near this boundary point, every diﬀerential formω∈C∞(,∧l_{) can be decomposed as}

ω(x) =ωT(x) +ωN(x), where

ωT(x) =

≤i<···<il<n

ωi,...,il(x)dxi∧ · · · ∧dxil

and

ωN(x) =

≤i<···<il=n

ωi,...,il(x)dxi∧ · · · ∧dxil

are called the tangential and the normal part ofω, respectively. Now, the duality between dandd∗is expressed by the integration by parts formula

du,v=

u,d∗v (.)

for allu∈C∞(,∧l_{) and}_v_∈_C∞_(,_∧l+_{), provided}_u

T=  orvN = . The symbol ·,·

denotes the inner product,i.e., letα=_IαI(x)dxI andβ =

IβI(x)dxI, then α,β=

IαI(x)βI(x).

Due to (.), extended deﬁnitions fordandd∗can be introduced as the introduction of weak derivatives.

Deﬁnition .[] Suppose thatω∈L_loc(,∧l) andv∈L_loc(,∧l+). If

ω,d∗η=

v,η

for every test formη∈C∞_(,∧l+), we say thatωhas generalized exterior derivativevand writev=d˜ω.

The notion of the generalized exterior coderivatived˜∗can be deﬁned analogously.

Deﬁnition .[] Suppose thatω∈L

loc(,∧l) andv∈Lloc(,∧l–). If

ω,dη=

v,η

for every test formη∈C_∞(,∧l–_{), we say that}_ω_{has generalized exterior coderivative}_v

and writev=d˜∗ω.

Remark[] (i) The generalized exterior derivative has many properties similar to those of the weak derivative. For example, (i) if the generalized exterior derivative exists, it is

unique; (i) ifωis diﬀerentiable in the conventional sense, then its generalized exterior

derivatived˜ωis identical to its classical exterior diﬀerentialdω. Analogous results hold for generalized exterior coderivative.

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(iii)dandd˜have analogous expressions,i.e., forω(x) =_IωI(x)dxI, we have

dω(x) =

I n

k=

∂ωI

∂xk

dxk∧dxI

and

˜ dω(x) =

I n

k= ˜ ∂ωI

∂xk

dxk∧dxI,

where∂denotes the ordinary derivative and∂˜ the weak derivative. So next, we usedto represent the action instead ofd˜, similar ford∗andd˜∗.

Next, we present brieﬂy some spaces of diﬀerential forms

Lq,∧l-the space ofl- formsωwith coeﬃcients inLq();

Lq_,∧l- the space ofl-formsωsuch that∇ωis a regular

distribution ofLq,∧l;

W,q,∧l- the Sobolev space ofl-formsωdeﬁned byLq,∧l∩Lq_,∧l,

which are used throughout this paper.

Finally, we introduce the deﬁnition of weak convergence for sequences in spaces of dif-ferential forms as needed. Throughout the paper, let  <q,q<∞be the Hölder conjugate pair,q+q=qq.

Deﬁnition .[] We say thatϕjweakly converges toϕinLq(,∧l) if

ϕj∧h→

ϕ∧h

wheneverh∈Lq_(,_∧n–l_{) and write}_ϕ

j ϕinLq(,∧l).

We also need some properties of spaces of diﬀerential forms.

Proposition .[] For <q<∞,Lq_(,_∧l₎_{is reﬂexive}_.

Proposition .[] ϕj ϕin W,q(,∧l)if and only ifϕj ϕin Lq(,∧l)and∇ϕj

∇ϕin Lq_(,_∧l_),_where_∇_ϕ_{= (}∂ϕ ∂x, . . . ,

∂ϕ

∂xn),and the partial diﬀerentiation is applied to the

coeﬃcients ofϕ.

3 Weak lower semicontinuity

In many variational problems, weak lower semicontinuity is an essential condition for the existence of minimizers, using the minimization method. This will motivate the study of various notions of convexity conditions. In this section, we study the conditions for weak lower semicontinuity of the integral functionalI[F] in the Sobolev spaceW,p(,∧l–×

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Deﬁnition . Iis called weak lower semicontinuous onW,p_(,_∧l–× · · · × ∧lm–_{) if}

I[F]≤lim inf ν→∞ I[Fν]

wheneverFν= (f

ν, . . . ,fνm)F= (f, . . . ,fm) inW,p(,∧l–× · · · × ∧lm–). 3.1 The convex case

Theorem . For a given g= (g, . . . ,gm)∈W,p_(,_∧l× · · · × ∧lm_),_{let W}≥__be

contin-uous and convex.Then I is weakly lower semicontinuous on W,p_(,_∧l–× · · · × ∧lm–_). Proof Suppose thatFν= (f

ν, . . . ,fνm)F= (f, . . . ,fm) inW,p(,∧l–× · · · × ∧lm–), we then need to show

I[F]≤lim inf ν→∞ I[Fν].

We may assume that{I[Fν]}is ﬁnite and convergent.

SincedFνdFinLp_(,_∧l× · · · × ∧lm_{), then, according to Mazur lemma [], for any}

ν∈N, there exists a sequence{Fl}l≥νof convex linear combinations

Fl₌ l

ν=ν

λl_νdFν, ≤λl_ν≤,

l

ν=ν

λl_ν= , l≥ν

such thatFl_→_dF₍_l_{→ ∞}_{) in}_Lp_(,_∧l× · · · × ∧lm_{) and pointwise almost everywhere.}

SinceWis continuous, we have

W(g+Fl)→W(g+dF)

asl→ ∞. On the other hand, we have by the convexity ofWthat

W(g+Fl) =W _l

ν=ν

λlν(g+dFν)

≤

l

ν=ν

λl_νW(g+dFν).

Therefore, we obtain from Fatou lemma that

W(g+dF) =

lim

l→∞W

g+Fl

=

lim inf

l→∞ W

g+Fl

≤lim inf

l→∞

Wg+Fl

≤lim inf

l→∞ l

ν=ν

λl_ν

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≤lim inf

l→∞ ν≤supν≤l

W(g+dFν)

= lim

l→∞infk≥lν≤supν≤k

W(g+dFν)

= lim

l→∞ν≤supν≤l

W(g+dFν)

= sup ν≥ν

W(g+dFν).

Letν→ ∞in the inequality above, we obtain

W(g+dF)≤ lim ν→∞_νsup_≥_ν



W(g+dFν)

=lim sup ν→∞

W(g+dFν)

=lim inf ν→∞

W(g+dFν),

that is,I[F]≤lim infν→∞I[Fν]. The theorem follows.

3.2 The non-convex case

This section considers the case whenWis not convex. Fruitful results have been obtained for the weak lower semicontinuity of integral functionalsIwhenWfails to be convex, see [–]. We consider the counterpart in the case of diﬀerential forms.

.. Quasiconvex

Deﬁnition . LetW:∧l× · · · × ∧lm→ ∧n_{be continuous. Then}_W_{is said to be}

quasi-convex if

D

Wξ+dφ(x)

≥

D

W(ξ) (.)

holds for every bounded open setD⊂Rn_with_mea_∂_D_{=  for every ﬁxed}_ξ

∈ ∧l× · · · × ∧lm_{and for all}_φ∈_W,p

 (,∧l–× · · · × ∧lm–).

Remark (i) This deﬁnition is actually the so-calledW,p_-_quasiconvex_{, see []. For the}

general notion ofquasiconvex(relatively isW,∞_-_quasiconvex_{) and other results, readers}

can refer to [, , , ] and references therein.

(ii) From Jensen’s inequality, we see that every convex function is quasiconvex. In fact, we have by Jensen’s inequality that

 ||

Wξ+dφ(x)

≥W  ||

ξ+dφ(x)

=W

ξ+

 ||

dφ(x)

.

Sinceφ∈W_,p(,∧l–× · · · × ∧lm–_{), the Stokes theorem [] yields}

dφ(x) =

∂

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Therefore, we have

 ||

Wξ+dφ(x)

≥W(ξ).

(iii) Quasiconvexity implies that the functionWis convex with respect to eachξi∈ ∧li_,

i= , . . . ,m, which together with (.) yields a Lipschitz-type condition

W(ξ+ζ) –W(ξ)≤C(p)

m

i=

ζiξi+ζipi–. (.)

Lemma . Letξ∈ ∧l×· · ·×∧lmand W be quasiconvex satisfying the growth condition

(.).Suppose that

Fν=fν, . . . ,fνm



in W,p(,∧l–× · · · × ∧lm–_)._Then

lim inf ν→∞

W(ξ+dFν)≥

W(ξ). (.)

Proof Letν={x∈:dist(x,∂) > /ν}, then|–ν| → asν→ ∞. We choose cut-oﬀ functionsην∈C∞_() such that ≤ην≤,ην|ν=  and|∇ην| ≤Cν<∞. Let

φν(x) =ην(x)Fν(x),

thenφν∈W_,p(,∧l–× · · · × ∧lm–) and dφν(x) =dην(x)∧Fν(x) +ην(x)dFν(x).

We takeφνas test functions in the deﬁnition of quasiconvexity to obtain

W(ξ)≤

Wξ+dφν(x)

=

ν

Wξ+dFν(x)

+

\ν

Wξ+dφν(x)

=

Wξ+dFν(x)

+

\ν

Wξ+dφν(x)

–Wξ+dFν(x)

. (.)

SinceW is continuous, thus,W is bounded on bounded sets, together with that|– ν| → asν→ ∞, we then have the last integral in the right hand side of (.) converges to zero asν→ ∞. Therefore, we have

W(ξ)≤lim inf ν→∞

W(ξ+dFν),

which is the desired result.

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Proof Suppose thatFν= (f

ν, . . . ,fνm)F= (f, . . . ,fm) inW,p(,∧l–× · · · × ∧lm–), we then need to show that

I[F]≤lim inf ν→∞ I[Fν].

We may assume that{I[Fν]}is ﬁnite and convergent,i.e.,

lim inf

ν→∞ I[Fν] =νlim→∞I[Fν].

LetQk⊂be disjoint cubes parallel to the axes, and edge-length is /k. Denotek=

Qk, and we then have that

|–k| →

ask→ ∞. Let

dF(x) =

I

(dF)I(x)dxI,

with (dF)I(x)∈Lp(,∧l× · · · × ∧lm). Forx∈Qk, set

(ξk)I=

 |Qk|

Qk

(dF)I(y)dy.

Then (ξk)I is constant onQkand writeξk=

I(ξk)IdxI∈ ∧l× · · · × ∧lm. Then we have dF(x) –ξk_p_,_Q_k=

I

Qk

(dF)I(x) – (ξk)I p

dx



p

−→

ask→ ∞. Therefore, for arbitraryε> , we can chooseklarge enough such that

|–k|<ε and dF(x) –ξk_p_,_Q

k<ε.

Note that Fν = (f

ν, . . . ,fνm)F = (f, . . . ,fm) in W,p(,∧l–× · · · × ∧lm–), we have Fν,p,is uniformly bounded. We then consider

I[Fν] –I[F] =

W(g+dFν) –W(g+dF)

J+J+J+J, (.)

where

J=

\k

W(g+dFν) –W(g+dF),

J=

k

Qk

W(g+dFν) –W(g+ξk+dFν–dF)

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J=

k

Qk

W(g+ξk+dFν–dF) –W(g+ξk)

,

J=

k

Qk

W(g+ξk) –W(g+dF)

.

ForJ: SinceW≥,W is continuous andis bounded, we have

J =

\k

W(g+dFν) –W(g+dF)≥–

\k

W(g+dF)

≥–sup

|W| · |–k|

≥–εsup

|W|= –εC; (.)

ForJ: From (.) and the Hölder inequality, we see that

|J| ≤

k

Qk

W(g+dFν) –W(g+ξk+dFν–dF)

≤C(p)

k m

i=

Qk

dfi–ξ_kigi+df_νi+dfi–ξ_kipi–

≤C(p)

k m

i=

Qk

dfi–ξ_kigipi–

+dfi–ξ_kidf_νipi–

+

Qk

dfi–ξ_kipi

≤C(p,m)gpp–,Qk+dFν

p–

p,Qk

dF–ξkp,Qk+dF–ξk

p p,Qk

≤εC; (.)

ForJ: Applying Lemma . forξ=g+ξk, yields

lim inf ν→∞

Qk

W(g+ξk+dFν–dF)≥

Qk

W(g+ξk),

which implies that

lim inf

ν→∞ J≥; (.)

ForJ: It follows from (.) and the Hölder inequality that

|J|=

k

Qk

W(g+ξk) –W(g+dF)

≤C(p)

k m

i=

Qk

dfi–ξ_kigi+dfi+ξ_kipi–

≤C(p,m)gpp–,Qk+dF

p–

p,Qk+ξk

p–

p,Qk

dF–ξkp,Qk

≤εC. (.)

Substituting (.)-(.) into (.), we have by lettingν→ ∞that

lim inf

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Letε→ in the above inequality, we obtain

lim inf

ν→∞ I[Fν]≥I[F].

The theorem follows.

.. Polyconvex

Polyconvexity is also an important condition for the weak lower semicontinuity ofI, which is quasiconvex, but not necessarily convex. We give below the deﬁnition of polyconvex, and we want to establish the weakly lower semicontinuity ofI onW,p_(,_∧l–× · · · ×

∧lm–_{), provided}_W_{is polyconvex.}

Deﬁnition . LetN=N(m) be the number of elements of the set

T(ξ) =ξi∧ · · · ∧_ξik|_≤_i

<· · ·<ik≤m,k= , . . . ,m

forξ= (ξ_{, . . . ,}_ξm₎_{∈ ∧}l× · · · × ∧lm_{. A function}_W_:∧l× · · · × ∧lm→ ∧n_{is called}

poly-convex if there existsg:RN→ ∧nbe convex in each of its variables, such that

W(ξ) =gT(ξ).

Proceeding with the proof in much the same way as [], the equivalent deﬁnition for polyconvexity is given by Iwaniec and Lutoborski.

Deﬁnition .[] A continuous functionW:∧l× · · · × ∧lm→ ∧n_{is called polyconvex}

if

W(ξ) –W(ζ)≥

m

k=≤i<···<ik≤m

Ai···ik(ζ)∧

ξi∧ · · · ∧_ξik_–_ζi∧ · · · ∧_ζik_, _(.)

where ξ= (ξ_{, . . . ,}_ξm_{) and}_ζ _{= (ζ}_{, . . . ,}_ζm_{) are variable points from} _∧l× · · · × ∧lm _and

Ai···ik:∧

l× · · · × ∧lm→ ∧n–l_,_l₌_l

i+· · ·+lik, are functions ofζ only.

Remark (i) Note that (.) implies convexity ofW in each variable, and we have from the deﬁnitions that

Convex ⇒ Polyconvex ⇒ Quasiconvex.

(ii) The non-zero terms in the right side of (.) imply that

k≤m≤l+· · ·+lm≤n.

(iii) ForF= (f_{, . . . ,}_fm₎_∈_W,p_(,_∧l–× · · · × ∧lm–_{), and assume that}

fij₌

I

fij

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then

dfi∧ · · · ∧_dfik

=

I n

j=

∂(fi₎ I

∂xj

dxj∧dxI

∧ · · · ∧

I n

j=

∂(fik₎

I

∂xj

dxj∧dxI

=

Iv,...,Ivk

j,...,jk

(–)∂((f

i₎ Iv, . . . , (f

ik₎

I_vk)

∂(xj, . . . ,xjk)

dxj∧dxIv∧ · · · ∧dxjk∧dxI_vk, (.)

where (–)=  or (–). Observe that the coeﬃcients of (.) arek×ksubdeterminant of the Jacobian off :→RN,N=Cl–

n +· · ·+Cnlm–, where

f =f_I

, . . . ,

f_I

C_nl–

, . . . ,fm_I

, . . . ,

fm_I

Clm_n –

.

We apply the following lemma.

Lemma .[, Corollary .] LetJk(Du)be any k×k subdeterminant,and let q>k be

a number.Let{uj}be any sequence weakly convergent to u in W,q(,RN)as j→ ∞.Then Jk(Duj)Jk(Du)weakly in L

q k_().

By adding the condition

˜

pmin{p, . . . ,pm}>m,

we have that fors= , . . . ,m,

dfi

ν ∧ · · · ∧dfνisdfi∧ · · · ∧dfis

inLp˜s(,∧li×· · ·×∧lis_{) as}_ν→ ∞_when_Fν_F_in_W,p_(,∧l–×· · ·×∧lm–_{). Therefore,}

we obtain the following theorem for the weakly lower semicontinuity ofI.

Theorem . Let W≥be polyconvex,and letp˜min{p, . . . ,pm}>m.Then I is weakly

lower semicontinuous,provided functions Ai···ikin(.)belong to L

˜

p

˜

p–m_(,∧n–l_).

Proof Suppose thatFν= (f

ν, . . . ,fνm)F= (f, . . . ,fm) inW,p(,∧l–× · · · × ∧lm–). We may assume that{I[Fν]}is ﬁnite and convergent as before. We ﬁrst obtain from (.) that

W(g+dFν) –W(g+dF)

≥

≤k≤m

≤i<···<ik≤m

Ai···ik(g+dF)

∧gi₊_dfi

ν

∧ · · · ∧gik₊_dfik

ν

–gi₊_dfi∧ · · · ∧_gik₊_dfik

=

≤k≤m

≤i<···<ik≤m

Ai···ik∧

dfi

ν ∧ · · · ∧dfνik–dfi∧ · · · ∧dfik

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+

≤k≤m

≤i<···<ik≤m

Ai···ik∧g

i∧_dfi

ν ∧ · · · ∧dfνik–dfi∧ · · · ∧dfik

+

≤k≤m

≤i<···<ik≤m

Ai···ik∧g

i∧_gi∧_dfi

ν ∧ · · · ∧dfνik–dfi∧ · · · ∧dfik

+· · ·

+

≤k≤m

≤i<···<ik≤m

Ai···ik∧g

i∧ · · · ∧_gik–∧_dfik

ν –dfik

. (.)

Next, observe that fork= , . . . ,m,

dfi

ν ∧ · · · ∧dfνisdfi∧ · · · ∧dfis inL ˜

p

s _for_s_{= , . . . ,}_k_,

and sinceAi···ik ∈L

˜

p

˜

p–m_and_gs∈_Lp˜_,_s_{= , . . . ,}_m_{, we then have by the Hölder inequality that}

Ai···ik∈L

˜

p

˜

p–k_,

Ai···ik∧g

i∈_L

˜

p

˜

p–(k–)←− p˜–k

˜

p– (k– )+  ˜

p– (k– )= ,

Ai···ik∧g

i∧_gi∈_L

˜

p

˜

p–(k–)←− p˜–k

˜

p– (k– )+  ˜

p– (k– )= ,

.. .

Ai···ik∧g

i∧ · · · ∧_gik–∈_L

˜

p

˜

p– ←−p˜–k

˜ p– +

 ˜

p– +· · ·+  ˜ p– 

k–

= 

fork= , . . . ,m.

Therefore, integrating both sides of (.) in, and then lettingν→ ∞, we obtain

lim

ν→∞I[Fν]≥I[F],

that is, we have

lim inf

ν→∞ I[Fν] =νlim→∞I[Fν]≥I[F],

which is the desired result.

4 Existence of minimizers

To prove the existence of minimizers ofI, another important point is requiring its coer-civity condition. The counterpart in the variational problem of diﬀerential forms is intro-duced by Iwaniec and Lutoborski [] as follows.

Deﬁnition .[] Suppose thatWsatisﬁes (.), we say thatWis coercive in the mean if

Rn

Wζ, . . . ,ζm≥ρ

Rn

ζp+· · ·+ζmpm (.)

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Theorem . In addition to the hypotheses,which ensure I to be weakly lower semicon-tinuous,assume that W satisﬁes the growth condition(.)and mean-coercivity condition (.).Then the minimizer of I is achieved at some F∈M.

Proof Set

m=inf

MI[F].

It follows from (.) and (.) thatmis finite. By the definition of infimum, there exists a minimizing sequence{Fν} ⊂Msuch that

I[Fν]→m

asν→ ∞. Then the mean coerciveness (.) yields{Fν}is bounded inW,p_(,_∧l–×

· · · × ∧lm–_{), which implies that}_{f_ν}i _{is bounded in}_W,pi_(,∧li–_),_i_{= , . . . ,}_m_{. From}

Propo-sition ., we see that{fi

ν}has a weakly convergent subsequence, denoted by{fνi}again, in W,pi_(,∧li–_{). Then we may assume that}

fνifi inW,pi

,∧li–

asν→ ∞. WriteF= (f_{, . . . ,}_fm_{), then we get}

FνF inW,p_,_∧l–× · · · × ∧lm–

andF∈M. SinceIis weakly lower semicontinuous, it follows that

m≤I[F]≤lim inf

ν→∞ I[Fν] =m,

i.e.,I[F] =m=inf_MI[F] as desired.

5 Applications

In this section, we present two examples to illustrate our main results. The ﬁrst example is to explain the existence result Theorem . for the stored-energy functionW, which is polyconvex.

Example  We consider a special case ofn= ,l= ,m=  andg= . Letξ= (ξ_,_ξ₎_∈ ∧₍_R₎_{× ∧}₍_R_{) with}

ξ=a_dx+a_dx, ξ=a_dx+a_dx

and assume thatW:∧₍_R₎_{× ∧}₍_R₎_{→ ∧}₍_R_{) is deﬁned by}

W(ξ) =Wξ,ξ=|ξ|∗|ξ|– ξ∧ξ,

where∗is the Hodge star duality operator. Step . Direct computation implies

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then we see thatW satisﬁes the growth condition (.) withp= (p,p) = (, ). Thus, we

havep˜=p=p=  >m= ,_p_˜_–p˜_m =  andM=W,(,∧× ∧).

Step . Note thatξ∧ξ= (a

a–aa)dx∧dxand

a_a_–a_a_ ≤a_a_+a_a_

≤ 

a

 



+a_+ a

 



+a_

=  ξ

₊_ξ₌

|ξ|

_,

then (a

a–aa)∗≤ |ξ|∗,ai.e.,

ξ∧ξ≤ |ξ|∗.

Hence we haveW≥.

Step . Applying Proposition . in [], yieldsWsatisﬁes (.). Step . Suppose thatζ= (ζ_,_ζ₎_∈_W,_(,_∧_{× ∧}_{) and}

ζ=b_dx+b_dx,

ζ=b_dx+b_dx.

Let

A=

a_ a_ a

 a

, B=

b_ b_ b

 b

.

Then|ξ|=|A|,|ζ|=|B|andξ_∧_ξ₌_det_A_∗_,_ζ_∧_ζ₌_det_B_∗_{. Thus, we have}

W(ξ) –W(ζ) =|ξ|–|ζ|∗ – |ξ|ξ∧ξ–|ζ|ζ∧ζ

=F(A) –F(B)∗, (.)

whereF(X) =|X|_{– }_|_X_|_det_X_{. Write}

A⊗B=a_b_+a_b_+a_b_+a_b_.

Then by applying

F(A) –F(B)≥|B|–detBB⊗(A–B) – |B|(detA–detB)[,],

we get from (.) that

W(ξ) –W(ζ) =F(A) –F(B)∗

≥|B|–detBB⊗(A–B) – |B|(detA–detB)∗

= |B|–detBb_a_–b_+b_a_–b_

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Observe that

∗ζ∧ξ–ζ=ξ–ζ∧ ∗ζ

= ξ–ζ|ζ ∗

=b_a_–b_+b_a_–b_∗. (.)

Similarly, we have

∗ζ∧ξ–ζ=ξ–ζ∧ ∗ζ

= ξ–ζ|ζ ∗

=b_a_–b_+b_a_–b_∗. (.)

By substituting (.) and (.) into (.), we obtain that

W(ξ) –W(ζ)≥|B|–detB∗ζ∧ξ–ζ

+ |B|_–_detB_∗_ζ_∧_ξ_–_ζ

– |ζ|ξ∧ξ–ζ∧ζ. (.)

Then we have from (.) that the corresponding terms in (.) are

A(ζ) = 

|B|–detB∗ζ,

A(ζ) = 

|B|–detB∗ζ

and

A,(ζ) = |ζ|.

Sinceζ= (ζ_,_ζ₎_∈_W,_(,_∧_{× ∧}_{) and}_{is bounded, then we get that}_A

,A,A,∈L.

Therefore, it follows from Theorem . that the minimizer ofIcan be achieved.

Finally, we give an example for a stored-energy functionW, which is quasiconvex.

Example  We consider the casel= ,m=nandg= . Letξ= (ξ, . . . ,ξn)∈ ∧(Rn)× · · · × ∧₍_Rn_{) and}

ξi=ai_dx+· · ·+ai_ndxn

fori= , . . . ,n. WriteA= (ai

j), ≤i,j≤n, whereaij denotes the element ofith-row and

jth-column inA. Fix ≤k≤n, then we have by simple calculation that

ξi∧ · · · ∧_ξik₌

≤j<···<jk≤n

det

⎛ ⎜ ⎜ ⎝

ai

j . . . a i jk

..

. . .. ... aik

j . . . a ik

jk

⎞ ⎟ ⎟

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for all orderedl-tuplesI= (i, . . . ,ik), ≤i<· · ·<ik≤n. LetJi,...,ik={, . . . ,n}–I, then it

follows from (.) that

ξi∧ · · · ∧_ξik∧_dx

Ji,...,ik=det

⎛ ⎜ ⎜ ⎝

ai

i . . . a i ik

..

. . .. ... aik

i . . . a ik

ik

⎞ ⎟ ⎟

⎠dx. (.)

Here,

Ai,...,ik

i,...,ikdet

⎛ ⎜ ⎜ ⎝

ai

i . . . a i ik

..

. . .. ... aik

i . . . a ik

ik

⎞ ⎟ ⎟ ⎠

denotes thek×kprincipal minors ofA. Let [A]kdenote the sum of thek×kprincipal

minors ofA, then

W(ξ)

n

k=≤i<···<ik≤n

ξi∧ · · · ∧_ξik∧_dx

Ji,...,ik

=

n

k=

[A]kdx.

Suppose thatAis positive semideﬁnite, thenW≥, and we have from [] thatW is quasiconvex.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this paper. They read and approved the ﬁnal manuscript.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11071048).

Endnote

a _{Inequality between two volume forms should be understood as inequality between their coeﬃcients with respect}

to the standard basis, that is to say, we say that ann-formαonRn_{is nonnegative if}_α₌_λ_dx_{for some nonnegative}

functionλ.

Received: 6 March 2013 Accepted: 31 July 2013 Published: 22 August 2013 References

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