Relaxation theorems in nonlinear elasticity

www.elsevier.com/locate/anihpc

Relaxation theorems in nonlinear elasticity

Omar Anza Hafsa

, Jean-Philippe Mandallena

aLMGC (Laboratoire de Mécanique et Génie Civil), UMR-CNRS 5508, Université Montpellier II, Place Eugène Bataillon, 34090 Montpellier, France

bEquipe AVA (Analyse Variationnelle et Applications), Centre Universitaire de Formation et de Recherche de Nîmes, Site des Carmes, Place Gabriel Péri, Cedex 01, 30021 Nîmes, France

cI3M (Institut de Mathématiques et Modélisation de Montpellier), UMR-CNRS 5149, Université Montpellier II, Place Eugène Bataillon, 34090 Montpellier, France

Received 21 September 2006; accepted 30 November 2006 Available online 28 December 2006

Abstract

Relaxation theorems which apply to one, two and three-dimensional nonlinear elasticity are proved. We take into account the fact an infinite amount of energy is required to compress a finite line, surface or volume into zero line, surface or volume. However, we do not prevent orientation reversal.

©2006 Elsevier Masson SAS. All rights reserved.

Keywords:Relaxation; Nonlinear elasticity; Determinant condition

1. Main results 1.1. Introduction

Consider an elastic material occupying in a reference configuration Ω ⊂R^N with N =1,2 or 3, whereΩ is bounded and open with Lipschitz boundary∂Ω. The mechanical properties of the material are characterized by a stored-energy functionW:M^3×N→ [0,+∞](assumed to be Borel measurable) in terms of which the total stored- energy is the integral

I (u):=

Ω

W

∇u(x)

dx (1)

with∇u(x)∈M^3×Nthe gradient ofuatx, whereM^3×Ndenotes the space of all real 3×Nmatrices. In order to take into account the fact that an infinite amount of energy is required to compress a finite line (N=1), surface (N=2) or volume (N=3) into zero line, surface or volume, i.e.,

* Corresponding author.

E-mail addresses:anza@lmgc.univ-montp2.fr (O. Anza Hafsa), jean-philippe.mandallena@unimes.fr (J.-P. Mandallena).

0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2006.11.005

W (ξ )→ +∞ as

|ξ| →0 ifN=1,

|ξ1∧ξ2| →0 ifN=2,

|detξ| →0 ifN=3,

(2) we consider the following conditions:

(C1)there existα, β >0such that for everyξ∈M^3×1, if |ξ|αthenW (ξ )β

1+ |ξ|^p forN=1;

(C2)there existα, β >0such that for everyξ=(ξ1|ξ2)∈M^3×2, if |ξ1∧ξ2|αthenW (ξ )β

1+ |ξ|^p

forN=2, whereξ₁∧ξ₂denotes the cross product of vectorsξ₁, ξ₂∈R³; (C3)for everyδ >0, there existsc_δ>0such that for everyξ∈M^3×3,

if |detξ|δthenW (ξ )c_δ

1+ |ξ|^p , where detξ denotes the determinant ofξ, and

(C4)W (P ξ Q)=W (ξ )for allξ∈M^3×3and allP , Q∈SO(3)

for N=3, with SO(3):= {Q∈M^3×3: Q^TQ=QQ^T=I₃ and detQ=1}, whereI₃denotes the identity matrix inM^3×3andQ^Tis the transposed matrix ofQ. (In fact, (C4) is an additional condition which is not related to (2).

However, it means thatWis frame-indifferent, i.e.,W (P ξ )=W (ξ )for allξ ∈M^3×3and allP ∈SO(3), and isotropic, i.e.,W (ξ Q)=W (ξ )for allξ∈M^3×3and allQ∈SO(3), see for example [12] for more details.)

Fixp∈ ]1,+∞[, setW_g^1,p(Ω;R³):= {u∈W^1,p(Ω;R³): u=gon∂Ω}, wheregis given continuous piecewise affine function fromΩtoR³, define the integral

QI (u):=

Ω

QW

∇u(x) dx,

whereQW:M^3×N→ [0,+∞]denotes the quasiconvex envelope ofW, and consider the following assertions:

(R1) inf{I (u): u∈W_g^1,p(Ω;R³)} =inf{QI (u): u∈W_g^1,p(Ω;R³)};

(R2) if un u with {un}n1 minimizing sequence for I in W_g^1,p(Ω;R³), then u¯ is a minimizer for QI in W_g^1,p(Ω;R³);

(R3) if u¯ is a minimizer for QI in W_g^1,p(Ω;R³), then there exists a minimizing sequence {u_n}n1 for I in W_g^1,p(Ω;R³)such thatu_nu,¯

where “” denotes the weak convergence inW^1,p(Ω;R³). In this paper we prove (see Section 1.3) the following relaxation theorems:

Theorem 1.1.(N=1)If(C1)holds and ifW is coercive, i.e.,W (ξ )C|ξ|^pfor allξ∈M^3×Nand someC >0, then (R1),(R2)and(R3)hold.

Theorem 1.2.(N=2)If (C2)holds and ifW is coercive, then(R1),(R2)and(R3)hold.

Theorem 1.3.(N=3)If (C3)and(C4)hold and ifWis coercive, then(R1),(R2)and(R3)hold.

Typically, these theorems can be applied with stored-energy functionsWof the form W (ξ ):= |ξ|^p+

⎧⎪

⎨

⎪⎩ h

|ξ|

ifN=1, h

|ξ1∧ξ2|

ifN=2, h

|detξ|

ifN=3,

for allξ∈M^3×N, whereh:[0,+∞[ → [0,+∞]is Borel measurable and such that for everyδ >0, there existsr_δ>0 such thath(t )r_δfor alltδ(for example,h(0)= +∞andh(t )=1/t^s ift >0 withs >0).

1.2. Outline of the paper

LetI:W^1,p(Ω;R³)→ [0,+∞]be defined by I(u):=

⎧⎨

⎩

Ω

W

∇u(x)

dx ifu∈W_g^1,p Ω;R³

,

+∞ otherwise,

letQI:W^1,p(Ω;R³)→ [0,+∞]be defined by QI(u):=

⎧⎨

⎩

Ω

QW

∇u(x)

dx ifu∈W_g^1,p Ω;R³

,

+∞ otherwise,

and letI:W^1,p(Ω;R³)→ [0,+∞]be the lower semicontinuous envelope (or relaxed functional) ofI with respect to the weak topology ofW^1,p(Ω;R³), i.e.,

I(u):=inf

lim inf

n→+∞I(u_n):u_n u

.

Set Y := ]0,1[^N and Aff0(Y;R³):= {φ∈Aff(Y;R³): φ=0 on∂Y}, where Aff(Y;R³)denotes the space of all continuous piecewise affine functions fromY toR³, and considerZW:M^3×N→ [0,+∞]given by

ZW (ξ ):=inf

Y

W

ξ+ ∇φ(y)

dy: φ∈Aff0

Y;R³ .

Here is the central theorem of the paper:

Theorem 1.4.IfZW is ofp-polynomial growth, i.e.,ZW (ξ )c(1+ |ξ|^p)for allξ∈M^3×Nand somec >0, then I=Q I.

Herem=3 andN=1,2 or 3, but the proof of Theorem 1.4 (given in Section 3) does not depend on the integers mandN. This immediately gives the following relaxation result:

Corollary 1.5.Under the hypotheses of Theorem1.4, ifW is coercive, then(R1),(R2)and(R3)hold.

Such results was proved by Dacorogna in [8] whenWis continuous and ofp-polynomial growth. The distinguish- ing feature here is that Theorem 1.4 (and so Corollary 1.5) is compatible with (2). More precisely, in Section 4 we prove the following propositions:

Proposition 1.6.(N=1)If (C1)holds thenZW is ofp-polynomial growth.

Proposition 1.7.(N=2)If (C2)holds thenZW is ofp-polynomial growth.

Proposition 1.8.(N=3)If (C3)and(C4)hold thenZW is ofp-polynomial growth.

Theorem 1.4 follows from Propositions 1.9 and 1.10 below whose proofs are given in Section 3:

Proposition 1.9.IfZW is finite thenQW=Q[ZW] =ZW. Furthermore, forN=1 we haveZW=W^∗∗, where W^∗∗denotes the lower semicontinuous convex envelope ofW.

Proposition 1.10.J0=J1withJ0,J1:W^1,p(Ω;R³)→ [0,+∞]respectively defined by J0(u):=inf

lim inf

n→+∞I (u_n): Affg

Ω;R³

u_n u

and

J1(u):=inf

lim inf

n→+∞ZI (un): Affg

Ω;R³

un u

,

whereAffg(Ω;R³):= {u∈Aff(Ω;R³): u=gon∂Ω}and ZI (u):=

Ω

ZW

∇u(x) dx.

Taking Proposition 1.9 into account, from Propositions 1.6, 1.7 and 1.8, we see that stored-energy functionsW satisfying (C1) forN=1, (C2) forN=2 and (C3) and (C4) forN=3, are not quasiconvex, so that the integralI (u) in (1) is not weakly lower semicontinuous onW^1,p(Ω;R³)(see [4, Corollary 3.2]). Thus, the Direct Method of the Calculus of Variations cannot be applied to study the existence of minimizers ofI inW_g^1,p(Ω;R³). For this reason, in the present paper we establish relaxation theorems instead of existence theorems. (In fact, the term “relaxation”

means “generalized existence”, see [10,9,6] for a deeper discussion.)

Other related results can be found in [7,5] where we refer the reader. The present work improves our previous one [1] (see also [2,3]). The main new contribution of the present paper is the treatment of the caseN=3.

The plan of the paper is as follows. The proofs of Theorems 1.1, 1.2 and 1.3 are given in Section 1.3 (although these can be easily deduced from the previous discussion). Section 2 presents some preliminaries. In Section 3 we prove Propositions 1.9 and 1.10 and Theorem 1.4. Finally, Section 4 contains the proofs of Propositions 1.6, 1.7 and 1.8.

1.3. Proof of Theorems 1.1, 1.2 and 1.3

According to Corollary 1.5, we see that Theorems 1.1, 1.2 and 1.3 are immediate consequences of respectively Propositions 1.6, 1.7 and 1.8.

2. Preliminaries

In this section we recall some (classical) definitions and results. These will be used throughout the paper.

Letm, N 1 be two integers. For any bounded open set D⊂R^N, we denote by Aff(D;R^m)the space of all continuous piecewise affine functions fromD toR^m, i.e.,u∈Aff(D;R^m)if and only ifu is continuous and there exists a finite family(Di)i∈Iof open disjoint subsets ofDsuch that|D\

i∈IDi| =0 and for everyi∈I,∇u(x)=ξi

inD_i withξ_i∈M^m×N (where| · |denotes the Lebesgue measure inR^N). For anyg∈W^1,p(D;R^m)withp >1, we set

Affg

D;R^m :=

u∈Aff D;R^m

: u=gon∂D (Aff0(D;R^m)corresponds to Affg(D;R^m)withg=0) and

W_g^1,p D;R^m

:=

u∈W^1,p D;R^m

: u=gon∂D ,

(where∂Ddenotes the boundary ofD). Fixg∈W^1,p(Ω;R^m)whereΩ⊂R^Nis bounded and open with Lipschitz boundary. The following density theorem will play an essential role in the proof of Theorem 1.4:

Theorem 2.1.(Ekeland and Temam[10]) Affg(Ω;R^m)is dense inW_g^1,p(Ω;R^m)with respect to the strong topology ofW^1,p(Ω;R^m).

Letf:M^m×N→ [0,+∞]be Borel measurable and letZf:M^m×N→ [0,+∞]be defined by Zf (ξ ):=inf

Y

f

ξ+ ∇φ(x)

dx: φ∈Aff0

Y;R^m

withY := ]0,1[^N. To prove Propositions 1.6, 1.7 and 1.8, we will use the following properties ofZf:

Proposition 2.2.(Fonseca[11])

(i) For every bounded open setD⊂R^N with|∂D| =0and everyξ∈M^m×N, Zf (ξ )=inf

1

|D|

D

f

ξ+ ∇φ(x)

dx: φ∈Aff0

D;R^m .

(ii) IfZf is finite thenZf is rank-one convex, i.e., for everyξ, ξ∈M^m×Nwithrank(ξ−ξ)1, Zf

λξ+(1−λ)ξ

λZf (ξ )+(1−λ)Zf (ξ).

(iii) IfZf is finite thenZf is continuous.

(iv) For every bounded open setD⊂R^N with|∂D| =0, everyξ ∈M^m×Nand everyφ∈Aff0(D;R^m), Zf (ξ ) 1

|D|

D

Zf

ξ+ ∇φ(x) dx.

Note that Proposition 2.2 is not exactly the one that can found in Fonseca. Nevertheless, it can be proved using the same arguments than the one given by Fonseca (for more details see [2, Remark A.2]).

Quasiconvexity is the correct concept to deal with multiple integral problems in the Calculus of Variations. For the convenience of the reader, we recall its definition:

Definition 2.3.(Morrey [13]) We say thatf is quasiconvex if for everyξ∈M^m×N, every bounded open setD⊂R^N with|∂D| =0 and everyφ∈W₀^1,∞(D;R^m),

f (ξ ) 1

|D|

D

f

ξ+ ∇φ(x) dx.

Remark 2.4.Clearly, iff is quasiconvex thenZf =f.

By the quasiconvex envelope off, that we denoteQf, we mean the greatest quasiconvex function which less than or equal tof. (Thus,f is quasiconvex if and only ifQf =f.) To prove Proposition 1.9 we will need Theorem 2.5:

Theorem 2.5.(Dacorogna[8,9])Iff is continuous and finite thenQf =Zf. LetF:W^1,p(Ω;R^m)→ [0,+∞]be defined by

F(u):=

⎧⎪

⎨

⎪⎩

Ω

f

∇u(x)

dx ifu∈Wg^1,p

Ω;R^m ,

+∞ otherwise,

letQF:W^1,p(Ω;R^m)→ [0,+∞]be given by QF(u):=

⎧⎪

⎨

⎪⎩

Ω

Qf

∇u(x)

dx ifu∈W_g^1,p Ω;R^m

,

+∞ otherwise,

and letF:W^1,p(Ω;R^m)→ [0,+∞]be the lower semicontinuous envelope (or relaxed functional) ofIwith respect to the weak topology ofW^1,p(Ω;R^m), i.e.,

F(u):=inf

lim inf

n→+∞F(u_n):u_n u

,

where “” denotes the weak convergence inW^1,p(Ω;R^m). We close this section with the following integral repre- sentation theorem that we will use in the proof of Theorem 1.4:

Theorem 2.6.(Dacorogna [8,9])Iff is continuous and ofp-polynomial growth, i.e., f (ξ )c(1+ |ξ|^p)for all ξ∈M^m×N and somec >0, thenF=QF.

3. Proof of Propositions 1.9 and 1.10 and Theorem 1.4

3.1. Proof of Proposition 1.9

Since ZW is finite, ZW is continuous by Proposition 2.2(iii). From Theorem 2.5, we deduce that Q[ZW] = Z[ZW]. ButZ[ZW] =ZW by Proposition 2.2(iv), henceQ[ZW] =ZW. On the other hand, asZWWwe have Q[ZW]QW. Moreover, asQW is quasiconvex, from Remark 2.4 we see thatZ[QW] =QW, and consequently QWQ[ZW]. It follows thatQW=Q[ZW] =ZW.

Assume thatN =1. Then, quasiconvexity is equivalent to convexity (see [9, Theorem 1.1, p. 102]). Thus,ZW is convex (resp.ZW^∗∗=W^∗∗) sinceZW =QW (resp.W^∗∗is convex). ButZW is continuous (resp.W^∗∗W), henceZWW^∗∗(resp.W^∗∗ZW). It follows thatZW=W^∗∗.

3.2. Proof of Proposition 1.10

ClearlyJ1J0. We are thus reduced to prove that

J0J1. (3)

We need the following lemma:

Lemma 3.1.Ifu∈Affg(Ω;R³)thenJ0(u)ZI (u).

Proof of Lemma 3.1. Letu∈Affg(Ω;R³). By definition, there exists a finite family(Ωi)i∈I of open disjoint subsets ofΩ such that|Ω\ ∪i∈IΩi| =0 and, for everyi∈I,∇u(x)=ξi inΩi withξi∈M^3×N. Given anyδ >0 and any i∈I, we considerφi∈Aff0(Y;R³)such that

Y

W

ξi+ ∇φi(y)

dyZW (ξi)+ δ

|Ω|. (4)

Fix any integern1. By Vitali’s covering theorem, there exists a finite or countable family (a_i,j +ε_i,jY )_j∈Ji of disjoint subsets ofΩ_i, wherea_i,j∈R^Nand 0< ε_i,j<¹_n, such that|Ω_i\

j∈J_i(a_i,j+ε_i,jY )| =0 (and so

j∈J_iε_i,j^N =

|Ω_i|). Defineψ_n:Ω→R³by ψn(x):=εi,jφi

x−a_i,j ε_i,j

ifx∈ai,j+εi,jY.

Clearly, for everyn1,ψn∈Aff0(Ω;R³), ψ_nL∞(Ω;R³)1

nmax

i∈I φ_iL∞(Y;R³) and ∇ψ_nL∞(Ω;M^3×N)max

i∈I ∇φ_iL∞(Y;M^3×N), hence (up to a subsequence)ψn ∗

0 inW^1,∞(Ω;R³), where “” denotes the weak^{∗ ∗}convergence inW^1,∞(Ω;R³).

Consequently,ψn0 inW^1,p(Ω;R³). Moreover, I (u+ψn)=

i∈I

Ω_i

W

ξi+ ∇ψn(x) dx

=

i∈I

j∈Ji

ε^N_i,j

Y

W

ξ_i+ ∇φ_i(y) dy

=

i∈I

|Ω_i|

Y

W

ξ_i+ ∇φ_i(y) dy.

Asu+ψn∈Affg(Ω;R³)for alln1 andu+ψn uinW^1,p(Ω;R³), from (4) we deduce that J0(u)lim inf

n→+∞I (u+ψ_n)

i∈I

|Ω_i|ZW (ξ_i)+δ=ZI (u)+δ,

and the lemma follows. 2

Fix anyu∈W^1,p(Ω;R³)and any sequenceu_n uwithu_n∈Affg(Ω;R³). Using Lemma 3.1 we haveJ0(u_n) ZI (u_n)for alln1. Thus,

J0(u)lim inf

n→+∞J0(u_n)lim inf

n→+∞ZI (u_n), and (3) follows.

3.3. Proof of Theorem 1.4

Since ZW is of p-polynomial growth, ZW is finite, and so ZW is continuous by Proposition 2.2(iii). From Theorem 2.1 it follows that

J1(u)=inf

lim inf

n→+∞ZI (un):W_g^1,p Ω;R³

un u

.

But QW =Q[ZW] by Proposition 1.9, hence J1=QI by Theorem 2.6. On the other hand, given any u∈ W^1,p(Ω;R³)and anyu_n uwithu_n∈W_g^1,p(Ω;R³), we haveZI (u_n)I (u_n)for alln1. Thus,

J1(u)lim inf

n→+∞ZI (u_n)lim inf

n→+∞I (u_n),

and soJ1I. ButIJ0andJ0=J1by Proposition 1.10, henceI=J1, and the theorem follows.

4. Proof of Propositions 1.6, 1.7 and 1.8

4.1. CaseN=1

In this section we prove Proposition 1.6.

Proof of Proposition 1.6. By (C1) it is clear that if|ξ|αthenZW (ξ )β(1+ |ξ|^p). Fix anyξ∈M^3×1such that

|ξ|αand considerφ∈Aff0(Y;R³)given by φ(x):=

xν ifx∈ ]0,¹₂], (1−x)ν ifx∈ ]¹₂,1[ withν∈R³such that|ν| =2α. Then,

ξ+ ∇φ(x)=

|ξ+ν| ifx∈ ]0,¹₂[,

|ξ−ν| ifx∈ ]¹₂,1[,

hence|ξ+ ∇φ(x)|min{|ξ+ν|,|ξ−ν|}|ν| − |ξ|αfor allx∈ ]0,¹₂[ ∪ ]¹₂,1[, and from (C1) we deduce that ZW (ξ )

Y

W

ξ+ ∇φ(x) dxβ

Y

1+ξ+ ∇φ(x)^p

dxβ2^2pmax 1, α^p

1+ |ξ|^p .

It follows thatZW (ξ )β2^2pmax{1, α^p}(1+ |ξ|^p)for allξ∈M^3×1. 2

4.2. CaseN=2

In this section we prove Proposition 1.7. We begin with the following lemma.

Lemma 4.1.Under(C2)there existsγ >0such that for allξ=(ξ₁|ξ₂)∈M^3×2, if min

|ξ₁+ξ₂|,|ξ₁−ξ₂|

αthenZW (ξ )γ

1+ |ξ|^p .

Proof. Letξ=(ξ₁|ξ₂)∈M^3×2be such that min{|ξ₁+ξ₂|,|ξ₁−ξ₂|}α(withα >0 given by (C2)). Then, one the three possibilities holds:

(i) |ξ1∧ξ2| =0;

(ii) |ξ1∧ξ2| =0 withξ1=0;

(iii) |ξ1∧ξ2| =0 withξ2=0.

SetD:= {(x₁, x₂)∈R²:x₁−1< x₂< x₁+1 and−x₁−1< x₂<1−x₁}and, for eacht∈R, defineϕ_t∈Aff0(D;R) by

ϕ_t(x1, x2):=

⎧⎪

⎨

⎪⎩

−t x₁+t (x₂+1) if(x₁, x₂)∈Δ₁, t (1−x₁)−t x₂ if(x₁, x₂)∈Δ₂, t x₁+t (1−x₂) if(x₁, x₂)∈Δ₃, t (x1+1)+t x2 if(x1, x2)∈Δ4

with

Δ₁:=

(x₁, x₂)∈D: x₁0 andx₂0

; Δ₂:=

(x₁, x₂)∈D: x₁0 andx₂0

; Δ₃:=

(x₁, x₂)∈D: x₁0 andx₂0

; Δ₄:=

(x₁, x₂)∈D: x₁0 andx₂0 . Considerφ∈Aff0(D;R³)given by

φ:=(ϕ_ν1, ϕ_ν2, ϕ_ν3) with

⎧⎪

⎪⎨

⎪⎪

⎩

ν= ξ₁∧ξ₂

|ξ₁∧ξ₂| if (i) is satisfied,

|ν| =1 andξ₁, ν =0 if (ii) is satisfied,

|ν| =1 andξ₂, ν =0 if (iii) is satisfied (ν1,ν2,ν3are the components of the vectorν). Then,

ξ+ ∇φ(x)=

⎧⎪

⎨

⎪⎩

(ξ₁−ν|ξ₂+ν) ifx∈int(Δ1), (ξ₁−ν|ξ₂−ν) ifx∈int(Δ2), (ξ₁+ν|ξ₂−ν) ifx∈int(Δ3), (ξ₁+ν|ξ₂+ν) ifx∈int(Δ4)

(where int(E)denotes the interior of the setE). Taking Proposition 2.2(i) into account, it follows that ZW (ξ )1

4

W (ξ1−ν|ξ2+ν)+W (ξ1−ν|ξ2−ν)+W (ξ1+ν|ξ2−ν)+W (ξ1+ν|ξ2+ν)

. (5)

But|(ξ1−ν)∧(ξ2+ν)|²= |ξ1∧ξ2+(ξ1+ξ2)∧ν|²= |ξ1∧ξ2|²+ |(ξ1+ξ2)∧ν|²|(ξ1+ξ2)∧ν|²,and so (ξ₁+ν)∧(ξ₂−ν)(ξ₁+ξ₂)∧ν= |ξ₁+ξ₂|.

Similarly, we obtain:

(ξ₁−ν)∧(ξ₂−ν)|ξ₁−ξ₂|;

(ξ₁+ν)∧(ξ₂−ν)|ξ₁+ξ₂|;

(ξ₁+ν)∧(ξ₂+ν)|ξ₁−ξ₂|.

Thus,|(ξ1−ν)∧(ξ2+ν)|α,|(ξ1−ν)∧(ξ2−ν)|α,|(ξ1+ν)∧(ξ2−ν)|αand|(ξ1+ν)∧(ξ2+ν)|α, because min{|ξ1+ξ2|,|ξ1−ξ2|}α. Using (C2) it follows that

W (ξ1−ν|ξ2+ν)β

1+(ξ1−ν|ξ2+ν)^p β2^p

1+(ξ₁|ξ2)^p+(−ν|ν)^p β2^2p+1

1+ |ξ|^p . In the same manner, we have:

W (ξ₁−ν|ξ₂−ν)β2^2p+1

1+ |ξ|^p

; W (ξ₁+ν|ξ₂−ν)β2^2p+1

1+ |ξ|^p

; W (ξ₁+ν|ξ₂+ν)β2^2p+1

1+ |ξ|^p ,

and, from (5), we conclude thatZW (ξ )β2^2p+1(1+ |ξ|^p). 2

Proof of Proposition 1.7. Letξ=(ξ₁|ξ₂)∈M^3×2. Then, one the four possibilities holds:

(i) |ξ1∧ξ2| =0;

(ii) |ξ1∧ξ2| =0 withξ1=ξ2=0;

(iii) |ξ1∧ξ2| =0 withξ1=0;

(iv) |ξ1∧ξ2| =0 withξ2=0.

For eacht∈R, defineϕt∈Aff0(Y;R)by

ϕ_t(x₁, x₂):=

⎧⎪

⎨

⎪⎩

t x₂ if(x₁, x₂)∈Δ₁, t (1−x₁) if(x₁, x₂)∈Δ₂, t (1−x₂) if(x₁, x₂)∈Δ₃, t x₁ if(x₁, x₂)∈Δ₄ with

Δ1:=

(x1, x2)∈Y:x2x1−x2+1

; Δ2:=

(x1, x2)∈Y:−x1+1x2x1

; Δ3:=

(x1, x2)∈Y:−x2+1x1x2

; Δ₄:=

(x₁, x₂)∈Y:x₁x₂−x₁+1 .

Considerφ∈Aff0(Y;R³)given by

φ:=(ϕν₁, ϕν₂, ϕν₃) with

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ν=α(ξ₁∧ξ₂)

|ξ₁∧ξ₂| if (i) is satisfied,

|ν| =α if (ii) is satisfied,

|ν| =αandξ₁, ν =0 if (iii) is satisfied,

|ν| =αandξ₂, ν =0 if (iv) is satisfied (ν1,ν₂,ν₃are the components of the vectorνandα >0 is given by (C2)). Then,

ξ+ ∇φ(x)=

⎧⎪

⎨

⎪⎩

(ξ1|ξ2+ν) ifx∈int(Δ1), (ξ₁−ν|ξ₂) ifx∈int(Δ2), (ξ₁|ξ₂−ν) ifx∈int(Δ3), (ξ₁+ν|ξ₂) ifx∈int(Δ4)

(where int(E)denotes the interior of the setE). Taking Proposition 2.2(iv) into account, it follows that ZW (ξ )1

4

ZW (ξ₁|ξ₂+ν)+ZW (ξ₁−ν|ξ₂)+ZW (ξ₁|ξ₂−ν)+ZW (ξ₁+ν|ξ₂)

. (6)

But ξ₁+(ξ₂+ν)²=(ξ₁+ξ₂)+ν²= |ξ₁+ξ₂|²+ |ν|²= |ξ₁+ξ₂|²+α²α², hence|ξ₁+(ξ₂+ν)|α.Similarly, we obtain|ξ₁−(ξ₂+ν)|α,and so

minξ₁+(ξ₂+ν),ξ₁−(ξ₂+ν)α.

In the same manner, we have:

min(ξ₁−ν)+ξ₂,(ξ₁−ν)−ξ₂α; minξ₁+(ξ₂−ν),ξ₁−(ξ₂−ν)α; min(ξ₁+ν)+ξ₂,(ξ₁+ν)−ξ₂α.

Using Lemma 4.1 it follows that ZW (ξ₁|ξ₂+ν)γ

1+(ξ₁|ξ₂+ν)^p γ2^p

1+(ξ1|ξ2)^p+(0|ν)^p

max

1, α^p γ2^p+1

1+ |ξ|^p . Similarly, we obtain:

ZW (ξ₁−ν|ξ₂)max 1, α^p

γ2^p+1

1+ |ξ|^p

; ZW (ξ₁|ξ₂−ν)max

1, α^p γ2^p+1

1+ |ξ|^p

; ZW (ξ₁+ν|ξ₂)max

1, α^p γ2^p+1

1+ |ξ|^p ,

and, from (6), we conclude thatZW (ξ )max{1, α^p}γ2^p+1(1+ |ξ|^p). 2 4.3. CaseN=3

In this section we prove Proposition 1.8. We begin with three lemmas.

Lemma 4.2.If (C3)holds thenZWis finite.

Proof. Clearly, ifξ∈M³_∗^×3thenZW (ξ ) <+∞withM³_∗^×3:= {ξ∈M^3×3: detξ=0}. We are thus reduced to prove that ZW (ξ ) <+∞ for allξ ∈M^3×3\M³_∗^×3. Fix ξ =(ξ₁|ξ₂|ξ₃)∈M^3×3\M³_∗^×3 whereξ₁, ξ₂, ξ₃∈R³ are the columns ofξ. Then rank(ξ )∈ {0,1,2}(where rank(ξ )denotes the rank of the matrixξ).

Step1.We prove that ifrank(ξ )=2thenZW (ξ ) <+∞. Without loss of generality we can assume that there exist λ, μ∈Rsuch thatξ3=λξ1+μξ2. Given anys∈R^∗, considerD⊂R³given by

D:=int ₈

i=1

Δ^s_i

(7) (where int(E)denotes the interior of the setE) with:

Δ^s₁:=

(x1, x2, x3)∈R³: x10, x20, x30 andx1+x2+sx31

; Δ^s₂:=

(x1, x2, x3)∈R³: x10, x20, x30 and −x1+x2+sx31

; Δ^s₃:=

(x1, x2, x3)∈R³: x10, x20, x30 and −x1−x2+sx31

; Δ^s₄:=

(x1, x2, x3)∈R³: x10, x20, x30 andx1−x2+sx31

; Δ^s₅:=

(x₁, x₂, x₃)∈R³: x₁0, x20, x30 andx₁+x₂−sx₃1

; Δ^s₆:=

(x₁, x₂, x₃)∈R³: x₁0, x20, x30 and −x₁+x₂−sx₃1

; Δ^s₇:=

(x₁, x₂, x₃)∈R³: x₁0, x20, x30 and −x₁−x₂−sx₃1

; Δ^s₈:=

(x₁, x₂, x₃)∈R³: x₁0, x20, x30 andx₁−x₂−sx₃1 .

Clearly,Dis bounded, open and|∂D| =0. For eacht∈R, defineϕs,t∈Aff0(D;R)by

ϕs,t(x1, x2, x3):=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

−t (x₁+1)−t x₂−t sx₃ if(x₁, x₂, x₃)∈Δ^s₁, t x₁−t (x₂+1)−t sx₃ if(x₁, x₂, x₃)∈Δ^s₂, t (x₁−1)+t x₂−t sx₃ if(x₁, x₂, x₃)∈Δ^s₃,

−t x₁+t (x₂−1)−t sx₃ if(x₁, x₂, x₃)∈Δ^s₄,

−t (x₁+1)−t x₂+t sx₃ if(x₁, x₂, x₃)∈Δ^s₅, t x₁−t (x₂+1)+t sx₃ if(x₁, x₂, x₃)∈Δ^s₆, t (x₁−1)+t x₂+t sx₃ if(x₁, x₂, x₃)∈Δ^s₇,

−t x₁+t (x₂−1)+t sx₃ if(x₁, x₂, x₃)∈Δ^s₈.

(8)

(Note thatϕ_s,tis simply the only function, affine on everyΔ^s_i, null on∂D, such thatϕ_s,t(0)= −t.) Fix s∈R^∗\

λ−μ,−(λ−μ), λ+μ,−(λ+μ) and considerφ∈Aff0(D;R³)given by

φ:=(ϕs,ν₁, ϕs,ν₂, ϕs,ν₃) withν:= ξ₁∧ξ₂

|ξ₁∧ξ₂|², (ν1,ν₂,ν₃are the components of the vectorν). Then,

ξ+ ∇φ(x)=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

(ξ1−ν|ξ2−ν|ξ3−sν) ifx∈int Δ^s₁

, (ξ1+ν|ξ2−ν|ξ3−sν) ifx∈int

Δ^s₂ , (ξ1+ν|ξ2+ν|ξ3−sν) ifx∈int

Δ^s₃ , (ξ1−ν|ξ2+ν|ξ3−sν) ifx∈int

Δ^s₄ , (ξ₁−ν|ξ₂−ν|ξ₃+sν) ifx∈int

Δ^s₅ , (ξ₁+ν|ξ₂−ν|ξ₃+sν) ifx∈int

Δ^s₆ , (ξ₁+ν|ξ₂+ν|ξ₃+sν) ifx∈int

Δ^s₇ , (ξ₁−ν|ξ₂+ν|ξ₃+sν) ifx∈int

Δ^s₈ . As detξ=0,ξ₁∧ξ₃=μ(ξ₁∧ξ₂)andξ₂∧ξ₃=λ(ξ₂∧ξ₁)we have

det

ξ+ ∇φ(x)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

s+(λ−μ) ifx∈int Δ^s₁

∪int Δ^s₇ s−(λ+μ) ifx∈int ,

Δ^s₂

∪int Δ^s₈ s−(λ−μ) ifx∈int ,

Δ^s₃

∪int Δ^s₅ s+(λ+μ) ifx∈int ,

Δ^s₄

∪int Δ^s₆

.

It follows that for a.e.x∈D,|det(ξ+ ∇φ(x)|min{|s+(λ−μ)|,|s−(λ+μ)|,|s−(λ−μ)|,|s+(λ+μ)|} =:δ (δ >0). Taking Proposition 2.2(i) into account and using (C3), we see that there existsc_δ>0 such that

ZW (ξ ) 1

|D|

D

W

ξ+ ∇φ(x)

cδ+ c_δ

|D|ξ+ ∇φ^p_Lp(D;R³), which implies thatZW (ξ ) <+∞.

Step2. We prove that ifrank(ξ )=1 thenZW (ξ ) <+∞. Without loss of generality we can assume that there existλ, μ∈Rsuch thatξ2=λξ1andξ3=μξ1. ConsiderD⊂R³ given by (7) withs∈R^∗\ {−μ, μ}, and define φ∈Aff0(D;R³)byφ:=(ϕs,ν₁, ϕs,ν₂, ϕs,ν₃)withν=(ν1, ν2, ν3)∈R³\ {0}such thatν, ξ1 =0, where, for every i∈ {1,2,3},ϕs,ν_i is defined by (8) witht=νi. By Proposition 2.2(iv) we have

ZW (ξ )1 8

ZW (ξ₁−ν|ξ₂−ν|ξ₃−sν)+ZW (ξ₁+ν|ξ₂−ν|ξ₃−sν) +ZW (ξ₁+ν|ξ₂+ν|ξ₃−sν)+ZW (ξ₁−ν|ξ₂+ν|ξ₃−sν) +ZW (ξ₁−ν|ξ₂−ν|ξ₃+sν)+ZW (ξ₁+ν|ξ₂−ν|ξ₃+sν) +ZW (ξ₁+ν|ξ₂+ν|ξ₃+sν)+ZW (ξ₁−ν|ξ₂+ν|ξ₃+sν)

.

Noticing thats∈R^∗\ {−μ, μ}it is easy to see that:

rank(ξ1−ν|ξ₂−ν|ξ₃−sν)=2;

rank(ξ1+ν|ξ2−ν|ξ3−sν)=2; rank(ξ1+ν|ξ2+ν|ξ3−sν)=2;

rank(ξ1−ν|ξ₂+ν|ξ₃−sν)=2;

rank(ξ1−ν|ξ₂−ν|ξ₃+sν)=2; rank(ξ1+ν|ξ2−ν|ξ3+sν)=2;

rank(ξ1+ν|ξ₂+ν|ξ₃+sν)=2;

rank(ξ1−ν|ξ₂+ν|ξ₃+sν)=2, and using Step 1 we deduce thatZW (ξ ) <+∞.

Step 3.We prove thatZW (0) <+∞. This follows from Step 2 by using Proposition 2.2(iv) withD⊂R³given by (7) withs∈R^∗, andφ∈Aff0(D;R³)defined byφ:=(ϕs,ν₁, ϕs,ν₂, ϕs,ν₃)with(ν1, ν2, ν3)∈R³\ {0}, where, for everyi∈ {1,2,3},ϕs,ν_i is defined by (8) witht=νi. 2

Lemma 4.3.Under(C3)there existsc >0such that for everyξ∈M^3×3, if ξis diagonal thenZW (ξ )c

1+ |ξ|^p .

Proof. Combining Lemma 4.2 with Proposition 2.2(ii), we deduce thatZWis continuous, and so there existsc0>0 such thatZW (ξ )c0for all ξ ∈M^3×3with|ξ|²3. Moreover, it is obvious thatZW (ξ )c1(1+ |ξ|^p)for all ξ∈M^3×3such that|detξ|1, wherec1>0 is given by (C3) withδ=1. We are thus led to considerξ∈M^3×3such thatξ is diagonal,|detξ|1 and|ξ|²3, i.e.,ξ=(ξij)withξij=0 ifi=j,

|ξ₁₁ξ₂₂ξ₃₃|1 and |ξ₁₁|²+ |ξ₂₂|²+ |ξ₃₃|²3.

Then, one the six possibilities holds:

(i) |ξ₁₁|1,|ξ₂₂|1 and|ξ₃₃|1;

(ii) |ξ₂₂|1,|ξ₃₃|1 and|ξ₁₁|1;

(iii) |ξ₃₃|1,|ξ₁₁|1 and|ξ₂₂|1;

(iv) |ξ₁₁|1,|ξ₂₂|1 and|ξ₃₃|1;

(v) |ξ₂₂|1,|ξ₃₃|1 and|ξ₁₁|1;

(vi) |ξ₃₃|1,|ξ₁₁|1 and|ξ₂₂|1.

Claim 1.There exists c₂>0 such that if ξ is diagonal with |detξ|1 and satisfies either (i),(ii) or (iii), then ZW (ξ )c₂(1+ |ξ|^p).

ConsiderD⊂R³given by (7) withs=1, and defineφ∈Aff0(D;R³)byφ:=(ϕ_1,ν1, ϕ_1,ν2, ϕ_1,ν3), where (ν1, ν2, ν3)=

(2,0,0) if (i) is satisfied, (0,2,0) if (ii) is satisfied, (0,0,2) if (iii) is satisfied,

and, for everyi∈ {1,2,3},ϕ_1,νi is defined by (8) withs=1 andt=ν_i. It is then easy to see that for a.e.x∈D, det

ξ+ ∇φ(x)

⎧⎪

⎨

⎪⎩

2|ξ22||ξ33| − |detξ| if (i) is satisfied, 2|ξ11||ξ33| − |detξ| if (ii) is satisfied, 2|ξ11||ξ22| − |detξ| if (iii) is satisfied,

so that|det(ξ+ ∇φ(x))|1. Taking Proposition 2.2(i) into account, using (C3) and noticing that|∇φ(x)| =2√ 3 for a.e.x∈D, we deduce thatZW (ξ )c₂(1+ |ξ|^p)withc₂:=c₁2^p(1+(2√

3)^p).