A-quasiconvexity: weak-star convergence and the gap

Irene Fonseca Department of Mathematical Sciences, Carnegie-Mellon University, Pittsburgh, PA, U.S.A.

Giovanni Leoni Department of Mathematical Sciences, Carnegie-Mellon University, Pittsburgh, PA, U.S.A and Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy

Stefan Müller Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany

Abstract

Lower semicontinuity results with respect to weak-∗ convergence in the sense of measures and with respect to weak convergence in L^{p} are obtained for functionals

v ∈ L^{1}(Ω; ℝ^{m}) ↦ ∫_{Ω} f(x, v(x)) dx,

where admissible sequences {v_{n}} satisfy a first order system of PDEs Av_{n} = 0. We suppose that A has constant rank, f is A-quasiconvex and satisfies the non standard growth conditions

\frac{1}{C} (|v|^{p} − 1) ≤ f(v) ≤ C(1 + |v|^{q})

with q ∈ [p, pN/(N − 1)) for p ≤ N − 1, q ∈ [p, p + 1) for p > N − 1. In particular, our results generalize earlier work where Av = 0 reduced to v = ∇^{s}u for some s ∈ ℕ.

Keywords. A-quasiconvexity, gap, non standard growth conditions, lower semicontinuity, Sobolev Embedding Theorem, Radon-Nikodym Decomposition Theorem

AMS Subject Classification. 35E99, 49J45, 74B20

1 Introduction

It is well known that quasiconvexity is a necessary and sufficient condition for lower semicontinuity with respect to strong convergence in L^{1} of functionals of the form

u ∈ W^{1,1} (Ω; ℝ^{m}) ↦ ∫_{Ω} f (∇u (x)) dx,  (1.1)

where the integrand f = f (∇u) is nonnegative and has linear growth. More precisely, the following result holds:

Theorem 1.1 Let Ω ⊂ ℝ^{N} be an open bounded set, and let f : ℝ^{m×N} → [0, ∞) be a quasiconvex function such that

0 ≤ f (ξ) ≤ C (1 + |ξ|)  (1.2)

for all ξ ∈ ℝ^{m×N} and for some constant C > 0. Then

∫_{Ω} f (∇u (x)) dx ≤ lim inf_{n→∞} ∫_{Ω} f (∇u_{n} (x)) dx

for any sequence {u_{n}} ⊂ W^{1,1} (Ω; ℝ^{m}) strongly convergent in L^{1} (Ω; ℝ^{m}) to some u ∈ BV (Ω; ℝ^{m}) if and only if f is quasiconvex.

The proof of the necessity is due to Morrey [39], while the sufficiency relies on De Giorgi’s Slicing Lemma (see e.g. [6]; see also [25], [26], [35]). In the Appendix we present a another argument based on Gagliardo’s Trace Theorem for W^{1,1} (Ω; ℝ^{m}) (see [29]). It is interesting to observe that the idea behind the proofs using either De Giorgi’s Slicing Lemma or Gagliardo’s Trace Theorem is actually the same.

In the scalar case, that is when m = 1, it has been proved by Serrin [44] that Theorem 1.1 continues to hold without assuming the upper bound in (1.2). This is due to the fact that when m = 1 quasiconvexity reduces to convexity. Since any nonnegative convex function is the supremum of a sequence of linear functions, trivially satisfying (1.2), lower semicontinuity results for this type of integrands do not require apriori growth conditions. The situation is completely different in the vectorial case m > 1, where Theorem 1.1 fails in general if f has superlinear growth. Indeed, Acerbi, Buttazzo and Fusco [2] proved that when N = m = 2 the functional

u ∈ W^{1,2} (Ω; ℝ^{2}) ↦ ∫_{Ω} |det ∇u| dx

is not lower semicontinuous with respect to strong convergence in L^{p} (Ω; ℝ^{2}) for any 1 ≤ p < ∞.

This striking difference in lower semicontinuity properties between functionals with integrands with linear growth of the type (1.2) and integrands with superlinear growth such as

0 ≤ f (ξ) ≤ C (1 + |ξ|^{q}) , q > 1,

maybe explained in part by the profound disparity in the characterization and properties of the trace space of W^{1,q} (Ω; ℝ^{m}) when q = 1 and q > 1. If Ω is a Lipschitz domain then the trace space of W^{1,1} (Ω; ℝ^{m}) is L^{1} (∂Ω; ℝ^{m}), and thus strong convergence in L^{1} (Ω; ℝ^{m}) implies (up to a subsequence) strong convergence of the traces in L^{1} (∂Ω_{t}; ℝ^{m}) where Ω_{t} is a smooth domain arbitrarily “close” do Ω and hence there exists an extension which converges in W^{1,1} (Ω; ℝ^{m}). On the other hand, when q > 1 the trace space of W^{1,q} (Ω; ℝ^{m}) is the fractional Sobolev space W^{1− \frac{1}{q} ,q} (∂Ω_{t}; ℝ^{m}), therefore strong convergence alone in L^{p} (Ω; ℝ^{m}) for any 1 ≤ p < ∞ does not necessarily entail strong convergence of the traces in W^{1− \frac{1}{q} ,q} (∂Ω; ℝ^{m}). By virtue of Sobolev’s Imbedding Theorem this is guaranteed, however, if the integrand f satisfies a coercivity condition of the form

f (ξ) ≥ \frac{1}{C} (|ξ|^{p} − 1) ,

with

1 ≤ p ≤ q < \frac{N}{N − 1}p.  (1.3)

As a consequence, the following result holds:

Theorem 1.2 Assume that p, q satisfy (1.3). Let Ω ⊂ ℝ^{N} be an open bounded set, and let f : ℝ^{m×N} → [0, ∞) be a quasiconvex function such that

\frac{1}{C} (|ξ|^{p} − 1) ≤ f (ξ) ≤ C (1 + |ξ|^{q})  (1.4)

for all ξ ∈ ℝ^{m×N}. Then

∫_{Ω} f(∇u) dx ≤ lim inf_{n→∞} ∫_{Ω} f(∇u_{n}) dx

for any sequence {u_{n}} ⊂ W^{1,q}(Ω; ℝ^{m}) which converges to u ∈ BV (Ω; ℝ^{m}) strongly in L^{1}(Ω; ℝ^{m}).

In this generality Theorem 1.2 was proved by Fonseca e Malý [23] for p > 1, and by Kristensen [32] when p = 1 (see the bibliography therein for previous partial results). For the convenience of the reader we present a proof of Theorem 1.2 in the Appendix.

Observe that we take admissible converging sequences {u_{n}} in the space W^{1,q}(Ω; ℝ^{d}), otherwise not only we would be unable to guarantee finiteness of the energy, but also, since f is quasiconvex and f (ξ) ≤ C (1 + |ξ|^{q}) , f is W^{1,q}-quasiconvex but it may fail to be W^{1,p}-quasiconvex (see [8]). In addition, note that by (1.4) if p > 1 and if lim inf_{n→∞} ∫_{Ω} f(∇u_{n}) dx < ∞ then, necessarily,

u ∈ W^{1,p}(Ω; ℝ^{m}).

The proof of Theorem 1.2 strongly hinges on the properties of a linear, compact, lifting operator

E : W^{1,p}(∂Ω; ℝ^{m}) → W^{1,q}(Ω; ℝ^{m})
v ↦ E (v)

such that v is the trace of E (v). The existence of such an operator follows from standard Sobolev trace and compact embedding theorems when q < \frac{N}{N−1} p. The exponent

q_{c} = \frac{N}{N − 1} p

is critical for the existence of the operator E, and, not surprisingly, also for lower semicontinuity of functionals of the type (1.1). Indeed, Malý [33] proved that the functional

u ∈ W^{1,N} (Ω; ℝ^{N}) ↦ ∫_{Ω} |det ∇u| dx

is not lower semicontinuous with respect to weak convergence in W^{1,p} (Ω; ℝ^{N}) for any p < N − 1.

Lower semicontinuity of (1.1) in the borderline case where (1.4) holds for q = \frac{N}{N−1} p is still unknown (see [24], [32], [34] for some partial results), except for the special case where m = N and

f (ξ) := |ξ|^{N−1} + g (det ξ) .  (1.5)

Theorem 1.3 Let Ω ⊂ ℝ^{N} be an open bounded set, and let g : ℝ → [0, ∞] be a lower semicontinuos convex function such that g (0) < ∞. Let {u_{n}} be a sequence of functions in W^{1,N}(Ω; ℝ^{N}) which converges to u ∈ BV (Ω; ℝ^{N}) in L^{1}(Ω; ℝ^{N}), and such that

sup_{n} ∫_{Ω} |∇u_{n}|^{N−1} dx < ∞.

Then

∫_{Ω} g(det ∇u) dx ≤ lim inf_{n→∞} ∫_{Ω} g(det ∇u_{n}) dx.

Theorem 1.3 was proved by Celada and Dal Maso [13] using cartesian current (see also [22] for a new proof).

Functions of the form (1.5) may be viewed as prototypes of integrands f = f(x, u, ∇u) satisfying a “limiting” non standard growth condition (1.4) and whose importance stems from the study of cavitation and related issues in nonlinear elasticity and continuum mechanics. For further results in related subjects we refer the reader to [1], [3], [7], [9], [13], [17], [20], [23], [24], [32], [33], [34], [35], [36], [53], [54].

The purpose of this paper is to extend Theorems 1.1 and 1.2 to the general setting of A-quasiconvexity, which has been introduced by Dacorogna [14] and further developed by Fonseca and Müller in [27] (see also [10]). Here, and following [40],

A : L^{q}(Ω; ℝ^{d}) → W^{−1,q}(Ω; ℝ^{l}), Av := ∑_{i=1}^{N} A^{(i)} \frac{∂v}{∂x_{i}} ,

is a constant–rank (see (2.1)), first order linear partial differential operator, with A^{(i)} : ℝ^{d} → ℝ^{l} linear transformations, i = 1, …, N. We recall that a function f : ℝ^{d} → ℝ is said to be A-quasiconvex if

f(ξ) ≤ ∫_{Q} f(ξ + w(y)) dy  (1.6)

for all ξ ∈ ℝ^{d} and all w ∈ C_{per}^{∞}(ℝ^{N}; ℝ^{d}) such that Aw = 0 and ∫_{Q} w(y) dy = 0, where Q denotes the unit cube in ℝ^{N}, and the space C_{per}^{∞}(ℝ^{N}; ℝ^{d}) is introduced in Section 2.

The relevance of this general framework, as emphasized by Tartar (see [47, 48, 49, 50, 51, 52]), lies on the fact that in continuum mechanics and in electromagnetism PDEs other than curl v = 0 arise naturally and are physically relevant, and this calls for a relaxation theory which encompasses PDE constraints of the type Av = 0. Some important examples included in this general setting are given by:

(a) [Unconstrained Fields]

Av ≡ 0.

Here, due to Jensen’s inequality, A-quasiconvexity reduces to convexity.

(b) [Divergence Free Fields]

Av = div v = 0,

where v : Ω ⊂ ℝ^{N} → ℝ^{N} (see [41]).

(c) [Maxwell’s Equations]

A ( \begin{matrix}m \\ h \end{matrix} ) := ( \begin{matrix}div(m + h) \\ curl h \end{matrix} ) = 0,

where m : ℝ^{3} → ℝ^{3} is the magnetization and h : ℝ^{3} → ℝ^{3} is the induced magnetic field (see [18, 51]).

(d) [Gradients]

Av = curl v = 0.

Note that w ∈ C_{per}^{∞}(ℝ^{N}; ℝ^{d}) is such that curl w = 0 and ∫_{Q} w(y) dy = 0 if and only if there exists φ ∈ C_{per}^{∞}(ℝ^{N}; ℝ^{m}) such that ∇φ = v, where d = m × N. In this case, (1.6) reduces to the well-known notion of quasiconvexity introduced by Morrey [39].

(e) [Higher Order Gradients] Replacing the target space ℝ^{d} by an appropriate finite dimensional vector space E_{s}^{m} of m-tuples of symmetric s-linear maps on ℝ^{N}, it is possible to find a first order linear partial differential operator A such that v ∈ L^{p}(Ω; E_{s}^{m}) and Av = 0 if and only if there exists φ ∈ W^{s,q}(Ω; ℝ^{m}) such that v = ∇^{s}φ (see Theorem 1.8). In this case, (1.6) reduces to the notion of s−quasiconvexity introduced by Meyers [38].

The first main result of the paper is given by the following theorem:

Theorem 1.4 Let

1 ≤ q < \frac{N}{N − 1}.  (1.7)

Let Ω ⊂ ℝ^{N} be an open bounded set, and let f : Ω × ℝ^{d} → [0, ∞) be a Borel measurable, A-quasiconvex function such that

|f (x, ξ) − f (x, ξ_{1})| ≤ C (1 + |ξ|^{q−1} + |ξ_{1}|^{q−1}) |ξ − ξ_{1}|  (1.8)

for all x ∈ Ω and all ξ, ξ_{1} ∈ ℝ^{d}, and for some C > 0. Assume further that for all x_{0} ∈ Ω and ε > 0 there exists δ > 0 such that

f(x_{0}, ξ) − f(x, ξ) ≤ ε(1 + f(x, ξ))  (1.9)

for all x ∈ Ω with |x − x_{0}| ≤ δ and for all ξ ∈ ℝ^{d}. Then

∫_{Ω} f (x, \frac{dλ}{dL^{N}} (x)) dx ≤ lim inf_{n→∞} ∫_{Ω} f (x, v_{n} (x)) dx

for any sequence {v_{n}} ⊂ L^{q} (Ω; ℝ^{d}) ∩ ker A weakly-∗ converging in the sense of measures to some ℝ^{d}−valued Radon measure λ ∈ M (Ω; ℝ^{d}) .

Lower semicontinuity properties of the constrained functional

∫_{Ω} f (x, v (x)) dx with Av = 0,

have been proved by Fonseca and Müller in [27] with respect to weak convergence in L^{1} (Ω; ℝ^{d}) . Note, however, that for integrands with linear growth weak-∗ convergence in the sense of measures is more natural in view of the lack of reflexivity of the space L^{1}(Ω; ℝ^{d}).

Also, in the case (d) of gradients, that is, when

Av = curl v = 0,

Theorem 1.4 includes Theorem 1.1 for integrands which satisfy the additional coercivity assumption

f (ξ) ≥ \frac{1}{C} |ξ| − C.  (1.10)

Indeed, condition (1.10) implies that the sequence {∇u_{n}} is uniformly bounded in L^{1} (Ω; ℝ^{m×N}) , and thus a subsequence weakly-∗ converges in the sense of measures.

We do not know if Theorem 1.4 continues to hold under a convergence weaker than weak-∗ convergence in the sense of measures. On one hand, Theorem 1.1 certainly seems to point in that direction, but on the other hand, even for higher order gradients (also contemplated within the A-quasiconvexity framework; see example (e) above) the situation is far from clear. Indeed, it is still an open problem to determine whether the functional

u ∈ W^{2,1}(Ω; ℝ^{m}) ↦ ∫_{Ω} f(∇^{2}u) dx,

where f : E_{2}^{d} → [0, ∞) is a 2-quasiconvex function satisfying

0 ≤ f(ξ) ≤ C (1 + |ξ|)

for all ξ ∈ E_{2}^{d}, is lower semicontinuous with respect to strong convergence in W^{1,1}(Ω; ℝ^{m}). Note that if u ∈ W^{2,1}(Ω; ℝ^{m}), then (see [11], [19], [37])

(u, \frac{∂u}{∂ν})|_{∂Ω} ∈ B^{1,1} (∂Ω; ℝ^{m}) × L^{1} (∂Ω; ℝ^{m}) ,

and strong convergence in W^{1,1}(Ω; ℝ^{m}) implies strong convergence of the nor-

mal derivatives \{\frac{∂u_{n}}{∂ν} \} in L^{1} (∂Ω_{t}; ℝ^{m}) where Ω_{t} is a smooth domain arbitrarily

“close” do Ω. However, this does not necessarily guarantee strong convergence of the traces in the Besov space B^{1,1} (∂Ω_{t}; ℝ^{m}) . This suggests that lower semicontinuity might not hold under strong convergence in W^{1,1}(Ω; ℝ^{m}) and that a stronger notion of convergence is needed. We do not know how to prove or disprove this.

Condition (1.9) is satisfied in the important special case where the integrand f (x, ξ) is a decoupled product. Indeed we have the following

Corollary 1.5 Let 1 ≤ q < ∞ satisfy (1.7), let g : ℝ^{N} → [0, ∞) be an Aquasiconvex function such that

|g (ξ) − g (ξ_{1})| ≤ C (1 + |ξ|^{q−1} + |ξ_{1}|^{q−1}) |ξ − ξ_{1}|

for all ξ, ξ_{1} ∈ ℝ^{d}, and for some C > 0, and let h : Ω × ℝ → [0, ∞] be a lower semicontinuous function. Then

∫_{Ω} h(x)g (\frac{dλ}{dL^{N}} (x)) dx ≤ lim inf_{n→∞} ∫_{Ω} h(x)g (v_{n} (x)) dx

for any sequence {v_{n}} ⊂ L^{1} (Ω; ℝ^{d}) ∩ ker A weakly-∗ converging in the sense of measures to some ℝ^{d}−valued Radon measure λ ∈ M (Ω; ℝ^{d}) .

The second main result of the paper partially extends Theorem 1.2 to the realm of A-quasiconvexity:

Theorem 1.6 Let 1 < p ≤ q < ∞, and assume that

q < \{ \begin{matrix}\frac{N}{N − 1} p & if p ≤ N − 1, \\ p + 1 & if p > N − 1. \end{matrix}  (1.11)

Let Ω ⊂ ℝ^{N} be an open bounded set, and let f : Ω × ℝ^{d} → [0, ∞) be an Aquasiconvex function such that

|f (x, ξ) − f (x, ξ_{1})| ≤ C (1 + |ξ|^{q−1} + |ξ_{1}|^{q−1}) |ξ − ξ_{1}|  (1.12)

for all x ∈ Ω and all ξ, ξ_{1} ∈ ℝ^{d}, and for some C > 0. Assume that f satisfies condition (1.9) . Then

∫_{Ω} f (x, v (x)) dx ≤ lim inf_{n→∞} ∫_{Ω} f (x, v_{n} (x)) dx

for any sequence {v_{n}} ⊂ L^{q} (Ω; ℝ^{d}) ∩ ker A weakly converging in L^{p} (Q; ℝ^{d}) to some v ∈ L^{p} (Ω; ℝ^{d}) .

Note that, unlike the case where p = q (see [4], [15]), in general one may not take f to be a Carathéodory function, and some kind of regularity is needed in the x variable. Indeed, Gangbo [30] has proved that the functional

u ∈ W^{1,q} (Ω; ℝ^{N}) ↦ ∫_{Ω} χ_{K} (x) |det ∇u (x)| dx,

where K ⊂ ℝ^{N} is a compact set, is lower semicontinuous with respect to weak convergence in W^{1,p} (Ω; ℝ^{N}) for some N − 1 < p < N if and only if

L^{N} (∂K) = 0.

Here, again, one witnesss the intrinsic differences between the convex and the quasiconvex frameworks, as it has been shown by Acerbi, Bouchitté and Fonseca [1] that Theorem 1.6 still holds for Carathéodory functions f and with Av = 0 if and only if curl v = 0, provided f(x, ·) is convex, and without requiring condition (1.12).

The analog of Corollary 1.5 is now:

Corollary 1.7 Let 1 < p ≤ q < ∞ satisfy (1.11), let g : ℝ^{N} → [0, ∞) be an A-quasiconvex function such that

|g (ξ) − g (ξ_{1})| ≤ C (1 + |ξ|^{q−1} + |ξ_{1}|^{q−1}) |ξ − ξ_{1}|

for all ξ, ξ_{1} ∈ ℝ^{d}, and for some C > 0, and let h : Ω × ℝ → [0, ∞] be a lower semicontinuous function. Then

∫_{Ω} h(x)g (v (x)) dx ≤ lim inf_{n→∞} ∫_{Ω} h(x)g (v_{n} (x)) dx

for any sequence {v_{n}} ⊂ L^{q} (Ω; ℝ^{d}) ∩ ker A weakly converging in L^{p} (Q; ℝ^{d}) to some v ∈ L^{p} (Ω; ℝ^{d}).

In the case of first or higher order gradients (d) and (e), the Lipschitz condition (1.12) follows from the s–quasiconvexity of the integrand f(x, ξ) together with the growth condition (1.13) below. For first order gradients, this was shown by Marcellini [35]. The case s = 2 was treated by Guidorzi and Poggiolini [31], while the general case was studied by Santos and Zappale [43]. More generally, it can be shown that if the span of the characteristic cone

Λ := ⋃_{w∈S^{N−1}} 𝔸(w),

where 𝔸(w) := ∑_{i=1}^{N} w_{i}A^{(i)}, has dimension d then A-quasiconvexity, together with

(1.13) below, implies (1.12).

As a corollary of Theorem 1.6 we obtain the following result:

Theorem 1.8 Let 1 < p, q < ∞ satisfy (1.11), let s ∈ ℕ, and suppose that f : Ω × E_{s}^{m} → [0, ∞) is a Borel integrand satisfying (1.9) , and

0 ≤ f(x, ξ) ≤ C (1 + |ξ|^{q})  (1.13)

for a.e. x ∈ Ω and all ξ ∈ E_{s}^{m}, where C > 0. Assume that for a.e. x ∈ Ω the function f(x, ·) is s−quasiconvex, that is for all ξ ∈ E_{s}^{m}

f(x, ξ) = inf {∫_{Q} f(x, ξ + ∇^{s}w(y)) dy : w ∈ C_{per}^{∞}(ℝ^{N}; ℝ^{m})} .

If {u_{n}} ⊂ W^{s,q}(Ω; ℝ^{m}) converges weakly to u in W^{s,p}(Ω; ℝ^{m}) then

∫_{Ω} f(x, ∇^{s}u) dx ≤ lim inf_{n→∞} ∫_{Ω} f(x, ∇^{s}u_{n}) dx.

Here E_{s}^{m} stands for the space of m-tuples of symmetric s-linear maps on ℝ^{N}. Theorem 1.8 was proved by Esposito and Mingione (see Theorem 4.1 in [21]) under the assumptions

q < \frac{N (s − 1)}{N (s − 1) − 1} p

when s ≥ 2.

2 Preliminaries

We start with some notation. Here Ω is an open, bounded subset of ℝ^{N}, L^{N} is the N dimensional Lebesgue measure, S^{N−1} := {x ∈ ℝ^{N} : |x| = 1} is the unit sphere, and Q := (−1/2, 1/2)^{N} the unit cube centered at the origin. For r > 0 and x_{0} ∈ ℝ^{N} we set Q_{r} := rQ and Q(x_{0}, r) := x_{0} + rQ . A function w ∈ L_{loc}^{q}(ℝ^{N}; ℝ^{d}) is said to be Q–periodic if w(x + e_{i}) = w(x) for a.e. x ∈ ℝ^{N} and every i = 1, …, N, where {e_{1}, …, e_{N}} is the canonical basis of ℝ^{N}, and we write w ∈ L_{per}^{q}(Q; ℝ^{d}). Also C_{per}^{∞}(Q; ℝ^{d}) will stands for the space of Q–periodic functions in C^{∞}(ℝ^{N}; ℝ^{d}). The Fourier coefficients of a function w ∈ L_{per}^{q}(Q; ℝ^{d}) are defined by

ŵ (λ) := ∫_{Q} w (x) e^{−2πix·λ}dx, λ ∈ ℤ^{N}.

If 1 < q ≤ ∞ then W^{−1,q}(Ω; ℝ^{l}) is the dual of W_{0}^{1,q′}(Ω; ℝ^{l}), where q′ is the Hölder conjugate exponent of q, that is 1/q + 1/q′ = 1. It is well known that F ∈ W^{−1,q}(Ω; ℝ^{l}) if and only if there exist g_{1}, …, g_{N} ∈ L^{q}(Ω; ℝ^{l}) such that

⟨F, w⟩ = ∑_{i=1}^{N} ∫_{Ω} g_{i} · \frac{∂w}{∂x_{i}} dx for all w ∈ W_{0}^{1,q′}(Ω; ℝ^{l}).

Consider a collection of linear operators A^{(i)} : ℝ^{d} → ℝ^{l}, i = 1, …, N, and define the differential operator

A_{Ω} : L^{q}(Ω; ℝ^{d}) ⟶ W^{−1,q}(Ω; ℝ^{l})
v ⟼ Av

as follows:

⟨A_{Ω}v, w⟩ := ⟨∑_{i=1}^{N} A^{(i)} \frac{∂v}{∂x_{i}} , w⟩

= − ∑_{i=1}^{N} ∫_{Ω} A^{(i)}v \frac{∂w}{∂x_{i}} dx for all w ∈ W_{0}^{1,q′}(Ω; ℝ^{l}).

Even though the operator A_{Ω} so defined depends on Ω, we will omit reference to the underlying domain whenever it is clear from the context, and we will write simply A in place of A_{Ω}. In particular, if v ∈ L_{per}^{q}(Q; ℝ^{d}) then we will say that v ∈ ker A if Av = 0 in W_{per}^{−1,q}(Q; ℝ^{l}), i.e. we consider test functions w ∈ W_{per}^{1,q′}(Q; ℝ^{l}).

In the sequel we will assume that A satisfies the constant-rank property (see [40]), precisely there exists r ∈ ℕ such that

rank 𝔸 (w) = r for all w ∈ S^{N−1}  (2.1)

where

𝔸 (w) := ∑_{i=1}^{N} w_{i} A^{(i)}, w ∈ ℝ^{N}.

For each w ∈ ℝ^{N} the operator ℙ (w) : ℝ^{d} → ℝ^{d} is the orthogonal projection of ℝ^{d} onto ker 𝔸 (w), and 𝕊 (w) : ℝ^{l} → ℝ^{d} is defined by 𝕊 (w) 𝔸 (w) z := z − ℙ (w) z for z ∈ ℝ^{d} and 𝕊 ≡ 0 on (range(𝔸(w))^{⊥}. It may be shown that ℙ : ℝ^{N}\ {0} →Lin(ℝ^{d}; ℝ^{d}) is smooth and homogeneous of degree zero and 𝕊 : ℝ^{N}\ {0} →Lin(ℝ^{l}; ℝ^{d}) is smooth and homogeneous of degree −1 (see [27]).

For q > 1 we define the operator

S_{q} : L_{per}^{q} (Q; ℝ^{d}) → W_{per}^{1,q} (Q; ℝ^{l}) ,

by

S_{q}v (x) := ∑_{λ∈ℤ^{N} \{0}} 𝕊 (λ) v̂ (λ) e^{2πix·λ}  (2.2)

whenever v ∈ L_{per}^{q}(Q; ℝ^{d}) can be written as

v (x) := ∑_{λ∈ℤ^{N}} v̂ (λ) e^{2πix·λ}.  (2.3)

Using (2.2) and (2.3) we may write

S_{q}v (x) := ∫_{Q} K (x − y) v (y) dy,

where the periodic kernel K is given by the Fourier series

K (x) := ∑_{λ∈ℤ^{N} \{0}} 𝕊 (λ) e^{2πix·λ},

which converges in the sense of distributions.

For any function w defined on ℝ^{N} and for every k = 1, …, N and any positive integer s ∈ ℕ we define

\frac{(∂^{±})^{s} w}{∂x_{k}^{s}} (x) = \underbrace{\frac{∂^{±}}{∂x_{k}} (\frac{∂^{±}}{∂x_{k}} (⋯ (\frac{∂^{±}w}{∂x_{k}} )))}_{s times},

where the difference quotient \frac{∂^{±}w}{∂x_{k}} is given by

\frac{∂^{±}w}{∂x_{k}} (x) := w (x ± e_{k}) − w (x) .

Moreover for any multi-index α = (α_{1}, …, α_{N}) ∈ ℕ^{N}, we use the notation

(∂^{±})^{α} w (x) := \frac{(∂^{±})^{α_{1}}}{∂x_{1}^{α_{1}}} (\frac{(∂^{±})^{α_{2}}}{∂x_{2}^{α_{2}}} (⋯ (\frac{(∂^{±})^{α_{N}} w (x)}{∂x_{N}^{α_{N}}}))) .

Proposition 2.1 There exists C > 0 such that

|K (x)| ≤ C |x|^{1−N}  (2.4)

for all x ∈ ℝ^{N}\ {0} .

Proof. Although the result is well-known to experts, we include a proof for the convenience of the reader. It suffices to prove that

|∇K (x)| ≤ C |x|^{−N}  (2.5)

for all x ∈ ℝ^{N}\ {0} . Let l ∈ {1, ⋯, N} and let

K̃ (x) = \frac{∂K}{∂x_{l}} (x) := ∑_{λ∈ℤ^{N} \\{0\}} 2πiλ_{l}𝕊 (λ) e^{2πix·λ}

= ∑_{λ∈ℤ^{N} \{0}} m (λ) e^{2πix·λ} = ∑_{λ∈ℤ^{N}} m (λ) e^{2πix·λ}

with

m (λ) := 2πiλ_{l}𝕊 (λ)

and where we have used the fact m (0) = 0.

We consider the following dyadic decomposition (cf. [45], page 241). Let φ ∈ C_{c}^{∞} (B (0, 2) ; [0, 1]) , φ = 1 in B (0, 1) , and define δ (x) := φ (x) − φ (2x) . Observe that δ = 0 if |x| ≤ \frac{1}{2} and |x| ≥ 2. It turns out that

∑_{j=−∞}^{∞} δ (\frac{x}{2^{j}} ) = 1

for all x ∈ ℝ^{N}\ {0} . Hence

∑_{j=−n}^{n} K_{j} → K̃  (2.6)

in the sense of distributions, where

K_{j} (x) := ∑_{λ∈ℤ^{N}} m_{j} (λ) e^{2πix·λ}

and m_{j} (λ) := m (λ) δ (\frac{λ}{2^{j}} ) . Since

m_{j} (λ) ≠ 0 only if 2^{j−1} ≤ |λ| ≤ 2^{j+1},  (2.7)

it is clear that K_{j} reduces to a finite sum. Note that if j ≤ −2 clearly no integer satisfies 2^{j−1} ≤ |λ| ≤ 2^{j+1} and so m_{j} ≡ 0 for all j ≤ −2.

We claim that for every M ∈ ℕ

|K_{j} (x)| ≤ C_{M} \frac{1}{|x|^{M}} 2^{j(N−M)}  (2.8)

for all x ∈ ℝ^{N}\ {0} . This, together with (2.6), yields the result. Indeed, fix x ∈ ℝ^{N}\ {0} , and note first that (2.8) with M = 0 reduces to

|K_{j} (x)| ≤ C_{0}2^{jN}.  (2.9)

Choose M > N. We have

∑_{j=−∞}^{∞} |K_{j} (x)| ≤ ∑_{2^{j}≤|x|^{−1}} |K_{j} (x)| + ∑_{2^{j}>|x|^{−1}} |K_{j} (x)|

≤ C_{0} ∑_{2^{j}≤|x|^{−1}} 2^{jN} + C_{M} \frac{1}{|x|^{M}} ∑_{2^{j}>|x|^{−1}} 2^{j(N−M)}.  (2.10)

In the latter expression the first sum can be bounded above by

∑_{2^{j}≤|x|^{−1}} 2^{jN} ≤ \frac{(2^{N})^{1+log_{2}|x|^{−1}} − 1}{2^{N} − 1} ≤ \frac{1}{|x|^{N}} \frac{2^{N}}{2^{N} − 1},

while the second term in (2.10) may be estimated by

C_{M} \frac{1}{|x|^{M}} ∑_{2^{j}>|x|^{−1}} 2^{j(N−M)} ≤ C_{M} \frac{1}{|x|^{M}} \frac{1}{|x|^{N−M}} \frac{1}{1 − 2^{N−M}} \frac{1}{2^{N−M}} ≤ C_{M} \frac{1}{|x|^{N}} .

To conclude the proof, it remains to establish (2.8). By means of a summation by parts and by the Mean Value Theorem, for any k = 1, ⋯, N, we have

(e^{2πix_{k}} − 1) K_{j} (x) = ∑_{λ∈ℤ^{N}} m_{j} (λ) (e^{2πix·(λ+e_{k})} − e^{2πix·λ})

= ∑_{λ∈ℤ^{N}} (m_{j} (λ − e_{k}) − m_{j} (λ)) e^{2πix·λ}

= ∑_{λ∈ℤ^{N}} \frac{∂^{−}m_{j}}{∂λ_{k}} (λ) e^{2πix·λ}

= ∑_{λ∈ℤ^{N}} \frac{∂m_{j}}{∂λ_{k}} (λ + θ_{k}^{(1)}e_{k}) e^{2πix·λ},

for some θ_{k}^{(1)} ∈ (0, 1) . By replacing m_{j} with \frac{∂m_{j}}{∂λ_{l}} in the previous identity, we

obtain respectively

(e^{2πix_{l}} − 1) (e^{2πix_{k}} − 1) K_{j} (x) = ∑_{λ∈ℤ^{N}} \frac{∂^{2}m_{j}}{∂λ_{l}∂λ_{k}} (λ + θ_{k}^{(1)}e_{k} + θ_{l}^{(1)}e_{l}) e^{2πix·λ}

if l ≠ k,

(e^{2πix_{k}} − 1)^{2} K_{j} (x) = ∑_{λ∈ℤ^{N}} \frac{∂^{2}m_{j}}{∂λ_{k}^{2}} (λ + (θ_{k}^{(1)} + θ_{k}^{(2)}) e_{k}) e^{2πix·λ}

if l = k, where we have used the fact that partial derivatives and difference quotients commute, i.e.

\frac{∂^{−}}{∂λ_{l}} (\frac{∂m_{j}}{∂λ_{k}} ) = \frac{∂}{∂λ_{k}} (\frac{∂^{−}m_{j}}{∂λ_{l}} ) ,

and, once again, we have invoked the Mean Value Theorem.

In turn, if α is a multi-index with |α| = M, we have

∏_{k=1}^{N} (e^{2πix_{k}} − 1)^{α_{k}} K_{j} (x) = ∑_{λ∈ℤ^{N}} \frac{∂^{|α|}m_{j}}{∂λ^{α}} (λ + ∑_{k=1}^{N} (θ_{k}^{(1)} + ⋯ + θ_{k}^{(α_{k})}) e_{k}) e^{2πix·λ},

where θ_{k}^{(1)}, ⋯, θ_{k}^{(α_{k})} ∈ (0, 1) . By the Mean Value Theorem we derive

(2π)^{|α|} |x^{α}| |K_{j} (x)| ≤ ∑_{λ∈ℤ^{N}} |\frac{∂^{|α|}m_{j}}{∂λ^{α}} (λ + ∑_{k=1}^{N} (θ_{k}^{(1)} + ⋯ + θ_{k}^{(α_{k})}) e_{k})| ,

which, togethere with (2.7), yields

|K_{j} (x)| ≤ \frac{C}{|x|^{M}} ∑_{2^{j−1}−|α|≤|λ|≤2^{j+1}+|α|} |\frac{∂^{|α|}m_{j}}{∂λ^{α}} (λ + ∑_{k=1}^{N} (θ_{k}^{(1)} + ⋯ + θ_{k}^{(α_{k})}) e_{k})|

≤ \frac{C2^{−jM}}{|x|^{M}} ∑_{2^{j−1}−|α|≤|λ|≤2^{j+1}+|α|} 1

≤ C \frac{2^{−jM+jN}}{|x|^{M}} .

Note that here we have used the fact that

|\frac{∂^{|α|}m_{j} (λ)}{∂λ^{α}} | ≤ C_{α}2^{−jM},

which results directly from the homogeneity of degree zero of the function m, yielding

|\frac{∂^{|α|}m (λ)}{∂λ^{α}} | ≤ C_{α} |λ|^{−M} ,

and from

|\frac{∂^{|α|}}{∂λ^{α}} (δ (\frac{λ}{2^{j}} ))| = (\frac{1}{2^{j}} )^{M} |\frac{∂^{|α|}δ}{∂λ^{α}} (\frac{λ}{2^{j}} )| ≤ C_{α} |λ|^{−M} ,

where we took into account the fact supp δ (\frac{·}{2^{j}} ) ⊂ [2^{j−1}, 2^{j+1}].

It is clear that S_{q} may be extended as an homogeneous operator of degree −1 from W_{per}^{−1,q}(Q; ℝ^{d}) into L_{per}^{q}(Q; ℝ^{d}). Indeed, as it is usual, using duality principles, if L ∈ W_{per}^{−1,q}(Q; ℝ^{d}) and if φ ∈ L_{per}^{q′}(Q; ℝ^{d}) then

⟨S_{q}L, φ⟩ := ⟨L, S_{q′}^{∗}φ⟩ ,  (2.11)

where for f, g ∈ C_{per}^{∞}(Q; ℝ^{d}) the duality pair is defined by

⟨f, g⟩ := ∑_{λ∈ℤ^{N}} \overline{f̂(λ)} ĝ (λ) = ∫_{Q} \overline{f (x)} g (x) dx,

and where the operator

S_{q′}^{∗} : L_{per}^{q′} (Q; ℝ^{d}) → W_{per}^{1,q′} (Q; ℝ^{l})

is defined by

S_{q′}^{∗}v (x) := ∑_{λ∈ℤ^{N} \\{0\}} \overline{𝕊 (λ)}v̂ (λ) e^{2πix·λ}

whenever v ∈ L_{per}^{q′}(Q; ℝ^{d}).

In particular, consider

1 < q < \frac{N}{N − 1}.

Since the space of all Q-periodic ℝ^{l}-valued Radon measures M_{per} (Q; ℝ^{l}) is contained in W_{per}^{−1,q}(Q; ℝ^{d}), if μ ∈ M_{per} (Q; ℝ^{l}) then in view of (2.11), S_{q}μ is well defined, and using Fubini’s Theorem we may find the representation

S_{q}μ (x) = ∫_{Q} K (x − y) dμ (y) .  (2.12)

Indeed, if φ ∈ C_{per}^{∞}(Q; ℝ^{d}) we have

∫_{Q} \overline{(S_{q}μ) (x)} φ (x) dx = ⟨S_{q}μ, φ⟩ = ⟨μ, S_{q′}^{∗}φ⟩ = ∫_{Q} S_{q′}^{∗}φ (y) d\overline{μ} (y)

= ∫_{Q} ∑_{λ∈ℤ^{N} \\{0\}} \overline{𝕊 (λ)} φ̂ (λ) e^{2πiy·λ} d\overline{μ} (y)

= ∫_{Q} (∫_{Q} ∑_{λ∈ℤ^{N} \\{0\}} \overline{𝕊 (λ)} e^{2πi(y−x)·λ} d\overline{μ} (y)) φ (x) dx

= ∫_{Q} (\overline{∫_{Q} K (x − y) dμ (y)}) φ (x) dx,

thus asserting (2.12).

We can now define the operator

T_{q} : L_{per}^{q}(Q; ℝ^{d}) → L_{per}^{q}(Q; ℝ^{d})

as follows

T_{q}v (x) := v − S_{q}Av.

When there is no possibility of confusion we write simply S and T in place of S_{q} and T_{q}, respectively.

The following proposition may be found in [27].

Proposition 2.2 T : L_{per}^{q}(Q; ℝ^{d}) → L_{per}^{q}(Q; ℝ^{d}) is a bounded linear operator and S : W_{per}^{−1,q} (Q; ℝ^{l}) → L_{per}^{q} (Q; ℝ^{d}) is a pseudo differential bounded operator of order −1 such that

(i) if v ∈ L_{per}^{q}(Q; ℝ^{d}) then T ∘T v = T v and A (T v) = 0;

(ii) ||v−T v||_{L^{q}} ≤ C_{q}||A (v) ||_{W^{−1,q}} for all v ∈ L_{per}^{q}(Q; ℝ^{d}) such that ∫_{Q} v dx = 0, for some C_{q} > 0;

(iii) v − T v = SAv.

The next result is well-known to experts. We include a proof for the convenience of the reader.

Proposition 2.3 Let 1 ≤ p < ∞, let h ∈ L^{p} (∂Q_{r}; ℝ^{d}) , where r ∈ (\frac{3}{4}, 1) , and consider the measure

μ = h H^{N−1}⌊_{∂Q_{r}} .

Then for s ∈ (0, 1) , 0 < α ≤ 1, α ≠ \frac{N−1}{p}, we have

||S μ||_{L^{t}(∂Q_{s})} ≤ C |s − r|^{−α} ||h||_{L^{p}(∂Q_{r})},

where

t := \{ \begin{matrix}\frac{p(N−1)}{N−1−αp} & if \frac{1}{p} − \frac{α}{N−1} > 0, \\ ∞ & if \frac{1}{p} − \frac{α}{N−1} < 0. \end{matrix}

Proof. Consider now

μ = h H^{N−1}⌊_{∂Q_{r}} .

We have

Sμ (x) = ∫_{Q} K (x − y) dμ (y) = ∫_{∂Q_{r}} K (x − y) h (y) dH^{N−1} (y) .

For any α ∈ (0, 1] there exists a constant C > 0 such that

|x − y|^{N−1} ≥ C |r − s|^{α} |ξ − ξ′|^{N−1−α}

for all x = sξ ∈ ∂Q_{s} and y = rξ′ ∈ ∂Q_{r}, where ξ, ξ′ ∈ ∂Q (recall that r ∈ (\frac{3}{4}, 1)). Thus for x = sξ ∈ ∂Q_{s} we have

|Sμ (x)| ≤ C |r − s|^{−α} ∫_{∂Q} \frac{|h_{r} (ξ′)|}{|ξ − ξ′|^{N−1−α}} dH^{N−1} (ξ′) ,

where

h_{r} (ξ′) = r^{N−1}h (rξ′) ,

and we used (2.4). The conclusion follows from the standard convolution inequality for fractional integrals applied to the (N − 1)-dimensional Lipschitz manifold ∂Q equipped with the distance induced by ℝ^{N}; see [45], I§8.21, for a very general version of fractional integration. For the case at hand one can of course use the classical argument on local charts (see also Hardy-Littlewood-Sobolev inequality in ℝ^{N−1} [45], page 354).

A function f : ℝ^{d} → ℝ is said to be A-quasiconvex if

f(ξ) ≤ ∫_{Q} f(ξ + w(y)) dy

for all ξ ∈ ℝ^{d} and all w ∈ C_{per}^{∞}(ℝ^{N}; ℝ^{d}) such that Aw = 0 and ∫_{Q} w(y) dy = 0.

As it is usual, the regularity of the test function w maybe relaxed if f satisfies appropriate growth conditions.

Proposition 2.4 Let f : ℝ^{N} → ℝ be an upper semicontinuous, A-quasiconvex function, such that

f (ξ) ≤ C (1 + |ξ|^{q})  (2.13)

for all ξ ∈ R^{d}, and for some 1 < q < ∞ and C > 0. Then

f(ξ) ≤ ∫_{Q} f(ξ + w(y)) dy

for all ξ ∈ ℝ^{d} and all w ∈ L_{per}^{q}(ℝ^{N}; ℝ^{d}) such that Aw = 0 and ∫_{Q} w(y) dy = 0.

Proof. Fix ξ ∈ ℝ^{d} and let w ∈ L_{per}^{q}(ℝ^{N}; ℝ^{d}) be such that Aw = 0 and ∫_{Q} w(y) dy = 0. Then the functions

w_{ε} := ρ_{ε} ∗ w − ∫_{Q} ρ_{ε} ∗ w dy

are in C_{per}^{∞}(ℝ^{N}; ℝ^{d}), Aw_{ε} = 0 and ∫_{Q} w_{ε}(y) dy = 0. In view of (2.13), Fatou’s Lemma and the upper semicontinuity of f imply

lim inf_{ε→0^{+}} ∫_{Q} [C (1 + |w_{ε}|^{q}) − f(ξ + w_{ε})] dy ≥ ∫_{Q} [C (1 + |w|^{q}) − f(ξ + w)] dy.

Since f is A-quasiconvex it follows that

∫_{Q} f(ξ + w(y)) dy ≥ lim sup_{ε→0^{+}} ∫_{Q} f(ξ + w_{ε}) dy ≥ f(ξ),

and the proof is complete.

3 Proof of Theorem 1.4

In this section we prove Theorem 1.4 using the blow-up method. As it is usual, the main effort will target the case where the limit function v reduces to a constant.

Proposition 3.1 Let g : ℝ^{N} → [0, ∞) be an A-quasiconvex function such that

|g (ξ) − g (ξ_{1})| ≤ C (1 + |ξ|^{q−1} + |ξ_{1}|^{q−1}) |ξ − ξ_{1}| ,  (3.1)

for all ξ, ξ_{1} ∈ ℝ^{d} and for some C > 0, where 1 ≤ q < ∞ satisfies (1.7). Then

g (0) ≤ lim inf_{n→∞} ∫_{Q} g (v_{n} (x)) dx

for any sequence {v_{n}} ⊂ L^{1} (Q; ℝ^{d}) ∩ ker A converging weakly-∗ to zero in the sense of measures.

Proof. By a simple mollification argument and by passing to a subsequence if necessary, without loss of generality we may assume that \{v_{n}\} ⊂ C^{∞} (\overline{Q}; ℝ^{d})∩ ker A,

C_{0} := sup_{n} ∫_{Q} |v_{n}(x)| dx < ∞,  (3.2)
lim inf_{n→∞} ∫_{Q} g (v_{n} (x)) dx = lim_{n→∞} ∫_{Q} g (v_{n} (x)) dx,  (3.3)

and there exists a nonnegative Radon measure μ such that

|v_{n}(x)| L^{N}⌊ Q ⇀^{∗} μ  (3.4)

as n → ∞, weakly ∗ in the sense of measures. Fix δ > 0. By (3.2)_{1} there exist

E_{n} ⊂ (1 − 2δ, 1 − δ) , L^{1} (E_{n}) = \frac{δ}{2}

such that

μ(∂Q_{r}) = 0  (3.5)

and

∫_{∂Q_{r}} |v_{n}| dH^{N−1} ≤ C_{1} (δ) := \frac{4C_{0}}{δ}  (3.6)

for all r ∈ E_{n}. Fix r ∈ E_{n} and let

w_{n,r} := χ_{r}v_{n} − ∫_{Q_{r}} v_{n} dy

By the A-quasiconvexity of g and as g ≥ 0, we have

∫_{Q} g (v_{n}) dx ≥ ∫_{Q} g (T (w_{n,r})) dx + ∫_{Q_{r}} g (v_{n}) dx − ∫_{Q} g (T (w_{n,r})) dx
≥ g (0) |Q| + ∫_{Q_{r}} g (v_{n}) dx − ∫_{Q} g (T (w_{n,r})) dx  (3.7)
= g (0) |Q_{r}| + ∫_{Q} [g (χ_{r}v_{n}) − g (T (w_{n,r}))] dx,

where χ_{r} is the characteristic function of the set Q_{r} and where we have used Proposition 2.4. By (3.1) we have

∫Q [g (χ_{r}v_{n}) − g (T (w_{n,r}))] dx
≤ C ∫_{Q} (1 + |χ_{r}v_{n}|^{q−1} + |T (w_{n,r})|^{q−1}) |χ_{r}v_{n} − T (w_{n,r})| dx

≤ C ∫_{Q} |χ_{r}v_{n} − T (w_{n,r})|^{q} dx + C ∫_{Q} (1 + |χ_{r}v_{n}|^{q−1}) |χ_{r}v_{n} − T (w_{n,r})| dx.

Hence from (3.7) we have

∫_{Q} g (v_{n}) dx ≥ g (0) |Q_{1−2δ}| − C ∫_{Q} |χ_{r}v_{n} − T (w_{n,r})|^{q} dx
− C ∫_{Q} (1 + |χ_{r}v_{n}|^{q−1}) |χ_{r}v_{n} − T (w_{n,r})| dx.

Multiply the previous inequality by χ_{E_{n}} and integrate in r to obtain

\frac{δ}{2} ∫_{Q} g (v_{n}) dx ≥g (0) |Q_{1−2δ}| \frac{δ}{2}

− C ∫_{1−2δ}^{1−δ} χ_{E_{n}} ∫_{Q} |χ_{r}v_{n} − T (w_{n,r})|^{q} dx dr  (3.8)

− C ∫_{1−2δ}^{1−δ} χ_{E_{n}} ∫_{Q} (1 + |χ_{r}v_{n}|^{q−1}) |χ_{r}v_{n} − T (w_{n,r})| dx dr,

where we have used the fact that L^{1} (E_{n}) = \frac{δ}{2}. By (1.7) we may choose q_{1} such that

q_{1} := \{ \begin{matrix}q & if q > 1, \\ ∈ (1, \frac{N}{N−1}) & if q = 1. \end{matrix}  (3.9)

We claim that

lim_{n→∞} ∫_{1−2δ}^{1−δ} χ_{E_{n}} |∫_{Q_{r}} v_{n} dy|^{q_{1}} dr = 0.  (3.10)

Indeed, fix r ∈ (1 − 2δ, 1 − δ) . If μ(∂Q_{r}) > 0 then χ_{E_{n}} (r) = 0 for all n by (3.5), while if μ(∂Q_{r}) = 0 then ∫_{Q} χ_{r}v_{n}dy → 0 by Theorem 1.62 in [5], and because

v_{n} ⇀^{∗} 0 in the sense of measures. The claim now follows from (3.2) by Lebesgue Dominated Convergence Theorem.

Next we show that

lim_{n→∞} ∫_{1−2δ}^{1−δ} χ_{E_{n}} ∫_{Q} |χ_{r}v_{n} − T (w_{n,r})|^{q_{1}} dx dr = 0,  (3.11)

or, equivalently by (3.10),

lim_{n→∞} ∫_{1−2δ}^{1−δ} χ_{E_{n}} ∫_{Q} |w_{n,r} − T (w_{n,r})|^{q_{1}} dx dr = 0.  (3.12)

By Proposition 2.2(ii) we have

∥w_{n,r} − T (w_{n,r})∥_{L^{q_{1}} (Q)} ≤ C ∥A (v_{n}χ_{r})∥_{W^{−1,q_{1}} (Q)} ,

and thus to prove (3.12) it suffices to show that

∫_{1−2δ}^{1−δ} χ_{E_{n}} ∥A (v_{n}χ_{r})∥_{W^{−1,q_{1}} (Q)}^{q_{1}} dr → 0.  (3.13)

Fix ψ ∈ C_{c}^{∞}(Q; ℝ^{l}). Using (3.6), if r ∈ E_{n} then we deduce that

|⟨A (χ_{r}v_{n}) , ψ⟩| = |∑_{i=1}^{N} ∫_{Q} A^{(i)}χ_{r}v_{n} \frac{∂ψ}{∂x_{i}} dx|

= |− ∫_{∂Q_{r}} 𝔸 (ν_{r}) v_{n}ψ dH^{N−1}|  (3.14)
≤ C ∫_{∂Q_{r}} |v_{n}| dH^{N−1} ∥ψ∥_{L^{∞}(Q;ℝ^{l})} ≤ C ∥ψ∥_{L^{∞}(Q;ℝ^{l})} .

Hence

χ_{E_{n}} ∥A (χ_{r}v_{n})∥_{M(Q;ℝ^{l})} ≤ C  (3.15)

for all n and r ∈ (1 − 2δ, 1 − δ) . We now show that

χ_{E_{n}}A (χ_{r}v_{n}) ⇀^{∗} 0

in the sense of measures. Fix ψ ∈ C_{c}^{∞}(Q; ℝ^{l}) and r ∈ (1 − 2δ, 1 − δ) . If μ(∂Q_{r}) > 0 then χ_{E_{n}} (r) = 0 for all n by (3.5). Thus assume μ(∂Q_{r}) = 0. Since v_{n} ∈ ker A, we have, by Theorem 1.62 in [5] and the fact that v_{n} ⇀^{∗} 0 in the sense of measures,

⟨A (χ_{r}v_{n}) , ψ⟩ = − ∑_{i=1}^{N} ∫_{Q} A^{(i)}χ_{r}v_{n} \frac{∂ψ}{∂x_{i}} dx → 0.  (3.16)

Therefore, {χ_{E_{n}}A (χ_{r}v_{n})} is a bounded sequence of ℝ^{l}-valued Radon measures converging weak-∗ to zero. Since M (Q; ℝ^{l}), the space of all ℝ^{l}-valued Radon measures, is compactly embedded in W^{−1,q_{1}}(Q; ℝ^{l}), we deduce that

χ_{E_{n}}A (χ_{r}v_{n}) → 0 in W^{−1,q_{1}}(Q; ℝ^{l}) as n → ∞

for all r ∈ (1 − 2δ, 1 − δ), with

χ_{E_{n}} ∥A (v_{n}χ_{r})∥_{W^{−1,q_{1}} (Q)} ≤ C

for all n and r ∈ (1 − 2δ, 1 − δ) . By Lebesgue Dominated Convergence Theorem, we obtain (3.13), and, in turn, (3.11).

Finally, we prove that

lim_{n→∞} ∫_{1−2δ}^{1−δ} χ_{E_{n}} ∫_{Q_{r}} |v_{n}|^{q−1} |v_{n} − T (w_{n,r})| dx dr = 0.  (3.17)

If q = 1 then this is a consequence of (3.11). Thus, without loss of generality, we may assume that q > 1. We begin by showing that

lim_{n→∞} ∫_{1−2δ}^{1−δ} χ_{E_{n}} (∫_{Q_{r}} |v_{n}|^{q−1} dx) |∫_{Q_{r}} v_{n} dy| dx dr = 0.  (3.18)

Indeed, since q ≤ 2, by (3.2)_{1} we have

∫_{1−2δ}^{1−δ} χ_{E_{n}} (∫_{Q_{r}} |v_{n}|^{q−1} dx) |∫_{Q_{r}} v_{n} dy| dx dr ≤ C ∫_{1−2δ}^{1−δ} χ_{E_{n}} |∫_{Q_{r}} v_{n} dy| dx dr,

and thus (3.18) follows from (3.10). In view of (3.18), proving (3.17) is equivalent to showing that

lim_{n→∞} ∫_{1−2δ}^{1−δ} χ_{E_{n}} ∫_{Q_{r}} |v_{n}|^{q−1} |w_{n,r} − T (w_{n,r})| dx dr = 0.  (3.19)

Now, if ε ∈ (0, 1) then we have

∫E_{n} ∫Q_{r} |v_{n}|^{q−1} |w_{n,r} − T (w_{n,r})| dx dr
= ∫_{E_{n}} ∫_{Q_{r}} |v_{n}|^{q−1}|w_{n,r} − T (w_{n,r}) |^{1−ε}|w_{n,r} − T (w_{n,r}) |^{ε} dx dr (3.20)

≤ (∫_{E_{n}} ∫_{Q_{r}} |v_{n}|^{\frac{q−1}{1−ε}} |w_{n,r} − T (w_{n,r}) | dx dr)^{1−ε}
× (∫_{E_{n}} ∫_{Q_{r}} |w_{n,r} − T (w_{n,r}) | dx dr)^{ε} ,

where we used Hölder’s inequality with exponents 1/(1 − ε) and 1/ε. By (3.12) the second factor on the right hand side of the previous inequality converges to zero as n → ∞, hence to prove (3.19), and thus (3.17), it remains to show that

sup_{n} ∫_{E_{n}} ∫_{Q_{r}} |v_{n}|^{\frac{q−1}{1−ε}} |w_{n,r} − T (w_{n,r}) | dx dr < ∞.

In light of (3.14), and since A (v_{n}) = 0, we may identify A (w_{n,r}) = A (χ_{r}v_{n}) with the measure

μ_{r,n} := −𝔸 (ν_{r}) v_{n} H^{N−1}⌊_{∂Q_{r}} .

Hence by Proposition 2.2(iii)

w_{n,r} − T (w_{n,r}) = SA (χ_{r}v_{n}) = Sμ_{r,n}.  (3.21)

Note that 0 < (q − 1) (N − 1) < 1 and let

α ∈ ((q − 1) (N − 1) , 1) , t := \frac{N − 1}{N − 1 − α}.  (3.22)

Then

t > \frac{N − 1}{N − 1 − (q − 1) (N − 1)} = \frac{1}{2 − q} , t′ < \frac{1}{q − 1}.

Using Hölder’s inequality, Proposition 2.3 with p = 1, (3.6), and (3.21), we have

∫E_{n} ∫Q_{r}|v_{n}|^{\frac{q−1}{1−ε}} |w_{n,r} − T (w_{n,r}) | dx dr
= ∫_{E_{n}} ∫_{0}^{r} ∫_{∂Q_{s}} |v_{n}|^{\frac{q−1}{1−ε}} |w_{n,r} − T (w_{n,r}) | dH^{n−1} ds dr
≤ ∫_{E_{n}} ∫_{0}^{r} ∥|v_{n}|^{\frac{q−1}{1−ε}} ∥_{L^{t′} (∂Q_{s})} ∥w_{n,r} − T (w_{n,r})∥_{L^{t}(∂Q_{s})} ds dr
≤ C ∫_{E_{n}} ∫_{0}^{r} ∥|v_{n}|^{\frac{q−1}{1−ε}} ∥_{L^{t′} (∂Q_{s})} (r − s)^{−α}||v_{n}||_{L^{1}(∂Q_{r})} dsdr
≤ C ∫_{0}^{1} ∥|v_{n}|^{\frac{q−1}{1−ε}} ∥_{L^{t′} (∂Q_{s})} ∫_{E_{n}∩[s,1]}(r − s)^{−α} dr ds

≤ C ∫_{0}^{1} ∥|v_{n}|^{\frac{q−1}{1−ε}} ∥_{L^{t′} (∂Q_{s})} ds

≤ C (∫_{0}^{1} ∥|v_{n}|^{\frac{q−1}{1−ε}} ∥_{L^{t′} (∂Q_{s})}^{t′} ds)^{1/t′} = C (∫_{Q} |v_{n}|^{\frac{q−1}{1−ε} t′} dx)^{1/t′} ,

which remains bounded as n → ∞, since (q − 1) t′ < 1 we may choose ε := 1 − (q − 1) t′. Hence (3.17) holds.

By (3.11) and (3.17), letting n → ∞ in (3.8) yields

\frac{δ}{2} lim inf_{n→∞} ∫_{Q} g (v_{n}) dx ≥ g (0) |Q_{1−2δ}| \frac{δ}{2},

and to conclude the proof it suffices to divide the previous inequality by \frac{δ}{2} and then let δ → 0^{+}.

We are now ready to prove Theorem 1.4.

Proof of Theorem 1.4. Without loss of generality we may assume that

lim inf_{n→∞} ∫_{Ω} f(x, v_{n}(x)) dx = lim_{n→∞} ∫_{Ω} f(x, v_{n}(x)) dx < ∞.

Passing to a subsequence, if necessary, we find a nonnegative Radon measure μ such that

f(x, v_{n}(x)) L^{N}⌊ Ω ⇀^{∗} μ

as n → ∞, weakly ∗ in the sense of measures. We claim that

\frac{dμ}{dL^{N}} (x_{0}) = lim_{r→0^{+}} \frac{μ(Q(x_{0}, r))}{r^{N}} ≥ f(x_{0}, v(x_{0}))  for L^{N} a.e. x_{0} ∈ Ω. (3.23)

If (3.23) holds, then the conclusion of the theorem follows immediately. Indeed, let φ ∈ C_{c}(Ω; ℝ), 0 ≤ φ ≤ 1. Since

μ = \frac{dμ}{dL^{N}} L^{N} + μ_{s}

where μ_{s} ≥ 0, we have

lim_{n→∞} ∫_{Ω} f(x, v_{n}) dx ≥ lim inf_{n→∞} ∫_{Ω} φ (x) f(x, v_{n}) dx = ∫_{Ω} φ dμ

≥ ∫_{Ω} φ \frac{dμ}{dL^{N}} dx ≥ ∫_{Ω} φ f(x, v) dx.

By letting φ → 1^{−}, and using Lebesgue Dominated Convergence Theorem, we obtain the desired result. Thus, to conclude the proof of the theorem, it suffices to show (3.23).

Let

v := \frac{dλ}{dL^{N}} ∈ L^{1} (Ω; ℝ^{d}) ,

and fix x_{0} ∈ Ω such that

\frac{dμ}{dL^{N}} (x_{0}) = lim_{r→0^{+}} \frac{μ(Q(x_{0}, r))}{r^{N}} < ∞,  (3.24)

lim_{r→0^{+}} \frac{1}{r^{N}} ∫_{Q(x_{0},r)} |v(x) − v(x_{0})| dx = 0, lim_{r→0^{+}} \frac{|λ_{s}| (Q(x_{0}, r))}{r^{N}} = 0.

Choosing r_{k} ↘ 0 such that μ(∂Q(x_{0}, r_{k})) = 0, we have

\frac{dμ}{dL^{N}} (x_{0}) = lim_{k→∞} \frac{μ(Q(x_{0}, r_{k}))}{r_{k}^{N}} = lim_{k→∞} lim_{n→∞} \frac{1}{r_{k}^{N}} ∫_{Q(x_{0},r_{k})} f(x, v_{n}) dx

= lim_{k→∞^{+}} lim_{n→∞} ∫_{Q} f(x_{0} + r_{k}y, v(x_{0}) + w_{n,k}(y)) dy,

where w_{n,k}(y) := v_{n}(x_{0} + r_{k}y) − v(x_{0}). Clearly w_{n,k} ∈ ker A, and we claim that w_{n,k} ⇀^{∗} 0 weakly-∗ in the sense of measures if we first let n → ∞ and then k → ∞. Indeed, fix φ ∈ C_{c}(Q; ℝ^{d}). After a change of variables, we get

∫_{Q} φ(y) w_{n,k}(y) dy = ∫_{Q} φ(y)(v_{n}(x_{0} + r_{k}y) − v(x_{0})) dy

= \frac{1}{r_{k}^{N}} ∫_{Q(x_{0},r_{k})} φ (\frac{x − x_{0}}{r_{k}} ) (v_{n}(x) − v(x_{0})) dx.

If we now let n → ∞, and use the facts that v_{n} ⇀^{∗} λ weakly-∗ in the sense of measures and that

λ = v L^{N} ⌊_{Ω} + λ_{s},

we obtain that

lim_{n→∞} ∫_{Q} φ(y) w_{n,k}(y) dy =\frac{1}{r_{k}^{N}} ∫_{Q(x_{0},r_{k})} φ (\frac{x − x_{0}}{r_{k}} ) (v(x) − v(x_{0})) dx

+ \frac{1}{r_{k}^{N}} ∫_{Q(x_{0},r_{k})} φ (\frac{x − x_{0}}{r_{k}} ) dλ_{s}.

Hence

| lim_{n→∞} ∫_{Q} φ(y) w_{n,k}(y) dy| ≤||φ||_{L^{∞}(Q)} \frac{1}{r_{k}^{N}} ∫_{Q(x_{0},r_{k})} |v(x) − v(x_{0})| dx

+ ||φ||_{L^{∞}(Q)} \frac{|λ_{s}| (Q(x_{0}, r_{k}))}{r_{k}^{N}}.

The claim then follows by letting k → ∞ and by using (3.24). Diagonalize to get w_{k} ∈ L^{1}(Q; ℝ^{d})∩ker A such that w_{k} ⇀^{∗} 0 weakly-∗ in the sense of measures, and

\frac{dμ}{dL^{N}} (x_{0}) = lim_{k→∞} ∫_{Q} f(x_{0} + r_{k}y, v(x_{0}) + w_{k}(y)) dy

where r_{k} → 0. Fix ε > 0. By (1.9) and Proposition 3.1 we have

\frac{dμ}{dL^{N}} (x_{0}) ≥ \frac{1}{1 + ε} lim_{k→∞} ∫_{Q} f(x_{0}, v(x_{0}) + w_{k}(y)) dy − \frac{ε}{1 + ε}

≥ \frac{1}{1 + ε}f(x_{0}, v(x_{0})) − \frac{ε}{1 + ε}.

It now suffices to let ε → 0^{+}.

4 Proof of Theorem 1.6

In this section we prove Theorem 1.6. We begin with the following

Proposition 4.1 Let g : ℝ^{N} → [0, ∞) be an A-quasiconvex function such that

|g (ξ) − g (ξ_{1})| ≤ C (1 + |ξ|^{q−1} + |ξ_{1}|^{q−1}) |ξ − ξ_{1}| ,  (4.1)

for all ξ, ξ_{1} ∈ ℝ^{d}, and for some 1 < q < ∞ and C > 0. If {v_{n}} ⊂ L^{q} (Q; ℝ^{d}) ∩ ker A converges weakly to zero in L^{p} (Q; ℝ^{d}), where 1 < p < ∞ satisfies (1.11), then

g (0) ≤ lim inf_{n→∞} ∫_{Q} g (v_{n} (x)) dx.

Proof. The proof of this proposition follows closely that of Proposition 3.1, therefore we indicate only the main modifications. Condition (3.2)_{1} should be replaced by

C_{0} := sup_{n} ∫_{Q} |v_{n}(x)|^{p} dx < ∞,

and, correspondingly, (3.6) by

∫_{∂Q_{r}} |v_{n}|^{p} dH^{N−1} ≤ C_{1} (δ)  (4.2)

for all r ∈ E_{n}. Conditions (3.4) and (3.5) are no longer needed, while the exponent q_{1} in (3.9) is set to be equal to q. Equality (3.10) now follows immediately since ∫_{Q} χ_{r}v_{n}dy → 0 as v_{n} ⇀ 0 in L^{p} (Q; ℝ^{d}) for any r ∈ (1 − 2δ, 1 − δ) . To prove (3.13), fix ψ ∈ C_{c}^{∞}(Q; ℝ^{l}). Since v_{n} ∈ ker A and v_{n} ⇀ 0 in L^{p} (Q; ℝ^{d}), we have

⟨A (χ_{r}v_{n}) , ψ⟩ = ∑_{i=1}^{N} ∫_{Q} A^{(i)}χ_{r}v_{n} \frac{∂ψ}{∂x_{i}} dx → 0,  (4.3)

and if r ∈ E_{n}, and by (4.2),

|⟨A (χ_{r}v_{n}) , ψ⟩| = |∑_{i=1}^{N} ∫_{Q} A^{(i)}χ_{r}v_{n} \frac{∂ψ}{∂x_{i}} dx| = |− ∫_{∂Q_{r}} 𝔸 (ν_{r}) v_{n}ψ dH^{N−1}|

≤ C (∫_{∂Q_{r}} |v_{n}|^{p} dH^{N−1})^{1/p} ∥ψ∥_{L^{p′} (∂Q_{r};ℝ^{l})} ≤ C ∥ψ∥_{L^{p′} (∂Q_{r};ℝ^{l})} .

Hence,

χ_{E_{n}} ∥A (χ_{r}v_{n})∥_{L^{p}(∂Q_{r};ℝ^{l})} ≤ C  (4.4)

for all n and r ∈ (1 − 2δ, 1 − δ) .

We recall that Sobolev Compact Embedding Theorem we have

W_{0}^{1,q′} (Q; ℝ^{l}) ↪ L^{s} (∂Q_{r}; ℝ^{l}) ,

where

s < \{ \begin{matrix}\frac{(N − 1) q′}{N − q′} & if q′ < N, \\ ∞ & if q′ ≥ N. \end{matrix}

Thus, (1.11) yields that the Sobolev space W_{0}^{1,q′} (Q; ℝ^{l}) is compactly embedded in L^{p′} (∂Q_{r}; ℝ^{l}), and by duality we have L^{p} (∂Q_{r}; ℝ^{l}) compactly embedded in W^{−1,q} (Q; ℝ^{l}), which, together with (4.3) and (4.4) implies that

χ_{E_{n}}A (χ_{r}v_{n}) → 0 in W^{−1,q}(Q; ℝ^{l}) as n → ∞

for all r ∈ (1 − 2δ, 1 − δ), with

χ_{E_{n}} ∥A (v_{n}χ_{r})∥_{W^{−1,q}(Q)} ≤ C

for all n and r ∈ (1 − 2δ, 1 − δ) . By Lebesgue Dominated Convergence Theorem we obtain (3.13), and, in turn, (3.11).

To prove (3.17), in place of (3.22) we take

α ∈ (\frac{(q − p) (N − 1)}{p}, 1) , t := \frac{p (N − 1)}{N − 1 − αp},

where, without loss of generality, we are assuming q > p (see [27] for the case p = q). Then

t > \frac{p}{p − q + 1}, t′ < \frac{p}{q − 1},

where we have used the fact that p − q + 1 > 0 by (1.11).

We may now proceed exactly as before, with the only exception that now we have (q − 1) t′ < p. Hence taking

1 − ε = \frac{q − 1}{p}t′

we conclude.

We are now ready to prove Theorem 1.6.

Proof of Theorem 1.6. We proceed as in the proof of Theorem 1.4, until (3.24) which should be replaced by

\frac{dμ}{dL^{N}} (x_{0}) = lim_{r→0^{+}} \frac{μ(Q(x_{0}, r))}{r^{N}} < ∞, lim_{r→0^{+}} \frac{1}{r^{N}} ∫_{Q(x_{0},r)} |v(x) − v(x_{0})|^{p}dx = 0.

(4.5)

As in Theorem 1.4 we let w_{n,k}(y) := v_{n}(x_{0} + r_{k}y) − v(x_{0}). We claim that w_{n,k} ⇀ 0 in L^{p}(Q; ℝ^{d}) if we first let n → ∞ and then k → ∞. Indeed, fix φ ∈ L^{p′}(Q; ℝ^{d}), where p′ is the Hölder conjugate exponent of q. Using Hölder’s inequality and then making a change of variables, we get

|∫_{Q} φ(y) w_{n,k}(y) dy| ≤ |∫_{Q} φ(y)(v_{n}(x_{0} + r_{k}y) − v(x_{0} + r_{k}y)) dy|

+ |∫_{Q} φ(y)(v(x_{0} + r_{k}y) − v(x_{0})) dy|

≤ |\frac{1}{r_{k}^{N}} ∫_{Q(x_{0},r_{k})} φ (\frac{x − x_{0}}{r_{k}} ) (v_{n}(x) − v(x)) dx|

+ ||φ||_{L^{p′} (Q)} (\frac{1}{r_{k}^{N}} ∫_{Q(x_{0},r_{k})} |v(x) − v(x_{0})|^{p} dx)^{1/p} .

If we now let n → ∞ the first integral tends to zero due to the fact that v_{n} ⇀ v in L^{p}(Q(x_{0}, r_{k}); ℝ^{d}). The claim then follows by letting k → ∞ and by using (4.5). Diagonalize to get w_{k} ∈ L^{q}(Q; ℝ^{d}) ∩ ker A such that w_{k} ⇀ 0 in L^{q}(Q; ℝ^{d}) and

\frac{dμ}{dL^{N}} (x_{0}) = lim_{k→∞} ∫_{Q} f(x_{0} + r_{k}y, v(x_{0}) + w_{k}(y)) dy

where r_{k} → 0. We may now continue as in the proof of Theorem 1.4 using Proposition 4.1 in place of Proposition 3.1.

5 Proof of Theorem 1.8

Finally, we prove Theorems 1.8.

Proof. For any function v ∈ L^{p}(Ω; E_{s}^{m}) consider the differential operator A

Av := (\frac{∂}{∂x_{i}} v_{i_{1}...i_{h}ji_{h+2}...i_{s}} − \frac{∂}{∂x_{j}} v_{i_{1}...i_{h}ii_{h+2}...i_{s}})_{0≤h≤s−1, 1≤i,j,i_{1}...i_{s}≤N} .

Here h = 0 and h = s − 1 correspond to the multi-indexes ji_{2} … i_{s} and i_{1} … i_{s−1}j. As shown in [27],

{w ∈ C_{per}^{∞}(ℝ^{N}; E_{s}^{m}) : Aw = 0, ∫_{Q} w dx = 0} = {∇^{s}φ : φ ∈ C_{per}^{∞}(ℝ^{N}; ℝ^{m})} .

(5.1)

Since for a.e. x ∈ Ω and for all v ∈ E_{s}^{m},

f(x, v) = inf {∫_{Q} f(x, v + ∇^{s}φ(y)) dy : φ ∈ C_{per}^{∞}(ℝ^{N}; ℝ^{N})} ,

it follows from (5.1) that

f(x, v) = inf {∫_{Q} f(x, v + w(y)) dy :w ∈ C_{per}^{∞}(ℝ^{N}; E_{s}^{m}) ∩ ker A,
∫_{Q} w(y) dy = 0}

and thus f is A-quasiconvex. Let {u_{k}} ⊂ W^{s,q}(Ω; ℝ^{m}) be any sequence such that u_{k} ⇀ u in W^{s,p}(Ω; ℝ^{m}). Again by (5.1) A∇^{s}u_{k} = 0, and so we may apply Theorem 1.6, where the target space ℝ^{d} is replaced by the finite dimensional Euclidean vector space E_{s}^{m}, to obtain

∫_{Ω} f(x, ∇^{s}u) dx ≤ lim inf_{k→∞} ∫_{Ω} f(x, ∇^{s}u_{k}) dx.

6 Appendix

Proof of Theorem 1.1. Using the blow-up method as in Theorem 1.4, we may assume, without loss of generality, that

Ω = Q := (−\frac{1}{2}, \frac{1}{2})^{N} and u (x) ≡ 0.

As u_{n} → 0 in L^{1} (Q; ℝ^{d}) , by Egoroff’s and Fubini’s Theorems for any δ ∈ (0, 1) we may find a subsequence of {u_{n}} (not relabelled) such that for a.e. r ∈ (δ, 1)

lim_{n→∞} ∫_{∂Q_{r}} |u_{n}| dH^{N−1} = 0.

Since L^{1} (∂Q_{r} ∪ ∂Q; ℝ^{d}) is the trace space of W^{1,1} (Q\Q_{r}; ℝ^{d}) , we may find {v_{n}} ⊂ W^{1,1} (Q\Q_{r}; ℝ^{d}) such that v_{n} = u_{n} on ∂Q_{r} and v_{n} = 0 on ∂Q (in the sense of traces) and

∥v_{n}∥_{W^{1,1}(Q\Q_{r};ℝ^{d})} ≤ K_{r} ∥u_{n}∥_{L^{1}(∂Q_{r};ℝ^{d})}

for some constant K_{r} > 0. We have

∫_{Q\Q_{r}} f(∇v_{n}) dx ≤ ∫_{Q\Q_{r}} C (1 + |∇v_{n}|) dx
≤ CL^{N} (Q\Q_{r}) + CK_{r} ∥u_{n}∥_{L^{1}(∂Q_{r};ℝ^{d})} .

If we define v_{n} to be u_{n} in Q_{r} then {v_{n}} ⊂ W_{0}^{1,1} (Q; ℝ^{d}) and thus by the quasiconvexity of f we have

f(0) ≤ ∫_{Q} f(∇v_{n}) dx = ∫_{Q\Q_{r}} f(∇v_{n}) dx + ∫_{Q_{r}} f(∇u_{n}) dx
≤ CL^{N} (Q\Q_{r}) + CK_{r} ∥u_{n}∥_{L^{1}(∂Q_{r};ℝ^{d})} + ∫_{Q} f(∇u_{n}) dx

and letting n → ∞ we conclude that

f(0) ≤ CL^{N} (Q\Q_{r}) + lim inf_{n→∞} ∫_{Q} f(∇u_{n}) dx.

It now suffices to let δ → 1^{−} (and hence r).

Proof of Theorem 1.2. We consider only the case 1 < p < ∞. As in the previous proof, we may assume, without loss of generality, that

Ω = Q := (−\frac{1}{2}, \frac{1}{2})^{N} and u (x) ≡ 0,

and

lim inf_{n→∞} ∫_{Q} f(∇u_{n}) dx = lim_{n→∞} ∫_{Q} f(∇u_{n}) dx < ∞,

so that by condition (1.4)

K := sup_{n} ∫_{Q} |∇u_{n}|^{p} dx < ∞.

Fix δ ∈ (0, 1) . By Egoroff’s and Fubini’s Theorems, we may find a subsequence (not relabeled) such that for a.e. r ∈ (0, 1)

lim_{n→∞} ∫_{∂Q_{r}} |u_{n}|^{p} dH^{N−1} = 0.

Define

R := {r ∈ (δ, 1) : lim_{n→∞} ∫_{∂Q_{r}} |u_{n}|^{p} dH^{N−1} = 0,

lim inf_{n→∞} ∫_{∂Q_{r}} |∇u_{n}|^{p} dH^{N−1} ≤ \frac{2K}{1 − δ} \} .

Note that by Fatou’s Lemma

K ≥ lim inf_{n→∞} ∫_{(δ,1)\R} ∫_{∂Q_{r}} |∇u_{n}|^{p} dH^{N−1}dr

≥ ∫_{(δ,1)\R} lim inf_{n→∞} ∫_{∂Q_{r}} |∇u_{n}|^{p} dH^{N−1}dr

≥ L^{1}((δ, 1) \ R)\frac{2K}{1 − δ} ,

and so L^{1}(R) ≥ (1 − δ)/2.

Fix r ∈ R. Since p > \frac{N−1}{N}q, standard Sobolev trace and compact embedding theorems guarantee the existence of a lifting linear and compact operator

E :W^{1,p}(∂Q_{r}; ℝ^{d}) → W^{1,q}(Q; ℝ^{d})
v ⟼ E (v)

such that v is the trace of E (v) . Define {v_{n}} ⊂ W_{0}^{1,p}(Q; ℝ^{d}) by

v_{n} (x) := \{ \begin{matrix}u_{n} (x) & if x ∈ Q_{r} \\ φ (x) E (u_{n}) (x) & if x ∈ Q\Q_{r}, \end{matrix}

where φ ∈ C_{c}^{1} (Q; [0, 1]) is such that φ (x) ≡ 1 in Q_{r} and |∇φ| ≤ \frac{C}{1−r} . As E (u_{n}) → 0 in W^{1,q}(Q\Q_{r}; ℝ^{d}), by condition (1.4) we have

lim_{n→∞} ∫_{Q\Q_{r}} f(∇v_{n}) dx = ∫_{Q\Q_{r}} f(0) dx.

Hence, using the quasiconvexity of f at 0 we obtain

lim inf_{n→∞} ∫_{Q} f(∇u_{n}) dx ≥ lim inf_{n→∞} ∫_{Q_{r}} f(∇v_{n}) dx

= lim inf_{n→∞} ∫_{Q} f(∇v_{n}) dx − lim_{n→∞} ∫_{Q\Q_{r}} f(∇v_{n}) dx

≥ f(0) − L^{N}(Q \ Q_{r})f(0)

= L^{N}(Q_{r})f(0),

and the proof is complete if we let δ → 1^{−} (and hence r).

Acknowledgments

The research of I. Fonseca was partially by the National Science Foundation under Grants No. DMS–9731957 and DMS–0103799, that of G. Leoni by MURST and by GNAFA. I.F. and S. M. were also supported by the TMR....

G. Leoni would like to thank G. Savarè for many stimluating conversations on Sobolev spaces and for pointing out the reference [11].

I. Fonseca and G. Leoni thank the Center for Nonlinear Analysis (NSF Grant No. DMS–9803791), Carnegie Mellon University, Pittsburgh, PA, USA, for its support during the preparation of this paper.

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