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A relaxation result in BV×Lp for integral functionals depending on chemical composition and elastic strain

Abstract

An integral representation result is obtained for the relaxation of a class of energy functionals depending on two vector fields with different behaviors which appear in the context of thermochemical equilibria and are related to image decomposition models and directors theory in nonlinear elasticity.

1.Introduction

In this paper we consider energies depending on two vector fields with different behaviors: uW1,1(Ω;Rn) and vLp(Ω;Rm), Ω being a bounded open subset of RN.

Let 1p and for every (u,v)W1,1(Ω;Rn)×Lp(Ω;Rm) define the functional

(1.1)J(u,v):=Ωf(v,u)dx,
where f:Rm×Rn×N[0,) is a continuous function.

Minimization of energies depending on two independent vector fields have been introduced to model several phenomena. For instance the case of thermochemical equilibria among multiphase multicomponent solids and Cosserat theories in the context of elasticity: we refer to [7,9] and the references therein for a detailed explanation about this kind of applications.

In the Sobolev setting, after the pioneer works [7,9], relaxation with a Carathéodory density ff(x,u,u,v), and homogenization for density of the type f(xε,u,v) have been considered in [5] and [6], respectively.

In the present paper we are interested in studying the lower semicontinuity and relaxation of (1.1) with respect to the L1-strong ×Lp-weak convergence (p>1). Clearly, bounded sequences {uh}W1,1(Ω;Rn) may converge in L1, up to a subsequence, to a BV function.

In the BV-setting this question has been already addressed in [8], only when the density f is convex–quasiconvex (see (2.2)) and the vector field vL(Ω;Rm).

Here we allow v to be in Lp(Ω;Rm), p>1 and f is not necessarily convex–quasiconvex. We provide an argument alternative to the one in [8, Section 4], devoted to clarify some points in the lower semicontinuity result therein.

We also emphasize that under specific restrictions on the density f, i.e. f(x,u,v,u)W(x,u,u)+φ(x,u,v), the analysis in the case 1<p< was considered already in [10] in order to describe image decomposition models. In [11] a general f was taken into account when the target u is in W1,1(Ω;Rn).

In this manuscript we consider ff(b,ξ), (b,ξ)Rm×Rn×N and the target uBV(Ω;Rn).

We study separately the cases 1<p<, p= and discuss briefly the case p=1 in the Appendix. Comparing the results in [5] (where the assumptions allow to invoke De La Vallé–Poussin Criterion) with the true linear growth setting as in [3].

To this end, we introduce for 1<p< the functional

Jp(u,v):=inf{liminfhJ(uh,vh):uhW1,1(Ω;Rn),vhLp(Ω;Rm),(1.2)uhu in L1,vhv in Lp},
for any pair (u,v)BV(Ω;Rn)×Lp(Ω;Rm) and, for p= the functional
J(u,v):=inf{liminfhJ(uh,vh):uhW1,1(Ω;Rn),vhL(Ω;Rm),(1.3)uhu in L1,vhv in L},
for any pair (u,v)BV(Ω;Rn)×L(Ω;Rm).

Since bounded sequences {uh} in W1,1(Ω;Rn) converge in L1 to a BV function u and bounded sequences {vh} in Lp(Ω;Rm) if 1<p< (in L(Ω;Rm) if p=), weakly converge to a function vLp(Ω;Rm) (weakly ∗ in L), the relaxed functionals Jp and J will be composed by an absolutely continuous part and a singular one with respect to the Lebesgue measure (see (2.12)). On the other hand, as already emphasized in [8], it is crucial to observe that v, regarded as a measure, is absolutely continuous with respect to the Lebesgue one, besides it is not defined on the singular sets of u, namely in those sets where the singular part with respect the Lebesgue measure of the distributional gradient of u, Dsu, is concentrated. Thus specific features of the density f will come into play to ensure a proper integral representation.

The integral representation of (1.2) will be achieved in Theorem 1.1 under the following hypotheses:

  • (H1)p There exists C>0 such that

    1C(|b|p+|ξ|)Cf(b,ξ)C(1+|b|p+|ξ|),
    for (b,ξ)Rm×Rn×N.

  • (H2)p There exists C>0, L>0, 0<τ1 such that

    t>0,ξRn×N,with t|ξ|>L|f(b,tξ)tf(b,ξ)|C(|b|p+1t+|ξ|1τtτ),
    where f is the recession function of f defined for every bRm as
    (1.4)f(b,ξ):=lim suptf(b,tξ)t.

In order to characterize the functional J introduced in (1.3) we will replace assumptions (H1)p and (H2)p by the following ones:

  • (H1) Given M>0, there exists CM>0 such that, if |b|M then

    1CM|ξ|CMf(b,ξ)CM(1+|ξ|),
    for every ξRn×N.

  • (H2) Given M>0, there exist CM>0, L>0, 0<τ1 such that

    |b|M,t>0,ξRn×N,with t|ξ|>L|f(b,tξ)tf(b,ξ)|CM|ξ|1τtτ.

Section 2 is devoted to notations, preliminaries about measure theory and some properties of the energy densities. In particular, we stress that a series of results is presented in order to show all the properties and relations among the relaxed energy densities involved in the integral representation and that can be of further use for the interested readers since they often appear in the integral representation context. Section 3 contains the arguments necessary to prove the main results stated below.

Theorem 1.1.

Let J be given by (1.1), with f satisfying (H1)p and (H2)p and let Jp be given by (1.2) then

Jp(u,v)=ΩCQf(v,u)dx+Ω(CQf)(0,dDsud|Dsu|)d|Dsu|,
for every (u,v)BV(Ω;Rn)×Lp(Ω;Rm).

We denote by CQf the convex–quasiconvex envelope of f in (2.5) and (CQf) represents the recession function of CQf, defined according to (1.4), which coincides, under suitable assumptions (see assumptions (2.6), (2.7), Proposition 2.12 and Remark 2.13), with the convex–quasiconvex envelope of f, CQ(f), and this allows us to remove the parenthesis.

For the case p= we have the following.

Theorem 1.2.

Let J be given by (1.1), with f satisfying (H1) and (H2) and let J be given by (1.3) then

J(u,v)=ΩCQf(v,u)dx+Ω(CQf)(0,dDsud|Dsu|)d|Dsu|,
for every (u,v)BV(Ω;Rn)×L(Ω;Rm).

For the case 1<p<, the proof of the lower bound is presented in Theorem 3.1 while the upper bound is in Theorem 3.2, both under the extra hypothesis

  • (H0) f is convex–quasiconvex.

The case p= is discussed in Section 3.2. Furthermore, we observe that Proposition 2.14 in Section 2.3 is devoted to remove the convexity-quasiconvexity assumption on f.

2.Notations, preliminaries and properties of the energy densities

In this section, we start by establishing notations, recalling some preliminary results on measure theory that will be useful through the paper and finally we recall the space of functions of bounded variation.

Then we deduce the main properties of convex–quasiconvex functions, recession functions and related envelopes.

If νSN1 and {ν,ν2,,νN} is an orthonormal basis of RN, Qν denotes the unit cube centered at the origin with its faces either parallel or orthogonal to ν,ν2,,νN. If xRN and ρ>0, we set Q(x,ρ):=x+ρQ and Qν(x,ρ):=x+ρQν, Q is the cube (12,12)N.

Let Ω be a generic open subset of RN, we denote by M(Ω) the space of all signed Radon measures in Ω with bounded total variation. By the Riesz Representation Theorem, M(Ω) can be identified to the dual of the separable space C0(Ω) of continuous functions on Ω vanishing on the boundary Ω. The N-dimensional Lebesgue measure in RN is designated as LN.

If μM(Ω) and λM(Ω) is a nonnegative Radon measure, we denote by dμdλ the Radon–Nikodým derivative of μ with respect to λ. By a generalization of the Besicovich Differentiation Theorem (see [1, Proposition 2.2]), it can be proved that there exists a Borel set EΩ such that λ(E)=0 and

dμdλ(x)=limρ0+μ(x+ρC)λ(x+ρC)
for all xSuppλE and any open bounded convex set C containing the origin.

We recall that the exceptional set E above does not depend on C. An immediate corollary is the generalization of Lebesgue–Besicovitch Differentiation Theorem given below.

Theorem 2.1.

If μ is a nonnegative Radon measure and if fLloc1(RN,μ) then

limε0+1μ(x+εC)x+εC|f(y)f(x)|dμ(y)=0
for μ-a.e. xRN and for every, bounded, convex, open set C containing the origin.

Definition 2.2.

A function uL1(Ω;Rn) is said to be of bounded variation, and we write uBV(Ω;Rn), if all its first distributional derivatives, Djui, belong to M(Ω) for 1in and 1jN.

The matrix-valued measure whose entries are Djui is denoted by Du and |Du| stands for its total variation. We observe that if uBV(Ω;Rn) then u|Du|(Ω) is lower semicontinuous in BV(Ω;Rn) with respect to the Lloc1(Ω;Rn) topology.

By the Lebesgue Decomposition Theorem we can split Du into the sum of two mutually singular measures Dau and Dsu, where Dau is the absolutely continuous part and Dsu is the singular part of Du with respect to the Lebesgue measure LN. By u we denote the Radon–Nikodým derivative of Dau with respect to the Lebesgue measure so that we can write

Du=uLN+Dsu.

Proposition 2.3.

If uBV(Ω;Rn) then for LN-a.e. x0Ω

(2.1)limε0+1ε{1εNQ(x0,ε)|u(x)u(x0)u(x0)·(xx0)|NN1dx}N1N=0.

For more details regarding functions of bounded variation we refer to [2].

2.1.Convex-quasiconvex functions

We start by recalling the notion of convex–quasiconvex function, presented in [8] (see also [7] and [9]).

Definition 2.4.

A Borel measurable function f:Rm×Rn×NR is said to be convex–quasiconvex if, for every (b,ξ)Rm×Rn×N, there exists a bounded open set D of RN such that

(2.2)f(b,ξ)1|D|Df(b+η(x),ξ+φ(x))dx,
for every ηL(D;Rm), with Dη(x)dx=0, and for every φW01,(D;Rn).

Remark 2.5.

  • i) It can be easily seen that, if f is convex–quasiconvex then condition (2.2) is true for any bounded open set DRN.

  • ii) A convex–quasiconvex function is separately convex.

  • iii) By [11, Proposition 3], the growth condition from above in (H1)p, ii), entail that there exists γ>0 such that

    (2.3)|f(b,ξ)f(b,ξ)|γ(|ξξ|+(1+|b|p1+|b|p1+|ξ|1p+|ξ|1p)|bb|)
    for every b,bRm, ξ,ξRn×N, where p>1 and p its conjugate exponent.

  • iv) By [11, Proposition 4]), under the growth assumptions in (H1), ii) entails that, given M>0 there exists a constant β(M,n,m,N) such that

    (2.4)|f(b,ξ)f(b,ξ)|β(1+|ξ|+|ξ|)|bb|+β|ξξ|
    for every b, bRm, such that |b|M and |b|M, for every ξ, ξRn×N.

We introduce the notion of convex–quasiconvex envelope of a function, which is crucial to deal with the relaxation procedure.

Definition 2.6.

Let f:Rm×Rn×NR be a Borel measurable function bounded from below. The convex–quasiconvex envelope is the largest convex–quasiconvex function below f, i.e.,

CQf(b,ξ):=sup{g(b,ξ):gf,g convex–quasiconvex}.

By Theorem 4.16 in [9], the convex–quasiconvex envelope coincides with the so called convex-quasiconvexification

CQf(b,ξ)=inf{1|D|Df(b+η(x),ξ+φ(x))dx:ηL(D;Rm),(2.5)Dη(x)dx=0,φW01,(D;Rn)}.
As for convexity-quasiconvexity, condition (2.5) can be stated for any bounded open set DRN. It can also be showed that if f satisfies a growth condition of type (H1)p then in (2.2) and (2.5) the spaces L and W01, can be replaced by Lp and W01,1, respectively.

The following proposition, that will be exploited in the sequel, can be found in [11, Proposition 5]. The proof is omitted since it is very similar to [10, Proposition 2.1].

Proposition 2.7.

Let f:Rm×Rn×N[0,) be a continuous function satisfying (H1)p. Then CQf is continuous and satisfies (H1)p. Consequently, CQf satisfies (2.3).

In order to deal with vL(Ω;Rm) and to compare with the result in BV×Lp, 1<p<, one can consider a different setting of assumptions on the energy density f.

Namely, following [11, Proposition 6 and Remark 7], if α:[0,)[0,) is a convex and increasing function, such that α(0)=0 and if f:Rm×Rn×N[0,) is a continuous function satisfying

(2.6)1C(α(|b|)+|ξ|)Cf(b,ξ)C(1+α(|b|)+|ξ|)
for every (b,ξ)Rm×Rn×N, then CQf satisfies a condition analogous to (2.6). Moreover, CQf is a continuous function.

Analogously, one can assume that f satisfies the following variant of (H2): there exist c>0, L>0, 0<τ1 such that

t>0,ξRn×N,(2.7)with t|ξ|>L|f(b,tξ)tf(b,ξ)|c(α(|b|)+1t+|ξ|1τtτ).

We observe that, if from one hand (2.6) and (2.7) generalize (H1)p and (H2)p respectively, from the other hand they can be regarded also as a stronger version of (H1) and (H2), respectively.

2.2.The recession function

Let f:Rm×Rn×N[0,[, and let f:Rm×Rn×N[0,[ be its recession function, defined in (1.4).

The following properties are an easy consequence of the definition of recession function and conditions (H0), (H1)p and (H2)p, when 1<p<.

Proposition 2.8.

Provided f satisfies (H0), (H1)p and (H2)p, then

  • 1. f is convex–quasiconvex;

  • 2. there exists C>0 such that

    (2.8)1C|ξ|f(b,ξ)C|ξ|;

  • 3. f(b,ξ) is constant with respect to b for every ξRn×N;

  • 4. f is continuous.

Remark 2.9.

We emphasize that not all the assumptions (H1)p and (H2)p in Proposition 2.8 are necessary to prove items above. In particular, one has that:

  • i) The proof of 2. uses only the fact that f satisfies (H1)p.

  • ii) To prove 3. it is necessary to require that f satisfies only (H0) and (H1)p. Indeed if it satisfies (2.3) one can avoid to require (H0).

Proof.

  • 1. The convexity-quasiconvexity of f can be proven exactly as in [8, Lemma 2.1], exploiting the growth condition (H1)p and the estimate given by (H2)p.

  • 2. By definition (1.4) we may find a subsequence {tk} such that

    f(b,ξ)=limtkf(b,tkξ)tk.
    By (H1)p one has
    f(b,ξ)limtkC(1+|b|p+|tkξ|)tk=C|ξ|
    and
    f(b,ξ)limtk1C(|b|p+|tkξ|)Ctk1C|ξ|.
    Hence (H1)p holds for f.

  • 3. Let ξRn×N, and let b,bRm, up to a subsequence, by (1.4) and the fact that f satisfies (2.3) it results that,

    f(b,ξ)f(b,ξ)limtkf(b,tkξ)f(b,tkξ)tklimtkγ(1+|b|p1+|b|p1+|tkξ|1p)|bb|tk=0.
    By interchanging the role of b and b, it follows that f(·,ξ) is constant and this concludes the proof.

  • 4. The continuity is a consequence of the growth conditions and the convexity-quasiconvexity of f.

 □

Remark 2.10.

Under assumptions (H0), (H1) and (H2), f satisfies properties analogous to those at the beginning of Section 2.2. In particular in [8, Lemma 2.1 and Lemma 2.2] it has been proved that

  • i) f is convex–quasiconvex;

  • ii) 1CM|ξ|f(b,ξ)CM|ξ|, for every b, with |b|M;

  • iii) If rankξ1, then f(b,ξ) is constant with respect to b.

Remark 2.11.

We observe that, if f:Rm×Rn×N[0,) is a continuous function satisfying (H1)p and (H2)p, then the function (CQf):Rm×Rn×N[0,[, obtained first taking the convex-quasiconvexification in (2.5) of f and then its recession through formula (1.4) applied to CQf, satisfies the following properties:

  • 1. (CQf) is convex–quasiconvex;

  • 2. there exists C>0 such that 1C|ξ|(CQf)(b,ξ)C|ξ|, for every (b,ξ)Rm×Rn×N;

  • 3. for every ξRn×N, (CQf)(·,ξ) is constant, i.e. (CQf) is independent on b;

  • 4. (CQf) is Lipschitz continuous in ξ.

Assuming that f satisfies (H1)p, one can prove that the convex-quasiconvexification of f, CQ(f), satisfies the following conditions:

  • 5. CQ(f) is convex–quasiconvex;

  • 6. there exists C>0 such that 1C|ξ|CQ(f)(b,ξ)C|ξ|, for every (b,ξ)Rm×Rn×N;

  • 7. for every ξRn×N, and assuming that f satisfies (2.3), CQ(f)(·,ξ) is constant, i.e. CQ(f) is independent on b;

  • 8. CQ(f) is Lipschitz continuous in ξ.

The above properties are immediate consequences of Propositions 2.7, 2.8 and (2.3). In particular 8. follows from 3. of Proposition 2.8, without requiring (H2)p.

On the other hand, Proposition 2.12 below entails that CQ(f) is independent on b, without requiring that f is Lipschitz continuous, but replacing this assumption with (H2)p.

We also observe that (CQf) and CQ(f) are only quasiconvex functions, since they are independent of b. In particular, in our setting, these functions coincide as it is stated below.

Proposition 2.12.

Let f:Rm×Rn×N[0,) be a continuous function satisfying (H1)p and (H2)p.

Then

CQ(f)(b,ξ)=(CQf)(b,ξ)for every (b,ξ)Rm×Rn×N.

Proof.

The proof will be achieved by double inequality.

For every (b,ξ)Rm×Rn×N the inequality

(2.9)(CQf)(b,ξ)CQ(f)(b,ξ)
follows by Definition 2.6, and the fact that CQf(b,ξ)f(b,ξ). In fact, (1.4) entails that the same inequality holds when, passing to (·). Finally, 1. in Proposition 2.8, guarantees (2.9).

In order to prove the opposite inequality, fix (b,ξ)Rm×Rn×N and, for every t>1, take ηtL(Q;Rm), with 0 average, and φtW01,(Q;Rn) such that

(2.10)Qf(b+ηt,tξ+φt(y))dyCQf(b,tξ)+1.
By (H1)p and Proposition 2.7, we have that b+ηtLp(Q), (1tφt)L1(Q)C for a constant independent on t. Defining ψt:=1tφt, one has ψtW01,(Q;Rn) and thus
CQ(f)(b,ξ)Qf(b+ηt,ξ+ψt(y))dy.

Let L be the constant appearing in condition (H2)p. We split the cube Q into the set {yQ:t|ξ+ψt(y)|L} and its complement in Q. Then we apply condition (H2)p and (2.8) to get

CQ(f)(b,ξ)Q(C1+|b+ηt|pt+C|ξ+ψt|1τtτ+f(b+ηt,tξ+φt)t+CLt)dy.

Applying Hölder inequality and (2.10), we get

CQ(f)(b,ξ)Ctτ(Q|ξ+ψt|dy)1τ+CQf(b,tξ)+1t+CLt+Ct,
and the desired inequality follows by definition of (CQf) and using the fact that ψt is bounded in L1 norm, letting t go to . □

Remark 2.13.

It is worth to observe that inequality

(CQf)(b,ξ)CQ(f)(b,ξ)for every (b,ξ)Rm×Rn×N,
has been proven without requiring neither (H1)p and (H2)p on f, nor (H1) and (H2).

Furthermore, we emphasize that the proof of Proposition 2.12 cannot be performed in the same way in the case p=, with assumptions (H1)p and (H2)p replaced by (H1) and (H2). Indeed, an L bound on b+ηt analogous to the one in Lp cannot be obtained from (H1). On the other hand, it is possible to deduce the equality between CQf and (CQf), when f satisfies (2.6) and (2.7).

2.3.Auxiliary results

Here we prove that assumption (H0) on f is not necessary to provide an integral representation of Jp as in (1.2). Indeed, we can assume that f:Rm×Rn×N[0,[ is a continuous function and satisfies assumptions (H1)p and (H2)p, (p(1,]). First we extend, with an abuse of notation, the functional J in (1.1), to L1(Ω;Rn)×Lp(Ω;Rm), p(1,], as

(2.11)J(u,v):=Ωf(v,u)dx if (u,v)W1,1(Ω;Rn)×Lp(Ω;Rm), otherwise.
Then we define the functional
JCQf(u,v):=ΩCQf(v,u)dx if (u,v)W1,1(Ω;Rn)×Lp(Ω;Rm),otherwise,
(p(1,]) where CQf is given by Definition 2.6 and,
JCQfp(u,v):=inf{liminfhJCQf(uh,vh):uhW1,1(Ω;Rn),vhLp(Ω;Rm),uhu in L1,vhv in Lp},
for any pair (u,v)BV(Ω;Rn)×Lp(Ω;Rm),p(1,). Analogously, one can consider
JCQf(u,v):=inf{liminfhJCQf(uh,vh):uhW1,1(Ω;Rn),vhLp(Ω;Rm),uhu in L1,vhv in L},
for any pair (u,v)BV(Ω;Rn)×L(Ω;Rm).

Clearly, it results that for every (u,v)BV(Ω;Rn)×Lp(Ω;Rm),

JCQfp(u,v)Jp(u,v),
but, as in [11, Lemma 8 and Remark 9], the following proposition can be proven.

Proposition 2.14.

Let p(1,] and consider the functionals J and JCQf and their corresponding relaxed functionals Jp and JCQfp. If f satisfies conditions (H1)p and (H2)p if p(1,), and both f and CQf satisfy (H1) and (H2) if p=, then

Jp(u,v)=JCQfp(u,v)
for every (u,v)BV(Ω;Rn)×Lp(Ω;Rm),p(1,].

Remark 2.15.

The proof is omitted since it can be performed as in [11, Lemma 8 and Remark 9]. In [11] it is not required that f satisfies (H2)p, (p(1,]). Indeed, the equality Jp=JCQfp holds independently on this assumption on f, but in order to remove hypothesis (H0) from the representation theorem we need to assume that CQf inherits the same properties as f, which is the case as it has been observed in Proposition 2.7. It is also worth to observe that, when p=, (2.7) is equivalent to

|f(b,ξ)f(b,ξ)|C(1+α(|b|)+|ξ|)
for every (b,ξ)Rm×Rn×N, and this latter property is inherited by CQf and CQf as it can be easily verified arguing as in [10, Proposition 2.3]. Thus Proposition 2.14 holds when p= just requiring that f satisfies (2.6) and (2.7).

The following result can be deduced in full analogy with [11, Theorem 13], where it has been proven for J.

Proposition 2.16.

Let Ω be a bounded and open set of RN and let f:Rm×Rn×NR be a continuous function satisfying (H1)p and (H2)p, 1<p. Let J be the functional defined in (1.1), then Jp in (1.2) (1<p<), (1.3) (p=) is a variational functional, namely it is lower semicontinuous with respect to the first arguments and for every (u,v)BV(Ω;Rn)×Lp(Ω;Rm), one can define Jp(u,v;·), (p(1,]) (in analogy with (1.2) and (1.3)) as a set function on the open subsets of Ω, and it turns out to be the restriction of a Radon measure to these subsets of Ω.

By virtue of this result Jp can be decomposed as the sum of two terms

(2.12)Jp(u,v;·)=Jpa(u,v;·)+Jps(u,v;·),
where Jpa(u,v;·) and Jps(u,v;·) denote the absolutely continuous part and the singular part with respect to the Lebesgue measure, respectively. Next proposition deals with the scaling properties of Jp.

Proposition 2.17.

Let f:Rm×Rn×NR be a continuous and convex–quasiconvex function, let J and Jp be the functionals defined respectively by (1.1) and (1.2) when p(1,], respectively ((1.3), when p=). Then the following scaling properties are satisfied

(2.13)Jp(u+η,v;Ω)=Jp(u,v;Ω)for every ηRn,Jp(u(·x0),v(·x0);x0+Ω)=Jp(u(·),v(·);Ω)for every x0RN,Jp(uϱ,vϱ;Ωx0ϱ)=ϱNJp(u,v;Ω),
where uϱ(y):=u(x0+ϱy)u(x0)ϱ and vϱ(y):=v(x0+ϱy), for yΩx0ϱ.

The following result will be exploited in the sequel. The proof is omitted since it develops along the lines of [2, Lemma 5.50], the only differences being the presence of v and the convexity-quasiconvexity of f.

Lemma 2.18.

Let f:Rm×RN×nR be a continuous and convex–quasiconvex function, and let J and Jp be the functionals defined respectively by (1.1) and (1.2). Let νSN1, ηSn1 and ψ:RR, bounded and increasing. Denoted by Q the cube Qν, let uBV(Q;Rn) be representable in Q as

u(y)=ηψ(y·ν),
and let wBV(Q;Rn) be such that supp(wu)Q. Let vLp(Q;Rm). Then
Jp(w,v;Q)f(Qvdy,Dw(Q)).

3.Main results

This section is devoted to deduce the results stated in Theorems 1.1 and 1.2. We start by proving the lower bound in the case 1<p<. For what concerns the upper bound we present, for the reader’s convenience, a self contained proof in Theorem 3.2. For the sake of completeness we observe that the upper bound, in the case 1<p<, could be deduced as a corollary from the case p= (see Theorem 1.2), which, in turn, under slightly different assumptions, is contained in [8].

3.1.Lower semicontinuity in BV×Lp, 1<p<

Theorem 3.1.

Let Ω be a bounded open set of RN, let f:Rm×Rn×N[0,) be a continuous function satisfying (H0), (H1)p and (H2)p, and let Jp be the functional defined in (1.2). Then

(3.1)Jp(u,v;Ω)Ωf(v,u)dx+Ωf(0,dDsud|Dsu|)d|Dsu|
for any (u,v)BV(Ω;Rn)×Lp(Ω;Rm).

Proof.

The proof will be achieved, in two steps, namely by showing that

(3.2)limϱ0+Jp(u,v;Q(x0;ϱ))LN(Q(x0,ϱ))f(v(x0),u(x0)),for LN-a.e. x0Ω,(3.3)limϱ0+Jp(u,v;Q(x0,ϱ))|Du|(Q(x0,ϱ))f(0,dDsud|Dsu|(x0)),for |Dsu|-a.e. x0Ω.

Indeed, if (3.2) and (3.3) hold then, by virtue of (2.12), and [2, Theorem 2.56], (3.1) follows immediately.

Step 1.

Inequality (3.2) is obtained through an argument entirely similar to [2, Proposition 5.53] and exploiting [11, Theorem 11].

For LN-a.e. x0Ω it results that u is approximately differentiable (see (2.1)) and

limϱ0+1LN(Q(x0,ϱ))Q(x0,ϱ)|v(x)v(x0)|dx=0.

Consequently, given ϱ>0, and defined uϱ and vϱ as in Proposition 2.17, it results that uϱu0 in L1(Ω;Rn), where u0:=u(x0)x and vϱv(x0) in Lp(Ω;Rm). Then the scaling properties (2.13), and the lower semicontinuity of Jp entail that

(3.4)lim infϱ0+Jp(u,v;Q(x0,ϱ))ϱN=lim infϱ0+Jp(uϱ,vϱ;Q)Jp(u0,v(x0);Q).
Then the lower semicontinuity result proven in [11, Theorem 11], when u is in W1,1(Ω;Rn) and vLp(Ω;Rm), allows us to estimate the last term in (3.4) as follows
Jp(u0,v(x0);Q)f(v(x0),u(x0)),
and that provides (3.2).

Step 2. Here we present the proof of (3.3). To this end we exploit techniques very similar to [1] (see [2, Proposition 5.53]). Let Du=z|Du| be the polar decomposition of Du (see [2, Corollary 1.29]), for zSN×n1, and recall that for |Dsu|-a.e. x0, z(x0) admits the representation η(x0)ν(x0), with η(x0)Sn1 and ν(x0)SN1 (see [2, Theorem 3.94]). In the following, we will denote the cube Qν(x0,1) by Q.

To achieve (3.3) it is enough to show that

limϱ0+Jp(u,v;Q(x0,ϱ))|Du|(Q(x0,ϱ))f(0,z(x0))
at any Lebesgue point x0 of z relative to |Du| such that the limit on the left hand side exists and
(3.5)z(x0)=η(x0)ν(x0),limϱ0+|Du|(Q(x0,ϱ))ϱN=,(3.6)0=limϱ0+Q(x0,ϱ)|v|pdx|Du|(Q(x0,ϱ))=limϱ0+Q(x0,ϱ)|v|dx|Du|(Q(x0,ϱ)).
The above requirements are, indeed, satisfied at |Dsu|-a.e. x0Ω, by Besicovitch’s derivation theorem and Alberti’s rank-one theorem (see [2, Theorem 3.94]). Set ηη(x0) and νν(x0), for ϱ<N12dist(x0,Ω), define
uϱ(y):=u(x0+ϱy)u˜ϱϱϱN|Du|(Q(x0,ϱ)),yQ,
where u˜ϱ is the average of u in Q(x0,ϱ). Analogously define, as in Proposition 2.17,
(3.7)vϱ(y):=v(x0+ϱy),yQ.
Let us fix t(0,1). By [2, formula (2.32)], there exists a sequence {ϱh} converging to 0 such that
(3.8)limh|Du|(Q(x0,tϱh))|Du|(Q(x0,ϱh))tN.
Denote uϱh by uh, then |Duh|(Q)=1 and, passing to a not relabelled subsequence, {uh} converges in L1(Q;Rn) to a BV function u. Correspondingly, denote vϱh by vh. Then, arguing as in [2, Proof of Proposition 5.53] we have
(3.9)|Du|(Q)1and|Du|(Qt)tN,
where Qt:=tQ. It results that u(y)=ηψ(y·ν), for some bounded increasing function ψ in (12,12). Take φCc1(Q) such that φ=1 on Qt and 0φ1, and let us define wh:=φuh+(1φ)u. The functions wh converge to u in L1(Q;Rn) and moreover we have
|D(whuh)|(Q)|D(uhu)|(QQt)+Q|φ||uhu|dy|Duh|(QQt)+|Du|(QQt)+Q|φ||uhu|dy.
Therefore, by (3.8) and (3.9), one has
(3.10)lim suph|D(whuh)|(Q)2(1tN).
Similarly,
|Dwh|(QQt)|Duh|(QQt)+|Du|(QQt)+Q|φ||uhu|dy,
consequently
(3.11)lim suph|Dwh|(QQt)2(1tN).

Setting ch:=|Du|(Q(x0,ϱh))ϱhN, by the scaling properties of Jp in Proposition 2.17 and by the growth conditions (H1)p, we have

Jp(u,v;Q(x0,ϱh))|Du|(Q(x0,ϱh))=Jp(chuh,vh;Q)chJp(chwh,vh;Qt)chJp(chuh,vh;Q)chC(ch1|QQt|+|Dwh|(QQt)+ch1QQt|vh|pdy).
By (3.5), ch, moreover taking into account (3.7) and (3.6), by (3.11), it results that
limϱ0+Jp(u,v;Q(x0,ϱ))|Du|(Q(x0,ϱ))lim suphJp(chuh,vh;Q)ch2C(1tN).
On the other hand, Lemma 2.18 entails that, for every hN,
Jp(chwh,vh;Q)f(Qvhdy,chDwh(Q))f(Qvhdy,chDuh(Q))chγ|D(whuh)|(Q),
where γ is the constant appearing in (2.3). Then by (3.10), we have that
limϱ0+Jp(u,v;Q(x0,ϱ))|Du|(Q(x0,ϱ))lim suphf(Qvhdy,chDuh(Q))ch2(C+γ)(1tN).
By the definition of uh, Duh(Q)=Du(Q(x0,ϱh))|Du|(Q(x0,ϱh)), hence Duh(Q)z(x0), since x0 is a Lebesgue point of z. Now, taking into account (2.3) and (H2)p, we have
lim suphf(Qvhdy,chDuh(Q))ch=limhf(Qvhdy,chz(x0))ch=limh(f(Qvhdy,z(x0))C|Qvhdy|p+1ch)=f(0,z(x0)),
where it has been exploited the fact that ch, 3. of Proposition 2.8, the nondecreasing behaviour of the Lp norm in the unit cube with respect to p (i.e. |Qvhdy|pQ|vh|pdy), and (3.6). □

3.2.Relaxation

We start by observing that Theorem 1.2 is contained in [8] under a uniform coercivity assumption. We do not propose the proof in our setting, since it develops along the lines of Theorems 3.1 and 3.2.

On the other hand, several observations about Theorem 1.2 are mandatory:

  • i) If f satisfies (H1)p and (H2)p then Jp(u,v)J(u,v) for every (u,v)BV(Ω;Rn)×L(Ω;Rm).

  • ii) For the reader’s convenience we observe that the proof of the lower bound in Theorem 1.2 develops exactly as that of Theorem 3.1, using the L bound on v to deduce (3.6) and the uniform bound on vϱ in (3.7), (H2) and (2.4) in order to estimate lim suphf(Qvhdy,chDuh(Q))ch.

    Regarding the upper bound, the bulk part follows from [11, Theorems 12 and 14], while for the singular part we can argue exactly as proposed in the proof of the upper bound in [8] just considering conditions (H1) and (H2) in place of (H1)p and (H2)p.

  • iii) The above arguments remain true under assumptions (2.6) and (2.7).

We are now in position to prove the upper bound for the case BV×Lp, for 1<p<. We emphasize that an alternative proof could be obtained via a truncation argument from the case p= as the one presented in [11, Theorem 12], but we prefer the self contained argument below.

Theorem 3.2.

Let Ω be a bounded open set of RN and let f:Rm×Rn×N[0,) be a continuous function. Then, assuming that f satisfies (H0),(H1)p and (H2)p,

Jp(u,v)Ωf(v,u)dx+Ωf(0,dDsud|Dsu|(x))d|Dsu|(x),
for every (u,v)BV(Ω;Rn)×Lp(Ω;Rm).

Proof.

First we observe that Proposition 2.16 entails that Jp is a variational functional. Thus the inequality can be proved analogously to [2, Proposition 5.49]. For what concerns the bulk part, it is enough to observe that given uBV(Ω;Rn) and vLp(Ω;Rm), taking a sequence of standard mollifiers {ϱεk}, where εk0, it results that uk=uϱεk+Dsuϱεk, where uk:=uϱεk. The local Lipschitz behaviour of f in (2.3) gives

Af(v,uk)dxAf(v,uϱεk)dx+γ|Dsu|(Iεk(A))
for every kN, where Iεk(A) denotes the εk neighborhood of A. Then if |Dsu|(A)=0, letting εk0, we obtain
Jp(u,v;A)Af(v,u)dx+γ|Dsu|(A),
for every open subset A of Ω. Thus we can conclude that
Jpa(u,v;B)Bf(v(x),u(x))dx
for every (u,v)BV(Ω;Rn)×Lp(Ω;Rm) and B Borel subset of Ω.

To achieve the result, it will be enough to show that

Jps(u,v;B)Bf(0,dDsud|Dsu|)d|Dsu|for every B Borel subset of Ω.
For every ξRn×N and bRm, define the function
g(b,ξ):=supt0f(t1pb,tξ)f(0,0)t.

It is easily seen that g is (p,1)-positively homogeneous, i.e. tg(b,ξ)=g(t1pb,tξ) for every t>0, (b,ξ)Rm×Rn×N, g is continuous and, since f satisfies (2.3), g inherits the same property. Moreover, the monotonicity property of difference quotients of convex functions ensures that, whenever rank ξ1, g(b,ξ)=fp(b,ξ), where the latter is defined as

fp(b,ξ):=lim suptf(t1pb,tξ)t.
In particular g(0,ξ)=f(0,ξ)=fp(0,ξ), whenever rank ξ1.

Then for every open set AΩ such that |Du|(A)=0, defining for every hN, uh:=uϱεh and vh:=v where {ϱεh} is a sequence of standard mollifiers and εh0. Then uhu in L1. Also [2, Theorem 2.2] entails that |Duh||Du| weakly ∗ in A and |Duh|(A)|Du|(A). Thus, since ff(0,0)+g,

Jp(u,v;A)lim infhAf(v,uh)dx(3.12)lim suphAf(0,0)dx+lim infhAg(v,uh)dx.
Since the first term in the right hand side is bounded by CLN(A), taking the Radon–Nikodým derivative with respect to |Dsu| we obtain 0.

Regarding the second term in the right hand side of (3.12), we have

lim infhAg(v(x),Duϱh)dxlim suphAg(0,Duϱh)dx+CA|v(x)|pdx+lim infh+A|v(x)||Duϱh|1pdx.

Taking the Radon–Nikodým derivative, the last two terms disappear, since |Duϱh||Du|, |v|pLN is singular with respect to |Dsu| and the Hölder inequality can be applied, i.e.

A|v(x)||Duϱh|1pdx(A|v(x)|pdx)1p(A|Duϱn|dx)1p.
Then the thesis is achieved via the same arguments as in [2, Proposition 5.49]. □

Remark 3.3.

It is worth to observe that an alternative argument to the one presented above, concerning the upper bound inequality for the singular part, can be provided by means of approximation. In fact, one can prove that Jps(u,v;B)Bf(0,dDsud|Dsu|)d|Dsu| for every B Borel subset of Ω, when uBV(Ω;Rn) and vC(Ω;Rm), and then a standard approximation argument via mollification allows to reach every vLp(Ω;Rm).

For what concerns the case vC(Ω;Rm) it is enough to consider the function g(b,ξ):=supt0f(b,tξ)f(b,0)t, exploit its properties of positive 1-homogeneity in the second variable, i.e. tg(b,ξ)=g(b,tξ), for every t>0, (b,ξ)Rm×Rn×N, (2.4), and the fact that when rank ξ1, then g(b,ξ) is constant with respect to b and f(b,ξ)=g(b,ξ)=f(0,ξ). To conclude it is enough to apply Reshetnyak continuity theorem.

Proof of Theorem 1.1.

The result follows from Theorems 3.1 and 3.2, applying Proposition 2.14 to remove assumption (H0). □

Acknowledgements

The research of the authors has been partially supported by Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through UTA-CMU/MAT/0005/2009 and CIMA-UE.

The second author is a member of INdAM-GNAMPA, whose support is gratefully acknowledged.

Appendices

Appendix

Consider the functional

J1(u,v):=inf{liminfhJ(uh,vh):uhW1,1(Ω;Rn),vhL(Ω;Rm),(A.1)uhu in L1,vhv in M},
for any pair (u,v)BV(Ω;Rn)×M(Ω;Rm), where this latter set denotes the set of signed Radon measures and the weak ∗ convergence denotes the one in the sense of measures.

The integral representation of (A.1) will be stated in Theorem A.1 under the following hypotheses:

  • (H1)1 There exists C>0 such that

    1C(|b|+|ξ|)Cf(b,ξ)C(1+|b|+|ξ|),
    for (b,ξ)Rm×Rn×N.

  • (H2)1 There exists C>0,L>0,0<τ1 such that

    t>0,ξRn×N,with t|(b,ξ)|>L|f(tb,tξ)tf(b,ξ)|C(|(b,ξ)|1τtτ),
    where f1 is the recession function of f defined for every bRm as
    (A.2)f1(b,ξ):=lim suptf(tb,tξ)t.

Theorem A.1.

Let J be given by (1.1), with f satisfying (H1)1 and (H2)1 and let J1 be given by (A.1) then

J1(u,v)=ΩCQf(va,u)dx+Ω(CQf)1(dvd|Dsu|,dDsud|Dsu|)d|Dsu|+Ω(CQf)1(dvd|vs|,0)d|vs|
for every (u,v)BV(Ω;Rn)×M(Ω;Rm), where va is the absolutely continuous part of the Radon measure v with respect to the Lebesgue measure and vs is the singular part of v with respect to |Du|.

Proof.

The same arguments which lead to Proposition 2.14, allow to assume without loss of generality that f is convex–quasiconvex, i.e. to replace f by CQf in (A.1).

Arguing as in [3, Lemma 4.7] one can prove that the set function J1(v,u,·), defined as in (A.1) for all the open subsets of Ω, is the trace on these latter sets of a Radon measure absolutely continuous with respect to LN+|Du|+|v|.

Then the rest of the proof can be obtained as in [3] by providing a lower bound and an upper bound, that we will sketch in the sequel, just emphasizing the main differences.

For what concerns the lower bound, it is enough to prove that

(A.3)limϱ0+J1(u,v;Q(x0;ϱ))LN(Q(x0,ϱ))f(va(x0),u(x0)),for LN-a.e. x0Ω,(A.4)limϱ0+J1(u,v;Q(x0,ϱ))|Du|(Q(x0,ϱ))f(dvd|Dsu|(x0),dDsud|Dsu|(x0)),for |Dsu|-a.e. x0Ω,(A.5)limϱ0+J1(u,v;Q(x0,ϱ))|Du|(Q(x0,ϱ))f(dvd|vs|(x0),0),for |vs|-a.e. x0Ω.

All the inequalities can be proven following arguments analogous to the ones in [3, Lemma 5.1], taylored for thin structures. Indeed we observe that the density Q(·) appearing therein coincides with CQ(·), as proven in [4]. Then the proofs in [3, Lemma 5.1] can be repeated line by line but with easier constructions, in particular it suffices to replace the unique sequence {un} (and its average), by the couple {(un,vn)} with unu in L1, and vnv in M.

For what concerns the upper bound, i.e proving inequalities opposite to (A.3)–(A.5), one can argue as in the proof of Theorem 3.2, using as a recovery sequence the couple {(vk,uk)} with uk:=uϱεk and vk:=vϱεk, where {ϱεk} is a sequence of standard mollifiers. Then for the bulk term it suffices to exploit the standard Lipschitz property of f with respect to the couple (b,ξ) (i.e. (2.3), when p=1), and the fact that vϱεk=vaϱεk+vsϱεk, and finally taking the Radon–Nikodým derivative around points of absolute continuity with respect to the Lebesgue measure of u,u and va.

For what concerns the other two terms in the limiting energy they can be reached through the function g(b,ξ):=supt0f(tb,tξ)t, which is positively 1 homogeneous and coincides with f1(b,ξ), whenever rankξ1. Finally the conclusion can be achieved differentiating with respect to |Dsu| or vs. □

Remark A.2.

We observe that in [5] it was considered for the vs the weak convergence in L1 thus leading to a target function still in L1. In the present case the limiting function is a measure, which can be decomposed in three terms, the first two absolutely continuous with respect to LN and |Dsu| respectively, and the third possibly singular with respect to |Du| and this entails the presence of a third integrand in the energy above.

We also observe that an argument entirely similar to Proposition 3.6 warranties that (CQf)1(b,ξ)=CQ(f1)(b,ξ) for every bRm,ξRn×N.

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