On lower semicontinuity in BH^{p} and 2-quasiconvexification

Irene Fonseca Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213 fonseca@andrew.cmu.edu

Giovanni Leoni Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy 15100, leoni@unipmn.it

Roberto Paroni Dipartimento di Ingegneria Civile, Università degli Studi di Udine, Udine, Italy 33100, roberto.paroni@dic.uniud.it

September 13, 2001

Abstract

It is proved that if u ∈ BH^{p}(Ω; ℝ^{d}), with p > 1, if {u_{n}} is bounded in BH^{p}(Ω; ℝ^{d}), |D_{s}^{2}u_{n}|(Ω) → 0, and if u_{n} → u in W^{1,1}(Ω; ℝ^{d}), then

∫_{Ω} f(x, u(x), ∇u(x), ∇^{2}u(x)) dx ≤ lim inf_{n→+∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x), ∇^{2}u_{n}(x)) dx

provided f(x, u, ξ, ·) is 2-quasiconvex and satisfies some appropriate growth and continuity condition. Characterizations of the 2-quasiconvex envelope when admissible test functions belong to BH^{p} are provided.

2000 AMS Mathematics Classification Numbers: 49J45, 49Q20 Keywords: lower semicontinuity, 2-quasiconvexity, functions with bounded Hessian, maximal function

1 Introduction

In a recent paper Ambrosio [5] proved the following result:

Theorem 1.1 [5, Thm. 4.3] Let Ω ⊂ ℝ^{N} be an open set and let

f : Ω × ℝ^{N} × ℝ^{d×N} → [0, +∞) be a Carathéodory function, quasiconvex in ξ, and such that

|ξ|^{p} ≤ f(x, u, ξ) ≤ a (x) + Ψ (|u|) (1 + |ξ|^{p})

for all (x, u, ξ) ∈ Ω × ℝ^{N} × ℝ^{d×N}, where p > 1, a ∈ L^{1}(Ω) and Ψ : [0, ∞) → [0, ∞) is continuous. Then for every u ∈ SBV (Ω; ℝ^{d}) and any sequence {u_{n}} ⊂ SBV (Ω; ℝ^{d}) converging to u in L_{loc}^{1}(Ω; ℝ^{d}) and such that

sup_{n} H^{N−1} (S(u_{n})) < ∞  (1.1)

we have

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

Theorem 1.1 extends to the space SBV a classical result obtained by Acerbi and Fusco [1] in the Sobolev space W^{1,p}. The approach is based on a careful articulation of the blow-up method developed by Fonseca and Müller [27] together with a Lusin type theorem which allows to approximate SBV functions with Lipschitz functions. This approximation result was first established by Liu [34] in the Sobolev setting and is the key ingredient also in the argument used in [1].

Theorem 1.1 was later improved by Kristensen [32] who studied normal integrands f, possibly unbounded from below, and weakened condition (1.1) to read

sup_{n} ∫_{S(u_{n})} θ(|u_{n}^{+} − u_{n}^{−}|) dH^{N−1} < ∞,

where θ : [0, ∞) → [0, ∞) is a concave, nondecreasing function such that

lim_{t→0^{+}} \frac{θ (t)}{t} = ∞.

The proof in [32] is based on Young measures and on the Hodge Decomposition Theorem.

The aim of this paper is to extend Theorem 1.1 to functionals depending also on the second gradient, i.e., functionals of the form

J(u) := ∫_{Ω} f(x, u, ∇u, ∇^{2}u) dx  (1.2)

where f is a 2-quasiconvex integrand. We recall that a function f : 𝕊^{d×N×N} → ℝ is 2-quasiconvex if

∫_{Q} f(Λ + ∇^{2}ψ) dx ≥ f(Λ)

for every Λ ∈ 𝕊^{d×N×N}, and for every ψ ∈ W_{0}^{2,∞}(Ω; ℝ^{d}), where Q := (−1/2, 1/2)^{N} and 𝕊^{d×N×N} is the space of third order symmetric tensors.

The notion of 2-quasiconvexity was introduced by Meyers [36] and extends the concept of quasiconvexity, which is due to Morrey [37], to integrands which depend on second orders gradients.

Our main motivation to study this class of functionals when ∇u may have jumps derived from the theory of second order structured deformations (SOSD). The notion of first order structured deformation is due to Del Piero and Owen [19], and the energetics of these deformations was studied by Choksi and Fonseca [15]. Owen and Paroni [38] extended this framework to encompass second order derivatives and they introduced SOSD: a SOSD is a quadruple (κ, g, G, Σ) satisfying some technical conditions preventing interpenetration of matter, as well as regularity properties of the fields away from κ, where κ is the disarrangement site, g is the transplacement and G and Σ are tensor fields with properties similar to those of ∇g and ∇^{2}g, respectively. A pair (κ, u) is a simple deformation from a region Ω if κ ⊂ Ω has zero volume, u is injective and is a “piece-wise” classical deformation from Ω\κ. Owen and Paroni [38] showed that a SOSD (κ, g, G, Σ) may be approximated by simple deformations (κ_{n}, u_{n}) i.e., κ = lim inf κ_{n}, u = lim u_{n}, g = lim ∇u_{n} and f = lim ∇^{2}u_{n}, where the latter three limits are taken in the L^{∞} sense, while lim inf κ_{n} := ∪_{n=1}^{∞} ∩_{m=n}^{∞} κ_{m} (for an analog of this approximation result for first order structured deformations and within SBV see [15]). In light of this approximation, we may view G(x) as the local deformation at x without including the effects of discontinuities of the transplacement u_{n} at the disarrangement site κ_{n} for the approximating simple deformation (κ_{n}, u_{n}). A similar interpretation holds for Σ. The energy associated to a simple deformation is

E(u, κ) := ∫_{Ω} W (x, u, ∇u, ∇^{2}u) dx + ∫_{κ} ψ(x, [u], ν) dH^{N−1}
+ ∫_{κ} η(x, [∇u], ν) dH^{N−1},  (1.3)

where the first term is the bulk energy of the material in the placement Ω, the second and third terms take into account the contribution of the surface energy due to slips and separation and to interfaces between two phases of material, respectively. In (1.3) ν represents the normal to κ and [·] denotes the jump, while H^{N−1} stands for N − 1 Hausdorff measure. By means of this energy we define the energy of a SOSD as the energetically most economical way to attain it from simple deformations, i.e.,

I(κ, g, G, Σ) := inf{lim inf E(u_{n}, κ_{n}) : (k_{n}, u_{n}) approximates (κ, g, G, Σ)}.  (1.4)

The characterization of (1.4) by means of an integral representation requires a good handle of the lower semicontinuity properties of the bulk contribution (1.2) (see [15] for the case with only first derivatives). Here, we search for conditions ensuring that if u_{n}, u ∈ BH^{p}(Ω; ℝ^{d}) and if u_{n} → u in W^{1,1}(Ω; ℝ^{d}) then J(u) ≤ lim inf_{n→+∞} J(u_{n}). We recall that the space of Bounded Hessian functions, BH, was introduced by Demengel [20], [21] (see also Temam [41], Carriero, Leaci and Tomarelli [13]), and is defined as the set of functions in W^{1,1} whose Hessian, in the sense of distributions, is a finite Radon measure. Also BH^{p}, for p > 1, is the space of functions u ∈ BH such that ∇^{2}u ∈ L^{p}.

One of the principal results of this paper is the following theorem:

Theorem 1.2 Let Ω ⊂ ℝ^{N} be an open bounded set and let

f : Ω × ℝ^{N} × ℝ^{d×N} × 𝕊^{d×N×N} → [0, +∞)

be a normal integrand, 2−quasiconvex in Λ, and such that

|Λ|^{p} ≤ f(x, u, ξ, Λ) ≤ a (x, u, ξ) (1 + |Λ|^{p})  (1.5)

for L^{N} a.e. x ∈ Ω and all (u, ξ, Λ) ∈ ℝ^{N} × ℝ^{d×N} × 𝕊^{d×N×N}, where a(x, u, ξ) is a non-negative constant, and p > 1. Then for every u ∈ W^{1,1}(Ω; ℝ^{d}) and any sequence {u_{n}} ⊂ BH(Ω; ℝ^{d}) converging to u in W^{1,1}(Ω; ℝ^{d}) and such that

|D_{s}^{2}u_{n}| (Ω) → 0  (1.6)

we have

∫_{Ω} f(x, u, ∇u, ∇^{2}u) dx ≤ lim inf_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}, ∇^{2}u_{n}) dx.

We remark that no smoothness nor integrability properties are required from the function (x, u, ξ) ↦ a(x, u, ξ) that appears in the upper bound (1.5). All that is needed is that a(x, u, ξ) be defined, finite, and non-negative for L^{N} a.e. x ∈ Ω and all (u, ξ, Λ) ∈ ℝ^{N} × ℝ^{d×N} × 𝕊^{d×N×N}. In this generality, Theorem 1.2 is new even in the Sobolev setting. Indeed, we have

Corollary 1.3 Let Ω ⊂ ℝ^{N} be an open bounded set and let

f : Ω × ℝ^{N} × ℝ^{d×N} × 𝕊^{d×N×N} → [0, +∞)

be a normal integrand, 2−quasiconvex in Λ, and such that

0 ≤ f(x, u, ξ, Λ) ≤ a (x, u, ξ) (1 + |Λ|^{p})

for L^{N} a.e. x ∈ Ω and all (u, ξ, Λ) ∈ ℝ^{N} × ℝ^{d×N} × 𝕊^{d×N×N}, where a(x, u, ξ) is a non-negative constant, and p > 1. Then for every u ∈ W^{2,p}(Ω; ℝ^{d}) and any sequence {u_{n}} ⊂ W^{2,p}(Ω; ℝ^{d}) weakly converging to u in W^{2,p}(Ω; ℝ^{d}) we have

∫_{Ω} f(x, u, ∇u, ∇^{2}u) dx ≤ lim inf_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}, ∇^{2}u_{n}) dx.

Corollary 1.3 was first proved by Meyers [36] for integrands satisfying strong continuity conditions, which essentially exclude genuine dependence on lower order terms, and using results of Agmon, Douglis and Nirenberg [2] concerning Poisson kernels for elliptic equations. Fusco [30], under the same set of hypotheses, later gave a simpler proof using De Giorgi’s Slicing Lemma.

More recently Guidorzi and Poggiolini [31] established Corollary 1.3 for Carathéodory integrands under the Lipschitz condition

|f(x, u, ξ, Λ) − f(x, u, ξ, Λ_{1})| ≤ C(1 + |Λ|^{p−1} + |Λ_{1}|^{p−1})|Λ − Λ_{1}|.

As proved in [31], this Lipschitz type condition is automatically satisfied for 2−quasiconvex integrands which satisfy the growth condition

0 ≤ f(x, u, ξ, Λ) ≤ C(1 + |Λ|^{p})

(see also [39] for a significantly simpler proof valid within the realm of k-quasiconvexity for any k ∈ ℕ).

The lower semicontinuity result in [31] is based on the approximation of the integrand f with a nondecreasing sequence of 2−quasiconvex integrands each of them independent of (x, u, ξ) and convex for large values of Λ. This technique was developed by Marcellini [35] and relies on regularity theory.

Yet another proof of Corollary 1.3 was given by Braides, Fonseca and Leoni in [12], who obtained a general relaxation result in W^{2,p}(Ω; ℝ^{d}) with respect to weak convergence (see also [24] for some related results). The methods used there are based on the general setting of A-quasiconvexity, as introduced by Fonseca and Müller in [29]. The argument exploits the use of Young measures (see also the work of Balder [6] and of Kristensen [32]) together with the blow-up method introduced by Fonseca and Müller in [27].

Both approaches in [12] and [31] present difficulties when one tries to extend them to the space BH. Indeed, Marcellini’s method allows one to replace general integrands f = f(x, u, ∇u, ∇^{2}u) with integrands of the form f = f(∇^{2}u), but then one is still left with the problem of approximating the admissible sequences in BH(Ω; ℝ^{d}) with sequences in W^{2,p}(Ω; ℝ^{d}).

On the other hand, the A−quasiconvexity method used in [12] strongly relies on the underlying PDEs which characterize the space W^{2,p}(Ω; ℝ^{d}). However, in a recent paper extending a result of Alberti [3] we have shown that in the passage from the Sobolev spaces to the space BH the Hessian matrix D^{2}u remains symmetric but it may loose, in general, the PDE constraint curl= 0 More precisely, we have proved that Theorem 1.4 [26, Thm. 1.4] Let Ω be an open subset of ℝ^{N} and let f ∈ L^{1} (Ω; ℝ_{sym}^{N×N}) . Then there exists u ∈ BH (Ω) and a constant C > 0 depending only on N such that

D^{2}u = f L^{N} + [∇u] ⊗ ν_{∇u} H^{N−1}⌊ S(∇u),

and

∫_{Ω} |u| + |∇u| dx + ∫_{S(∇u)∩Ω} |[∇u]| dH^{N−1} ≤ C ∫_{Ω} |f| dx.

In spite of this fact, and in view of condition (1.6), it is still possible to approximate BH^{p} functions by W^{2,∞} functions. We establish these approximation results in Section 3 following Ambrosio [5] lead and Acerbi and Fusco [1] ideas, via maximal functions. In Section 4 we prove Theorem 1.2 and, as a consequence, in Section 6 we find a new characterization of the quasiconvexification of an integrand with p−growth, namely Theorems 6.2, 6.3.

Ongoing work addresses the relaxation problem described in (1.4) in the space BH. A function u ∈ BH has no jump discontinuities, hence the second integral on the right hand side of (1.3) is equal to zero, and thence the disarrangement site κ can be taken to be the jump set of the gradient of u. A relaxed version (suitable for BH functions) of the approximation of SOSD by means of simple deformations will rest on Theorem 1.4. An integral representation of (1.4) will be obtained using the global method of relaxation introduced by Bouchitté, Fonseca and Mascarenhas [9]. Generalizations to the space SBV^{2}, introduced by Carriero, Leaci and Tomarelli [14], will take into account also the jumps in the function, thus completing the study of the full energy (1.3).

2 Preliminaries

Let Ω be an open bounded subset, let B denote the open ball centered at the origin with radius 1, and let B(x, r) be the open ball centered at x and with radius r, i.e. B(x, r) := x + rB. Q stands for the open cube (−\frac{1}{2}, \frac{1}{2})^{N}, and Q(x, r) is the cube centered at x with side length r. For a given set U ⊂ ℝ^{N} we denote by H^{N−1}(U) its (N − 1)-dimensional Hausdorff measure, and by L^{N}(U) its Lebesgue outer measure; we denote ω_{N} := L^{N}(B). 𝕊^{d×N×N} is the set of third-order symmetric constant tensors, i.e., Λ ∈ 𝕊^{d×N×N} if its components Λ_{ijk} satisfy Λ_{ijk} = Λ_{ikj} for every i = 1, …, d and every j, k = 1, …, N.

We recall the definition of the space of functions of bounded variation in Ω with values in ℝ^{d},

BV (Ω; ℝ^{d}) := {u ∈ L^{1}(Ω; ℝ^{d}) : Du is a finite Radon measure},

where Du = (D_{i}u_{j})_{i=1,... ,N,j=1,...d} is the distributional derivative of u. For general BV space theory we refer to Braides [11], Evans and Gariepy [23], Ziemer [42]. We represent by ∇u the density of the absolutely continuous part of Du with respect to the Lebesgue measure (or Radon-Nikodym derivative), and S(u) is the jump set, i.e., the set of points x where the approximate upper limit u_{j}^{+}(x) is different from the approximate lower limit u_{j}^{−}(x) (see Evans and Gariepy [23], Section 5.9); the approximate upper limits are oriented with respect to a chosen normal ν_{u} to S(u). We set [u](x) := u^{+}(x) − u^{−}(x), the difference between the trace of u at x ∈ S(u), and we denote by C(u) the Cantor part of the measure Du. The following decomposition holds:

Du = ∇u L^{N}⌊Ω + D_{s}u = ∇u L^{N}⌊Ω + [u] ⊗ ν_{u} H^{N−1}⌊S(u) + C(u),  (2.1)

where D_{s}u denotes the singular part of the measure Du with respect to the Lebesgue measure. Following De Giorgi and Ambrosio [18], we define the space of special functions of bounded variation, SBV (Ω; ℝ^{d}), as the space of all functions u ∈ BV (Ω; ℝ^{d}) with C(u) = 0.

We now introduce the space of functions with bounded Hessian

BH(Ω; ℝ^{d}) := {u ∈ W^{1,1}(Ω; ℝ^{d}) : D^{2}u is a finite Radon measure}
= {u ∈ L^{1}(Ω; ℝ^{d}) : Du ∈ BV (Ω; ℝ^{d×N})},

where D^{2}u denotes the distributional Hessian of u. For various properties of the space BH, we refer to Demengel [20], [21], Carriero, Leaci and Tomarelli [13] and Temam [41]. We recall that if u ∈ BH(Ω; ℝ^{d}) then Du = ∇u and [u](x) = 0 for H^{N−1}-a.e. x ∈ Ω. Moreover,

Theorem 2.1 [Demengel [20], [21]] Let Ω ⊂ ℝ^{N} be a Lipschitz, bounded open set. Then

BH(Ω) ⊂ W^{1,p}(Ω)

with continuous embedding if p ≤ \frac{N}{N−1}; the embedding is compact if p < \frac{N}{N−1}.

We shall also have occasion to use the following two theorems.

Theorem 2.2 (Interpolation inequality) Let Ω ⊂ ℝ^{N} be a Lipschitz, bounded open set. Then for every ε > 0 there is a constant C = C(ε) such that

∥∇u∥_{L^{1}(Ω)} ≤ C∥u∥_{L^{1}(Ω)} + ε|D^{2}u|(Ω)

for all u ∈ BH(Ω).

Proof. The proof follows easily from Theorem 2.1 and a standard argument by contradiction. See, for instance, Lemma 4.2.2 of Ziemer [42], where the result is proven for Sobolev functions.

In view of (2.1), if u ∈ BH(Ω; ℝ^{d}) then

D^{2}u = ∇^{2}u L^{N}⌊Ω + [∇u] ⊗ ν_{∇u} H^{N−1}⌊S(∇u) + C(∇u),

where ∇^{2}u is the density of the absolutely continuous part of D^{2}u with respect to L^{N} and [∇u] = (∇u)^{+} − (∇u)^{−}. Since D^{2}u is a symmetric distribution, it follows that ∇^{2}u ∈ 𝕊^{d×N×N} and [∇u] = a ⊗ ν_{∇u} for some a ∈ L^{1}(Ω; ℝ^{d}). We now define for 1 < p < +∞

BH^{p}(Ω; ℝ^{d}) := {u ∈ BH(Ω; ℝ^{d}) : ∇^{2}u ∈ L^{p}(Ω; 𝕊^{d×N×N})}.

Theorem 2.3 If u ∈ BH(Ω; ℝ^{d}) then

lim_{ε→0^{+}} \frac{1}{ε}∫_{B(x,ε)} |∇u(y) − ∇u(x) − ∇^{2}u(x)(y − x)| dy = 0,

and

lim_{ε→0^{+}} \frac{1}{ε^{2}} ∫_{B(x,ε)} |u(y) − u(x) − ∇u(x)(y − x) − \frac{1}{2}∇^{2}u(x)(y − x, y − x)| dy = 0,

for L^{N} a.e. x ∈ Ω.

Proof. The first identity follows immediately by applying Theorem 1 of Section 6.1 of Evans and Gariepy [23] to ∇u, while the second identity can be proved with obvious extensions to second order derivatives of the arguments used in that theorem.

Next we recall a well known criteria for equi-integrability.

Theorem 2.4 [Dunford and Pettis] Let Ω ⊂ ℝ^{N} be a bounded, measurable set, and let {f_{j}} be a bounded sequence in L^{1}(Ω). The following statements are equivalent:

1. There exists a subsequence of {f_{j}} which is weakly converging in L^{1}(Ω);

2.

lim_{k→+∞} sup_{j} ∫_{Ω∩{|f_{j}|>k}} |f_{j}| dx = 0;

3. for all ε > 0 there exists δ > 0 such that

sup_{j} ∫_{E} |f_{j}| dx ≤ ε

for all measurable sets E ⊂ Ω with L^{N}(E) ≤ δ.

The lemma below was proved by Fonseca and Müller [29] using Young measures.

Theorem 2.5 Let 1 < p < +∞, let {u_{n}} be a bounded sequence in L^{p}(Ω; ℝ^{d}). For λ > 0 consider the truncation τ_{λ} : ℝ^{N} → ℝ^{d} given by

τ_{λ}(x) := \{ \begin{matrix}x & if |x| ≤ λ, \\ λ\frac{x}{|x|} & if |x| > λ. \end{matrix}

Then there exists a subsequence of {u_{n}} (not relabeled) and an increasing sequence λ_{n} → +∞ such that the truncated sequence {τ_{λ_{n}} ∘ u_{n}} is p-equi-integrable and

∥τ_{λ_{n}} ∘ u_{n} − u_{n}∥_{L^{q}(Ω)} → 0 for all 1 ≤ q < p.

Moreover, if ε_{n} > 0 is any decreasing sequence approaching zero, we may choose the sequence {λ_{n}} so that

lim_{n→+∞} λ_{n}ε_{n} = 0.

Meyers [36] extended the notion of quasiconvexity introduced by Morrey [37] to functions depending on derivatives of order higher than one. In particular for second order derivatives the definition reads as

Definition 2.6 Let U ⊂ ℝ^{N} be an open, bounded set with L^{N}(∂U) = 0. A function f : 𝕊^{d×N×N} → ℝ is said to be 2-quasiconvex if

∫_{U} f(Λ + ∇^{2}ψ) dx ≥ L^{N}(U)f(Λ)

for every constant Λ ∈ 𝕊^{d×N×N} and for every ψ ∈ W_{0}^{2,∞}(U; ℝ^{d}).

As in the case of first order gradients (see [17]), it can be shown that this definition does not depend on the choice of the open set U (see [36]).

3 Approximation of BH^{p}, p > 1, functions by W^{2,∞} functions

Throughout this section we will use ideas of Acerbi and Fusco [1] and Liu [34] (see also Ambrosio [5]).

If μ is a (nonnegative) Radon measure in ℝ^{N} we define the maximal function of μ by

M(μ)(x) := sup_{r>0} \frac{μ(B(x, r))}{ω_{N}r^{N}}, for x ∈ ℝ^{N}.

If the measure μ = u L^{N}, with u ≥ 0, then we simply write M(u) in place of M(uL^{N}). It is well known that if u ∈ L^{p}(ℝ^{N}) for p > 1 then

∥M(u)∥_{L^{p}(ℝ^{N})} ≤ C∥u∥_{L^{p}(ℝ^{N})},  (3.1)

for some constant C which does not depend of u. Hereafter for u ∈ BH(ℝ^{N}) and for λ > 0, we set

H_{λ} := {x : M(|D^{2}u|)(x) ≤ 2λ}.

The following lemmas are extensions of classical results (see [5] and [40]).

Lemma 3.1 Let u ∈ BH^{p}(ℝ^{N}), for p > 1, and let A be a measurable set in ℝ^{N}. Then there exists a constant C, which depends only on N, such that

L^{N}(A\H_{λ}) ≤ ∫_{A∩\{|M(∇^{2}u)|>λ\}} \frac{1}{λ^{p}} M(|∇^{2}u|)^{p} dx + \frac{C}{λ} |D_{s}^{2}u|(ℝ^{N}).

Proof. For every x ∈ {M(|D_{s}^{2}u|) > λ} there exists a radius r_{x} > 0 such that |D_{s}^{2}u|(B(x, r_{x})) ≥ λω_{N}r_{x}^{N}. Hence, by Vitali’s Covering Theorem we may find a countable family of mutually disjoint closed balls \{\overline{B}(x_{i}, r_{i})\} ⊂ \{\overline{B}(x, r_{x}) : x ∈ \{M(|D_{s}^{2}u|) > λ\}\} such that \{x : M(|D_{s}^{2}u|)(x) > λ\} ⊂ ∪_{i}\overline{B}(x_{i}, 5r_{i}). Thus

λL^{N}({x : M(|D_{s}^{2}u|)(x) ≥ λ}) ≤ λ ∑_{i} L^{N}(B(x_{i}, 5r_{i})) = 5^{N} ∑_{i} λω_{N}r_{i}^{N}
≤ 5^{N}ω_{N} ∑_{i} |D_{s}^{2}u| (B(x_{i}, r_{i})) ≤ 5^{N}ω_{N}|D_{s}^{2}u|(ℝ^{N}). (3.2)

Since |D^{2}u| = |∇^{2}u|L^{N} + |D_{s}^{2}u| we have that M(|D^{2}u|) ≤ M(|∇^{2}u|) + M(|D_{s}^{2}u|), and thus

A ∩ {M(|D^{2}u|) > 2λ} ⊂ (A ∩ {M(|∇^{2}u|) > λ}) ∪ {M(|D_{s}^{2}u|) > λ}.

Hence, by (3.2) we find

L^{N}(A\H_{λ}) ≤ L^{N}(A ∩ {M(|∇^{2}u|) > λ}) + L^{N}({M(|D_{s}^{2}u|) > λ})
≤ ∫_{A∩\{M(|∇^{2}u|)>λ\}} \frac{1}{λ^{p}} M(|∇^{2}u|)^{p} dx + \frac{5^{N}ω_{N}}{λ} |D_{s}^{2}u|(ℝ^{N}).

Lemma 3.2 Let u ∈ BH(ℝ^{N}) be given. There exist a constant K, depending only on N, and a set E = E(u) with L^{N}(E) = 0 such that if λ > 0 then

\frac{|u(x) − u(y) − ∇u(y)(x − y)|}{|x − y|^{2}} ≤ Kλ

and

\frac{|∇u(x) − ∇u(y)|}{|x − y|} ≤ Kλ

for every x, y ∈ H_{λ}\E with x ≠ y.

Proof. Fix λ > 0 and let η be the standard symmetric mollifier. Let E be the complement in Ω of the intersection of the set of Lebesgue points of u and ∇u, i.e., E := Ω\(L_{u} ∩ L_{∇u}), where L_{u} and L_{∇u} are the set of Lebesgue points of u and ∇u, respectively. Define η_{ε} := η(x/ε)/ε^{N} and u_{ε} := η_{ε} ∗ u. Applying the identity for C^{∞} functions

g(1) − g(0) − g′(0) = ∫_{0}^{1} ∫_{0}^{s} g′′(t) dt ds

to g(t) := u_{ε}(y + t(x − y)), we find

u_{ε}(x) − u_{ε}(y) − ∇u_{ε}(y)(x − y) = ∫_{0}^{1} ∫_{0}^{s} ∇^{2}u_{ε}(y + t(x − y))(x − y) · (x − y) dt ds,

from which we conclude that

\frac{|u_{ε}(x) − u_{ε}(y) − ∇u_{ε}(y)(x − y)|}{|x − y|^{2}} ≤ ∫_{0}^{1} ∫_{0}^{s} |∇^{2}u_{ε}|(y + t(x − y)) dt ds.

Now integrating and applying Fubini’s theorem we find

∫_{B(y,r)} \frac{|u_{ε}(x) − u_{ε}(y) − ∇u_{ε}(y)(x − y)|}{|x − y|^{2}} dx ≤ ∫_{0}^{1} ∫_{0}^{s} ∫_{B(y,r)}|∇^{2}u_{ε}|(y + t(x − y)) dx dt ds

= ∫_{0}^{1} ∫_{0}^{s} ∫_{B(y,tr)}|∇^{2}u_{ε}|(x) dx dt ds.

Note that from the inequality ∫_{B(y,s)} |∇^{2}u_{ε}| dx ≤ |D^{2}u|(B(y, s + ε)) it follows that

lim sup_{ε→0+} ∫_{B(y,s)} |∇^{2}u_{ε}| dx ≤ |D^{2}u|(\overline{B}(y, s)).

In view of the lower semicontinuity of the variation we deduce that

|D^{2}u|(B(y, s)) ≤ lim inf_{ε→0+} ∫_{B(y,s)} |∇^{2}u_{ε}| dx

and hence ∫_{B(y,s)} |∇^{2}u_{ε}| dx → |D^{2}u|(B(y, s)) for every s > 0 such that |D^{2}u|(∂B(y, s)) = 0. Therefore, letting ε go to zero and applying Dominated Convergence Theorem we obtain

∫_{B(y,r)} \frac{|u(x) − u(y) − ∇u(y)(x − y)|}{|x − y|^{2}} dx ≤ ∫_{0}^{1} ∫_{0}^{s} \frac{|D^{2}u|(B(y, tr))}{L^{N}(B(y, tr))} dt ds

for every y ∉ E, where we have used the fact that u_{ε} → u in W^{1,1}(ℝ^{N}). Similarly, it can be shown that

∫_{B(y,r)} \frac{|∇u(x) − ∇u(y)|}{|x − y|} dx ≤ ∫_{0}^{1} \frac{|D^{2}u|(B(y, tr))}{L^{N}(B(y, tr))} dt if y ∉ E.

For every positive number k, define

S_{k,r}(y) := \{x ∈ B(y, r) : \frac{|u(x) − u(y) − ∇u(y)(x − y)|}{|x − y|^{2}} ≥ kλ\} ,

and

T_{k,r}(y) := \{x ∈ B(y, r) : \frac{|∇u(x) − ∇u(y)|}{|x − y|} ≥ kλ\} .

If y ∈ H_{λ}\E then we have

kλ\frac{L^{N}(S_{k,r}(y))}{L^{N}(B(y, r))} ≤ \frac{1}{L^{N}(B(y, r))} ∫_{S_{k,r}(y)} \frac{|u(x) − u(y) − ∇u(y)(x − y)|}{|x − y|^{2}} dx
≤ ∫_{B(y,r)} \frac{|u(x) − u(y) − ∇u(y)(x − y)|}{|x − y|^{2}} dx

≤ ∫_{0}^{1} ∫_{0}^{s} \frac{|D^{2}u|(B(y, tr))}{L^{N}(B(y, tr))} dt ds ≤ ∫_{0}^{1} ∫_{0}^{s} 2λ dt ds = λ,

and so

L^{N}(S_{k,r}(y)) ≤ \frac{1}{k} L^{N}(B(y, r)).

Similarly, for y ∈ H_{λ}\E we can show that

L^{N}(T_{k,r}(y)) ≤ \frac{2}{k} L^{N}(B(y, r)).

Denote by γ_{N} the measure of the intersection of two balls of radius 1 in ℝ^{N} whose centers are also at distance 1. Fix x, y ∈ H_{λ}\E and set r := |y − x|. Choose k := 8ω_{N}/γ_{N}, so that

L^{N}(S_{k,r}(x) ∪ S_{k,r}(y) ∪ T_{k,r}(x) ∪ T_{k,r}(y)) ≤ \frac{6}{k} ω_{N}r^{N} = \frac{3}{4}γ_{N}r^{N} < L^{N}(B(x, r) ∩ B(y, r)).

We may, therefore, choose a z ∈ (B(x, r) ∩ B(y, r))\(S_{k,r}(x) ∪ S_{k,r}(y) ∪ T_{k,r}(x) ∪ T_{k,r}(y)), and conclude that

\frac{|u(z) − u(y) − ∇u(y)(z − y)|}{|z − y|^{2}} ≤ kλ,  \frac{|u(z) − u(x) − ∇u(x)(z − x)|}{|z − x|^{2}} ≤ kλ,
\frac{|∇u(z) − ∇u(y)|}{|z − y|} ≤ kλ,  \frac{|∇u(z) − ∇u(x)|}{|z − x|} ≤ kλ.

Whence, by the triangle inequality we have

|∇u(x) − ∇u(y)| ≤ |∇u(z) − ∇u(y)| + |∇u(z) − ∇u(x)| ≤ 2kλ|x − y|,  (3.3)

and

|u(x) − u(y) − ∇u(y)(x − y)| ≤ |u(z) − u(y) − ∇u(y)(z − y)|
+ |u(z) − u(x) − ∇u(x)(z − x)|
+ |∇u(y)(z − y) − ∇u(x)(z − x) − ∇u(y)(x − y)|
≤ 2kλ|x − y|^{2} + |∇u(y) − ∇u(x)||z − x| ≤ 4kλ|x − y|^{2},

where we have used (3.3).

Definition 3.3 [Stein [40] p. 176] Let F ⊂ ℝ^{N}. We say that a function u : F → ℝ belongs to Lip(2, F) if there exists a function U : F → ℝ^{N} and a constant K > 0 such that

|u(x)| ≤ K, |U(x)| ≤ K,  (3.4)

and

\frac{|U(x) − U(y)|}{|x − y|} ≤ K, \frac{|u(x) − u(y) − U(y)(x − y)|}{|x − y|^{2}} ≤ K,  (3.5)

for every x, y ∈ F with x ≠ y.

Note that, unless F is open, a function u does not determine uniquely the function U and so when we speak of an element of Lip(2, F) we refer to the couple (u, U). Also the smallest constant K > 0 for which (3.4) and (3.5) hold can be taken as norm in the space Lip(2, F) and will be denoted

∥u∥_{Lip(2,F)} .

It can be shown that with this norm the space Lip(2, F) becomes a Banach space. Moreover, if F is open then the space Lip(2, F) can actually be identified with the Sobolev space W^{2,∞} (F).

We now state and prove an extension theorem of Whitney type.

Theorem 3.4 Let F ⊂ ℝ^{N} be a Lebesgue measurable set. Then for every u ∈ Lip(2, F) there exists v ∈ W^{2,∞}(ℝ^{N}) such that v(x) = u(x) for L^{N} a.e. x ∈ F and

∥v∥_{W^{2,∞}(ℝ^{N})} ≤ C∥u∥_{Lip(2,F)},

where C > 0 depends only on N.

Proof. In view of the inner regularity of the Lebesgue measure, we may find a sequence of closed sets {F_{n}} such that F_{n} ⊂ F_{n+1} ⊂ F and

L^{N} (F \ ⋃_{n=1}^{∞} F_{n}) = 0.

By Theorem 4 of Section 2.3 of Chapter VI in [40] for each n ∈ ℕ there exists a continuous operator E_{n} of Lip(2, F_{n}) into W^{2,∞}(ℝ^{N}) such that for every u ∈ Lip(2, F_{n}) we have E_{n}(u) = u on F_{n}. The norm of the operator E_{n} has a bound independent of F_{n}.

Fix u ∈ Lip(2, F) and for each n ∈ ℕ let v_{n} := E_{n} (u|_{F_{n}}). Then

∥v_{n}∥_{W^{2,∞}(ℝ^{N})} ≤ ∥E_{n}∥ ∥u|_{F_{n}}∥_{Lip(2,F_{n})} ≤ C∥u∥_{Lip(2,F)}.

Thus we may extract a subsequence (not relabelled) of {v_{n}} such that v_{n} ⇀^{∗} v in W^{2,∞}(ℝ^{N}) and v_{n} converges to v pointwise almost everywhere in ℝ^{N}. Without loss of generality, we may assume that v_{n} converges to v pointwise everywhere in ℝ^{N}. For any fixed

x ∈ ⋃_{n=1}^{∞} F_{n}

we may find n_{x} ∈ ℕ such that x ∈ F_{n} for all n ≥ n_{x}. Since v_{n}(x) = u(x) for all n ≥ n_{x} by letting n → ∞ we obtain that

v(x) = u(x) for all x ∈ ⋃_{n=1}^{∞} F_{n}

and hence almost everywhere in F. Moreover, by the lower semicontinuity of the norm

∥v∥_{W^{2,∞}(ℝ^{N})} ≤ lim inf_{n→∞} ∥v_{n}∥_{W^{2,∞}(ℝ^{N})} ≤ C∥u∥_{Lip(2,F)}.

This completes the proof.

We now state and prove an approximation theorem.

Theorem 3.5 Let p > 1 and consider a function u ∈ BH^{p}(ℝ^{N}; ℝ^{d}), with supp u ⊂ B(0, R) for some R > 0. Then there exists a constant α > 0 depending only on N, such that for every

λ > max \{1, \frac{α}{R^{N}} (∫_{ℝ^{N}} |∇^{2}u|^{p} dx + |D_{s}^{2}u|(ℝ^{N}))\}

there exists a function v_{λ} ∈ W^{2,∞}(ℝ^{N}; ℝ^{d}) such that

∥v_{λ}∥_{W^{2,∞}(ℝ^{N};ℝ^{d})} ≤ Cλ, u(x) = v_{λ}(x) for L^{N} a.e. x ∈ H_{λ}

where the constant C > 0 depends only on R and N.

Proof. By arguing component by component, we may reduce to the scalar-valued case where u ∈ BH(ℝ^{N}; ℝ). Let A_{R} := B(0, 2R)\B(0, R). By Lemma 3.1 and (3.1), for every λ > 1 we have that

L^{N}(A_{R}\H_{λ}) ≤ ∫_{ℝ^{N}} \frac{1}{λ^{p}} M(|∇^{2}u|)^{p} dx + \frac{C}{λ} |D_{s}^{2}u|(ℝ^{N})

≤ \frac{C}{λ} (∫_{ℝ^{N}} |∇^{2}u|^{p} dx + |D_{s}^{2}u|(ℝ^{N}))

< L^{N}(A_{R})

where C = C(N), provided

λ > \frac{C}{L^{N}(A_{R})} (∫_{ℝ^{N}} |∇^{2}u|^{p} dx + |D_{s}^{2}u|(ℝ^{N})) .  (3.6)

Fix any λ > 1 which satisfies (3.6). Then L^{N}(A_{R} ∩ H_{λ}) > 0, and by Lemma 3.2 we may find y ∈ (A_{R} ∩ H_{λ})\E(u) and K = K(N) such that

\frac{|u(x) − u(y) − ∇u(y)(x − y)|}{|x − y|^{2}} ≤ Kλ, \frac{|∇u(x) − ∇u(y)|}{|x − y|} ≤ Kλ

for every x ∈ H_{λ}\E(u) with x ≠ y. Since supp u ⊂ B(0, R) we have that u(y) = 0 and ∇u(y) = 0, and thus

|u(x)| ≤ Kλ|x − y|^{2} ≤ 9KR^{2}λ, |∇u(x)| ≤ Kλ |x − y| ≤ 3KRλ

for every x ∈ (H_{λ}\E)∩B(0, R). As u ≡ 0 outside B(0; R) the previous inequalities actually hold in H_{λ}\E. Again by Lemma 3.2, and in view of Definition 3.3, this implies that u ∈ Lip(2, H_{λ}\E(u)) and thus the desired result follows from Theorem 3.4 and Lemma 3.1.

Theorem 3.6 Let Ω be an open, bounded subset in ℝ^{N} with smooth boundary, and let {u_{n}} ⊂ BH^{p}(Ω; ℝ^{d}), p > 1, be a sequence converging in W^{1,1}(Ω; ℝ^{d}) to some function u, and such that

sup_{n} ∫_{Ω} |∇^{2}u_{n}|^{p} dx < +∞

and

|D_{s}^{2}u_{n}|(Ω) → 0 as n → +∞.  (3.7)

Then there exists a subsequence {u_{n_{k}}} and a sequence {v_{k}} ⊂ W^{2,∞}(Ω; ℝ^{d}) such that {v_{k}} converges to u weakly in W^{2,p}(Ω; ℝ^{d}), {|∇^{2}v_{k}|^{p}} is equi-integrable, and

L^{N}({x ∈ Ω : v_{k}(x) ≠ u_{n_{k}}(x)}) → 0 as k → +∞.

If, in addition, u ∈ W^{2,∞}(Ω; ℝ^{d}) then v_{k} = u in a neighborhood of ∂Ω. Moreover, if ε_{k} ↘ 0, as k → +∞, it is possible to construct {v_{k}} so that ∥v_{k}∥_{W^{1,∞}(ℝ^{N};ℝ^{d})} ≤ CΛ_{k} for some sequence Λ_{k} ↗ +∞ such that Λ_{k}ε_{k} → 0 as k → +∞.

Proof. Let Ω ⊂⊂ B(0, R), for some R > 0. Since Ω has smooth boundary we can extend the functions u_{n} in such a way that the extensions U_{n} ∈ BH^{p}(ℝ^{N}; ℝ^{d}) have all compact support supp U_{n} ⊂ B(0, R), and (see [13])

∫_{ℝ^{N}} (|U_{n}|+|∇U_{n}|+|∇^{2}U_{n}|^{p}) dx+|D_{s}^{2}U_{n}|(ℝ^{N}) ≤ C(N, d) ∫_{Ω}(|u_{n}|+|∇u_{n}|+|∇^{2}u_{n}|^{p}) dx+|D_{s}^{2}u_{n}|(Ω),

|D_{s}^{2}U_{n}|(ℝ^{N}) → 0 as n → +∞.

Since p > 1, by (3.1) we have that

sup_{n} ∫_{ℝ^{N}} M(|∇^{2}U_{n}|)^{p} dx ≤ C sup_{n} ∫_{ℝ^{N}} |∇^{2}U_{n}|^{p} dx < +∞,  (3.8)

and thus if {ε_{n}} is a sequence of real numbers such that ε_{n} ↘ 0, by Theorem 2.5 we may find a sequence λ_{n} ↗ +∞ and a subsequence (not relabeled) of {U_{n}} such that

λ_{n}^{p−1}|D_{s}^{2}U_{n}|(ℝ^{N}) → 0, λ_{n}^{2}ε_{n} → 0,

as n → +∞, and {τ_{λ_{n}} ∘ M(|∇^{2}U_{n}|)} is p-equi-integrable. Moreover, without loss of generality, we may assume that for every n ∈ ℕ

λ_{n} > max \{1, \frac{α}{R^{N}} sup_{j} (∫_{ℝ^{N}} |∇^{2}U_{j}|^{p} dx + |D_{s}^{2}U_{j}|(ℝ^{N}))\} ,

and so by Theorem 3.5 and Lemma 3.1 for every n there exists a w_{n} ∈ W^{2,∞}(ℝ^{N}; ℝ^{d}) such that

∥w_{n}∥_{W^{2,∞}(ℝ^{N};ℝ^{d})} ≤ Cλ_{n}, U_{n}(x) = w_{n}(x) for L^{N} a.e. x ∈ H_{n},

and

L^{N}(ℝ^{N} \ H_{n}) ≤ ∫_{\{M(|∇^{2}U_{n}|)>λ_{n}\}} \frac{1}{λ_{n}^{p}} M(|∇^{2}U_{n}|)^{p} dx + \frac{C}{λ_{n}} |D_{s}^{2}U_{n}|(ℝ^{N}),  (3.9)

where

H_{n} := {x ∈ ℝ^{N} : M(|D^{2}U_{n}|) ≤ 2λ_{n}}

and C = C(N, R). Since ∥w_{n}∥_{W^{2,∞}(ℝ^{N};ℝ^{d})} ≤ Cλ_{n}, on {M(|∇^{2}U_{n}|) ≥ λ_{n}} we have

|∇^{2}w_{n}| ≤ Cλ_{n} ≤ C τ_{λ_{n}} ∘ M(|∇^{2}U_{n}|),

while on H_{n} ∩ {M(|∇^{2}U_{n}|) < λ_{n}}

|∇^{2}w_{n}| = |∇^{2}U_{n}| ≤ M(|∇^{2}U_{n}|) = τ_{λ_{n}} ∘ M(|∇^{2}U_{n}|)|.

Hence to prove the equi-integrability of {|∇^{2}w_{n}|^{p}} it remains to study the sequence on the set {M(|∇^{2}U_{n}|) < λ_{n}}\H_{n}. By (3.2) we find

∫_{{M(|∇^{2}U_{n}|)<λ_{n}}\H_{n}} |∇^{2}w_{n}|^{p} dx ≤ Cλ_{n}^{p}L^{N}({M(|∇^{2}U_{n}|) < λ_{n}}\H_{n})
≤ Cλ_{n}^{p}L^{N}({M(|D_{s}^{2}U_{n}|) ≥ λ_{n}}
≤ Cλ_{n}^{p−1}|D_{s}^{2}U_{n}|(ℝ^{N}) → 0.

Thus {|∇^{2}w_{n}|^{p}} is equi-integrable. We now show that {w_{n}} converges weakly in W^{2,p}(Ω; ℝ^{d}) to u. By (3.7), (3.8) and (3.9) we have

lim_{n→∞} λ_{n}L^{N} ({x ∈ ℝ^{N} : w_{n}(x) ≠ U_{n}(x)}) = 0,  (3.10)

and so

∫_{Ω} |w_{n}| dx = ∫_{{w_{n}=u_{n}}} |u_{n}| dx + ∫_{{w_{n}≠u_{n}}} |w_{n}| dx ≤ ∫_{Ω} |u_{n}| dx + Cλ_{n}L^{N}(Ω ∩ {u_{n} ≠ w_{n}}) ≤ C,

and since ∇w_{n} = ∇u_{n} L^{N} a.e. on Ω∩{w_{n} = u_{n}}, we deduce, in a similar manner, that also {∇w_{n}} is bounded in L^{1}(Ω, ℝ^{d×N}). We then have that {w_{n}} is bounded in W^{2,p}(Ω; ℝ^{d}) and therefore, up to a subsequence, it converges weakly in W^{2,p}(Ω; ℝ^{d}) to some function w ∈ W^{2,p}(Ω; ℝ^{d}). In view of (3.10) we conclude that w = u L^{N} a.e. on Ω.

Finally, if u ∈ W^{2,∞}(Ω; ℝ^{d}), in order to match the boundary conditions define

r_{n} := max \{(∥w_{n} − u∥_{W^{1,p}(Ω;ℝ^{d×N})})^{\frac{1}{3}} , 1/λ_{n}\}

and note that r_{n} → 0 as n → +∞. Let

A_{n} := {x ∈ Ω : dist(x, ∂Ω) ≤ r_{n}},

and

B_{n} := {x ∈ Ω\A_{n} : dist(x, ∂A_{n}) ≤ r_{n}}.

For every n, let ϕ_{n} ∈ C_{c}^{∞}(Ω), 0 ≤ ϕ_{n} ≤ 1, ϕ_{n} = 1 on Ω\(A_{n} ∪ B_{n}), ϕ_{n} = 0 in A_{n},

∥∇ϕ_{n}∥_{L^{∞}(Ω)} ≤ \frac{2}{r_{n}} and ∥∇^{2}ϕ_{n}∥_{L^{∞}(Ω)} ≤ \frac{4}{r_{n}^{2}} .

Define

v_{n} := ϕ_{n}w_{n} + (1 − ϕ_{n})u.

Clearly v_{n} = u in a neighborhood of ∂Ω, and for n large enough

∥v_{n}∥_{L^{∞}(Ω)} ≤ ∥w_{n}∥_{L^{∞}(Ω)} + ∥u∥_{L^{∞}(Ω)} ≤ Cλ_{n},
∥∇v_{n}∥_{L^{∞}(Ω)} ≤ ∥∇ϕ_{n}∥_{L^{∞}(Ω)}∥w_{n} − u∥_{L^{∞}(Ω)} + ∥∇w_{n} − ∇u∥_{L^{∞}(Ω)} ≤ Cλ_{n}^{2} + Cλ_{n} ≤ Cλ_{n}^{2}.

Moreover

∥v_{n} − u∥_{L^{p}(Ω)} ≤ ∥w_{n} − u∥_{L^{p}(Ω)} → 0,
∥∇v_{n} − ∇u∥_{L^{p}(Ω)} ≤ ∥∇ϕ_{n}∥_{L^{∞}(Ω)}∥w_{n} − u∥_{L^{p}(Ω)} + ∥∇w_{n} − ∇u∥_{L^{p}(Ω)}
≤ C (∥w_{n} − u∥_{L^{p}(Ω)})^{2/3} + ∥∇w_{n} − ∇u∥_{L^{p}(Ω)} → 0,
∥∇^{2}v_{n} − ∇^{2}w_{n}∥_{L^{p}(Ω)} ≤ ∥∇^{2}ϕ_{n}∥_{L^{∞}(Ω)}∥w_{n} − u∥_{L^{p}(Ω)} + C∥∇ϕ_{n}∥_{L^{∞}(Ω)}∥∇w_{n} − ∇u∥_{L^{p}(Ω)}
+ (∫_{A_{n}∪B_{n}} |∇^{2}w_{n} − ∇^{2}u|^{p} dx)^{1/p}

≤ C (∥w_{n} − u∥_{W^{1,p}(Ω)})^{1/3} + (∫_{A_{n}∪B_{n}} |∇^{2}w_{n} − ∇^{2}u|^{p} dx)^{1/p} → 0,

where we have used the equi-integrability of {|∇^{2}w_{n}|^{p}}. Hence {v_{n}} is the sequence we were looking for with Λ_{n} := λ_{n}^{2}.

4 Lower semicontinuity in BH^{p}, p > 1, for 2-quasiconvex integrands

In this Section we prove Theorem 1.2.

The following Proposition may be found in [10]. For the convenience of the reader we present its proof.

Proposition 4.1 Let f : Ω × ℝ^{d} × ℝ^{d×N} × 𝕊^{d×N×N} → [0, +∞) be a normal integrand. Then for each ε_{0} > 0 there exists a compact set K ⊂ Ω such that |Ω \ K| ≤ ε_{0} and f : K × ℝ^{d} × ℝ^{d×N} × 𝕊^{d×N×N} → [0, +∞) is lower semicontinuous. Moreover, if f is continuous in the variable Λ, then for each (x_{0}, u_{0}, ξ_{0}) ∈ K×ℝ^{d}×ℝ^{d×N}, ε > 0, and L > 0 there exists δ = δ(x_{0}, u_{0}, ξ_{0}, ε, K, L) ∈ (0, 1) such that

f(x_{0}, u_{0}, ξ_{0}, Λ) ≤ f(x, u, ξ, Λ) + ε

for all (x, u, ξ) ∈ K × ℝ^{d} × ℝ^{d×N}, with |x − x_{0}| + |u − u_{0}| + |ξ − ξ_{0}| ≤ δ and all |Λ| ≤ L.

Proof. The first part of the Proposition follows from Theorem 1.1 in Chapter 8 of [22]. To prove the second claim assume, for contradiction, that there exist (x_{0}, u_{0}, ξ_{0}) ∈ K × ℝ^{d} × ℝ^{d×N}, L > 0, ε̄ > 0, and a sequence

\{(x_{n}, u_{n}, ξ_{n}, Λ_{n})\} ⊂ K × \overline{B_{d}(u_{0}, 1)} × \overline{B_{d×N}(ξ_{0}, 1)} × \overline{B_{d×N×N}(0, L)},

such that

ε̄+ f(x_{n}, u_{n}, ξ_{n}, Λ_{n}) < f(x_{0}, u_{0}, ξ_{0}, Λ_{n})  (4.1)

and (x_{n}, u_{n}, ξ_{n}, Λ_{n}) → (x_{0}, u_{0}, ξ_{0}, Λ_{0}) as n → ∞, for some Λ_{0} ∈ \overline{B_{d×N×N}(0, L)}. Since the function f(x_{0}, u_{0}, ξ_{0}, ·) is continuous and f restricted to K × ℝ × ℝ^{d×N} × 𝕊^{d×N×N} is lower semicontinuous, for any ε < \frac{1}{2} ε̄ there exists δ > 0 such that

|f(x_{0}, u_{0}, ξ_{0}, Λ) − f(x_{0}, u_{0}, ξ_{0}, Λ_{0})| ≤ ε, f(x_{0}, u_{0}, ξ_{0}, Λ_{0}) ≤ f(x, u, ξ, Λ) + ε

for all (x, u, ξ, Λ) ∈ K × \overline{B_{d}(u_{0}, 1)} × \overline{B_{d×N}(ξ_{0}, 1)} × \overline{B_{d×N×N}(0, L)} with

|x − x_{0}| + |u − u_{0}| + |ξ − ξ_{0}| + |Λ − Λ_{0}| ≤ δ.

Thus for all n sufficiently large, also by (4.1), we have

ε̄+ f(x_{n}, u_{n}, ξ_{n}, Λ_{n}) < f(x_{0}, u_{0}, ξ_{0}, Λ_{n}) ≤ f(x_{0}, u_{0}, ξ_{0}, Λ_{0}) + ε ≤ f(x_{n}, u_{n}, ξ_{n}, Λ_{n}) + 2ε

which is a contradiction.

Proof of Theorem 1.2. Without loss of generality we may assume that

lim inf_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}, ∇^{2}u_{n}) dx = lim_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}, ∇^{2}u_{n}) dx < ∞.

Passing to a subsequence, if necessary, there exists a nonnegative Radon measure μ such that

f(x, u_{n}(x), ∇u_{n}(x), ∇^{2}u_{n} (x)) L^{N}⌊ Ω ⇀^{∗} μ

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

\frac{dμ}{dL^{N}} (x_{0}) ≥ f(x_{0}, u(x_{0}), ∇u(x_{0}), ∇^{2}u (x_{0}))  for L^{N} a.e. x_{0} ∈ Ω.  (4.2)

If (4.2) holds, then the conclusion of the theorem follows immediately. Indeed, let φ ∈ C_{c}(Ω; ℝ), 0 ≤ φ ≤ 1. We have

lim inf_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}, ∇^{2}u_{n}) dx ≥ lim inf_{n→∞} ∫_{Ω} φ f(x, u_{n}, ∇u_{n}, ∇^{2}u_{n}) dx
= ∫_{Ω} φ dμ ≥ ∫_{Ω} φ \frac{dμ}{dL^{N}} dx ≥ ∫_{Ω} φ f(x, u, ∇u, ∇^{2}u) dx.

By letting φ → 1, and using Lebesgue Monotone Convergence Theorem, we obtain the desired result. To show (4.2) for each j ∈ ℕ we apply Proposition 4.1 to obtain a compact set K_{j} ⊂ Ω, with |Ω \ K_{j}| ≤ 1/j, such that

f : K_{j} × ℝ^{N} × ℝ^{d×N} × 𝕊^{d×N×N} → [0, +∞)

is lower semicontinuous. Let K_{j}^{∗} be the set of Lebesgue points of χ_{K_{j}}, and set

ω := ⋃_{j}^{∞}(K_{j} ∩ K_{j}^{∗}).  (4.3)

Then

|Ω \ ω| ≤ |Ω \ K_{j}| ≤ \frac{1}{j} → 0 as j → ∞.

Fix x_{0} ∈ ω such that a (x_{0}, u (x_{0}) , ∇u (x_{0})) < ∞ and

\frac{dμ}{dL^{N}} (x_{0}) = lim_{ε→0^{+}} \frac{μ(Q(x_{0}, ε))}{ε^{N}} < ∞,  (4.4)

lim_{ε→0^{+}} \frac{1}{ε^{N+1}} ∫_{Q(x_{0},ε)} |∇u(x) − ∇u(x_{0}) − ∇^{2}u(x_{0})(x − x_{0})|dx = 0,

lim_{ε→0^{+}} \frac{1}{ε^{N+2}} ∫_{Q(x_{0},ε)} |u(x) − u(x_{0}) − ∇u(x_{0})(x − x_{0}) − \frac{1}{2}∇^{2}u(x_{0})(x − x_{0}) · (x − x_{0})|dx = 0. (4.5)

Choosing ε_{k} ↘ 0 such that μ(∂Q(x_{0}, ε_{k})) = 0, then

\frac{dμ}{dL^{N}} (x_{0}) = lim_{k→∞} \frac{μ(Q(x_{0}, ε_{k}))}{ε_{k}^{N}}

= lim_{k→∞} lim_{n→∞} \frac{1}{ε_{k}^{N}} ∫_{Q(x_{0},ε_{k})} f(x, u_{n}, ∇u_{n}, ∇^{2}u_{n}) dx  (4.6)

= lim_{k→∞} lim_{n→∞} ∫_{Q}f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}^{2}w_{n,k}(y) + ε_{k}∇u(x_{0})y, ∇u(x_{0}) + ε_{k}∇w_{n,k}(y), ∇^{2}w_{n,k}(y)) dy,

where

w_{n,k}(y) := \frac{u_{n}(x_{0} + ε_{k}y) − u(x_{0}) − ε_{k}∇u(x_{0})y}{ε_{k}^{2}} .

Clearly w_{n,k} ∈ BH^{p}(Q; ℝ^{d}), and by (4.5), lim_{k→∞} lim_{n→∞} ||w_{n,k} − w_{0}||_{W^{1,1}(Q;ℝ^{d})} = 0, where

w_{0}(y) := ∇^{2}u(x_{0})y · y.

Moreover, by (1.5), (4.4), and (4.6) we have that

lim sup_{k→∞} lim sup_{n→∞} ∫_{Q} |∇^{2}w_{n,k}|^{p} dy < ∞,

while (1.6) ensures that

lim_{k→∞} lim_{n→∞} |D_{s}^{2}w_{n,k}| (\overline{Q}) = 0.  (4.7)

By a standard diagonalization argument, we may extract a subsequence {w_{k}} with w_{k} := w_{n_{k},k} which converges to w_{0} in W^{1,1}(Q; ℝ^{d}) and such that

sup_{k} ∫_{Q} |∇^{2}w_{k}|^{p} dy < ∞, lim_{k→∞} |D_{s}^{2}w_{k}| (\overline{Q}) = 0,

and

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim_{k→∞} ∫_{Q} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}^{2}w_{k}(y) + ε_{k}∇u(x_{0})y, ∇u(x_{0}) + ε_{k}∇w_{k}(y), ∇^{2}w_{k}(y)) dy.

By Theorem 3.6 we may find a subsequence (not relabeled) of {w_{k}} and a sequence {v_{k}} ⊂ W^{2,∞}(Q; ℝ^{d}) weakly converging to w_{0} in W^{2,p}(Q; ℝ^{d}) such that v_{k} = w_{0} on ∂Q,

|{y ∈ Q : v_{k}(y) ≠ w_{k}(y)}| → 0 as k → ∞  (4.8)

and {|∇^{2}v_{k}|^{p}} is equi-integrable. Hence

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{k→∞} ∫_{\{v_{k}=w_{k}\}} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}^{2}v_{k}(y) + ε_{k}∇u(x_{0})y, ∇u(x_{0}) + ε_{k}∇v_{k}(y), ∇^{2}v_{k}(y)) dy,

where we have used the fact that f ≥ 0. As x_{0} ∈ ω there exists an integer j_{0} such that x_{0} ∈ K_{j_{0}} ∩ K_{j_{0}}^{∗}. By Proposition 4.1 for any fixed integer j ∈ ℕ there exists 0 < ρ_{j} < 1 such that

f(x_{0}, u(x_{0}), ∇u(x_{0}), Λ) ≤ f(x, u, ξ, Λ) + \frac{1}{j}  (4.9)

for all (x, u, v) ∈ K_{j_{0}} ×\overline{B_{d}(u(x_{0}), 1)}×\overline{B_{d×N}(∇u(x_{0}), 1)}, with |x−x_{0}|+|u−u(x_{0})|+|ξ−∇u(x_{0})| ≤ ρ_{j} and all Λ ∈ 𝕊^{d×N×N} with |Λ| ≤ j. Set

E_{k,j} := \{y ∈ Q : v_{k}(y) = w_{k}(y), |ε_{k}^{2}v_{k}(y)| ≤ \frac{ρ_{j}}{2} , |ε_{k}∇v_{k}(y)| ≤ ρ_{j}, |∇^{2}v_{k}(y)| ≤ j\} .

Since {v_{k}} is bounded in W^{2,1}(Q; ℝ^{d}), and in view of (4.8), we have that

lim_{j→∞} lim_{k→∞} |Q \ E_{k,j}| = 0.  (4.10)

Thus

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim_{j→∞} lim_{k→∞} ∫_{E_{k,j}} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}^{2}v_{k}(y) + ε_{k}∇u(x_{0})y,

∇u(x_{0}) + ε_{k}∇v_{k}(y), ∇^{2}v_{k}(y)) dy,

= lim_{j→∞} lim_{k→∞} \frac{1}{ε_{k}^{N}} ∫_{D_{k,j}} f (x, u(x_{0}) + ε_{k}^{2}v_{k} (\frac{x − x_{0}}{ε_{k}} ) + ∇u(x_{0})(x − x_{0}),

∇u(x_{0}) + ε_{k}∇v_{k} (\frac{x − x_{0}}{ε_{k}} ) , ∇^{2}v_{k} (\frac{x − x_{0}}{ε_{k}} )) dx,

where D_{k,j} := x_{0} + ε_{k}E_{k,j}. By (4.9)

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{j→∞} lim inf_{k→∞} \frac{1}{ε_{k}^{N}} ∫_{K_{j_{0}} ∩D_{k,j}} f (x, u(x_{0}) + ε_{k}^{2}v_{k} (\frac{x − x_{0}}{ε_{k}} ) + ∇u(x_{0})(x − x_{0}),

∇u(x_{0}) + ε_{k}∇v_{k} (\frac{x − x_{0}}{ε_{k}} ) , ∇^{2}v_{k} (\frac{x − x_{0}}{ε_{k}} )) dx

≥ lim inf_{j→∞} lim inf_{k→∞} \frac{1}{ε_{k}^{N}} ∫_{K_{j_{0}} ∩D_{k,j}} f (x_{0}, u(x_{0}), ∇u(x_{0}), ∇^{2}v_{k} (\frac{x − x_{0}}{ε_{k}} )) dx,

(4.11)

where we used the fact that for fixed j and for k large enough |∇u(x_{0})(x − x_{0})| < ρ_{j}/2 for every x ∈ D_{k,j}. Since |∇^{2}v_{k}| ≤ j in E_{k,j}, and by virtue of the growth condition on f, for each fixed j we have

\frac{1}{ε_{k}^{N}} ∫_{D_{k,j}\K_{j_{0}}} f (x_{0}, u(x_{0}), ∇u(x_{0}), ∇^{2}v_{k} (\frac{x − x_{0}}{ε_{k}} )) dx

≤ a (x_{0}, u(x_{0}), ∇u(x_{0})) (1 + j^{p})\frac{|Q(x_{0}, ε_{k}) \ K_{j_{0}}|}{ε_{k}^{N}} → 0

as k → ∞, because x_{0} is a Lebesgue point of χ_{K_{j_{0}}} . Consequently, from (4.11) we get

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{j→∞} lim inf_{k→∞} \frac{1}{ε_{k}^{N}} ∫_{D_{k,j}} f (x_{0}, u(x_{0}), ∇u(x_{0}), ∇^{2}v_{k} (\frac{x − x_{0}}{ε_{k}} )) dx

= lim_{j→∞} lim_{k→∞} ∫_{E_{k,j}} f(x_{0}, u(x_{0}), ∇u(x_{0}), ∇^{2}v_{k}(y)) dy

= lim_{k→∞} ∫_{Q} f(x_{0}, u(x_{0}), ∇u(x_{0}), ∇^{2}v_{k}(y)) dy,

where we have used the growth condition on f, the p-equi-integrability of {v_{k}}, and (4.10). Since v_{k} = w_{0} on ∂Q it now follows from the 2-quasiconvexity of f that

\frac{dμ}{dL^{N}} (x_{0}) ≥ f(x_{0}, u(x_{0}), ∇u(x_{0}), ∇^{2}u(x_{0})).

Note that the assumption |D_{s}^{2}u_{n}|(Ω) → 0, as n → +∞, made in Theorem 1.2 was used only to derive (4.7). Indeed, Theorem 1.2 holds even if we require that |D_{s}^{2}u_{n}| ⇀^{∗} τ and τ ⊥ L^{N}, i.e., dτ/dL^{N} = 0, L^{N}-a.e. in Ω instead of |D_{s}^{2}u_{n}|(Ω) → 0. The small modifications needed in the previous proof are as follows: choose x_{0} and ε_{k} satisfying the further requirement that

\frac{dτ}{dL^{N}} (x_{0}) = lim_{k→+∞} \frac{τ(Q(x_{0}, ε_{k}))}{L^{N}(Q(x_{0}, ε_{k})} = 0, τ(∂Q(x_{0}, ε_{k})) = 0.

Then

lim_{k→+∞} lim_{n→+∞} |D_{s}^{2}v_{n,k}|(Q̄) = lim_{k→+∞} lim_{n→+∞} \frac{|D_{s}^{2}u_{n}|(\overline{Q(x_{0}, ε_{k})})}{ε_{k}^{N}}
= lim_{k→+∞} \frac{τ(Q(x_{0}, ε_{k}))}{ε_{k}^{N}} = \frac{dτ}{dL^{N}} (x_{0}) = 0.

We record this result below.

Theorem 4.2 Let Ω ⊂ ℝ^{N} be an open bounded set and let

f : Ω × ℝ^{N} × ℝ^{d×N} × 𝕊^{d×N×N} → [0, +∞)

be a normal integrand, 2−quasiconvex in Λ, and such that (1.5) holds. Then for every u ∈ W^{1,1}(Ω; ℝ^{d}) and any sequence {u_{n}} ⊂ BH(Ω; ℝ^{d}) converging to u in W^{1,1}(Ω; ℝ^{d}) and such that

|D_{s}^{2}u_{n}| ⇀^{∗} τ ⊥ L^{N}

we have

∫_{Ω} f(x, u, ∇u, ∇^{2}u) dx ≤ lim inf_{n→∞} ∫_{Ω} f(x, u_{n}, ∇u_{n}, ∇^{2}u_{n}) dx.

Remark 4.3 If Ω is a Lipschitz, bounded, open set then the results of Theorems 1.2, 4.2, hold even if we replace the assumption u_{n} → u in W^{1,1}(Ω, ℝ^{d}) with u_{n} → u in L^{1}(Ω, ℝ^{d}). Indeed, by the interpolation inequality (see Theorem 2.2) we have for every ε > 0

∥∇u_{n} − ∇u∥_{L^{1}(Ω)} ≤ C(ε)∥u_{n} − u∥_{L^{1}(Ω)} + ε|∇^{2}u_{n}L^{N} − D^{2}u|(Ω);

hence first letting n → +∞ and then letting ε → 0^{+} we obtain the convergence of the gradients.

5 Lower semicontinuity in BH for 2-quasiconvex integrands

In this Section we prove that when p = 1 we still have a lower semicontinuity result for integrands essentially of the type f = f(x, Λ). Precisely, let Ω ⊂ ℝ^{N} be an open, bounded domain, and let

f : Ω × ℝ^{d} × ℝ^{d×N} × 𝕊^{d×N×N} → [0, +∞)

satisfy (H1) f is Borel measurable; (H2) f(x, u, ξ, ·) is 2-quasiconvex; (H3) there exists a positive constant C such that

0 ≤ f(x, u, ξ, Λ) ≤ C(1 + |Λ|)

for L^{N} a.e. x ∈ Ω and all (u, ξ, Λ) ∈ ℝ^{d} × ℝ^{d×N} × 𝕊^{d×N×N};

(H4) for all (x_{0}, u_{0}, ξ_{0}) ∈ Ω × ℝ^{d} × ℝ^{d×N} and ε > 0 there exist δ_{0} > 0 and a modulus of continuity ρ, with ρ(s) ≤ C_{0}(1 + s) for s > 0 and for some C_{0} > 0, such that

f(x_{0}, u_{0}, ξ_{0}, Λ) − f(x, u, ξ, Λ) ≤ ε(1 + f(x, u, ξ, Λ)) + ρ(|(u, ξ) − (u_{0}, ξ_{0})|)  (5.1)

for all x ∈ Ω with |x − x_{0}| ≤ δ_{0}, and for all (u, ξ, Λ) ∈ ℝ^{d} × ℝ^{d×N} × 𝕊^{d×N×N}.

The result below may be found in Fonseca, Leoni, Malý and Paroni [25].

Proposition 5.1 Let f : 𝕊^{d×N×N} → [0, ∞) be a 2-quasiconvex function such that

0 ≤ f(Λ) ≤ C (1 + |Λ|)

for all Λ ∈ 𝕊^{d×N×N}. If {u_{n}} is a sequence of functions in W^{2,1}(B; ℝ^{d}) converging to 0 in W^{1,1}(Q; ℝ^{d}), and {∥∇^{2}u_{n}∥_{L^{1}(Ω,𝕊^{d×N×N})}} is bounded, then

ω_{N}f(0) ≤ lim inf_{n→∞} ∫_{B} f(∇^{2}u_{n}) dx.

The following lemma was proved by Guidorzi and Poggiolini [31] (see also [39])

Lemma 5.2 Let h : Ω → 𝕊^{d×N×N} be a 2-quasiconvex function. If there exists a constant C such that

|h(Λ)| ≤ C(1 + |Λ|)

then

|h(Λ_{1}) − h(Λ_{2})| ≤ C_{∞}|Λ_{1} − Λ_{2}|

for some constant C_{∞} = C_{∞}(N, d, C).

Theorem 5.3 Assume that the conditions (H1)-(H4) hold. If u_{n} ∈ BH(Ω, ℝ^{d}), u ∈ W^{1,1}(Ω, ℝ^{d}),

{∥∇^{2}u_{n}∥_{L^{1}(Ω)}} is bounded, |D_{s}^{2}u_{n}| ⇀^{∗} τ ⊥ L^{N}, and if u_{n} → u in W^{1,1}(Ω, ℝ^{d}), then

∫_{Ω} f(x, u(x), ∇u(x), ∇^{2}u(x)) dx ≤ lim inf_{n→+∞} ∫_{Ω} f(x, u_{n}(x), ∇u_{n}(x), ∇^{2}u_{n}(x)) dx.

Proof. We proceed as in the proof of Theorem 1.2 until the diagonalization argument after (4.7), precisely until we find a sequence {v_{k}} of functions with bounded Hessian such that

lim_{k→+∞} ∥v_{k} − v∥_{W^{1,1}(Q)} = 0,  sup_{k} ∥∇^{2}v_{k}∥_{L^{1}(Q)} < +∞,  lim_{k→+∞} |D_{s}^{2}v_{k}|(\overline{Q}) = 0

and

\frac{dμ}{dL^{N}} (x_{0}) = lim_{k→+∞} ∫_{Q} f(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}^{2}v_{k}(y) + ε_{k}∇u(x_{0})y, ∇u(x_{0}) + ε_{k}∇v_{k}(y), ∇^{2}v_{k}(y)) dy,

where v(y) := \frac{1}{2}∇^{2}u(x_{0})y · y. Using (5.1), for all ε > 0 we find

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim inf_{k→∞} ∫_{Q} f(x_{0}, u(x_{0}), ∇u(x_{0}), ∇^{2}v_{k}(y)) dy

− lim sup_{k→∞} ∫_{Q} ρ(|(ε_{k}∇u(x_{0})y + ε_{k}^{2}v_{k}(y), ε_{k}∇v_{k}(y))|) dy.

But (ε_{k}∇u(x_{0})y + ε_{k}^{2}v_{k}(y), ε_{k}∇v_{k}(y)) → 0 in L^{1}(Q, ℝ^{d+d×N}) and hence by Lebesgue Dominated Convergence Theorem we arrive at

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim_{k→∞} ∫_{Q} f(x_{0}, u(x_{0}), ∇u(x_{0}), ∇^{2}v_{k}(y)) dy.

Hereafter we denote by w_{k} := v_{k} − v and by h : 𝕊^{d×N×N} → ℝ the function defined by

h(Λ) := f(x_{0}, u(x_{0}), ∇u(x_{0}), Λ + ∇^{2}u(x_{0})) = f(x_{0}, u(x_{0}), ∇u(x_{0}), Λ + ∇^{2}v(y)),

so that w_{k} → 0 in W^{1,1}(Q, ℝ^{d}), sup_{k} ∥∇^{2}w_{k}∥_{L^{1}(Q)} < +∞,

lim_{k→+∞} |D_{s}^{2}w_{k}|(\overline{Q}) = 0  (5.2)

and

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim_{k→∞} ∫_{Q} h(∇^{2}w_{k}(y)) dy.  (5.3)

Let η be the standard symmetric mollifier and define η_{h}(x) := η(x/h)/h^{N} and w_{k,h} := η_{h} ∗ w_{k}. Then ∇^{2}w_{k,h} = η_{h} ∗ ∇^{2}w_{k} + η_{h} ∗ D_{s}^{2}w_{k}, and by using Lemma 5.2 we find

∫_{Q} h(∇^{2}w_{k,h}) dx ≤ ∫_{Q} h(η_{h} ∗ ∇^{2}w_{k}) dx + C_{∞}|D_{s}^{2}w_{k}|(Q(0, 1 + h)),

and by the Lebesgue Dominated Convergence Theorem we obtain

lim inf_{h→0} ∫_{Q} h(∇^{2}w_{k,h}) dx ≤ lim inf_{h→0} ∫_{Q} h(η_{h} ∗ ∇^{2}w_{k}) dx + C_{∞}|D_{s}^{2}w_{k}|(\overline{Q})
≤ ∫_{Q} h(∇^{2}w_{k}) dx + C_{∞}|D_{s}^{2}w_{k}|(\overline{Q}).

This, together with (5.2) and (5.3) yields

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim_{k→∞} ∫_{Q} h(∇^{2}w_{k}(y)) dy ≥ lim sup_{k→∞} lim inf_{h→0} ∫_{Q} h(∇^{2}w_{k,h}) dx.

Clearly lim_{k→∞} lim_{h→0} ∥w_{k,h}∥_{W^{1,1}(Q;ℝ^{d})} = 0, and

lim sup_{k→∞} lim_{h→0} ∥∇^{2}w_{k,h}∥_{L^{1}(Q;𝕊^{d×N×N})} ≤ lim sup_{k→∞} ∥∇^{2}w_{k}∥_{L^{1}(Q;𝕊^{d×N×N})} + lim sup_{k→∞} |D_{s}^{2}w_{k}|(Q̄) < +∞.

By a standard diagonalization argument we may then find a sequence z_{k} ∈ C^{∞}(Q; ℝ^{d}) such that z_{k} → 0 in W^{1,1}(Q, ℝ^{d}), sup_{k} ∥∇^{2}z_{k}∥_{L^{1}(Q)} < +∞, and

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim_{k→∞} ∫_{Q} h(∇^{2}z_{k}(y)) dy.

Hence applying Proposition 5.1 we arrive at

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ h(0)

and thence the theorem is proved by letting ε ↘ 0.

6 Characterizations of the 2-quasiconvex envelope

Throughout this section f : 𝕊^{d×N×N} → ℝ, denotes a continuous function satisfying

\frac{1}{C} |Λ|^{p} ≤ f(Λ) ≤ C(1 + |Λ|^{p})

for every Λ ∈ 𝕊^{d×N×N} and some constant C > 0 and p ≥ 1. We define

Q_{2}f(F) := inf {∫_{Q} f(F + ∇^{2}ψ) dx : ψ ∈ W_{0}^{2,p}(Q; ℝ^{d})}

where F ∈ 𝕊^{d×N×N}. It may be shown that the function Q_{2}f is 2-quasiconvex (see [29]) and that (see [12])

Proposition 6.1

Q_{2}f(F) = inf{lim inf_{n→+∞} ∫_{Q} f(∇^{2}u_{n}) dx : u_{n} ∈ W^{2,p}(Q; ℝ^{d})

and u_{n} ⇀ \frac{1}{2}F (x, x) in W^{2,p}(Q; ℝ^{d})\}.

The analog results for first order gradients with Qf in place of Q_{2}f are well known (see Dacorogna [16]). Moreover, in this case, using a result of Fonseca and Müller [28], Larsen [33] proved that if g : ℝ^{d×N} → ℝ is a continuous function such that 1/C|ξ| ≤ g(ξ) ≤ C(1+|ξ|) for some C > 0, then

Qg(G) = inf{lim inf_{n→+∞} ∫_{Q} g(∇u_{n}) dx : u_{n} ∈ BV (Q; ℝ^{d}), u_{n} → Gx in L^{1}(Q; ℝ^{d})
and |D_{s}u_{n}|(Q) → 0}  (6.1)

for every G ∈ ℝ^{d×N}.

Heuristically, representation (6.1) indicates that as the singular part of the gradient of u_{n} is getting closer and closer to zero, then u_{n} is getting closer and closer to a W^{1,1}-function, and we may allow competing infimizing sequences to have small singular gradients.

We now show that in order to recover the 2-quasiconvexification via BH functions, it is not necessary to require that the singular part of the Hessian of the infimizing sequence to disappear in the limit, but it suffices to guarantee that it will not diffuse into the bulk part.

Theorem 6.2

Q_{2}f(F) = inf{lim inf_{n→+∞} ∫_{Q} f(∇^{2}u_{n}) dx : u_{n} ∈ BH(Q; ℝ^{d}), u_{n} → 1/2F (x, x) in W^{1,1}(Q; ℝ^{d})

and |D_{s}^{2}u_{n}| ⇀^{∗} τ ⊥ L^{N}}.

Proof. Denote by Q̃_{2}f the right hand side of the equation above. By Proposition 6.1, Q̃_{2}f ≤ Q_{2}f. Conversely, since Q_{2}f ≤ f and Q_{2}f is quasiconvex, by Theorems 4.2 and 5.3 we have that

Q_{2}f(F) ≤ lim inf_{n→+∞} ∫_{Q} Q_{2}f(∇^{2}u_{n}) dx ≤ lim inf_{n→+∞} ∫_{Q} f(∇^{2}u_{n}) dx,

for every sequence u_{n} ∈ BH(Q; ℝ^{d}) such that u_{n} → 1/2F (x, x) in W^{1,1}(Q; ℝ^{d}) and |D_{s}^{2}u_{n}| ⇀^{∗} τ ⊥ L^{N}. Thence Q_{2}f ≤ Q̃_{2}f.

Since all the arguments we have used work, with obvious modifications, also in the case of first order gradients, we also have

Theorem 6.3 If g : ℝ^{d×N} → ℝ is such that 1/C|ξ|^{p} ≤ g(ξ) ≤ C(1 + |ξ|^{p}) for some C > 0 and p ≥ 1, then

Qg(G) = inf{lim inf_{n→+∞} ∫_{Q} g(∇u_{n}) dx : u_{n} ∈ BV (Q; ℝ^{d}), u_{n} → Gx in L^{1}(Q; ℝ^{d})

and |D_{s}u_{n}| ⇀^{∗} τ ⊥ L^{N}}.

Acknowledgement The research of I. Fonseca was partially supported by the National Science Foundation under Grants No. DMS–9731957 and DMS–0103798. The research of G. Leoni was partially supported by MURST, by the Italian CNR, through the strategic project “Metodi e modelli per la Matematica e l’Ingegneria”, and by GNAFA. The authors thank the Center for Nonlinear Analysis (NSF Grant No. DMS–9803791), Carnegie Mellon University, Pittsburgh, PA, USA and the Mathematical Institute, University of Oxford, Oxford, UK, for their support and hospitality during the preparation of this paper.

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