On the representation of effective energy densities

URL:http://www.emath.fr/cocv/

ON THE REPRESENTATION OF EFFECTIVE ENERGY DENSITIES

Christopher J. Larsen

Abstract. We consider the question raised in [1] of whether relaxed energy densities involving both bulk and surface energies can be written as a sum of two functions, one depending on the net gradient of admissible functions, and the other on net singular part. We show that, in general, they cannot. In particular, if the bulk density is quasiconvex but not convex, there exists a convex and homogeneous of degree 1 function of the jump such that there is no such representation.

AMS Subject Classification. 49Q15, 49Q20, 73M25, 73V25.

Received January 31, 2000. Revised August, 2000.

Introduction

This paper was motivated by a conjecture in [1] concerning the representation of effective energy densities.

Though the statement of the conjecture is limited to densities corresponding to “structured deformations”, it raises an interesting question about all densities that result from relaxation inBV, as we explain below. The question can be described as follows: suppose we have an energy

E(u) :=

Z

Ω

W(∇u)dx+ Z

Su

φ([u], ν)dH^N−1,

where Ω⊂R^N is Lipschitz,u∈SBV(Ω;R^m),∇uis the Radon-Nikodym derivative ofDu with respect toL^N, Su is the complement of the set of Lebesgue points ofu, [u] is the jump in uacrossSu, ν is the normal toSu, H^N−1 is theN−1 dimensional Hausdorff measure, andW, φare continuous with some growth assumptions.

We seek to minimize this energy (perhaps subject to some boundary conditions), and so we consider a minimizing sequence{un}, which will converge to some u. u may not be a minimizer, but in a sense it does reflect energetic optimality, being the limit of an optimal sequence. We associate the limiting energy of the sequence{un}with u, and call this therelaxedor effective energyI(u). That is,I(u) = limn→∞E(un). I(u) can be defined similarly even ifuis not the limit of a minimizing sequence, as follows:

I(u) := infn lim inf

n→∞ E(un) :un→uo

·

Keywords and phrases:Relaxation, quasiconvexity, integral representation.

1Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, U.S.A.;

e-mail:cjlarsen@wpi.edu

c EDP Sciences, SMAI 2000

A basic question in relaxation theory is to characterize I(u), especially as an integral of new densities: an effective bulk densityW^∗, aneffective interfacial densityφ^∗, and perhaps others, so that, for example,

I(u) = Z

Ω

W^∗(∇u)dx+ Z

Su

φ^∗([u], ν)dH^N−1.

The model proposed in [4] concerns the fact that ∇u may differ from the appropriate limit of {∇un}. The reason for this is that the functions un may have a considerable proportion of their variation on their jump sets Su_n, so that ∇un does not reflect their variation, yet the limit u may be smooth, or at least smooth in regions in whichun are not smooth. The point is that when the functionsu: Ω→R^m represent deformations, the difference∇u−limn→∞∇un can be considered to be a measure of deformation due to “disarrangements”.

It is then natural to consider one energy density on these disarrangements and a separate density on smooth deformation, reflected in limn→∞∇un. Motivated by this point of view [1] formulated the relaxation explicitly in terms of the limit of{∇un}. With the functionGrepresenting a possible limit of{∇un}, they define

I(u, G) := infn lim inf

n→∞ E(un) :{un} ⊂SBV(Ω;R^m), un→uinL¹(Ω;R^m),∇un ∗

* Gas measureso , where we have ignored some technical requirements on the admissible sequences {un}. They show that the effective bulk term is

Z

Ω

H(∇u(x), G(x))dx

and ifφis homogeneous of degree one, the effective bulk densityH is given by H(A, B) := inf

E(u) :u∈SBV(Q;R^m), u|^∂Q=Ax, and Z

Q∇udx=B

, (0.1)

where A, B∈M^m×N, the space ofm×N matrices, andQis a unit cube inR^N.

The conjecture is that this density is a sum of a function ofG(x) (representing the limit of the elastic parts of the deformations{un}) and a function of∇u(x)−G(x) (representing the limit of the disarrangement parts of the deformations). Precisely,H(A, B) =F1(B) +F2(A−B) for some functions F1 andF2.

The consequences for general effective energies stem from the fact that typically, the effective bulk density W^∗ is of the form

W^∗(A) = inf{E(u) :u∈SBV(Q;R^m), u|^∂Q=Ax}, which is evidently the same as

inf{H(A, B) :B∈M^m×N}·

Since H is generally continuous, any representation for H would translate into a representation for W^∗. A similar result holds for effective interfacial densitiesφ^∗.

We show that in general, there cannot be such a representation. The idea is the following. From (0.1), we see thatAreflects the net total derivative of the admissible functions, whileB gives the net gradient of admissible functions. A−B is then the net singular part of admissible functions. The point is that the presence of some singular part ofDucan allow a decrease inR

QW(∇u)dx, even without a change in the net gradient. This is due to the fact that “gradients” of BV functions need not be curl-free – they are not true gradients. R

QW(∇u)dx can then approach the convex envelope ofW at B, whereas if there is no singular part, theBV “gradient” is a true gradient, and R

QW(∇u)dx cannot be less thanW(B) ifW is quasiconvex. ButW(B) may be strictly

greater than the convex envelope ofW atB. Once it makes sense to have some singular part ofDu, there is no reason to expect it to be energetically optimal to have zero net singular part, yet this is a consequence of the representation.

Consistent with the assumptions in [1], we assume that W, φ≥0,W(0) =φ(0, ν) = 0, and both functions are continuous. For simplicity, we take N =m= 2, though the proof is straightforward to extend to higher dimensions. The first step is to prove

Lemma 0.1. IfH(A, B) =F1(B) +F2(A−B) for some functionsF1,F2, then minH(·, B) =H(B, B)

for all B∈M^2×2.

The contradiction to the representation then follows from

Theorem 0.2. If W:M^2×2→R is quasiconvex but not convex and has growth p≥1, there exists B ∈M^2×2 and a functionφ:R²×S¹→R that is linearly coercive, positive homogeneous of degree 1, and subadditive in the first variable such that

minH(·, B)6=H(B, B).

These are proved in Section 3 below. Section 1 contains some preliminaries, and Section 2 develops some analysis on sequences inBV that we use to prove Theorem 0.2.

1. Preliminaries

We consider a bounded, open set Ω⊂R^N with Lipschitz boundary, and we define the spacesL^p(Ω;R^m), the Sobolev spacesW^1,p(Ω;R^m), and the spaceBV(Ω;R^m) in the usual way (see,e.g.[5]). Foru∈BV(Ω;R^m), we define |Du|(Ω) :=Pm

i=1|Dui|(Ω). From now on, we will usually just write L^p(Ω) forL^p(Ω;R^m), and similarly for W^1,pandBV.

For u∈ BV(Ω), we write Du =Dacu+Dsu, whereDacu andDsu stand for, respectively, the absolutely continuous and singular part ofDu with respect toL^N. We also consider the set Su of points which are not Lebesgue points for u, and recall thatSu isN−1-rectifiable, and so it has a normal,ν,H^N−1-a.e. We use the representationsDacu=∇uL^N and (Dsu)bSu= [u]⊗νH^N−1bSu, so we have the decomposition

Du=∇uL^N+ [u]⊗νH^N−1bSu+C(u),

where C(u) := (Dsu)b(Ω\Su), [u] is the jump in uacrossSu, i.e. [u] = u⁺−u⁻, where u⁺ and u⁻ are the traces ofuon either side ofSu. IfC(u) = 0, then we sayuis a special function of bounded variation, and we writeu∈SBV(Ω). This space was introduced in [3].

We denote the space ofm×N matrices byM^m×N, and we considerW:M^m×N→Randφ:R^m×S^N−1→R. Foru∈BV(Ω;R^m), we define

E(u) :=

Z

Ω

W(∇u)dx+ Z

Su

φ([u], ν)dH^N−1.

We sayW (and alsoE) has growthpif there exist positive constantsc1 andc2such that c1|A|^p− 1

c1 ≤W(A)≤c2(|A|^p+ 1) for allA∈M^m×N.

Recall that a functionW:M^m×N→R with growthpisquasiconvexif W(A)≤Z

S

W(∇ϕ)dx

for allϕ∈Ax+W₀^1,p(S;R^m) and allA∈M^m×N, whereS⊂R^N is open and satisfies L^N(S) = 1 (see [2]).

Throughout, we usec to designate a constant that may change from line to line, depending only onN, m, and perhaps established uniform bounds. We will also use the fact that foru, v∈BV(Ω),∇u=∇v L^N almost everywhere in the set{u=v}.

2. Sequences in BV

We begin with some analysis of sequences inBV that will be useful in proving Theorem 0.2. The following is an extension of Lemma 1.2 in [6] to certain sequences inBV.

Lemma 2.1. Let {ui}be a bounded sequence inBV(Ω) such that{∇ui}is bounded in L^p(Ω) for some p >1 and|Dsui|(Ω)→0. Then there exists a sequence{wi}bounded inW^1,p(Ω)such that{|∇wi|^p}is equiintegrable and

L^N({ui6=wi or ∇ui6=∇wi})→0.

Proof. The proof is based on ideas from Theorem 2 in Section 6.6.2 and Theorem 3 in Section 6.6.3 of [5], together with Lemma 1.2 of [6]. We first define

R^λi :=

(

x∈Ω : 1 L^N(B(x, r))

Z

B(x,r)

d|Dui| ≤λ ∀r >0 )

and

Si^λ:=Si∪

x∈Ω :|∇ui|(x)> λ 2

,

where Si has measure zero and satisfies|Dsui|(Si) =|Dsui|(Ω). For each i∈N, we see from Vitali’s covering theorem that there exists a countable set of disjoint ballsB(xj, rj) such that

Ω\R_i^λ⊂ ∪^∞j=1B(xj,5rj) and

1 L^N(B(xj, rj))

Z

B(x_j,r_j)

d|Dui|> λ.

We then have

λL^N(∪^∞j=1B(xj, rj))<|Dui|(∪^∞j=1B(xj, rj)). (2.2) Now write

|Dui|(∪^∞j=1B(xj, rj)) =|Dui|(S_i^λ∩ ∪^∞j=1B(xj, rj)) +|Dui|([Ω\S_i^λ]∩ ∪^∞j=1B(xj, rj)),

and it follows from the definition ofS_i^λ that

|Dui|([Ω\S_i^λ]∩ ∪^∞j=1B(xj, rj))≤ λ

2L^N(∪^∞j=1B(xj, rj)).

We therefore have, from (2.2),

λ≤ 1

L^N(∪^∞j=1B(xj, rj))|Dui|(S^λ_i ∩ ∪^∞j=1B(xj, rj)) +λ 2 and so

λ≤ 2

L^N(∪^∞j=1B(xj, rj))|Dui|(S_i^λ∩ ∪^∞j=1B(xj, rj)) and

L^N(∪^∞j=1B(xj, rj))≤ 2

λ|Dui|(S_i^λ∩ ∪^∞j=1B(xj, rj)).

We now have that

L^N(Ω\R^λ_i) ≤cL^N(∪^∞j=1B(xj, rj))≤ c

λ|Dui|(S_i^λ∩ ∪^∞j=1B(xj, rj))≤ c λ

"

|Dsui|(Ω) + Z

S_i^λ

|∇ui|dx

#

≤ c λ



|Dsui|(Ω) + Z

S^λ_i |∇ui|^pdx

!¹_p

L^N(S_i^λ)^1−1p



.

We are given that Z

S_i^λ

|∇ui|^pdx≤M for someM >0, and from the definition ofS_i^λ we have that Z

S_i^λ|∇ui|^pdx≥ L^N(Si^λ) λ

2 p

, so that

L^N(S_i^λ)≤ c λ^p· We now have

L^N(Ω\R^λ_i) ≤ c λ



|Dsui|(Ω) + Z

S_i^λ

|∇ui|^pdx

!¹_p 1 λ^p−1



≤ c

λ|Dsui|(Ω) + c λ^p· Hence,

λ^pL^N(Ω\R^λ_i)≤cλ^p−1|Dsui|(Ω) +c.

We then chooseλ(i) such that

λ(i)^p−1|Dsui|(Ω) = 1

and λ(i)^pL^N(Ω\R_i^λ(i)) is then bounded. From the proof of Theorem 2 in Section 6.6.2 of [5], we know that there exist Lipschitz functionsvi with Lip(vi)≤cλ(i) andvi =ui onR^λ(i)_i . Hence,

Z

Ω|∇vi|^pdx≤ Z

R^λ(i)_i |∇ui|^pdx+cλ(i)^pL^N(Ω\R^λ(i)_i ) and {∇vi}is bounded in L^p(Ω). Note that since{ui6=vi} ⊂Ω\R_i^λ(i), we have

L^N({ui6=vi})→0.

It now follows from the fact that {ui}is bounded inL¹(Ω) and the Poincar´e inequality that {vi}is bounded inW^1,p(Ω). To see this, suppose||vi||^p→ ∞. Setyi:= _||v^vi

i||p, so||yi||^p= 1 and||∇yi||^p→0. Hence, yi→y_∞ inL^p, andy_∞is a constant. Now setzi:= _||v^ui

i||p. ThenL^N({zi6=yi})→0 and||zi||¹→0. Therefore,y_∞= 0, contradicting||yi||^p = 1.

By Lemma 1.2 in [6], there is a sequence{wi}bounded in W^1,p(Ω) with L^N({vi6=wi})→0

and{|∇wi|^p}is equiintegrable. We then also haveL^N({ui6=wi})→0, and{wi}satisfies the conclusion of the lemma.

The following is an easy consequence. Note that the conclusion below is false when p= 1, the idea being that concentrations in{|∇ui|}can serve a purpose – they can cause nonsmooth variations in the limit, whereas effects of concentrations only in{|∇ui|^p}disappear in the limit.

Remark 2.2. If{ui}is a sequence inBV(Ω)with|Dsui|(Ω)→0and minimizing an energy with growthp >1, then{|∇ui|^p}is equiintegrable.

We first note that a sequence{ui}inW^1,p(Ω) minimizing an energy with growthp >1 satisfies the hypotheses of the remark, but equiintegrability for that case could be proved directly from Lemma 1.2 in [6], with no need for Lemma 2.1 here.

To see the conclusion of the remark, suppose we have a sequence{ui}such that{|∇ui|^p}is not equiintegrable.

Then there existsδ >0 and a sequence of sets{Ei}such thatR

E_i|∇ui|^pdx≥δandL^N(Ei)→0. We then use Lemma 2.1 to find{wi}equiintegrable and such that, forTi:={ui6=wi}, we haveL^N(Ti)→0. Then

Z

Ω

W(∇ui)dx = Z

Ω\(E_i∪T_i)

W(∇wi)dx+ Z

E_i∪T_i

W(∇ui)dx

= Z

Ω

W(∇wi)dx− Z

E_i∪T_i

W(∇wi)dx+ Z

E_i∪T_i

W(∇ui)dx

≥ Z

Ω

W(∇wi)dx−c2

Z

E_i∪T_i|∇wi|^pdx−c2L^N(Ei∪Ti) +c1δ− 1

c1L^N(Ei∪Ti) so that, since{|∇wi|^p}is equiintegrable,

lim inf

i→∞

Z

Ω

W(∇ui)dx≥lim inf

i→∞

Z

Ω

W(∇wi)dx+c1δ and{ui}is not minimizing.

We now have the following, which is key to proving Theorem 0.2.

Lemma 2.3. Letβ >0,W quasiconvex with growth p≥1, andA∈M^m×N be given. Then there existsC >0 such that if u∈BV(Q)and

i) u|^∂Q=Ax ii)

Z

Q

W(∇u)dx≤W(A)−β, then|Dsu|(Q)> C.

Proof. We suppose to the contrary that fori∈Nthere existsui∈BV(Q) satisfying i) and ii), but|Dsui|(Q)<

1

i. If p > 1, then {ui} satisfies the hypotheses of Lemma 2.1, and so we choose a sequence {wi} as allowed by that lemma. Since {wi}is bounded in W^1,p(Ω), there is a subsequence such thatwi * w in W^1,p(Ω) for somew. Since a subsequence of {ui}converges inL¹(Ω) to the same function with |Dsui|(Q)→0 and{∇ui} bounded inL^p(Ω), we have

w|^∂Q =Ax.

Yet, sinceL^N({ui6=wi})→0 and{|∇wi|^p}is equiintegrable, we have lim inf

i→∞

Z

Q

W(∇wi)dx≤lim inf

i→∞

Z

Q

W(∇ui)dx≤W(A)−β.

From the weak lower semicontinuity of integrals of quasiconvex functions, we then get W(A)≤Z

Q

W(∇w)dx≤lim inf

i→∞

Z

Q

W(∇wi)dx≤W(A)−β,

a contradiction. We note that, as pointed out by the referee, this case also follows from results in [7], without the need for Lemma 2.1.

For the casep = 1, we get the same contradiction using Lemma 3.1 in [8] and rescaling{ui} so that the rescaled sequence converges toAxin L¹(Ω) without changingR

QW(∇ui)dx.

3. The representation theorem

We now use the previous analysis ofBV sequences to prove Theorem 0.2. The contradiction to the repre- sentation will come from the following.

Lemma 0.1 IfH(A, B) =F1(B) +F2(A−B)for some functionsF1,F2, then

minH(·, B) =H(B, B) (3.3)

for all B∈M^2×2.

Proof. Suppose there are suchF1 andF2. It is immediate from the definition (0.1) and the fact thatW, φ≥0 with W(0) =φ(0, ν) = 0 thatH ≥0 andH(0,0) = 0. It follows that F1(0) +F2(0) = 0, or F1(0) =−F2(0).

We can then assume that

F1(0) =F2(0) = 0

since by considering

F₁⁰(A) :=F1(A)−F1(0) and correspondingF₂⁰, we have the representation

H(A, B) =F₁⁰(B) +F₂⁰(A−B) withF₁⁰(0) =F₂⁰(0) = 0.

Now, since

H ≥0,

H(B, B) =F1(B) +F2(0) =F1(B), and

H(A,0) =F1(0) +F2(A) =F2(A), it follows thatF1≥0 andF2≥0. Therefore,

H(A, B) =F1(B) +F2(A−B)≥F1(B) =H(B, B) which gives (3.3).

Below we show that (3.3) is not, in general, true. We first note that from [9], we know that there exists a quasiconvex functionW that is homogeneous of degree 1 but not convex, and soW satisfies the hypotheses of Theorem 0.2.

Theorem 0.2. If W :M^2×2 →R is quasiconvex but not convex and has growth p ≥ 1, then there exists B ∈M^2×2 and a continuous functionφ:R²×S¹→Rthat is linearly coercive, positive homogeneous of degree1, and subadditive in the first variable such that

minH(·, B)6=H(B, B).

Proof. Forε, K >0 to be chosen later, set φ(ξ, ν) :=

X2 j=1

"

ε X2 k=1

(ξkν·ej)⁺+K X2 k=1

(ξkν·ej)⁻

# ,

which is linearly coercive, homogeneous of degree one, and subadditive inξ. We then choose B ∈M^2×2 such that CW(B)< W(B), whereCW is the convex hull ofW. Then we can writeB as a convex combination of Bi’s, B =P

λiBi, with λi rational and such thatP

λiW(Bi)< W(B). We will now use thisB to construct matricesA andB⁰ and a functionv⁰ admissible forH(A, B⁰) with energy less thanH(B⁰, B⁰).

Setγk := maxi||B_i^kx||^L∞(Q), where B_i^k is the kth row of Bi, k= 1,2, andQ is the cube (0,1)². We then set γ:= maxkγk. Put ann×n grid onQ, wherenis a multiple of all the denominators of the λi and will be chosen later. We now definev ∈SBV(Q;R²). Forl= 1, . . . , n, define v on (^l−_n¹,_n^l)×(0,_n¹) by∇v=Bil and,

writingv in terms of its components, v = (v1, v2), we have minv1= minv2= ^(l−_n¹⁾γ. Here theBi_l are chosen so that for l = 1, . . . , λ1nwe have Bi_l =B1, and so on, so that eachBi is the gradient of v in λin of then squares (^l−_n¹,_n^l)×(0,_n¹),l = 1, . . . , n. We then extendv to (0,1)×(_n¹,1) byvk(x) :=vk(x−^l−n¹e2) + ^l−_n¹γ, k= 1,2 forx∈(0,1)×(^l−_n¹,_n^l),l= 2, . . . , n.

From the definition ofγ and the construction ofv, it follows that [vk]ν·ej ≥0 forj, k= 1,2. Notice that, setting A to be the 2×2 matrix with every entry γ, we have ||v−Ax||^L∞(∂Q) ≤ 2_n^γ. It also follows that

|Dsv|(Q)<8γ. To see this, notice that the variation of each componentvk (by which we mean supvk−infvk) in each square of the grid is at most ^γ_n, and by construction ofv, the variation over two adjacent squares is at most 2_n^γ. Therefore,|[vk]| ≤2^γ_n between adjacent squares. Furthermore, since

Sv_k⊂

{1/n,2/n, . . . ,(n−1)/n} ×(0,1)

∪

(0,1)× {1/n,2/n, . . . ,(n−1)/n} , we haveH¹(Sv_k)≤2(n−1). Hence,|Dsvk|(Q)≤2^γ_n2(n−1)<4γ and|Dsv|(Q)<8γ.

We extendv outsideQbyv(x) =Ax, and for δ >0, define vδ(x) :=v((1 +δ){x−x¯}+ ¯x), where ¯xis the center ofQ. ThenW(R

Q∇vδdx)→W(B) andR

QW(∇vδ)dx→ P

iλiW(Bi) asδ→0. We chooseδ so that W(R

Q∇vδdx)>R

QW(∇vδ)dx and rename thisvδ v⁰. Setα:=W(R

Q∇v⁰dx)−R

QW(∇v⁰)dxandB⁰ :=R

Q∇v⁰dx. We now have thatv⁰ is admissible forH(A, B⁰)

and Z

Sv0

φ([v⁰], ν)dH¹<2ε|Dsv|(Q) + 8Kγ n, where the last term comes from the fact that ||v−Ax||L^∞(∂Q)≤2^γ_n.

We chooseε >0 such that

2ε|Dsv|(Q)<α 3·

Now, anyuadmissible forH(B⁰, B⁰) requires zero net jump, and so the jumps will be just as much in the minus ej directions as in the positive. That is,

Z

Q

dDsu= 0 implies

Z

Q

[uk]ν·ejdH¹= 0 so that

Z

Q

([uk]ν·ej)⁺dH¹= Z

Q

([uk]ν·ej)⁻dH¹ forj, k∈ {1,2}. Hence

E(u)> 1

2K|Dsu|(Q).

From Lemma 2.3 withβ =^α₃, there existsC >0 such that ifuis admissible forH(B⁰, B⁰) and satisfies Z

Q

W(∇u)dx≤W(B⁰)−α 3,

then|Dsu|(Q)≥C. An easy calculation shows that ifK > _C²(W(B⁰)−^α3), thenu∈SBV(Q;R²) with trace B⁰xandR

Q∇udx=B⁰ implies

E(u)> W(B⁰)−α 3·

Choosing n large enough so that 8K^γ_n < ^α₃ gives H(A, B⁰) < W(B⁰)− ^α₃ with H(B⁰, B⁰) ≥ W(B⁰)− ^α₃, contradicting minH(·, B⁰) =H(B⁰, B⁰) from Lemma 0.1.

References

[1] R. Choksi and I. Fonseca, Bulk and interfacial energy densities for structured deformations of continua.Arch. Rational Mech.

Anal.138(1997) 37–103.

[2] B. Dacorogna,Direct Methods in the Calculus of Variations.Springer-Verlag, Berlin (1989).

[3] E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni.Atti. Accad. Naz. Lincei Cl. Sci. Fis.

Mat. Natur. Rend. Lincei (9) Suppl.82(1988) 199–210.

[4] G. Del Piero and D.R. Owen, Structured deformations of continua.Arch. Rational Mech. Anal.124(1993) 99–155.

[5] L.C. Evans and R.F. Gariepy,Measure Theory and Fine Properties of Functions.CRC Press, Boca Raton (1992).

[6] I. Fonseca, S. M¨uller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients.SIAM J. Math.

Anal.29(1998) 736–756.

[7] J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions.Math. Ann.313(1999) 653–710.

[8] C.J. Larsen, Quasiconvexification inW^1,1and optimal jump microstructure inBV relaxation.SIAM J. Math. Anal.29(1998) 823–848.

[9] S. M¨uller, On quasiconvex functions which are homogeneous of degree 1.Indiana Univ. Math. J.41(1992) 295–301.