On the integral representation of relaxed functionals with convex bounded constraints

DOI:10.1051/cocv:2008063 www.esaim-cocv.org

ON THE INTEGRAL REPRESENTATION OF RELAXED FUNCTIONALS WITH CONVEX BOUNDED CONSTRAINTS

Omar Anza Hafsa

Abstract. We study the integral representation of relaxed functionals in the multi-dimensional cal- culus of variations, for integrands which are finite in a convex bounded set with nonempty interior and infinite elsewhere.

Mathematics Subject Classification.49J45.

Received February 14, 2008.

Published online October 21, 2008.

1. Introduction and main results

Letm, d≥1 be two integers. Let Ω⊂R^d be a nonempty open bounded domain with Lipschitz boundary.

In this paper we consider the problem of the integral representation of I(u) = inf

lim inf

n→+∞

Ωf(∇un(x))dx:W^1,∞(Ω;R^m)u_n→uinL¹

,

where f :M^m×d→[0,+∞] is a Borel measurable function and with M^m×d denotes the set ofm×dmatrices.

We denote by domf the effective domain off, i.e., domf ={ξ∈M^m×d :f(ξ)<+∞}. We are interested in integrands satisfying domf ⊂C, whereCis a convex bounded set with nonempty interior. The classical integral representation results of relaxed functionals in the vectorial case (i.e. when min{d, m} ≥2) require polynomial growth conditions (or at least integrands which are finite everywhere) on the integrands which do not allow us to deal with constraints on gradients. However, it is interesting for the applications in nonlinear elasticity to consider such constraints for problems, such as the elastic-plastic torsion problems and the modelling of rubber-like nonlinear elastomers as described by Carbone and De Arcangelis in [6]. In that book, we can find a detailed study of the problems of integral representation of relaxed functionals under constraints (not necessarily bounded) in the scalar case, i.e., min{d, m} = 1. In the vectorial case, and in the presence of some singular behaviors of the stored energy functions in nonlinear elasticity, we can find some relaxation results where the integrands can take the value +∞, see [2,3,5]. Moreover, recently, in connection with relaxation problems in optimal control, Wagner [16] studies the relaxation of integral functional with the assumption that f is continuous finite onC, and infinite elsewhere.

Keywords and phrases. Relaxation, convex constraints, integral representation.

1 Universit´e Montpellier II, UMR-CNRS 5508, LMGC, Place Eug`ene Bataillon, 34090 Montpellier, France.

[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2008

In this paper, we study the integral representation ofI for two classes of integrands (we will make precise the assumptions later). Firstly, we consider a class of integrands which are locally bounded on intC (the interior ofC) and which allow us to consider singular behavior of the type

f(ξ)→+∞ as ξ→∂C and domf ⊂C.

Secondly, we consider a class of integrands which are bounded on intC, which is in some sense a “complementary”

class of the previous one. Similar to the classical relaxation results in the vectorial case, we will deal with the quasiconvex envelope off. However, the definition of quasiconvex envelope is not obvious when f is not everywhere finite. We avoid the difficulties connected with this problem by studying the possibility of monotone nondecreasing approximation of the quasiconvex envelope of f by quasiconvex functions (which are finite by definition, see below).

1.1. Some preliminary notions

Following Morrey [12] we say that a functiong:M^m×d→[0,+∞[ isquasiconvex atξ∈M^m×d if it is Borel measurable and

g(ξ) = inf

Y

g(ξ+∇φ(x))dx:φ∈W₀^1,∞(Y;R^m)

,

where Y = ]0,1[^d is the unit cube inR^d. If g is quasiconvex at everyξ∈M^m×d then g is said quasiconvex. If g is quasiconvex then it is continuous (see for instance Dacorogna [7]).

Let us define byQf :M^m×d→[0,+∞] thequasiconvex envelopeoff defined by Qf(ξ) = sup

g(ξ) :g:M^m×d→[0,+∞[ is quasiconvex andg≤f .

Note thatQf is lower semicontinuous as pointwise supremum of continuous functions, and satisfies Qf(ξ) = inf

Y

Qf(ξ+∇φ(x))dx:φ∈W₀^1,∞(Y;R^m)

(1.1) for allξ∈M^m×d.

Leth:M^m×d →[0,+∞] be a Borel measurable function. We say thathis p-sup-quasiconvex if there exist p∈[1,+∞[ and a nondecreasing sequence{hn}_n∈N,h_n:M^m×d→[0,+∞[ such that

(i) h_n is quasiconvex for alln∈N;

(ii) for everyn∈Nthere existsα_n>0 such thath_n(ξ)≤α_n(1 +|ξ|^p) for allξ∈M^m×d; (iii) for everyξ∈M^m×d we have that

sup{hn(ξ) :n∈N}=h(ξ).

It is easy to see that if h is p-sup-quasiconvex then it is lower semicontinuous as pointwise supremum of continuous functions and satisfies

h(ξ) = inf

Y

h(ξ+∇φ(x))dx:φ∈W₀^1,∞(Y;R^m)

for allξ∈M^m×d.

DefineZf:M^m×d→[0,+∞] by Zf(ξ) = inf

Y

f(ξ+∇φ(x))dx:φ∈W₀^1,∞(Y;R^m)

.

In fact Zf does not depend onY, see LemmaA.1. If Zf is everywhere finite, thenZf =Qf (see Lem.A.2), andZf is called theDacorogna formulaof the quasiconvex envelope off.

In the rest of this paper we will use frequently some properties of convex sets which are summarized asline segment principle by Rockafellar and Wets [14], Theorem 2.33.

Line segment principle (l.s.p.):

Let C⊂M^m×d be a bounded convex set with0∈intC. Then

intC = intC, intC=C, and tC ⊂intC for all t∈[0,1[.

1.2. Main results

LetC⊂M^m×d be a bounded convex set with nonempty interior. To simplify the statements we will assume through the paper that 0∈intC. Letf :M^m×d→[0,+∞] be a Borel measurable function such that domf ⊂C.

We consider the following assertion:

(H₀) for everyε >0 there existsη >0 such that for everyξ∈intCand everyt∈[0,1[ we have 1−t≤η =⇒ Zf(tξ)≤ Zf(ξ) +ε.

1.2.1. Integral representation for integrands locally bounded on intC Consider the following assertions:

(H1) f is locally bounded on intC,i.e., sup{f(ξ) :ξ∈K}<+∞for all compact setsK⊂intC;

(H₂) for everya >0 there exists a compact set K_a⊂intCsuch that for everyξ∈C, ξ /∈K_a =⇒ Zf(ξ)≥a.

Theorem 1.1. Assume that (H0), (H1) and (H2) hold. Then Qf is 1-sup-quasiconvex and we have the representation formula

Qf(ξ) =

Zf(ξ) if ξ∈intC +∞ otherwise.

Theorem 1.2. Assume that (H₀), (H₁)and (H₂)hold. Then I(u) =

⎧⎨

⎩

ΩQf(∇u(x))dx if u∈W^1,∞(Ω;R^m)

+∞ if u∈L¹(Ω;R^m)\W^1,∞(Ω;R^m).

1.2.2. Integral representation for integrands bounded on intC Consider the following assertions:

(H3) f is bounded on intC, i.e., sup{f(ξ) :ξ∈intC}<+∞;

(H4) for everyξ∈∂Cwe have lim inf_[0,1[t→1f(tξ)≤f(ξ).

Theorem 1.3. Assume that (H0), (H3) and (H4) hold. Then Qf is 1-sup-quasiconvex and we have the representation formula

Qf(ξ) =Zf(ξ) =

⎧⎨

⎩

Zf(ξ) if ξ∈intC lim_[0,1[t→1Zf(tξ) if ξ∈∂C

+∞ otherwise.

(1.2) The above representation formula for Qf was found by Wagner in [15] (Zf denotes the lower semicontinuous envelope ofZf).

Theorem 1.4. Assume that (H₀), (H₃)and (H₄)hold. Then I(u) =

⎧⎨

⎩

ΩQf(∇u(x))dx if u∈W^1,∞(Ω;R^m)

+∞ if u∈L¹(Ω;R^m)\W^1,∞(Ω;R^m).

Remark 1.1. (i) In [6], Theorem 10.2.4, Carbone and De Arcangelis use similar assumption (H₀) for the problem of the integral representation of relaxed functionals, in the scalar case. They translate that assumption as a type of “uniform radial upper semicontinuity” on intC (see Rem. 10.1.1 in [6]).

The assertion (H0) is satisfied when for instance intC⊂domf ⊂C, and for everyε >0 there existsη >0 such that for everyξ∈domf and everyt∈[0,1[ it holds

1−t≤η =⇒ f(tξ)≤f(ξ) +ε 2·

Indeed, take ε >0, η >0 as above. Let t ∈[0,1[ be such that 1−t ≤η. Letξ ∈intC. Then we have that Zf(ξ)≤f(ξ)<+∞and there existsφ_ε∈W₀^1,∞(Y;R^m) such that

ε

2+Zf(ξ)≥

Y

f(ξ+∇φε(x))dx. (1.3)

Thusξ+∇φε(x)∈domf a.e. inY, and we have Zf(tξ)≤

Y

f(t(ξ+∇φ_ε(x)))dx≤ ε 2+

Y

f(ξ+∇φ_ε(x))dx.

By (1.3), we obtainZf(tξ)≤ Zf(ξ) +ε.

(ii) The assertion (H₂) is satisfied when for instancef ≥ψfor some convex function ψ satisfying for every a >0 there exists a compact setK_a ⊂intC such that for everyξ∈C,

ξ /∈K_a =⇒ ψ(ξ)≥a.

(iii) Letf :M^m×d→[0,+∞] be defined by

f(ξ) =

⎧⎨

⎩

g(ξ) + 1

1− |ξ| if|ξ|<1

+∞ otherwise,

whereg:M^m×d →[0,+∞[ is uniformly continuous. In view of (i) and (ii), we have that (H0), (H1) and (H2) are satisfied.

(iv) Note that (H3) and (H4) are satisfied if, for instance,f_C∈ C(C).

1.3. Comments on Zf

To our best knowledge, the formulaZf first appeared for arbitrary Borel measurable functionf in Ball and Murat [4], p. 240. Then Fonseca in [10] studied the rank-one convexity property of Zf for arbitrary Borel measurablef, in particular she showed that Zf is continuous on int(domf) (Thm. 2.17 and Prop. 2.3). Later Kinderlehrer and Pedregal in [11] considered Zf for functionsf satisfying

f_B∈ C(B) andf_Mm×d\B= +∞, (1.4)

withB ={ξ∈M^m×d:|ξ|<1} ⊂M^m×dis the unit ball. They showed, in particular, thatZf satisfies (1.4) and Z(Zf) =Zf in their Proposition 7.2. In the paper [8] Dacorogna and Marcellini studied Zf in Theorem 7.2 for the class of functions satisfying (1.4) whereB is replaced by an arbitrary compact convex set K⊂M^m×d with nonempty interior. Recently, the work of Wagner in [15] gives a detailed study ofZf withf satisfying the same assumptions as in Dacorogna and Marcellini [8].

1.4. Outline

An outline of the paper is shown as follows: We start by some preliminary lemmas, where we are mainly concerned with establishing some properties ofZf. The proofs of Theorems1.1and 1.3are achieved by using some arguments of M¨uller [13]. We give the proofs of Theorems1.2 and 1.4by dividing them into two steps.

The proof of the lower bound follows easily after we have shown that Qf is 1-sup-quasiconvex. To prove the upper bound, we use an approximation result due to Dacorogna and Marcellini [8]. In the appendix, we give some results concerningZf.

2. Preliminaries

Lemma 2.1.

(i) For every increasing sequence {tn}_n∈N^∗ ⊂[0,1[satisfying lim_n→+∞t_n= 1, it holds

intC=

t∈[0,1[

tC=

n∈N

t_nC, andintC=

t∈[0,1[

tintC=

n∈N

t_nintC.

(ii) If K⊂intC is compact thenK⊂tintC for somet∈[0,1[.

(iii) The functionf is locally bounded on intC if and only if sup{f(ξ) :ξ∈tC}<+∞for allt∈[0,1[.

(iv) Assume that f is locally bounded on intC. Then (H2) holds if and only if there exists an increasing sequence such that[0,1[t_n→1, and for everyn≥1

ξ∈C\tinfnCZf(ξ)≥n.

Proof. (i) Let {tn}_n∈N^∗ ⊂[0,1[ be an increasing sequence such that lim_n→+∞t_n = 1. Let ξ ∈ intC and set D=∪_n≥1t_nC, thent_nξ∈D for alln∈N^∗ by l.s.p. We deduce thatξ∈D and it holds

intD⊂D⊂intC⊂D. (2.1)

Note that firstlyDis convex since{tn}_n∈N^∗ is increasing, and secondly 0∈intD. Thus, by l.s.p., intD= intD, and by (2.1) it follows that intC= intD=D and we obtain

intC=

t∈[0,1[

tC=

n∈N

t_nC.

Now, note that by l.s.p., it holds t_nC ⊂ t_n+1intC for all n ∈ N^∗. Thus for every n ∈ N^∗ we have t_nC ⊂

∪_n∈N^∗t_nintC, hence intC⊂ ∪_n∈N^∗t_nintC.

(ii) Let K ⊂intC be a compact set. Assume that for everyn∈N^∗ there exists x_n ∈K andx_n ∈/ t_nintC where t_n= ⁿ⁻¹_n . By compactness, there exists a converging subsequenceKx_σ(n)→x∈K asn→+∞. By (i) we havet_σ(1)intC⊂ · · · ⊂t_σ(n)intC⊂ · · · ⊂ ∪n≥1t_σ(n)intC= intC. For everyk, n∈N^∗

x_σ(n+k)∈K\t_σ(n+k)intC⊂K\t_σ(n)intC.

Letting k→+∞, we obtainx∈K\ ∪n∈N^∗t_σ(n)intC=∅, which is impossible sinceK is compact.

(iii) Follows from (ii).

(iv) By (H₂), we can find a sequence of compact set{Kn}_n∈N^∗ ⊂intC such that K₁⊂K₂⊂ · · · ⊂K_n⊂ · · · ⊂

n∈N^∗

K_n =K_∞, and inf

C\Kn

Zf ≥nfor alln≥1.

Thus inf_C\K∞Zf = +∞. Assume thatK_∞= intC, then there existsξ₀∈intC\K_∞such thatZf(ξ₀) = +∞.

But by (i) and (iii) we obtain +∞=Zf(ξ₀)≤f(ξ₀)<+∞ which is impossible. ThusK_∞ = intC. By (ii), we can build an increasing sequence {t_n}n∈N^∗ ⊂ [0,1[ such that K_n ⊂ t_nC for all n ∈ N^∗. It follows that intC=K_∞=∪n≥1t_nCand therefore the sequencet_n→1 asn→+∞, indeed we cannot haveτ= sup_n≥1t_n= lim_n→+∞t_n <1, otherwise, by l.s.p. intC⊂τintC which is impossible since intC=∅. We also have for every n∈N^∗

ξ∈C\tinfnCZf(ξ)≥ inf

ξ∈C\Kn

Zf(ξ)≥n.

The other implication is easier. Leta >0. Let n≥aand choose K_a =t_nC then inf_C\K

aZf ≥n≥a. The

proof is complete.

Lemma 2.2. Assume that (H1)holds. Then

intC⊂domf ⊂domZf ⊂domZf ⊂domQf ⊂C and

int(domf) = int(domZf) = int(domZf) = int(domQf) = intC.

Proof. By definition ofQf andZf we have

Qf ≤ Zf ≤ Zf ≤f.

By Lemma2.1(i) and (ii), ifξ∈intC then there existst∈[0,1[ such thatξ∈tC and Qf(ξ)≤ Zf(ξ)≤ Zf(ξ)≤f(ξ)≤ sup

ζ∈tC

f(ζ)<+∞.

We deduce intC ⊂domf ⊂domZf ⊂domZf ⊂domQf. Now, we will show domQf ⊂C. For each n∈N, consider the function g_n :M^m×d →[0,+∞[ defined byg_n(ξ) =ndist(ξ, C). It is easy to see thatg_n is convex and then quasiconvex since Jensen inequality, andg_n≤f for alln∈N. Thusg_n ≤ Qf for alln∈N, and the inclusion domQf ⊂C follows by noticing that

supn∈Ng_n(ξ) =

0 ifξ∈C, +∞ otherwise.

The second sequence of equalities follows by applying l.s.p.The proof is finished.

Lemma 2.3. Assume that (H₁)holds. ThenZf is continuous in intC.

Proof. By Proposition 2.3 and Theorem 2.17 in [10], we deduce that Zf is continuous on every open set U ⊂domf. In particular,Zf is continuous on int(domf) which coincide with intCsince Lemma2.2.

The following lemma is essentially due to Wagner [15]. We prove it by borrowing some arguments of the proof of Theorem 3.12. 2) in [15].

Lemma 2.4. Assume that (H0)and(H1)hold. Thenlim_[0,1[t→1Zf(tξ)∈[0,+∞]for allξ∈∂C.

Proof. Letξ∈∂C. Setλ= lim sup_[0,1[t→1Zf(tξ) andμ= lim inf_[0,1[t→1Zf(tξ). Ifμ= +∞then λ=μ= lim

[0,1[t→1Zf(tξ) = +∞.

Assume thatμ <+∞. We have two possibilities, either λ= +∞orλ <+∞.

Suppose thatλ= +∞. Consider two sequences{t_n}n∈N^∗,{τ_n}n∈N^∗ ⊂[0,1[ such thatt_n→1 andτ_n →1 as n→+∞satisfying

λ= lim

n→+∞Zf(t_nξ) andμ= lim

n→+∞Zf(τ_nξ).

We can find two increasing functions σ, σ :N^∗→N^∗ such that for everyn∈N^∗ 1−1

n≤t_σ(n)< τ_σ(n)<1.

Letε >0. There existsN₀∈N^∗ such that for everyn≥N₀it holds

Zf(t_σ(n)ξ)≥1 +ε+μ and |Zf(τ_σ(n)ξ)−μ| ≤ ε

2· (2.2)

By (H0) there existsη >0 such that for everyξ∈intC and everyt∈[0,1[ it holds 1−t≤η =⇒ Zf(tξ)≤ Zf(ξ) +ε

2· (2.3)

Choose an integern≥max{2, N₀, η⁻¹}. Then it holds that τ_σ(n)>0 and 1− t_σ(n)

τ_σ(n) ≤η. (2.4)

Therefore, by (2.2), (2.4), (2.3) and l.s.p.we obtain

1 +ε+μ ≤ Zf(t_σ(n)ξ)− Zf(τ_σ(n)ξ) +Zf(τ_σ(n)ξ)

= Zf t_σ(n)

τ_σ(n)τ_σ(n)ξ

− Zf(τ_σ(n)ξ) +Zf(τ_σ(n)ξ)

≤ ε+μ,

which is impossible. It means that if μ <+∞thenλ <+∞.

Now, we will show that in this caseμ=λ. Consider{tn}_n∈N^∗,{τn}_n∈N^∗ ⊂[0,1[ such thatt_n →1 andτ_n→1 as n→+∞satisfying

λ= lim

n→+∞Zf(t_nξ) andμ= lim

n→+∞Zf(τ_nξ).

As above we can find two subsequences such that for everyn∈N^∗ 1−1

n≤t_σ(n)< τ_σ(n)<1.

Letε >0. There existsN₀∈N^∗ such that for everyn≥N₀it holds

|Zf(t_σ(n)ξ)−λ| ≤ ε

3 and |Zf(τ_σ(n)ξ)−μ| ≤ ε

3· (2.5)

By (H₀) there existsη >0 such that for everyξ∈intC and everyt∈[0,1[ it holds 1−t≤η =⇒ Zf(tξ)≤ Zf(ξ) +ε

3· (2.6)

Choose an integern≥max{2, N₀, η⁻¹}. Then it holds that τ_σ(n)>0 and 1− t_σ(n)

τ_σ(n) ≤η. (2.7)

Therefore, by (2.5), (2.7), (2.6) and l.s.p.we obtain

0≤λ−μ = λ− Zf(t_σ(n)ξ) +Zf(t_σ(n)ξ)− Zf(τ_σ(n)ξ) +Zf(τ_σ(n)ξ)−μ

≤ 2ε 3 +Zf

t_σ(n) τ_σ(n)τ_σ(n)ξ

− Zf(τ_σ(n)ξ)≤ε.

The proof is complete sinceε >0 is arbitrary.

Under (H0) and (H1) we will denote byZf :M^m×d→[0,+∞] the function defined by

Zf(ξ) =

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

Zf(ξ) ifξ∈intC

[0,1[limt→1Zf(tξ) ifξ∈∂C

+∞ otherwise.

Lemma 2.5. Assume that (H0)and (H1) hold. Then for everyε >0 there exists η >0 such that for every ξ∈C andt∈[0,1[it holds

1−t≤η =⇒ Zf(tξ)≤Zf(ξ) +ε.

Proof. Let{τn}_n∈N⊂[0,1[ be such thatτ_n→1 asn→+∞. Letε >0. By (H₀) and l.s.p., there exists η >0 such that for everyn∈N, everyt∈[0,1[ andξ∈C it holds

1−t≤η =⇒ Zf(tτ_nξ)≤ Zf(τ_nξ) +ε.

SinceZf is continuous in intC by Lemma2.3and using Lemma2.4, we deduce for everyt∈[0,1[ andξ∈C 1−t≤η =⇒ lim

n→+∞Zf(tτ_nξ) =Zf(tξ)≤ lim

n→+∞Zf(τ_nξ) +ε=Zf(ξ) +ε.

The proof is complete.

Lemma 2.6. Assume that (H0)and(H1)hold. Then for every ξ∈M^m×d Zf(ξ) = inf

Y

Zf(ξ+∇φ(x))dx:φ∈W₀^1,∞(Y;R^m)

.

Proof. It is enough to prove

Zf(ξ)≤inf

Y

Zf(ξ+∇φ(x))dx:φ∈W₀^1,∞(Y;R^m)

.

Letξ∈M^m×d andφ∈W₀^1,∞(Y;R^m) such that

Y

Zf(ξ+∇φ(x))dx <+∞.

Then ξ+∇φ(x) ∈C a.e. in Y, and soξ ∈ C since C is convex. Let n ≥1 and t_n = 1− ¹_n. Let Y x→ t_nξx+t_nφ(x)∈u_tnξ+W₀^1,∞(Y;R^m), whereu_tnξ(x) =t_nξxfor x∈Y. By l.s.p., it holds that∇u_n(x)∈t_nC a.e. in Y. Letε >0. By Lemma2.5, there existsη >0 such that for every ξ∈C andt ∈[0,1[, if 1−t < η thenZf(tξ)≤Zf(ξ) +ε. Choosen(ε)∈N^∗ such that 1−n⁻¹< ηfor alln≥n(ε). Thus, for everyn≥n(ε)

Y

Zf(t_nξ+t_n∇φ(x))dx≤ε+

Y

Zf(ξ+∇φ(x))dx.

Applying Lemma6.1, for everyn≥n(ε), we can find{v_kⁿ}k≥1⊂u_tnξ+W₀^1,∞(Y;R^m) such that lim sup

k→+∞

Y

f(∇v_kⁿ(x)−t_nξ+t_nξ)dx≤

Y

Zf(t_nξ+t_n∇φ(x))dx.

Thus, sinceZf ≤f andvⁿ_k −u_tnξ ∈W₀^1,∞(Y;R^m) we obtain Zf(t_nξ)≤ε+

Ω

Zf(ξ+∇φ(x))dx.

Letting n→+∞and using Lemmas2.3and2.4, we have Zf(ξ)≤ε+

Ω

Zf(ξ+∇φ(x))dx.

We obtain the desired result, sinceε >0 is arbitrary.

Lemma 2.7 (M¨uller [13], Thm. 4). Let K⊂M^m×d be a compact convex set. Let Ω⊂R^d be a bounded, open set with Lipschitz boundary. Let{un}_n∈N⊂W^1,∞(Ω;R^m) be a sequence satisfying

u_n →u_∞ in L¹(Ω;R^m) and

Ωdist(∇un(x), K)dx→0.

Then u_n u_∞ inW^1,1 and there exists{vn}_n∈N⊂W^1,∞(Ω;R^m)such that

⎧⎨

⎩

v_n=u_∞ on∂Ω

|{x∈Ω :∇u_n(x)=∇v_n(x)}| →0 dist(∇vn, K)_∞,Ω→0.

Lemma 2.8. Let r >0. Letρ >0be such that ρB⊂intC, withB={ξ∈M^m×d:|ξ| ≤1}. Then

ξ∈M^m×d: dist(ξ, C)≤ρr 2

⊂(1 +r)intC.

Proof. Note that by l.s.p. and the continuity of the norm we have dist(·,intC) = dist(·, C). Let r > 0. Let ξ∈M^m×dbe such that dist(ξ,intC)≤ρ^r₂. There existsF_r∈intCsuch thatρ^r₂ ≥dist(ξ,intC)≥ |ξ−F_r| −ρ^r₂. Now, if we writeξ=ξ−F_r+F_r thenξ∈rρB+ intC⊂(1 +r)intC.

3. Proof of Theorem 1.1

The following lemma gives a simplified formula forZf under (H0), (H1) and (H2).

Lemma 3.1. Assume that (H₀),(H₁) and(H₂)hold. Then Zf(ξ) =

⎧⎨

⎩

Zf(ξ) if ξ∈intC +∞ otherwise.

Proof. Letξ∈∂C. Lemma2.1(iv) and (H₂) give an increasing sequence{t_n}_n≥1⊂[0,1[ such thatt_n→1 as n→+∞and inf{Zf(ξ) :ξ∈C\t_nC} ≥nfor alln∈N^∗. Letn∈N^∗. By l.s.p., we havet_n+1ξ∈t_n+1C\t_nC, sincet_n < t_n+1. By Lemma2.4, it follows that

Zf(ξ) = lim

n→+∞Zf(t_n+1ξ)≥lim inf

n→+∞inf{Zf(ξ) :ξ∈C\t_nC} ≥lim inf

n→+∞n= +∞.

Proof of Theorem 1.1

We have to show that the functionZf is 1-sup-quasiconvex, and Qf =Zf =Zf.

Consider the sequence{tn}n∈N^∗ given by Lemma2.1(iv). For eachn∈N^∗, we setf_n =Qhn where h_n(ξ) =

Zf(ξ) ifξ∈t_nC n

1 + dist(ξ, C)

ifξ∈M^m×d\t_nC.

By (H₁), it holds for everyξ∈M^m×dand n∈N^∗

f_n(ξ)≤h_n(ξ)≤α_n(1 +|ξ|) whereα_n= max

sup_ξ∈t

nCf(ξ),2n(1 + diam(C))

<+∞, where diam(C) = sup{|ξ−ζ|:ξ, ζ ∈C}.

Letn∈N^∗, we will show thatf_n≤f_n+1. By l.s.p., we havet_nC⊂t_n+1C⊂intC. Letξ∈M^m×d, – ifξ∈t_nC thenh_n(ξ) =h_n+1(ξ) =Zf(ξ);

– ifξ∈t_n+1C\t_nC then, by Lemma2.1(iv), we obtain

h_n(ξ) =n(1 + dist(ξ, C)) =n≤ Zf(ξ) =h_n+1(ξ);

– ifξ /∈t_n+1C then

h_n(ξ) =n(1 + dist(ξ, C))≤(n+ 1)(1 + dist(ξ, C)) =h_n+1(ξ).

Hence h_n ≤h_n+1 and then f_n≤f_n+1. Note also that, by Lemma 2.2, we havef_n ≤ Zf ≤f. Thus, we have that {fn}n∈N^∗ is a nondecreasing sequence of quasiconvex functions satisfying

f_n(ξ)≤α_n(1 +|ξ|), and f_n(ξ)≤ Zf(ξ)≤f(ξ) (3.1) for all ξ∈M^m×d and n∈N^∗. Set f_∞ = sup_n∈N∗f_n, then, by the right hand side inequality in (3.1), it holds that f_∞≤Zf.

Now, we shall show thatf_∞ ≥Zf. Letξ∈M^m×d andk∈N^∗. Without loss of generality, we may assume that f_∞(ξ)<+∞. By LemmaA.2it holds that for everyn∈N^∗ there existsφ^k_n ∈W₀^1,∞(Y;R^m) such that

1

2k+f_∞(ξ)≥

A^k_n

Zf(ξ+∇φ^k_n(x))dx+n

Y\A^kn

dist(ξ+∇φ^k_n(x), C)dx+n|Y\A^k_n|

withA^k_n ={x∈Y :ξ+∇φ^k_n(x)∈t_nC}. We deduce that

n→+∞lim

Y

dist(ξ+∇φ^k_n(x), C)dx= 0. (3.2)

n→+∞lim |Y\A^k_n|= 0. (3.3)

Since for anyζ∈M^m×d it holds that|ζ| ≤diam(C) + dist(ζ, C), (3.2) implies that{ξ+∇φ^k_n}n∈N^∗ is bounded in L¹(Y;M^m×d). Using Poincar´e inequality and compact imbedding ofW^1,1(Y;R^m) inL¹(Y;R^m) we deduce that there exists a subsequence (not relabelled){φ^k_n}_n∈N^∗ converging inL¹. Applying Lemma2.7, we can find a sequence {ψ^k_n}_n∈N^∗⊂W₀^1,∞(Y;R^m) such that

n→+∞lim x∈Y :∇φ^k_n(x)=∇ψ^k_n(x)= 0, and (3.4)

n→+∞lim dist(ξ+∇ψ_n^k, C)∞,Y = 0. (3.5) Letρ >0 be such thatρB⊂intC, whereB={ξ∈M^m×d:|ξ| ≤1}. By (3.5), there existsσ(k)∈N^∗such that for everyn≥σ(k)

dist(ξ+∇ψ^k_n, C)∞,Y ≤ ρ 2k·

We constructσ:N^∗→N^∗in order to obtain an increasing function ofk. By Lemma2.8, we deduceξ+∇ψ_n^k(x)∈ (1 +_k¹)intCa.e. inY for alln≥σ(k).

For eachl ∈N^∗ we denote byM_l= sup{f(ζ) :ζ∈t_lC}which is finite since Lemma 2.1(iii). By (3.3) and (3.4), there exists δ(k)≥1 such that for everyn≥δ(k)

maxx∈Y :∇φ^k_n(x)=∇ψ_n^k(x),|Y\A^k_n|

≤ 1 4kM_σ(k)· Now, we taken≥max{δ(k), σ(k)} then

max

|B_n^k|,|Y\A^k_n|

≤ 1

4kM_σ(k) and τ_k(ξ+∇ψ_n^k(x))∈intC a.e. inY,

where B_n^k =x∈Y :∇φ^k_n(x)=∇ψ^k_n(x)and τ_k = (1 +k⁻¹)⁻¹. Set G^k_n =B_n^k∪(Y \(B_n^k∪A^k_n)), we have that |G^k_n| ≤(2kM_σ(k))⁻¹ sinceB_n^k∪A^k_n ⊃A^k_n. Then it holds

Y

Zf(t_σ(k)τ_k(ξ+∇ψ_n^k(x)))dx =

G^k_n

Zf(t_σ(k)τ_k(ξ+∇ψ_n^k(x)))dx +

(Y\Bn^k)∩A^kn

Zf(t_σ(k)τ_k(ξ+∇ψ_n^k(x)))dx

≤ |G_k| sup

ζ∈tσ(k)C

Zf(ζ) +

A^k_n

Zf(t_σ(k)τ_k(ξ+∇φ^k_n(x)))dx

≤ 1 2k+

A^k_n

Zf(t_σ(k)τ_k(ξ+∇φ^k_n(x)))dx.

By convexity of the distance function, we deduce from (3.2) thatξ∈C. The l.s.p.implies thatt_σ(k)τ_kξ∈intC.

Using Lemma 2.6, we deduce that for everyk∈N^∗and everyn≥max{δ(k), σ(k)}

Zf(t_σ(k)τ_kξ)≤ 1 2k+

A^k_n

Zf(t_σ(k)τ_k(ξ+∇φ^k_n(x)))dx, (3.6)

withA^k_n ={x∈Y :ξ+∇φ^k_n(x)∈t_nC}.

Let s ∈ N^∗. By (H₀), there exists η_s >0 such that for every t ∈ [0,1[ and ζ ∈ intC if 1−t ≤ η_s then Zf(tζ)≤ Zf(ζ) + ¹_s. There also exists an integer k_s ≥1 such that 1−t_σ(k)τ_k ≤η_s for all k≥k_s sinceσ is increasing. Thus, if we takek≥k_sthen for everyn≥max{σ(k), δ(k)}

A^k_n

Zf(t_σ(k)τ_k(ξ+∇φ^k_n(x)))dx ≤ 1 s+

A^k_n

Zf(ξ+∇φ^k_n(x))dx

≤ 1 s+ 1

2k+f_∞(ξ).

Hence, by (3.6)

Zf(t_σ(k)τ_kξ)≤ 1 s+1

k+f_∞(ξ). (3.7)

By Lemmas2.3and2.4, it follows

Zf(ξ) = lim

k→+∞Zf(t_σ(k)τ_kξ)≤ 1

s+f_∞(ξ).

Letting s→+∞, we obtainZf ≤f_∞.

Now, by (3.1), (1.1) and Lemma3.1, it follows

Qf ≤ Zf ≤Zf =f^∞≤ Q(Zf)≤ Qf.

The proof is complete.

4. Proof of Theorem 1.3

The proof follows essentially the same lines as the proof of Theorem1.1, however it is worth to write it here because of some differences due to the behavior ofZf near the boundary ofC.

We divide the proof into two steps. In the first step we show that Zf is 1-sup-quasiconvex. In the second step we show the representation formula forQf.

Step 1. Assume without loss of generality thatZf ≡0 in intC. For eachn∈N, we setf_n=Qgn where g_n(ξ) =

Zf(ξ) ifξ∈intC n

1 + dist(ξ, C)

ifξ∈M^m×d\intC.

Set M = sup_ξ∈intCZf(ξ) which is finite since (H₃). We have that {fn}_n∈N is a nondecreasing sequence of quasiconvex functions satisfying

f_n(ξ)≤max

M,2n(1 + diam(C))

(1 +|ξ|)

for allξ∈M^m×d andn∈N. For eachn∈N^∗ we havef_n ≤ Zf ≤f in intC, andf_n ≤Zf ≤f in C. Indeed, letξ∈∂C, by Lemma2.4there exists a sequence{tk}k∈N^∗ ⊂[0,1[ such that t_k →1 and lim_k→+∞Zf(t_kξ) =

lim_[0,1[t→1Zf(tξ). By l.s.p., t_kξ∈intCfor allk∈N^∗. Thus, by (H₄), we obtain for alln∈N^∗ f_n(ξ)≤ lim

k→+∞Zf(t_kξ) =Zf(ξ)≤lim inf

k→+∞f(t_kξ)≤f(ξ).

And so for everyξ∈M^m×d andn∈N^∗

f_n(ξ)≤f_∞(ξ)≤Zf(ξ)≤f(ξ) (4.1)

wheref_∞= sup_n≥1f_n.

Now, we shall show thatf_∞ ≥Zf. Letξ∈M^m×d andk∈N^∗. Without loss of generality, we may assume that f_∞(ξ)<+∞. By LemmaA.2it holds that for everyn∈Nthere existsφ^k_n ∈W₀^1,∞(Y;R^m) such that

1

2k+f_∞(ξ)≥

A^k_n

Zf(ξ+∇φ^k_n(x))dx+n

Y\A^k_ndist(ξ+∇φ^k_n(x), C)dx+n|Y\A^k_n| withA^k_n ={x∈Y :ξ+∇φ^k_n(x)∈intC}. We deduce that

n→+∞lim

Y

dist(ξ+∇φ^k_n(x), C)dx= 0. (4.2)

n→+∞lim |Y\A^k_n|= 0. (4.3)

Since for anyζ∈M^m×d it holds that|ζ| ≤diam(C) + dist(ζ, C), (4.2) implies that {ξ+∇φ^k_n}n∈N is bounded in L¹(Y;M^m×d). Using Poincar´e inequality and compact imbedding ofW^1,1(Y;R^m) inL¹(Y;R^m) we deduce that there exists a subsequence (not relabelled) {φ^k_n}n∈N converging inL¹. Applying Lemma2.7, we can find a sequence {ψ^k_n}n∈N⊂W₀^1,∞(Y;R^m) such that

n→+∞lim x∈Y :∇φ^k_n(x)=∇ψ_n^k(x)= 0 and

n→+∞lim dist(ξ+∇ψ^k_n(x), C)∞,Y = 0.

As in the proof of Theorem 1.1 we deduce that there exists an increasing sequence N^∗ l → σ(l) ∈N^∗ such that

max{|{x∈Y :∇φ^k_σ(k)(x)=∇ψ^k_σ(k)(x)}|,|Y\A^k_σ(k)|} ≤ 1 4kM τ_k(ξ+∇ψ_σ(k)^k (x))∈intC a.e. inY,

whereτ_k= (1 +k⁻¹)⁻¹. To simplify notation we writeφ_k,ψ_k andA_k for, respectively,φ^k_σ(k),ψ^k_σ(k) andA^k_σ(k). SetB_k ={x∈Y :∇ψk(x)=∇φk(x)}andG_k =B_k∪(Y \(B_k∪A_k)). By (H3) we have

Y

Zf(τ_k(ξ+∇ψk))dx ≤

Gk

Zf(τ_k(ξ+∇ψk))dx+

Ak

Zf(τ_k(ξ+∇φk))dx

≤ 1 2k +

Ak

Zf(τ_k(ξ+∇φ_k))dx.

Let s ∈ N^∗. By (H0), there exists η_s > 0 such that for every t ∈ [0,1[ and ξ ∈ intC if 1−t ≤ η_s then Zf(tξ)≤ Zf(ξ) +¹_s. Considerk∈N^∗ such thatk+ 1>(η_s)⁻¹, it follows that

Ak

Zf(τ_k(ξ+∇φk(x)))dx ≤ 1 s +

Ak

Zf(ξ+∇φk(x))dx

≤ 1 s + 1

2k+f_∞(ξ),

and then

Y

Zf(τ_k(ξ+∇ψ_k(x)))dx ≤ 1 s+1

k+f_∞(ξ). (4.4)

By convexity of the distance function, we deduce from (4.2) thatξ ∈ C. The l.s.p. implies that τ_kξ ∈intC.

According to Lemma2.6together with Lemmas2.4and2.3, we deduce from (4.4) that Zf(ξ)≤lim inf

k→+∞Zf(τ_kξ)≤ 1

s+f_∞(ξ).

Letting s→+∞, we obtainZf ≤f_∞, which finishes the proof of the first step.

Step 2. We shall show that

Qf =Zf =Zf .

We have thatQf ≤ Zf sinceQf satisfies inequality (1.1). Now, letξ∈∂C, by lower semicontinuity ofQf we have

Zf(ξ) = lim

[0,1[t→1Zf(tξ)≥ lim inf

[0,1[t→1Qf(tξ)≥ Qf(ξ).

We deduce thatQf ≤Zf. By (4.1) we haveZf =f_∞≤ Qf. Now, it holdsZf ≥ Zf onC, and by definition ofZf, we obtainZf ≥ Zf. SinceQf ≤ Zf, it follows thatQf ≤ Zf. Thus, we obtain

Qf ≤ Zf ≤Zf =f_∞≤ Qf

which finishes the proof.

5. Upper bound and approximation of Sobolev functions

continuous piecewise affine functions

5.1. Upper bound for continuous piecewise affine functions

Let us denote by Aff(Ω;R^m) the space of all continuous piecewise affine functions from Ω to R^m, i.e., u∈Aff(Ω;R^m) if and only ifuis continuous and there exists a finite family{Ui}i∈I of open disjoint subsets of U such that |U \ ∪i∈IU_i| = 0 and for every i ∈ I, ∇u(x) = ξ_i in U_i with ξ_i ∈ M^m×d. The proof of the following lemma follows by an easy adaptation of Lemma 3.1 in [3].

Lemma 5.1. Let p∈ [1,+∞]. Let g : M^m×d → [0,+∞] be a Borel measurable function and U ⊂ R^d be a bounded open set. Then for every u∈Aff(U;R^m)

inf

lim inf

n→+∞

U

g(∇u_n)dx:u+W₀^1,∞(U;R^m)u_nuin W^1,p

≤

U

Zg(∇u)dx

(replace “” by “” when^∗ p= +∞).

5.2. Approximation results

Let| · |1,p,A (resp. | · |p,A) stand for the norm ofW^1,p(A;R^m) (resp. L^p(A;R^m)) whereA⊂R^d is open and p∈[1,+∞]. LetB={ξ∈M^m×d:|ξ|<1}be the unit ball ofM^m×d.

The proof of the following lemma is omitted since it is a particular case of Theorem 10.16 in [9].