Lower semicontinuity in BV of quasiconvex integrals with subquadratic growth

DOI:10.1051/cocv/2012021 www.esaim-cocv.org

LOWER SEMICONTINUITY IN BV OF QUASICONVEX INTEGRALS WITH SUBQUADRATIC GROWTH

Parth Soneji

Abstract.A lower semicontinuity result inBV is obtained for quasiconvex integrals with subquadratic growth. The key steps in this proof involve obtaining boundedness properties for an extension operator, and a precise blow-up technique that uses fine properties of Sobolev maps. A similar result is obtained by Kristensen in [Calc. Var. Partial Differ. Equ.7(1998) 249–261], where there are weaker asssumptions on convergence but the integral needs to satisfy a stronger growth condition.

Mathematics Subject Classification. 49J45.

Received September 30, 2011. Revised March 2, 2012.

Published online March 14, 2013.

1. Introduction

We consider the variational integral

F(u;Ω) :=

Ω

f(Du(x)) dx, (1.1)

where Ω is a bounded open subset ofRⁿ,u:Ω → R^N is a vector-valued function, Du denotes the Jacobian matrix ofuandf is a nonnegative continuous function defined in the spaceR^N×n of allN×nmatrices.

The aim of this paper is to prove a lower semicontinuity result for a quasiconvex integral with an integrand f of subquadratic growth at infinity. Recall that a continuous function f is quasiconvex if for eachξ∈R^N×n we have

Rⁿ

[f(ξ+Dφ(x))−f(ξ)] dx≥0

for all test functionsφ∈C₀^∞(Rⁿ;R^N). We require thatf satisfies the following growth condition for 1< r <2:

0≤f(ξ)≤L(|ξ|^r+ 1) (1.2)

for a fixed finiteL >0 and allξ∈R^N×n. Note that (1.2) implies thatF is defined and continous on the Sobolev Space W^1,r(Ω;R^N).

Keywords and phrases.Lower semicontinuity, quasiconvex integrals, functions of bounded variation.

1 Mathematical Institute, University of Oxford, 24-29 St Giles’ Oxford, OX1 3LB, UK.soneji@maths.ox.ac.uk;

Parth.Soneji@maths.ox.ac.uk

Article published by EDP Sciences c EDP Sciences, SMAI 2013

P. SONEJI

The following theorem is our main result:

Theorem 1.1. Let f: R^N×n →R be a continuous quasiconvex function satisfying the growth condition (1.2) for some exponent1< r <2. LetΩ be an open subset of Rⁿ.

Let (u_j)be a sequence inW_loc^1,r(Ω;R^N)andu∈W_loc^1,p(Ω;R^N), wherep≥1 andp > ^r₂(n−1). Suppose

u_j u^∗ inBV_loc(Ω;R^N) (1.3)

and

(u_j)uniformly bounded in L^q_loc(Ω;Rⁿ), (1.4) where

q >max

1,r(n−1) 2−r

· (1.5)

Then

lim inf

j→∞ Ω

f(Du_j) dx≥

Ω

f(Du) dx. (1.6)

Note that whenn≥3, the conditions of this theorem require that the limit mapuis more regular than the maps (u_j). When n = 2, however, we can take u ∈ W_loc^1,1(Ω;R^N), so in this case ucan be less regular than theu_j. In this case, however, there is a result by Kristensen [19] even foru∈BV: see Theorem2.1. By (1.4) we mean simply that for any compact setK ⊂Ω, the sequence (u_j) is uniformly bounded in L^q(K;R^N),i.e.

sup_ju_jL^q_(K) ≤ C(K), where C(K) is a positive constant possibly depending on K. We may remark that this is a natural condition if, for example, we assume that the mapsu_j anduare constrained to remain on a compact manifold (in which case we would infer the stronger condition that the (u_j) are uniformly bounded in L^∞(Ω;R^N). Indeed, many problems in materials science involve such constrained variational problems – see for instance [10]).

The proof of this Theorem1.1 relies on the following lemma (which is essentially the theorem in the special case where the limit uis affine and whereΩ is the open unit ball B in Rⁿ) combined with a precise blow-up technique which will be detailed later in this paper.

Lemma 1.2. Let B denote the open unit ball inRⁿ. Suppose(u_j)⊂W^1,r(B;R^N),1< r <2, andf:R^N×n→ Ris as above. Suppose the following conditions hold:

(i)

u_j→0 strongly in L¹(B;R^N); (1.7)

(ii)

supj B|Du_j|dx <∞; (1.8)

(iii) There exists a setF ⊂(0,1)such that for all 0< δ <1,|F∩(δ,1)|>0 and supj sup

∈Fu_jL^q_(∂B)<∞, (1.9)

where

q >max

1,r(n−1) 2−r

· (1.10)

Then we have the following inequality:

lim inf

j→∞ B

f(Du_j) dx≥Lⁿ(B)·f(0). (1.11)

LOWER SEMICONTINUITY IN BV OF QUASICONVEX INTEGRALS WITH SUBQUADRATIC GROWTH

The proof of this lemma relies on a technique originating in works by Mal´y et al. (see [11,12,21,24]). A key step in this proof involves obtaining an integral estimate for a trace-preserving extension operator. The result, contained in the following lemma, involves adapting and generalising a result by Carozzaet al.[7]. In the statement of this result, as well as subsequently, we denote byB the open ball in Rⁿ with centre 0, radius. Lemma 1.3. Let 1< r <2. Then for qsatisfying

q≥max

1,r(n−1) 2−r

,

there exists a linear extension operator

E: (W^1,1∩L^q)(∂B;R^N)→W^1,r(B₂\B;¯ R^N) with the following properties:

1. ifg∈C¹(∂B;R^N)thenE(g)∈C^∞(B₂\B)¯ withE(g)|∂B=g;

2. if(z_j)⊂C^∞(∂B;R^N)andlim_j→∞

∂Bz_j·φdHⁿ⁻¹= 0for allφ∈C^∞(∂B;R^N), then for any multi-index α,∂^α[Ez_j]→0 locally uniformly inB₂\B;¯

3. there exist positive constantsc₁,c₂, dependent on n, N, r, such that:

(a)

B2\B|E(g)|^r≤c₁g^r_Lq(∂B); (b)

B2\B|DE(g)|^r≤c₂g_L^r2q(∂B)· Dg_L¹_(∂B) for allg∈C¹(∂B).

2. Preliminary remarks

2.1. Background

The classical lower semicontinuity result for quasiconvex integrands, essentially attributable to Morrey (see, for example, [9]) says that the functionalF(u;Ω) defined in (1.1) is (sequentially) weakly lower semicontinuous in W^1,p(Ω;R^N) forf satisfying the growth condition

0≤f(ξ)≤L(|ξ|^q+ 1) (2.1)

for a fixed finiteL >0, allξ∈R^N×n, and where 1≤q≤p <∞.

Remaining in the quasiconvex setting, a key refinement of this result is obtained by Mal´y in [22]: under the growth condition (2.1) (and imposing additional structure conditions),F(u;Ω) is proved to be lower semicon- tinuous inW^1,q(Ω;R^N) with respect to weak convergence inW^1,p(Ω;R^N) forp≥q−1. Other related works appear in [6,13,23].

For lower semicontinuity theorems in theBV context, we first refer to the monograph of Ambrosioet al.[3].

Firstly, note that it is not entirely clear how to define F(u;Ω) when uis aBV function. Following a method that was first used by Lebesgue for the area integral, and then adopted by Serrin [28,29] and, in the modern context, by Marcellini [23], we may consider the functional

F(u, Ω) := inf

(uj)

lim inf

j→∞ Ω

f(Du_j) dx

(u_j)⊂C¹(Ω,R^N) u_j→uinL¹_loc(Ω,R^N)

.

P. SONEJI

F is known as the Lebesgue–Serrin extension ofF and is an important quantity not only when we want to defineF(u;Ω) for a wider class of functionsubut also, for example, when there is a lack of quasiconvexity. In a paper of Ambrosio and Dal Maso [2], it is proved that iff has linear growth at infinity (soq = 1 in (2.1)), then for every open setΩ⊂Rⁿ and everyu∈BV(Ω;R^N), we have

F(u;Ω) =

Ω

f(∇u) dx+

Ω

f_∞ D^su

|D^su|

|D^su|,

where∇uis the density of the absolutely continuous part of the measureDuwith respect to Lebesgue measure, D^suis the singular part ofDu, _|D^Ds_s^u_u| is the Radon–Nikodym derivative of the measureD^suwith respect to its variation|D^su|, andf_∞:R^N×n→[0,∞) is the recession function off defined by

f_∞(ξ) := lim sup

t→∞

f(tξ) t ·

In this connection see also [15], where the case of general integrandsf =f(x, u,∇u) of linear growth is treated, and [26] for a proof that avoids the use of Alberti’s rank-one theorem.

For lower semicontinuity theorems in the superlinear growth case, we have the following result by Kris- tensen [19], which obtains a lower bound for a suitable Lebesgue–Serrin extension. This is similar to Theorem1.1 and indeed provided the initial motivation for this paper. Here there are weaker assumptions on convergence and on the mapsu_j,u, but the integrand f needs to satisfy a stronger growth condition.

Theorem 2.1. [19] Let f:R^N×n →Rbe a continuous quasiconvex function satisfying the growth condition 0≤f(ξ)≤L(|ξ|^q+ 1)

for a fixed finiteL >0, a fixed exponent q∈[1,_n−1ⁿ ), and all ξ∈R^N×n. Let Ω be an open, bounded subset of Rⁿ.

Suppose(u_j)⊂W_loc^1,q(Ω;R^N)converges tou∈BV_loc(Ω;R^N)inL¹_loc(Ω;R^N)and supj ω|Du_j|<+∞for all ω⊂⊂Ω.

Then

lim inf

j→∞ Ω

f(Du_j) dx≥

Ω

f(∇u) dx, where∇udenotes the approximate gradient ofu.

For related work in the compensated compactness set-up, see also [16]. Also of importance is the paper of Carozzaet al. [7] where a similar condition to (1.4) in Theorem 1.1 is used to obtain a lower semicontinuity result. Namely, for an integrandf satisfying (2.1), whereq=p+ 1 andp >1, we have lower semicontinuity with respect to sequences of functions (u_j) inW_loc^1,p+1(Ω;R^N) that converge weakly to uin W_loc^1,p(Ω;R^N), provided also thatu_j−uconverges to 0 strongly inL^∞_loc(Ω;R^N) andu∈C¹(Ω;R^N). Finally we mention that the theory discussed here is not vacuous. Indeed it has been shown by ˇSver´ak [30] that there exist quasiconvex functions onR^2×2that have subquadratic growth and are not polyconvex (and hence not convex). In this connection we also mention [25,31].

LOWER SEMICONTINUITY IN BV OF QUASICONVEX INTEGRALS WITH SUBQUADRATIC GROWTH

2.2. General remarks on the result

The results of this paper can actually be stated with the following more general subquadratic growth condi- tion:

Let Φ: [0,∞)→ [0,∞) be a convex, doubling, non-decreasing function such that Φ(0) = 0 and, for some σ_Φ>0 :

t→ Φ(t)

t^2−σΦ is nonincreasing on (0,∞) and

∞ 1

Φ(t)¹2

t² dt <∞. (2.2)

(Recall thatΦis doubling means that, for a fixed constantc,Φ(2t)≤c Φ(t) for allt≥0).

Assume thatf satisfies

0≤f(ξ)≤L(Φ(|ξ|) + 1) for a fixed finiteL >0 and allξ∈R^N×n.

This growth condition implies that F is defined and continous on the generalised Orlicz–Sobolev space W^1,Φ(Ω;R^N). For further information on the subject of such spaces, we refer to the book of Iwaniec and Martin [18]. We have focused in particular on the case whereΦ is of the formΦ(t) =t^r, which just puts us in the more familiar setting of the Sobolev SpaceW^1,r(Ω;R^N), noting that for suchΦ, (2.2) is satisfied whenever 1< r <2.

Our result for subquadratic quasiconvex integrands hinges on a result by Greco et al. in [17], concerning integral estimates of the Hardy–Littlewood Maximal function. In this connection the condition (2.2) is sharp and the proof provided cannot be weakened to include integrands of quadratic growth at infinity. Indeed, we do not know if the main theorem is true forf satisfying (1.2) forr= 2, even when Ω=B, u= 0, u_j ^∗ 0 in BV(B;R^N) and u_j →0 inL^∞(B;R^N). However, what is clear is that the proof of such a result, if it is true, needs to proceed by a different means. We may deduce from the following counterexample, established by Mal´y in [20], that the result is certainly not true whenf has at least cubic growth in some directions (andn, N ≥3).

Counterexample 2.2 ([20]).LetQbe the cube (0,1)ⁿ, and 1< r < n−1. There is a sequence of orientation- preservingC¹-diffeomorphisms (u_j) onQsuch thatu_j converge weakly to the identity in W^1,r(Q,Rⁿ) and

j→∞lim Q

detDu_jdx= 0.

A suitable counterexample for our purposes immediately follows by taking n= 3 (or, if we want a result in higher dimensions we can simply consider a suitable 3×3 minor of the Jacobian),Ω=Q, andf(ξ) =|detξ|.

f is polyconvex, hence quasiconvex, and satisfies the growth condition 0≤f(ξ)≤L(|ξ|³+ 1).

Moreover the u_j, being diffeomorphisms of Q onto Q, are clearly uniformly bounded in L^q(Q,Rⁿ) for any 1 ≤ q ≤ ∞, and weak convergence in W^1,r for 1 < r < 2 obviously implies weak* convergence in BV. And ifuis the identity map on Q, then_QdetDudx= 1. So all the conditions of Theorem1.1 except the growth condition are satisfied, but lower semicontinuity does not obtain.

It is interesting to note that we can also set the main result of this paper in the context ofW^1,1-quasiconvexity.

The condition ofW^1,p-quasiconvexity, which generalises in a natural way the quasiconvexity condition of Morrey, is defined and studied in the well-known paper of Ball and Murat [4]. Namely, a continuous integrandf is said to beW^1,p-quasiconvex (where 1≤p≤ ∞) if it is bounded below and satisfies

Rⁿ

[f(ξ+Dφ(x))−f(ξ)] dx≥0

for allξ∈R^N×n andφ∈W₀^1,p(Rⁿ;R^N). Ifp=∞ then this condition is just the usual definition of quasicon- vexity. In the Proof of Lemma 1.2, we use the well known result that the conditions onf in Theorem 1.1, in

P. SONEJI

particular its continuity, quasiconvexity and growth condition (1.2), imply that it isW^1,r-quasiconvex. Crucially, this is why we need the maps u_j in Theorem 1.1 to be in W_loc^1,r(Ω;R^N). However, if we assume the stronger condition that f is W^1,1-quasiconvex, we only require the (u_j) to be in W_loc^1,1(Ω;R^N). That is, we have the following result, which has virtually the same proof as Theorem1.1.

Theorem 2.3. Let f: R^N×n → R be a continuous W^1,1-quasiconvex function satisfying the growth condi- tion (1.2)for some exponent 1< r <2. LetΩ be an open subset of Rⁿ.

Let (u_j)be a sequence inW_loc^1,1(Ω;R^N) andu∈W_loc^1,p(Ω;R^N), wherep≥1 andp > ^r₂(n−1). Suppose u_j u^∗ inBV_loc(Ω;R^N)

and

(u_j)uniformly bounded in L^q_loc(Ω;Rⁿ), where

q >max

1,r(n−1) 2−r

· Then

lim inf

j→∞ Ω

f(Du_j) dx≥

Ω

f(Du) dx.

It is also worth briefly discussing more generally the regularity assumptions of the mapsu_j,uin the main result.

The increased regularity requirement onu, that it is inW_loc^1,p(Ω;R^N) wherep≥1 andp > ^r₂(n−1), is required to make use of the fact that the (u_j) are uniformly bounded in L^q_loc(Ω;R^N) for qsatisfying (1.5) when using the “blow-up argument” to obtain the proof of Theorem1.1from Lemma1.2.

Another issue is that if we just assume thatf is quasiconvex in the sense of Morrey, then we could consider whether lower semicontinuity still obtains if the maps (u_j) are less regular than W_loc^1,r(Ω;R^N). Even though it is still an open question whether lower semicontinuity obtains when f has quadratic growth, the following counterexample provided in [4] demonstrates (if we taken= 2) that in this case we would certainly require at least that the (u_j) are inW_loc^1,2(Ω;R^N).

Counterexample 2.4 ([4]). Letn >1 and Ω be an open bounded subset ofRⁿ. Let 1≤r < n. Then there exist (u_j)⊂W^1,r(Ω;Rⁿ) such thatu_j converge weakly inW^1,r(Ω;Rⁿ) to the identity map, as well as strongly in L^∞(Ω;Rⁿ) (hence the (u_j) are uniformly bounded inL^∞(Ω;Rⁿ)), but for allj, detDu_j(x) = 0 for almost allx∈Ω.

Let us now consider properties of a suitable Lebesgue–Serrin extension, and introduce the functional (for 1< r <2 andqsatisfying (1.5))

F(u, Ω) := inf

(uj)

⎧⎨

⎩lim inf

j→∞ Ω

f(Du_j) dx

(u_j)⊂W_loc^1,r(Ω,R^N)

(u_j) uniformly bounded inL^q_loc(Ω,R^N) u_j u^∗ weakly^∗ in BV_loc(Ω,R^N)

⎫⎬

⎭.

Note that Theorem1.1implies the following result:

Corollary 2.5. Let f:R^N×n→R, satisfy the conditions in Theorem1.1. Then

– If n≥3,p≥1andp > ^r₂(n−1), andqsatisfies (1.5),F(u;Ω) =F(u;Ω)for allu∈(W_loc^1,p∩L^q_loc)(Ω;R^N);

– if n= 2, then this equality holds for all u∈(W_loc^1,r∩L^q_loc)(Ω;R^N).

LOWER SEMICONTINUITY IN BV OF QUASICONVEX INTEGRALS WITH SUBQUADRATIC GROWTH

Proof. For anyn, Theorem1.1tells us that ifu∈W_loc^1,p(Ω;R^N) forpsatisfying the stated conditions, and (u_j) is a sequence satisfying the conditions given in the definition ofF(u;Ω), then

lim inf

j→∞ Ω

f(Du_j) dx≥F(u;Ω).

Taking the infimum of all such (u_j), we get

F(u;Ω)≥F(u;Ω) whenu∈W_loc^1,p(Ω;R^N).

Now note that ifu∈(W_loc^1,r∩L^q_loc)(Ω;R^N), then by simply takingu_j=ufor allj, we get a sequence satisfying the conditions inF(u;Ω), so certainly

F(u;Ω)≥F(u;Ω) whenu∈(W_loc^1,r∩L^q_loc)(Ω;R^N).

We conclude by noting that since 1< r <2, forn≥3 we have W_loc^1,p(Ω,R^N)⊂W_loc^1,r(Ω,R^N) , and forn= 2, since in this case we can takep= 1,

W_loc^1,r(Ω,R^N)⊂W_loc^1,p(Ω,R^N).

If we wish to describe F for an even wider class of functions u, things can be more difficult. Certainly, if u /∈ L^q_loc(Ω;R^N) then there can be no sequence (u_j) uniformly bounded inL^q_loc(Ω;R^N) satisfying u_j u^∗ in BV_loc(Ω;R^N) (or even just strongly in L¹_loc), since this would imply that uis itself in L^q_loc(Ω;R^N). Hence we have

F(u, Ω) = inf∅= +∞.

Results by Bouchitt´eet al.[5], Fonseca and Mal´y [11,12] indicate that, even forn≥3, a measure representation forF should exist foru∈(W_loc^1,r∩L^q_loc)(Ω;R^N), but we have been unable to prove this yet. A counterexample due to Acerbi and Dal Maso [1] shows that if r=n=N = 2 and u∈(BV_loc∩L^∞_loc)(Ω;R^N), then a measure representation does not exist at all. Although their conditions are slightly different from ours, it is not difficult to see from their paper that their counterexample also applies to our case. In fact, they present an example where the set functionω→F(u;ω) is not even subadditive (for an alternative proof, see [8]).

3. Proof of the main result

Proof of Lemma 1.2. By approximation we may assume (u_j) ⊂ C¹( ¯B;R^N). If the left hand side of (1.11) is infinite then there is nothing to prove, so suppose it is finite. Moreover, by extracting a subsequence if necessary, we can assume

l₀:= lim inf

j→∞ B

f(Du_j) dx= lim

j→∞ B

f(Du_j) dx.

With reference to (1.9), write M = sup_j sup_∈Fu_jL^q_(∂B). From (1.7), by the Fubini–Tonelli theorem and the Rellich–Kondrachoff compactness theorem we have

j→∞lim

1 0 ∂B

|u_j|dHⁿ⁻¹d= lim

j→∞ B|u_j|dx= 0.

This implies there exists a subsequence{u_j}j∈T such that

j→∞,limj∈T ∂B

|u_j|dHⁿ⁻¹= 0 (3.1)

P. SONEJI

for almost all∈(0,1). By Fatou’s Lemma and (1.8) we have

1 0

lim inf

j→∞,j∈T ∂B

|Du_j|dHⁿ⁻¹d≤ lim inf

j→∞,j∈T B|Du_j|dx <∞. Thus, for almost all∈(0,1)

lim inf

j→∞,j∈T ∂B

|Du_j|dHⁿ⁻¹<∞. (3.2)

Now fix 0< δ <1. By (3.1), (3.2) and (1.9) we can choose∈(δ,1) such that all the following hold:

1.

j→∞,limj∈T ∂B

|u_j|dHⁿ⁻¹= 0;

2.

lim inf

j→∞,j∈T ∂B

|Du_j|dHⁿ⁻¹<∞;

3.

supj∈Tu_jL^q_(∂B)≤M.

Now take a further subsequence{u_j}_j∈S, whereS⊆T, so that

j→∞,limj∈S ∂B

|Du_j|dHⁿ⁻¹= lim inf

j→∞,j∈T ∂B

|Du_j|dHⁿ⁻¹. Relabel the sequence (u_j) so thatS=N. Now define the sequence (gj)⊂W^1,1(∂B;R^N) as:

g_j(x) :=u_j|_∂B(x) forx∈∂B.

Take a cut-off functionη∈C¹(B;R) such that1_B ≤η≤1_B,|Dη| ≤ ₁₋² , and define (v_j)⊂W₀^1,r(B;R^N) as:

v_j(x) :=

η(x)·(E(g_j))(^x) if |x| ≥, u_j(x) if|x|< , whereE is the extension operator from Lemma1.3.

Since the functiont→t^r is convex, (s+t)^r≤2^r−1(s^r+t^r) for alls,t≥0. Hence from Lemma1.3we have

B\B

|Dv_j|^r≤

B\B

Dη·Eg_j(·/)+η·D[Eg_j(·/)]^r

≤2^r−1

B\B

|Dη|^r·Eg_j(·/)^r+ 2^r−1

B\B

|η|^r·D[Eg_j(·/)]^r

≤C

B\B

Eg_j(·/)^r+C

B\B

D[Eg_j(·/)]^r (3.3)

for some constant C. We estimate the two terms in (3.3) using Lemma1.3(3) as follows:

B\B

D[Eg_j(·/)]^r≤c₂g_jL^r2q(∂B)· Dg_jL¹_(∂B) by (1.9) ≤c₂M^r2 · Dg_jL¹_(∂B)

=C

∂B

|Du_j|dHⁿ⁻¹ (3.4)

LOWER SEMICONTINUITY IN BV OF QUASICONVEX INTEGRALS WITH SUBQUADRATIC GROWTH

for another constant C. Now note that we may obtain the same inequality (albeit for a different constantC) using Lemma1.3for any otherr such thatr < r <2, and also (with reference to (1.10)) satisfyingq > ^r_2−r⁽ⁿ⁻¹⁾ . Hence by (3.4) and Lemma1.3, since

supj ∂B

|Du_j|dHⁿ⁻¹<∞,

we can use the De la Vall´ee Poussin criterion to deduce that the sequence|D[Eg_j]|^ris equi-integrable onB\B. By Lemma1.3, since

supj ∂B

|u_j|dHⁿ⁻¹→0 as j→ ∞,

D[Eg_j]→0 locally uniformly onB\B, and hence so does|D[Eg_j]|^r. Thus, by Vitali’s Convergence Theorem,

B\B

D[Eg_j(·/)]^r→0 asj→ ∞. Similarly

B\B

[Eg_j(·/)]^r≤c₁g_j^r_Lq(∂B)

by (1.9) ≤c₁M^r, so|Eg_j|^ris equi-integrable onB\B, and using Lemma1.3and Vitali,

B\B

Eg_j(·/)^r→0 as j→ ∞.

Combining these estimates in (3.3), we have lim sup

j→∞ B\B

|Dv_j|^rdx= 0. (3.5)

Now we use the quasiconvexity and nonnegativity off to obtain

B

f(Du_j)≥

B

f(Du_j) =

B

f(Dv_j)−

B\B

f(Dv_j)≥Lⁿ(B)f(0)−

B\B

f(Dv_j)

≥Lⁿ(B)f(0)−L

B\B

1 +|Dv_j|^r . Letj→ ∞to get, using (3.5),

l₀≥Lⁿ(B)f(0)−LLⁿ(B\B).

Recall∈(δ,1) for fixed 0< δ <1. Hence we conclude by takingδarbitrarily close to 1.

In order to obtain the proof of Theorem 1.1 from Lemma 1.2, we use a technique originating in work by Foneseca and M¨uller, which was further developed by Fonseca and Marcellini (see [13,14]). However, this ”blow- up argument” still does not apply completely for our purposes. In order to use the fact the sequence (u_j) in Theorem1.1is uniformly bounded inL^q_loc(Ω;R^N) forqsatisfying (1.5), we need to be more careful in our choice of blow-up functions. This involves applying the following lemma.

P. SONEJI

Lemma 3.1. Let u∈W_loc^1,p(Ω;R^N), where1≤p < n−1. Then for almost allx₀∈Ωthe following holds: there exists a setE⊂(0,1) such that0 is a point of right density one ofE, and the difference quotient

∂B

u(x₀+z)−u(x₀)−[Du(x₀)]z

|z|

^(n−1)p_n−1−p

dHⁿ⁻¹(z) (3.6)

tends to 0 as→0 through the set E. Moreover, the setE has the following property: there exists a sequence t_k 0 and corresponding sets E_tk ⊂[¹₂,1]such that

E= ∞ i=1

t_iE_ti (3.7)

and, for any >0, we can choose t_k,E_tk such that

∞ i=1

E_ti >1

2 −. Proof of Lemma 3.1. Forx₀,y ∈Ω,t >0, define

v(t, y) := u(x₀+ty)−u(x₀)−[Du(x₀)](ty)

t ·

It is clear that, providedB(x₀, t)⊂Ω,v(t, y)∈W^1,p(B;R^N). Moreover, it is well known that

B|v(t, y)|^pdy→0 as t→0 for almost all x₀∈Ω(see, for example, [32]). In addition, by considering Lebesgue points ofDu, we have

B|Dv(t, y)|^pdy=

B

Du(x₀+ty)−Du(x₀)^pdy−→0 ast→0 for almost allx₀∈Ω. Fix such anx₀and, for 0< t <dist(x₀, ∂Ω), define

γ(t) :=

∂B|v(t, y)|^{(n−1)pn−1−p}dHⁿ⁻¹(y) , and

α(t) :=

B

|v(t, y)|^p+|Dv(t, y)|^p dy.

Note that by our choice of x₀ we haveα(t)→0 ast →0. Since v(t, y)∈W^1,p(B) for t sufficiently small, we havev(t, y)∈W^1,p(∂B;R^N) for almost all ∈ (0,1). It then follows by the Rellich–Kondrachoff embedding theorem that in factv(t, y)∈L^{(n−1)pn−1−p}(∂B;R^N) for almost all∈(0,1).

Now let

φ_t() :=

∂B

|v(t, y)|^p+|Dv(t, y)|^p

dHⁿ⁻¹(y) and let

E_t:=

1 2,1

∩ {:φ_t()< α(t)¹²}. (3.8) Note

α(t) = ¹

0

φ_t() d≥ ¹

12

φ_t() d≥

[¹₂,1]\Et

φ_t() d≥

1 2,1

\E_t

·α(t)¹²,

so

1 2,1

\E_t| ≤α(t)¹².

LOWER SEMICONTINUITY IN BV OF QUASICONVEX INTEGRALS WITH SUBQUADRATIC GROWTH

Next consider∈E_t. By the Sobolev Inequality we have forβ= ^(n−1)p_n−1−p and some constantM =M(p, n):

∂B

|v(y, t)|^βdHⁿ⁻¹(y) _β¹

≤M

∂B

|v(y, t)|^p+|Dv(t, y)|^pdHⁿ⁻¹(y) ¹_p

=Mφ_t()^1p ≤Mα(t)^2p1 . Hence, again for∈E_t, we have

γ(t) =

∂B

u(x₀+ty)−u(x₀)−[Du(x₀)](ty) t

β

dHⁿ⁻¹(y)

=^1−n−β

∂B

|v(t, y)|^βdHⁿ⁻¹(y)

≤M2^n+β+1·α(t)^{2(n−1−p)n−1} .

Now we may take any decreasing sequence (t_i)⊂(0,dist(x₀, ∂Ω)) such thatt_i 0 and let E_ti be defined as in (3.8). Note that we could also requiret_i+1< t_i/2, so that theE_tiare disjoint. Now defineEas stated in (3.7).

Thus 0 is a limit point ofE, and γ()→0 asr→0,∈E, so the main statement of the lemma is proved.

It remains to show that we can choose (t_i) such that^∞

i=1E_tiis arbitrarily close to¹₂. WriteE_t^cfor [¹₂,1]\E_t. SinceE_t^c0 ast0, for a given >0 we may chooset_i 0 such thatE_t^c

i<2⁻ⁱfor alli. Hence

∞ i=1

E_t^c_i ≤^∞

i=1

|E^c_ti|< .

So

∞ i=1

E_ti >1

2 −.

This completes the proof of the lemma.

Remark 3.2. Ifu∈W_loc^1,p(Ω;R^N) for p≥n−1, then obviouslyu∈W_loc^1,p(Ω;R^N) for any 1≤p ≤p, so we can still apply the above lemma for 1≤p < n−1 to prove Theorem1.1. In fact, if p > n−1, then we have a stronger result: namely that uhas aregular approximate total differential at almost allx₀∈Ω. This means that the difference quotient

u(x₀+z)−u(x₀)−[Du(x₀)]z

|z|

tends to 0 uniformly for z ∈ ∂B as → 0 through a set E for which 0 is a point of right density one. A scheme of a proof of this can be found in [32] (Chap. 3, Exercises), from which the Proof of Lemma3.1 has been adapted.

Proof of Theorem 1.1. We may assume that the left hand side of (1.6) is finite, as otherwise there is nothing to prove. Taking a subsequence if necessary, we can also assume that

Ωf(Du_j)→lim inf

j→∞ Ωf(Du_j).

For our purposes, for (1.3) we require only the following fact which characterizes weak* convergence in BV. That is, (1.3) is equivalent to:

u_j →u in L¹_loc(Ω;R^N) (3.9)

Du_j Du^∗ in M(Ω;R^N×n) , (3.10)

P. SONEJI

whereM(Ω;R^N×n) is the space ofN×nmatrix-valued Borel measures onΩ. By (3.10) and the Uniform Bound- edness Principle, sup_j

Ω|Du_j|dLⁿ <∞. SinceLⁿf(Du_j) andLⁿ|Du_j|are bounded inM( ¯Ω;R^N×n), we that have for some subsequence (for convenience not relabelled) there exist measuresμandν in ¯Ωsuch that

f(Du_j) μ^∗ and |Du_j| ν^∗

in M( ¯Ω;R^N×n).

Notice that, becausef ≥0, the proof of the theorem follows if we can prove that dμ

dLⁿ(x)≥f(Du(x)) (3.11)

holds for almost allx∈Ω.

LetΩ₀ denote the set of pointsx∈Ωsuch that:

1. dμ

dLⁿ(x) = lim

→0⁺

μ(B(x, ))

Lⁿ(B(x, )) exists and is finite;

2. dν

dLⁿ(x) = lim

→0⁺

ν(B(x, ))

Lⁿ(B(x, )) exists and is finite;

3.

→0lim⁺ 1

B(x,)|u(y)−u(x)−[Du(x)](x−y)|dy= 0;

4. Lemma3.1holds foruatx.

By standard results (see e.g. [27,32]) and Lemma 3.1, Ω₀ has full measure in Ω. Fix x₀ ∈ Ω₀. Let (r_k) ⊂ (0,dist(x₀, Ω)) be a sequence such that r_k 0 and define

v_j,k(y) := u_j(x₀+r_ky)−u(x₀)−[Du(x₀)](r_ky)

r_k , y∈B. (3.12)

Our aim is to pick a suitable sequence (r_k) so we may usev_j,k to define a sequence (z_k)⊂W^1,r(B;R^N), say, that will enable us to apply Lemma1.2 to obtain (3.11). In fact, we do not actually apply Lemma 1.2for the sameq as in Theorem1.1, but for an arbitrarily smallerq< q that nevertheless satisfies (1.10).

However, also note that in order to apply Lemmas1.2and3.1, from (1.10) and (3.6) we needu∈W_loc^1,p(Ω;R^N) for 1≤p < n−1 satisfying

p(n−1)

n−1−p >max

1,r(n−1) 2−r

·

It is straightforward to verify that this holds if and only if 1≤p < n−1 also satisfies p > ^r₂(n−1).

Recall from Lemma3.1 that, for any >0, we can choose a sequence t_k 0,t_k <dist(x₀, ∂Ω), such that

|E_tk|>¹₂−, withE_tk defined as in (3.8). By (1.4) and (3.9), using De la Vall´ee Poussin and Vitali, we have u_j→uinL^q_loc (Ω) for any 1≤q< q. Hence, for any fixedk:

j→∞lim 1

t^q_{k B}u_j(x₀+t_ky)−u(x₀+t_ky)^qdy= 0.

So, by Fubini–Tonelli,

j→∞lim 1 t^q_k

1 0 ∂B

u_j(x₀+t_ky)−u(x₀+t_ky)^qdHⁿ⁻¹(y) d= 0.

LOWER SEMICONTINUITY IN BV OF QUASICONVEX INTEGRALS WITH SUBQUADRATIC GROWTH

Hence, for everyk, there exists a subsequence (u_j)_j∈Sk,S_k⊆N, such that

j→∞,j∈Slim k

1

t^q_{k ∂B}u_j(x₀+t_ky)−u(x₀+t_ky)^qdH ⁿ⁻¹(y) = 0 (3.13) for almost all ∈(0,1). Now note that, by Egorov’s Theorem, for a given >0, there exists a setG_k ⊂(0,1) such that |(0,1)\G_k| < 2^−k and (3.13) holds uniformly for ∈ G_k. By discarding smaller elements of S_k if necessary, this implies that

j∈Ssupk

∈Gsupk

1

t^q_{k ∂B}u_j(x₀+t_ky)−u(x₀+t_ky)^qdHⁿ⁻¹(y)<1.

We can obtain such G_k and S_k for all k∈N. Now note that we have, similarly to the remark to Lemma 3.1,

|G_k|>1−, so|(¹₂,1)∩G_k|>¹₂−. Therefore ^∞

k=1

(G_k∩E_tk)> 1

2 −2 >0 ,

providedis small enough. This means that(G_k∩E_tk) contains a point of left density oneθ, say (soθ∈(¹₂,1)).

If we let

F=θ⁻¹ ∞ k=1

(G_k∩E_tk) ,

then 1 is a point of left density one ofF. Hence, for all 0< δ <1,|(δ,1)∩F|>0.

Now letr_k =θt_k. So

r_k)⊂(0,dist(x₀, Ω)

,r_k 0. Note also that the set ∈

0,dist(x₀, Ω)

: (μ+ν)

∂B(x₀, )

>0

is at most countable, so has measure 0. Since there are uncountably many points of left density one, like θ, above, we may assume in addition that (μ+ν)

∂B(x₀, r_k)

= 0 for allkfor our choice ofr_k. Definev_j,kin (3.12) using this choice ofr_k. Observe that we may also writev_j,k as follows:

v_j,k(y) = 1 r_k

u(x₀+r_ky)−u(x₀)−[Du(x₀)](r_ky) + 1

r_k

u_j(x₀+r_ky)−u(x₀+r_ky)

=I+II, say.

We now considerI andII separately.

Estimating I.If ∈F, thenθ∈E_tk, soθt_k ∈t_kE_tk for allk.i.e. r_k ∈t_kE_tk for allk. So, with reference to (3.7) in Lemma 3.1, we have thatr_k ∈E: so ify∈∂B, thenr_ky∈A. So by Lemma3.1we have

∈F ∂Bsup

u(x₀+r_ky)−u(x₀)−[Du(x₀)](r_ky)

|r_k|

^(n−1)p_n−1−p

dHⁿ⁻¹(y)

≤C·α(t_k)^{2(n−1−p)n−1} −→0 ask→0.

This implies that

sup

k sup

∈F

1

r_k

u(x₀+r_k·)−u(x₀)−[Du(x₀)](r_k·)

L^q(∂B)<∞

for any 1≤q≤ ^(n−1)p_n−1−p. Hence, as noted above, ifpsatisfies the conditions in Theorem1.1, then we can choose an appropriateq satisfying (1.10).