A NOTE ON MEYERS’ THEOREM IN W^{k,1}

IRENE FONSECA, GIOVANNI LEONI, JAN MALÝ, AND ROBERTO PARONI

Abstract. Lower semicontinuity properties of multiple-integrals

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

are studied when f grows at most linearly with respect to the highest order derivative, ∇^{k}u, and admissible W^{k,1}(Ω; ℝ^{d}) sequences converge strongly in W^{k−1,1}(Ω; ℝ^{d}). It is shown that under certain continuity assumptions on f, convexity, 1-quasiconvexity or k-polyconvexity of ξ ⟼ f(x_{0}, u(x_{0}), ⋯, ∇^{k−1}u(x_{0}), ξ) ensure lower semicontinuity. The case where f(x_{0}, u(x_{0}), ⋯, ∇^{k−1}u(x_{0}), ·) is k-quasiconvex remains open except in some very particular cases, such as when f(x, u(x), ⋯, ∇^{k}u(x)) = h(x)g(∇^{k}u(x)).

1. Introduction

In a classical paper Meyers [26] proved that k-quasiconvexity is a necessary and sufficient condition for (sequential) lower semicontinuity of a functional

u ↦ ∫_{Ω} f(x, u(x), ⋯, ∇^{k}u(x)) dx,

with respect to weak convergence (resp. weak* convergence if p = ∞) in W^{k,p}(Ω; ℝ^{d}) and under appropriate growth and continuity conditions on the integrand f, thus extending to the case k > 1 the notion of quasiconvexity introduced by Morrey when k = 1. Here Ω is an open, bounded subset of ℝ^{N}, with N ≥ 1, and k, d ∈ ℕ, 1 ≤ p ≤ ∞. Meyers’ theorem uses results of Agmon, Douglis and Nirenberg [1] concerning Poisson kernels for elliptic equations. Fusco [22] later gave a simpler proof using De Giorgi’s Slicing Lemma. He also extended the result to Carathéodory integrands when p = 1, while the case p > 1 has been recently established by Guidorzi and Poggiolini [24] under the Lipschitz condition

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

(note that this condition is automatically satisfied for k = 1 and k = 2, see [25] and [24]), and by Braides, Fonseca and Leoni in [8], who obtained a general relaxation result in W^{k,p}(Ω; ℝ^{d}) with respect to weak convergence.

In most applications, the lower semicontinuity results mentioned above are completely satisfactory when p > 1 since bounded sequences in W^{k,p}(Ω; ℝ^{d}) admit weakly convergent subsequences. However, when p = 1 due to loss of reflexivity of the space W^{k,1}(Ω; ℝ^{d}) one can only conclude that an energy bounded sequence {u_{n}} ⊂ W^{k,1}(Ω; ℝ^{d}) with

sup_{n} ∥u_{n}∥_{W^{k,1}} < ∞

admits a subsequence (not relabelled) such that u_{n} → u in W^{k−1,1}(Ω; ℝ^{d}), (1.1)

Date: November 26, 1999.

1991 Mathematics Subject Classification. 49J45, 49Q20.

Key words and phrases. k-quasiconvexity, higher order lower semicontinuity, gradient truncation.

The research of I. Fonseca was partially supported by the National Science Foundation under Grant No. DMS–9731957.

The research of G. Leoni was partially supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”, by the Italian CNR, through the strategic project “Metodi e modelli per la Matematica e l’Ingegneria”, and by GNAFA.

The research of J. Malý was supported by CEZ MSM 113200007, grants GA ČR 201/00/0768 and GA UK 170/99.

The authors wish to thank Guy Bouchitté for stimulating discussions on the subject of this work, and the Center for Nonlinear Analysis (NSF Grant No. DMS–9803791) for its support during the preparation of this paper.

where u ∈ W^{k−1,1}(Ω; ℝ^{d}) and ∇^{k−1}u is a vector-valued function of bounded variation. In this paper we seek to establish lower semicontinuity in the space W^{k,1}(Ω; ℝ^{d}) under this natural notion of convergence.

When k = 1 the scalar case d = 1 has been extensively treated, while the vectorial case d > 1 was first studied by Fonseca and Müller in [19] where it was proven (sequential) lower semicontinuity in W^{1,1}(Ω; ℝ^{d}) of a functional

u ↦ ∫_{Ω} f(x, u(x), ∇u(x)) dx,

with respect to strong convergence in L^{1}(Ω; ℝ^{d}) (see also [4], [20], [17], [18] and the references contained therein). The approach in [19] is based on blow–up and truncation methods.

Similar truncation techniques have been used quite successfully in the study of existence and qualitative properties of solutions of second order elliptic equations and systems (see e.g. the work of [7] and the references contained within). Their main drawback lies in the fact that they cannot be easily extended to truncate gradients or higher order derivatives. This may explain in part why several important results for second order elliptic equations have no analog for higher order equations.

The main result of this paper extends Meyers’ Theorem to the case where weak convergence in W^{k,1}(Ω; ℝ^{d}) is replaced by (0.1) together with a weak form of coercivity of the convex, 1-quasiconvex or k-polyconvex density f (see Theorems 2, 4 and 5 below). We start with the case where f depends essentially only on x and on the highest order derivatives, that is ∇^{k}u(x). This situation is significantly simpler than the general case, since it does not require to truncate the initial sequence {u_{n}} ⊂ W^{k,1}(Ω; ℝ^{d}). Using the notation and terminology introduced in Section 2, we state the following:

Theorem 1. Let f : Ω×E_{[k−1]}^{d}×E_{k}^{d} → [0, ∞) be a Borel integrand. Suppose that for all (x_{0}, v_{0}) ∈ Ω×E_{[k−1]}^{d} 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}, v_{0}, ξ) − f(x, v, ξ) ≤ ε(1 + f(x, v, ξ)) + ρ(|v − v_{0}|) (1.2)

for all x ∈ Ω with |x − x_{0}| ≤ δ_{0}, and for all (v, ξ) ∈ E_{[k−1]}^{d} × E_{k}^{d}. Assume also that one of the following three conditions is satisfied:

(a) f(x_{0}, v_{0}, ·) is k-quasiconvex in E_{k}^{d} and

\frac{1}{C_{1}} |ξ| − C_{1} ≤ f(x_{0}, v_{0}, ξ) ≤ C_{1}(1 + |ξ|) for all ξ ∈E_{k}^{d}, (1.3)

where C_{1} > 0;

(b) f(x_{0}, v_{0}, ·) is 1-quasiconvex in E_{k}^{d} and

0 ≤ f(x_{0}, v_{0}, ξ) ≤ C_{1}(1 + |ξ|) for all ξ ∈E_{k}^{d}, (1.4)

where C_{1} > 0;

(c) f(x_{0}, v_{0}, ·) is convex in E_{k}^{d}. Let u ∈ BV^{k}(Ω; ℝ^{d}) and let {u_{n}} be a sequence of functions in W^{k,1}(Ω; ℝ^{d}) converging to u in W^{k−1,1}(Ω; ℝ^{d}). Then

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

Here ∇^{k}u is the Radon–Nikodym derivative of the distributional derivative D^{k}u of ∇^{k−1}u, with respect to the N–dimensional Lebesgue measure L^{N}. An important class of integrands which satisfy (0.2) of Theorem 1 is given by

f = f(x, ξ) := h(x)g(ξ),

ON MEYERS’ THEOREM IN W^{k,1} 3

where h(x) is a nonnegative lower semicontinuous function and g is a nonnegative function which satisfies either (a) or (b) or (c). The case where h(x) ≡ 1 and g satisfies condition (a) was proved by Amar and De Cicco [2]. Theorem 1 extends a result of Fonseca and Leoni (Theorem 1.7 in [17]) to higher order derivatives, where the statement is exactly that of Theorem 1 setting k = 1 and excluding part (a). Related results when k = 1 where obtained previously by Serrin [28] in the scalar case d = 1 and by Ambrosio and Dal Maso [4] in the vectorial case d > 1 (see also Fonseca and Müller [19], [20]). Even in the simple case where f = f(ξ) it is not known if Theorem 1(a) still holds without the coercivity condition

f(ξ) ≥ \frac{1}{C_{1}} |ξ| − C_{1}.

The main tool in the proof of Theorem 1, used also in an essential way in subsequent results, is the blow–up method introduced by Fonseca and Müller [19], [20], which reduces the domain Ω to a ball and the target function u to a polynomial.

When the integrand f depends on the full set of variables in an essential way, the situation becomes significantly more complicated since one needs to truncate gradients and higher order derivatives in order to localize lower order terms.

The following theorem was proved for k = 1 by Fonseca and Leoni in [17] Theorem 1.8. Here we extend the result to the higher order case.

Theorem 2. Let f : Ω × E_{[k−1]}^{d} × E_{k}^{d} → [0, ∞) be a Borel integrand, with f(x, v, ·) 1-quasiconvex in E_{k}^{d}. Suppose that for all (x_{0}, v_{0}) ∈ Ω × E_{[k−1]}^{d} either f(x_{0}, v_{0}, ·) ≡ 0, or for every ε > 0 there exist C, δ_{0} > 0 such that

f(x_{0}, v_{0}, ξ) − f(x, v, ξ) ≤ ε(1 + f(x, v, ξ)), (1.5)

C|ξ| − \frac{1}{C} ≤ f(x_{0}, v_{0}, ξ) ≤ C(1 + |ξ|) (1.6)

for all (x, v) ∈ Ω × E_{[k−1]}^{d} with |x − x_{0}| + |v − v_{0}| ≤ δ_{0} and for all ξ ∈ E_{k}^{d}. Let u ∈ BV^{k}(Ω; ℝ^{d}), and let

{u_{n}} be a sequence of functions in W^{k,1}(Ω; ℝ^{d}) converging to u in W^{k−1,1}(Ω; ℝ^{d}). Then

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

A standing open problem is to decide whether Theorem 2 continues to hold under the weaker assumption that f(x, v, ·) is k-quasiconvex, which is the natural assumption in this context.

In the scalar case d = 1, that is when u is an ℝ-valued function, and for first order gradients, i.e. k = 1, condition (0.6) can be eliminated, see Theorem 1.1 in [17]. In particular in [17] Fonseca and Leoni have shown the following result

Proposition 1. (cf. [17], Corollary 1.2) Let g : ℝ^{N} → [0, ∞) be a convex function, and let h : Ω × ℝ → [0, ∞) be a lower semicontinuous function. If u ∈ BV (Ω; ℝ) and {u_{n}} ⊂ W^{1,1}(Ω; ℝ) converges to u in L^{1}(Ω; ℝ), then

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

It is interesting to observe that an analog of this result is false when k ≥ 2.

Theorem 3. Let Ω := (0, 1)^{N} , N ≥ 3, and let h be a smooth cut-off function on ℝ with 0 ≤ h ≤ 1, h(u) = 1 for u ≤ \frac{1}{2}, h(u) = 0 for u ≥ 1. There exists a sequence of functions \{u_{n}\} in W^{2,1}(Ω; ℝ) converging to zero in W^{1,1}(Ω; ℝ) such that {∥Δu_{n}∥_{L^{1}(Ω;ℝ)}} is uniformly bounded and

lim sup_{n→∞} ∫_{Ω} h(u_{n})(1 − Δu_{n})^{+} dx < ∫_{Ω} h(0) dx.

As in Theorem 1, conditions (0.5) and (0.6) can be considerably weakened if we assume that f(x, v, ·) is convex rather than 1-quasiconvex. Indeed we have the following result:

Theorem 4. Let f : Ω × E_{[k−1]}^{d} × E_{k}^{d} → [0, ∞] be a lower semicontinuous function, with f(x, v, ·) convex in E_{k}^{d}. Suppose that for all (x_{0}, v_{0}) ∈ Ω × E_{[k−1]}^{d} either f(x_{0}, v_{0}, ·) ≡ 0, or there exist C_{1}, δ_{0} > 0, and a continuous function g : B(x_{0}, δ_{0}) × B(v_{0}, δ_{0}) → E_{k}^{d} such that

f(x, v, g(x, v)) ∈ L^{∞} (B(x_{0}, δ_{0}) × B(v_{0}, δ_{0}); ℝ) , (1.7)

f(x, v, ξ) ≥ C_{1}|ξ| − \frac{1}{C_{1}} (1.8)

for all (x, v) ∈ Ω × E_{[k−1]}^{d} with |x − x_{0}| + |v − v_{0}| ≤ δ_{0} and for all ξ ∈ E_{k}^{d}. Let u ∈ BV^{k}(Ω; ℝ^{d}), and let {u_{n}} be a sequence of functions in W^{k,1}(Ω; ℝ^{d}) converging to u in W^{k−1,1}(Ω; ℝ^{d}). Then

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

Theorem 4 was obtained by Fonseca and Leoni for the case k = 1 in Theorem 1.1 [18]. It is interesting to observe that without a condition of the type (0.7) Theorem 4 is false in general. This has been recently proved by Černý and Malý in [12].

The proofs of Theorems 1(b) and (c), 2 and 4 can be deduced easily from the corresponding ones in [17],

[18] where k = 1. It suffices to write

∫_{Ω} f(x, u(x), …, ∇^{k}u(x)) dx =: ∫_{Ω} F (x, v(x), ∇v(x)) dx

with v := (u, …, ∇^{k−1}u), and then to perturb the new integrand F in order to recover the full coercivity conditions necessary to apply the results in [17], [18]. This approach cannot be used for k-polyconvex integrands and a new proof is needed to treat this case. Thus Theorem 1(a) and Theorem 5 below are the only truly genuine higher order results, in that they cannot be reduced in a trivial way to a first order problem.

For each ξ ∈ E_{k}^{d} let M(ξ) ∈ ℝ^{τ} be the vector whose components are all the minors of ξ.

Theorem 5. Let h : Ω × E_{[k−1]}^{d} × ℝ^{τ} → [0, ∞] be a lower semicontinuous function, with h(x, v, ·) convex in ℝ^{τ}. Suppose that for all (x_{0}, v_{0}) ∈ Ω × E_{[k−1]}^{d} either h(x_{0}, v_{0}, ·) ≡ 0, or there exist C, δ_{0} > 0, and a continuous function g : B(x_{0}, δ_{0}) × B(v_{0}, δ_{0}) → ℝ^{τ} such that

h(x, v, g(x, v)) ∈ L^{∞} (B(x_{0}, δ_{0}) × B(v_{0}, δ_{0}); ℝ) , (1.9)

h(x, v, v) ≥ C|v| − \frac{1}{C} (1.10)

for all (x, v) ∈ Ω × E_{[k−1]}^{d} with |x − x_{0}| + |v − v_{0}| ≤ δ_{0} and for all v ∈ ℝ^{τ}. Let u ∈ BV^{k}(Ω; ℝ^{d}), and let {u_{n}} be a sequence of functions in W^{k,p}(Ω; ℝ^{d}) which converges to u in W^{k−1,1}(Ω; ℝ^{d}), where p is the minimum between N and the dimension of the vectorial space E_{k−1}^{d}. Then

∫_{Ω} h(x, u, …, ∇^{k−1}u, M(∇^{k}u)) dx ≤ lim inf_{n→∞} ∫_{Ω} h(x, u_{n}, …, ∇^{k−1}u_{n}, M(∇^{k}u_{n})) dx.

Theorem 5 is closely related to a result of Ball, Currie and Olver [6], where it was assumed that

h(x, v, v) ≥ γ (|v|) − \frac{1}{C} ,

where

\frac{γ (s)}{s} → ∞ as s → ∞.

Also, as stated above and with k = 1, Theorem 5 was proved by Fonseca and Leoni in [18], Theorem 1.4.

2. Preliminaries

We start with some notation. Here Ω ⊂ ℝ^{N} is an open, bounded subset, L^{N} and H^{N−1} are, respectively, the N dimensional Lebesgue measure and the N − 1 dimensional Hausdorff measure in ℝ^{N}. Let Q be the the unit cube (−1/2, 1/2)^{N} and set Q(x_{0}, ε) := x_{0} + εQ.

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

∇^{α} := \frac{∂^{|α|}}{∂x_{1}^{α_{1}} … ∂x_{N}^{α_{N}}} , |α| = α_{1} + … + α_{N},

and for each j ∈ ℕ the symbol ∇^{j}u stands for the vector-valued function whose components are all the components of ∇^{α}u for |α| = j. If u is C^{∞} then for j ≥ 2 we have that ∇^{j}u(x) ∈ E_{j}^{d}, where E_{j}^{d} stands for the space of symmetric j-linear maps from ℝ^{ℕ} into ℝ^{d}. We set E_{0}^{d} := ℝ^{d}, E_{1}^{d} := ℝ^{d×N} and

E_{[j−1]}^{d} := E_{0}^{d} × ⋯ × E_{j−1}^{d}, E_{[0]}^{d} := E_{0}^{d}.

For any integer k ≥ 2 we define

BV^{k}(Ω; ℝ^{d}) := {u ∈ W^{k−1,1}(Ω; ℝ^{d}) : ∇^{k−1}u ∈ BV (Ω; E_{k−1}^{d})} ,

where ∇^{j}u is the Radon–Nikodym derivative of the distributional derivative D^{j}u of ∇^{j−1}u, with respect to the N–dimensional Lebesgue measure L^{N}.

We recall that a function f : E_{k}^{d} → ℝ is said to be k-quasiconvex if

f(ξ) ≤ ∫_{Q} f(ξ + ∇^{k}w(y)) dy

for all ξ ∈ E_{k}^{d} and all w ∈ C_{0}^{∞}(Q; ℝ^{d}) .

The following theorem was proved in the case k = 1 by Ambrosio and Dal Maso [4], while Fonseca and Müller [19] treated general integrands of the form f = f(x, u, ∇u), but their argument requires coercivity. The case k ≥ 2 is due to Amar and De Cicco [2]. For completeness we give a proof for all k ≥ 1.

Proposition 2. Let f : E_{k}^{d} → [0, ∞) be a k-quasiconvex function, such that

0 ≤ f(ξ) ≤ C (1 + |ξ|) , (2.1)

for all ξ ∈ E_{k}^{d}. Moreover, when k ≥ 2 assume that

(2.2) f(ξ) ≥ C_{1} |ξ| for |ξ| large.

Let {u_{n}} be a sequence of functions in W^{k,1}(Q; ℝ^{d}) converging to 0 in W^{k−1,1}(Q; ℝ^{d}). Then

f(0) ≤ lim inf_{n→∞} ∫_{Q} f(∇^{k}u_{n}) dx.

Proof. We start with the case k ≥ 2. Without loss of generality we may assume that

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

so that by condition (1.2)

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

Let ε > 0, M ∈ ℕ, and decompose L := Q \ (1 − ε) Q into M layers with mutually disjoint interiors, L_{i} := α_{i+1}Q \ α_{i}Q, so that 1 − ε = α_{1} < α_{2} < … < α_{M} < 1 =: α_{M+1}. Since

∑_{i=1}^{M} ∫_{L_{i}} (1 + |∇^{k}u_{n}|) dx ≤ 1 + K

for all n ∈ ℕ, there exists i_{ε} ∈ {1, …, M} and a subsequence of {u_{n}} (not relabelled) such that

∫_{L_{iε}} (1 + |∇^{k}u_{n}|) dx ≤ \frac{1 + K}{M} for all n ∈ ℕ. (2.3)

Let φ ∈ C_{c}^{∞} (Q; [0, 1]) with φ(x) = 1 in α_{i_{ε}}Q, φ(x) = 0 if x ∉ α_{i_{ε}+1}Q. Since f is k-quasiconvex

f(0) ≤ lim inf_{n→∞} ∫_{Q} f(∇^{k} (φu_{n})) dx

≤ lim inf_{n→∞} ∫_{Q} f(∇^{k}u_{n}) dx + ∫_{Q \ α_{iε+1}Q} f(0) dx

+ C lim sup_{n→∞} ∫_{L_{iε}} (1 + |∇^{k} (φu_{n}) |) dx,

where we have used (1.1). As u_{n} → 0 in W^{k−1,1}(Q; ℝ^{d}) strongly, we have

lim sup_{n→∞} ∫_{L_{iε}} (1 + |∇^{k} (φu_{n}) |) dx ≤ lim sup_{n→∞} ∫_{L_{iε}} (1 + |∇^{k}u_{n}|) dx ≤ \frac{1 + K}{M}

by (1.3). We conclude that

(1 − ε)^{N}f(0) ≤ α_{i_{ε}+1}^{N}f(0) ≤ lim inf_{n→∞} ∫_{Q} f(∇^{k}u_{n}) dx + \frac{1 + K}{M} ,

and the result now follows by letting first ε → 0^{+} and then M → ∞.

Next we consider the case where k = 1. Let ε > 0, fix n ∈ ℕ, set

M_{n} := [n ∫_{Q} (1 + |∇u_{n}|) dx] + 1,

where [·] denotes the integer part, and decompose L := Q \ (1 − ε) Q into M_{n} layers with mutually disjoint interiors, L_{i}^{(n)} := α_{i+1}^{(n)}Q \ α_{i}^{(n)}Q, so that 1 − ε = α_{1}^{(n)} < α_{2}^{(n)} < … < α_{M_{n}}^{(n)} < 1 =: α_{M+1}^{(n)} and, in addition α_{i+1}^{(n)} − α_{i}^{(n)} = \frac{ε}{M_{n}} , i = 1, …, M_{n}. Let φ_{i}^{(n)} ∈ C_{c}^{∞} (Q; [0, 1]) with φ_{i}^{(n)}(x) = 1 in α_{i}^{(n)}Q, φ_{i}^{(n)}(x) = 0 if

x ∉ α_{i+1}^{(n)}Q, ∥∇φ_{i}∥_{∞} ≤ \frac{2M_{n}}{ε} , i = 1, …, M_{n}. We have

∫_{Q} f (∇ (φ_{i}^{(n)}u_{n})) dx ≤ ∫_{Q} f(∇u_{n}) dx + ∫_{Q \ α_{i+1}^{(n)}Q} f(0) dx

+ C ∫_{L_{i}^{(n)}} (1 + |∇u_{n}|) dx + C \frac{2M_{n}}{ε} ∫_{L_{i}^{(n)}} |u_{n}| dx.

Thus

\frac{1}{M_{n}} ∑_{i=1}^{M_{n}} ∫_{Q} f(∇ (φ_{i}^{(n)}u_{n})) dx ≤ ∫_{Q} f(∇u_{n}) dx + ∫_{Q \ α_{1}^{(n)}Q} f(0) dx

+ \frac{C}{M_{n}} ∫_{Q \ α_{1}^{(n)}Q} (1 + |∇u_{n}|) dx + \frac{C}{ε} ∫_{Q \ α_{1}^{(n)}Q} |u_{n}| dx

≤ ∫_{Q} f(∇u_{n}) dx + O (ε) + \frac{C}{n} + \frac{C}{ε} ∫_{Q \ α_{1}^{(n)}Q} |u_{n}| dx.

We may, therefore, find i = i(n, ε) ∈ {1, …, M} such that, in view of the quasiconvexity of f,

f(0) ≤ ∫_{Q} f (∇ (φ_{i}^{(n)}u_{n})) dx ≤ ∫_{Q} f(∇u_{n}) dx + O (ε) + \frac{C}{n} + \frac{C}{ε} ∫_{Q} |u_{n}| dx,

and the conclusion follows by letting n → ∞ and then ε → 0^{+}.

Proposition 3. Let h : ℝ^{τ} → [0, ∞) be a convex function such that

h(v) → ∞ as |v| → ∞.

Let u ∈ W^{1,p}(Ω; ℝ^{d}), and let {u_{n}} be a sequence of functions in W^{1,p}(Ω; ℝ^{d}) which converges to u in L^{1}(Ω; ℝ^{d}), where p = min{d, N}. Then

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

Proposition 3 has been proved by Dal Maso and Sbordone (cf. Theorem 2.2 in [14]) using cartesian currents and by Fusco and Hutchinson (cf. Theorem 2.6 in [23]).

Next we present an approximation result for convex functions.

Proposition 4. Let M be a closed set of ℝ^{p}, let V be a reflexive and separable Banach space. Let f : M × V → (0, +∞] be a M × (w − V) sequentially lower semicontinuous function, convex in the last variable and such that there exists a continuous function v_{0} : M → V with

(f(·, v_{0}(·)))^{+} ∈ L_{loc}^{∞}(M; ℝ). (2.4)

Then there exist two sequences of continuous functions

a_{j} : M → ℝ, b_{j} : M → V^{∗},

where V^{∗} is the dual space of V, such that

f(t, v) = sup_{j} (a_{j}(t) + ⟨b_{j}(t), v⟩)^{+}

for all t ∈ M and v ∈ V.

Proposition 4 was proved by Fonseca and Leoni in [18], following closely the argument of Ambrosio in [3], who studied the case where (1.4) is replaced by the assumption that f(·, v_{0}(·)) is continuous.

3. Proof of Theorems 1 and 2

Proof of Theorem 1. Without loss of generality we may assume that

liminf_{n→∞} ∫_{Ω} f(x, u_{n}(x), …, ∇^{k}u_{n}(x)) dx = lim_{n→∞} ∫_{Ω} f(x, u_{n}(x), …, ∇^{k}u_{n}(x)) dx < ∞.

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

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

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

(3.1)  \frac{dμ}{dL^{N}} (x_{0}) = lim_{ε→0^{+}} \frac{μ(Q(x_{0}, ε))}{ε^{N}} ≥ f(x_{0}, u(x_{0}), …, ∇^{k}u(x_{0}))  for L^{N} a.e. x_{0} ∈ Ω.

If (2.1) holds, then the conclusion of the theorem follows immediately. Indeed, let φ ∈ C_{c}(Ω; ℝ), 0 ≤ φ ≤ 1. We have

∫_{Ω} f(x, u_{n}, …, ∇^{k}u_{n}) dx ≥ lim inf_{n→∞} ∫_{Ω} φ f(x, u_{n}, …, ∇^{k}u_{n}) dx

= ∫_{Ω} φ dμ ≥ ∫_{Ω} φ \frac{dμ}{dL^{N}} dx ≥ ∫_{Ω} φ f(x, u, …, ∇^{k}u) 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 (2.1).

Take x_{0} ∈ Ω such that

\frac{dμ}{dL^{N}} (x_{0}) = lim_{ε→0^{+}} \frac{μ(Q(x_{0}, ε))}{ε^{N}} < ∞,  lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} \frac{|u(x) − T_{k}(x)|}{|x − x_{0}|^{k}}dx = 0, (3.2)

where

T_{k}(x) := ∑_{|α|≤k} \frac{1}{α!}∇^{α}u(x_{0})(x − x_{0})^{α},

and set

v_{0} := (u(x_{0}), …, ∇^{k−1}u(x_{0})) .

8 IRENE FONSECA, GIOVANNI LEONI, JAN MALÝ, AND ROBERTO PARONI

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

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

= lim_{h→∞} lim_{n→∞} ∫_{Q} f(x_{0} + ε_{h}y, T_{k−1}(x_{0} + ε_{h}y) + ε_{h}^{k}w_{n,h}(y), ∇T_{k−1}(x_{0} + ε_{h}y)

+ ε_{h}^{k−1}∇w_{n,h}(y), ∇^{2}T_{k−1}(x_{0} + ε_{h}y) + ε_{h}^{k−2}∇^{2}w_{n,h}(y), …, ∇^{k}w_{n,h}(y)) dy,

where

w_{n,h}(y) := \frac{u_{n}(x_{0} + ε_{h}y) − T_{k−1}(x_{0} + ε_{h}y)}{ε_{h}^{k}}.

Clearly w_{n,h} ∈ W^{k,1}(Q; ℝ^{d}), and, by (2.2), lim_{h→∞} lim_{n→∞} ||w_{n,h} − w_{0}||_{W^{k−1,1}(Q;ℝ^{d})} = 0, where

w_{0}(y) := ∑_{|α|=k} \frac{1}{α!}∇^{α}u(x_{0})y^{α}.

By a standard diagonalization argument, we may extract a subsequence w_{h} := w_{n_{h},h} which converges to w_{0} in W^{k−1,1}(Q; ℝ^{d}), such that ∇^{j}w_{h} → ∇^{j}w pointwise a.e. for j = 0, …, k, and

\frac{dμ}{dL^{N}} (x_{0}) = lim_{h→∞} ∫_{Q} f(x_{0} + ε_{h}y, T_{k−1}(x_{0} + ε_{h}y) + ε_{h}^{k}w_{h}(y), …, ∇^{k}w_{h}(y)) dy. (3.3)

By condition (0.2) for all ε > 0 and for h large enough

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim_{h→∞} (∫_{Q} f(x_{0}, u(x_{0}), …, ∇^{k−1}u(x_{0}), ∇^{k}w_{h}(y)) dy − ∫_{Q} ρ(|z_{h}(y)|) dy) ,

where

z_{h}(y) := (T_{k−1}(x_{0} + ε_{h}y) + ε_{h}^{k}w_{h}(y), …, ∇^{k−1}T_{k−1}(x_{0} + ε_{h}y) + ε_{h}∇^{k−1}w_{h}(y)) − v_{0}.

By Fatou’s Lemma, and since ρ is continuous with ρ(0) = 0, we have

C_{0} − lim sup_{h→∞} ∫_{Q} ρ( |z_{h}(y)|) dy = lim inf_{h→∞} ∫_{Q}[C_{0}(1 + |z_{h}(y)|) − ρ(|z_{h}(y)|)] dy

≥ ∫_{Q} lim inf_{h→∞}[C_{0}(1 + |z_{h}(y)|) − ρ( |z_{h}(y)|)] dy = C_{0},

and so

∫_{Q} ρ(|z_{h}(y)|) dy → 0 as h → ∞.

Thus

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ lim_{h→∞} ∫_{Q} f(x_{0}, v_{0}, ∇^{k}w_{h}(y)) dy. (3.4)

If g(ξ) := f(x_{0}, v_{0}, ξ) satisfies condition (a) then use Proposition 2, and if either condition (b) or (c) hold then apply Theorem 1.7 in [17] to conclude that

(1 + ε)\frac{dμ}{dL^{N}} (x_{0}) + ε ≥ f(x_{0}, u(x_{0}), …, ∇^{k}u(x_{0})),

and it suffices to let ε → 0^{+}.

Proof of Theorem 2. Theorem 2 can be easily deduced from Theorem 1.8 in [17]. It suffices to write

∫_{Ω} f(x, u(x), …, ∇^{k}u(x)) dx =: ∫_{Ω} F (x, v(x), ∇v(x)) dx

with v := (u, …, ∇^{k−1}u). Note, however, that the coercivity condition (0.6) for F now reads

F (x_{0}, v_{0}, η) ≥ C|η_{k}| − \frac{1}{C} ,

where

η = (η_{1}, ⋯, η_{k}) ∈ E_{1}^{d} × ⋯ × E_{k}^{d}, and F (x, v, η) := f(x, v, η_{k}).

In order to be in position to apply Theorem 1.8 we need to ensure full coercivity. Due to the strong convergence of admissible sequences {u_{n}} in W^{k−1,1}(Ω; ℝ^{d}), and therefore of {v_{n}} in L^{1} (Ω; E_{[k−1]}^{d}) , it suffices to consider

F_{ε}(x, v, η) := F (x, v, η) + εχ_{A} (x, v) |(η_{1}, ⋯, η_{k−1})| ,

where A := {(x, v) ∈ Ω × E_{[k−1]}^{d} : f(x, v, ·) ≢ 0} . Theorem 1.8 in [17] now yields

lim inf_{n→∞} ∫_{Ω} f(x, u_{n}, …, ∇^{k}u_{n}) dx = lim inf_{n→∞} ∫_{Ω} F (x, v_{n}, ∇v_{n}) dx
≥ ∫_{Ω} F (x, v, ∇v) dx − ε ∫_{Ω} |(∇u, …, ∇^{k−1}u)| dx
= ∫_{Ω} f(x, u, …, ∇^{k}u) dx − ε ∫_{Ω} |(∇u, …, ∇^{k−1}u)| dx.

Let ε → 0^{+}.

4. Proof of Theorem 3

Throughout this section we assume that N ≥ 3.

Lemma 1. Let D be a cube with |D| ≤ 1. Then there exists constants C > 0, λ ∈ (0, 1) depending only on N, a function u ∈ W^{2,∞}(D; ℝ) with compact support in D, and sets A, E, G ⊂ D, with A ∪ E ∪ G = D and |E| ≤ λ|D|, such that

(4.1)  ∥Δu∥_{L^{1}(D;ℝ)} ≤ C|D|, ∥u∥_{W^{1,1}(D;ℝ)} ≤ C|D|^{1+ \frac{1}{N}} ,
(4.2)  Δu = 1 on A,
(4.3)  u = 0 on E, u ≥ 1 on G.

Proof. After a translation we may assume that there exists B(0, R) ⊂ D such that

C^{−1}R^{N} ≤ |D| ≤ CR^{N}, R ∈ (0, 1/2),

for some C > 0. We search for a radial function of type

u(x) := φ(|x|),

where φ is a C^{2}-function on (0, ∞) such that

φ(t) = 0 for t ≥ R, (4.4)

φ′(0+) = 0. (4.5)

Further we want that for some a < 0

Δu(x) = \{ \begin{matrix}−a & if |x| < r, \\ 1 & if r < |x| < R, \end{matrix} (4.6)

where r is determined by the equation

r^{2−N}R^{N} = 2N(N − 2). (4.7)

Note that r ∈ (0, R) because R < 1 and N ≥ 3. In order to find a and φ satisfying (3.4), (3.5) and (3.6), we note that

Δu(x) = φ′′(|x|) + |x|^{−1} (N − 1) φ′(|x|), for |x| ≠ 0,

or, equivalently,

Δu(x) = t^{1−N}(t^{N−1}φ′(t))′ , where t = |x|.

10 IRENE FONSECA, GIOVANNI LEONI, JAN MALÝ, AND ROBERTO PARONI

On the interval (r, R) (3.6) now yields

(t^{N−1}φ′(t))′ = t^{N−1},

and thus, by (3.4),

φ′(t) = \frac{t}{N} (1 − \frac{R^{N}}{t^{N}} ) . (4.8)

On the interval (0, r), and in view of (3.6), we have

(t^{N−1}φ′(t))′ = −at^{N−1}

which, together with (3.5), implies that

φ′(t) = −\frac{at}{N} . (4.9)

We have

−\frac{ar}{N} = φ′(r−) = φ′(r+) = \frac{r}{N} (1 − \frac{R^{N}}{r^{N}} )

and thus

a = (\frac{R^{N}}{r^{N}} − 1) . (4.10)

Now the function u is uniquely determined by its properties. Obviously we have (3.3)_{1} by setting

A := B(0, R) \ B(0, r), E := D \ B(0, r), G := B(0, r),

with |E| ≤ λ|D| and λ = λ (N) . In light of (3.6) and (3.10) we have

∥Δu∥_{L^{1}(D;ℝ)} ≤ |B(0, R) \ B(0, r)| + a |B(0, r)| = ω_{N} (R^{N} − r^{N} + (\frac{R^{N}}{r^{N}} − 1) r^{N}) = ω_{N}r^{N},

where ω_{N} := |B(0, 1)|. If x ∈ G, we have by (3.9), (3.8) and (3.7)

u(x) ≥ φ(r) = − ∫_{r}^{R} φ′(t) dt = \frac{1}{N} ∫_{r}^{R}(R^{N}t^{1−N} − t) dt

≥ \frac{1}{N(N − 2)}(r^{2−N}R^{N} − R^{2}) − \frac{R^{2}}{2N} = 2 − \frac{R^{2}}{2 (N − 2)} ≥ 1,

as R ≤ \frac{1}{2} as B(0, R) ≤ D and the side length of D does not exceed 1. This proves (3.3)_{2}. By (3.8) and (3.9),

∫_{D} |∇u| dx ≤ C ∫_{0}^{R} t^{N−1}|φ′(t)| dt ≤ C (∫_{0}^{r} r^{−N}R^{N}t^{N} dt + ∫_{r}^{R} R^{N} dt) ≤ CR^{N+1},

which, with the aid of the Poincaré inequality for zero boundary values, proves (3.1)_{2}. Proof of Theorem 3. We set Ω = (0, 1)^{N}, and we construct the \frac{1}{n}-periodic sequence \{u_{n}\} as follows: divide Ω into small cubes D_{α} of measure \frac{1}{n^{N}}, α ∈ I_{n} where the set of indices I_{n} has cardinality n^{N}. On each D_{α} we construct u_{n} as indicated in Lemma 1, and denote by A_{α}, E_{α}, G_{α} the corresponding sets. Then u_{n} → 0 in W^{1,1}(Ω; ℝ) because

∥u_{n}∥_{W^{1,1}(Ω;ℝ)} = ∑_{α∈I_{n}} ∥u_{n}∥_{W^{1,1}(D_{α};ℝ)} ≤ n^{N}C (\frac{1}{n^{N}} )^{1+ \frac{1}{N}} → 0 as n → ∞,

and {∥Δu_{n}∥_{L^{1}(Ω;ℝ)}} is uniformly bounded since

∥Δu_{n}∥_{L^{1}(Ω;ℝ)} = ∑_{α∈I_{n}} ∥Δu_{n}∥_{L^{1}(D_{α};ℝ)} ≤ n^{N}C \frac{1}{n^{N}} = C.

Consider the functional

F (v) := ∫_{Ω} h(v)(1 − Δv)^{+} dx.

For α ∈ I_{n} we have by (3.1)–(3.3)

h(u_{n}) = 1 and Δu_{n} = 0 on E_{α},
Δu_{n} = 1 on A_{α},
h(u_{n}) = 0 on G_{α},

and thus

∫_{D_{α}} h(u_{n})(1 − Δu_{n})^{+} dx = |E_{α}| ≤ λ|D_{α}|.

Summing up over α ∈ I_{n}, we conclude that

∫_{Ω} h(u_{n})(1 − Δu_{n})^{+} dx ≤ λ < 1 = F (0).

Remark. We cannot obtain an apriori bound on ∥u_{n}∥_{W^{2,1}(Ω;ℝ)} because the function

h(v)(1 − trace ξ)^{+}

is convex in the last variable, and, after adding a small multiple of |ξ|, the corresponding functional is lower semicontinuous on W^{1,1}(Ω; ℝ) according to Theorem 4. A direct heuristic computation using the notation of Lemma 1 yields

∫_{Q} |∇^{2}u| dx ∼ ∫_{0}^{R} (t^{N−1}|φ′′(t)| + t^{N−2}|φ′(t)|) dt

∼ (∫_{0}^{r} t^{N−1} \frac{R^{N}}{r^{N}} dt + ∫_{r}^{R} \frac{R^{N}}{t} dt)

∼ R^{N} log R,

hence an inequality of the type

∥∇^{2}u∥_{L^{1}(Q;E_{2}^{1})} ≤ C|Q|

will not hold.

5. Proof of Theorems 4 and 5

Proof of Theorem 4. As in the proof of Theorem 2, it is easy to obtain Theorem 4 from Theorem 1.1 in [18] by considering v := (u, …, ∇^{k−1}u) and the reformulated functionals

∫_{Ω} (F (x, v(x), ∇v(x)) + εχ_{A} (x, v (x)) |((∇v)_{1} , ⋯, (∇v)_{k−1})|) dx,

with

∇v : = ((∇v)_{1} , ⋯, (∇v)_{k}) ∈ E_{1}^{d} × ⋯ × E_{k}^{d}.

Remark that, as opposed to Theorems 1(b) and (c), 2 and 4, Theorem 5 cannot be deduced easily from the analogous result already obtained in the case where k = 1, precisely Theorem 1.4 in [18]. Indeed, there is no obvious way of perturbing the new integrand H (x, v, M(∇v)) := h(x, u, …, ∇^{k−1}u, M(∇^{k}u)), with v := (u, …, ∇^{k−1}u) , in such a way that (0.10) is satisfied for the perturbed integrand, i.e.

H_{ε} (x, v, M(∇v)) ≥ C_{ε} |M(∇v)| − \frac{1}{C_{ε}}

and

H_{ε} ≥ H, lim inf_{ε→0^{+}} lim inf_{n→∞} ∫_{Ω} H_{ε} (x, v_{n}, M(∇v_{n})) dx ≤ lim inf_{n→∞} ∫_{Ω} H (x, v_{n}, M(∇v_{n})) dx.

This is due to the fact that M(∇v) involves terms of the form ∇u∇^{k}u for which we have no bounds.

Proof of Theorem 5. Let f(x, v, ξ) := h(x, v, M(ξ)) . We proceed as in the proof of Theorem 1 until we reach (2.3), precisely

\frac{dμ}{dL^{N}} (x_{0}) = lim_{h→∞} ∫_{Q} f(x_{0} + ε_{h}y, T_{k−1}(x_{0} + ε_{h}y) + ε_{h}^{k}w_{h}(y), …, ∇^{k}w_{h}(y)) dy, (5.1)

where now w_{h} ∈ W^{k,p}(Q; ℝ^{d}), and ||w_{h} − w_{0}||_{W^{k−1,1}(Q;ℝ^{d})} → 0 as h → ∞, where

w_{0}(y) := ∑_{|α|=k} \frac{1}{α!}∇^{α}u(x_{0})y^{α}.

If f(x_{0}, v_{0}, ·) ≡ 0 then there is nothing to prove. Otherwise, let δ_{0} > 0 be given by (0.9) and (0.10). Setting

Q_{h} := {y ∈ Q : |(w_{h}(y), …, ∇^{k−1}w_{h}(y))| ≤ δ_{0}/(2ε_{h})} ,

by (4.1) and (0.10) we have

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim sup_{h→∞} ∫_{Q_{h}} f(x_{0} + ε_{h}y, T_{k−1}(x_{0} + ε_{h}y) + ε_{h}^{k}w_{h}(y), …, ∇^{k}w_{h}(y)) dy

≥ C lim sup_{h→∞} ∫_{Q_{h}} |M(∇^{k}w_{h}(y))| dy − 1/C,

and so there exists a constant K > 0 such that

∫_{Q_{h}} |M(∇^{k}w_{h}(y))| dy ≤ K for all h ∈ ℕ. (5.2)

By Proposition 4, with M = (x_{0} + ε_{1}\overline{Q}) × \overline{B(v_{0}, δ_{0}/2)} and V = ℝ^{τ}, in view of (0.9) there exist two sequences of continuous functions

a_{j} : M → ℝ, b_{j} : M → ℝ^{τ} ,

such that

h(x, v, η) = sup_{j} (a_{j}(x, v) + b_{j}(x, v) · η)^{+} (5.3)

for all (x, v) ∈ M and η ∈ ℝ^{τ}. Define

h_{j}(x, v, η) := (a_{j}(x, v) + b_{j}(x, v) · η)^{+} , f_{j}(x, v, ξ) := h_{j}(x, v, M (ξ)).

Clearly h_{j} is continuous, convex in η and

0 ≤ h_{j}(x, v, η) ≤ C_{j}(|η| + 1), (5.4)

for all (x, v) ∈ M and η ∈ ℝ^{τ}, where

C_{j} := max{|a_{j}(x, v)| + |b_{j}(x, v)| : (x, v) ∈ M}.

Fix ε > 0 and find 0 < δ_{j} ≤ δ_{0}/2 such that

|a_{j}(x, v) − a_{j}(x_{0}, v_{0})| + |b_{j}(x, v) − b_{j}(x_{0}, v_{0})| ≤ ε

for all (x, v) ∈ (x_{0} + δ_{j}\overline{Q}) × \overline{B(v_{0}, δ_{j})}. Since the function s ↦ s^{+} is Lipschitz continuous with Lipschitz constant 1, we have

|f_{j}(x, v, ξ) − f_{j}(x_{0}, v_{0}, ξ)| ≤ |a_{j}(x, v) − a_{j}(x_{0}, v_{0})| + |b_{j}(x, v) − b_{j}(x_{0}, v_{0})| |M (ξ) | ≤ ε(1 + |M (ξ) |) (5.5)

for all (x, v) ∈ (x_{0} + δ_{j}\overline{Q}) × \overline{B(v(x_{0}), δ_{j})} and all ξ ∈ E_{k}^{d}. By (4.1) and for any j ∈ ℕ we obtain

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{h→∞} ∫_{Q_{h}} f(x_{0} + ε_{h}y, T_{k−1}(x_{0} + ε_{h}y) + ε_{h}^{k}w_{h}(y), …, ∇^{k}w_{h}(y)) dy

≥ lim inf_{h→∞} ∫_{Q_{h}} f_{j}(x_{0} + ε_{h}y, T_{k−1}(x_{0} + ε_{h}y) + ε_{h}^{k}w_{h}(y), …, ∇^{k}w_{h}(y)) dy

≥ lim inf_{h→∞} (∫_{Q_{h}} f_{j}(x_{0}, v_{0}, ∇^{k}v_{h}(y)) dy − ε − ε ∫_{Q_{h}} |M (∇^{k}w_{h}(y))| dy)

≥ lim inf_{h→∞} ∫_{Q_{h}} f_{j}(x_{0}, v_{0}, ∇^{k}v_{h}(y)) dy − ε − εK,

(5.6)

where we have used (4.5) and (4.2). Define

z_{h}(y) := (ε_{h}^{k−1}w_{h}(y), …, ε_{h}∇^{k−2}w_{h}(y)) , u_{h} (y) := ∇^{k−1}w_{h}(y).

Fix an integer P ∈ ℕ such that e^{P} > 1 + ||∇^{k−1}w_{0}||_{∞}. For h sufficiently large, say h ≥ h_{P} , we have e^{2P +1} ≤ δ_{0}/(2ε_{h}), so in view of (4.2) we may find i_{h} ∈ {P + 1, ⋯, 2P } such that

{y ∈ Q : e^{i_{h}} ≤ |(z_{h}(y), u_{h} (y))| ≤ e^{i_{h}+1}} ⊂ Q_{h},

and

∫_{\{y∈Q: e^{i_{h}} ≤|(z_{h}(y),u_{h}(y))|≤e^{i_{h}+1}\}} (1 + |M(∇^{k}w_{h}(x))|) dx ≤ \frac{1 + K}{P} .

Since {P + 1, ⋯, 2P } is a finite set, we may find i_{P} ∈ {P + 1, ⋯, 2P } such that

∫_{\{y∈Q: e^{i_{P}} ≤|(z_{h}(y),u_{h}(y))|≤e^{i_{P} +1}\}} (1 + |M(∇^{k}w_{h}(x))|) dx ≤ \frac{1 + K}{P} . (5.7)

for infinitely many indices h ∈ ℕ. From now until the end of the proof we assume without loss of generality that the whole sequence satisfy (4.7).

Set

v_{h}(y) := g (|(z_{h}(y), u_{h} (y))|) u_{h} (y)

and

D_{h} := {y ∈ Q : |(z_{h}(y), u_{h} (y))| < e^{i_{P}} } ,
D_{h}^{−} := {y ∈ Q : e^{i_{P}} ≤ |(z_{h}(y), u_{h} (y))| ≤ e^{i_{P} +1}} ,
D_{h}^{+} := {y ∈ Q : |(z_{h}(y), u_{h} (y))| > e^{i_{P} +1}} ,

where

g(s) := \{ \begin{matrix}1 & if s < e^{i_{P}} , \\ \frac{e^{i_{P} +1} − s}{e^{i_{P} +1} − e^{i_{P}}} & if e^{i_{P}} ≤ s ≤ e^{i_{P} +1}, \\ 0 & if s > e^{i_{P} +1}. \end{matrix}

Note that

|D_{h}^{−} ∪ D_{h}^{+}| = |{y ∈ Q : |(z_{h}(y), u_{h} (y))| ≥ e^{i_{P}} }|
≤ |{y ∈ Q : | (z_{h}(y), u_{h} (y)) − (0, ∇^{k−1}w_{0}(y)) | ≥ 1}|
≤ ||z_{h}||_{L^{1}(Q)} + ||u_{h} − ∇^{k−1}w_{0}||_{L^{1}(Q)} → 0 as h → ∞,

(5.8)

where we have used the fact that e^{i_{P}} > 1 + ||∇^{k−1}w_{0}||_{∞}. Also,

|∇z_{h}| = |(ε_{h}^{k−1}∇w_{h}, …, ε_{h}∇^{k−1}w_{h})| ≤ ε_{h} |(ε_{h}^{k−1}w_{h}, ε_{h}^{k−2}∇w_{h}, …, ε_{h}∇^{k−1}w_{h})| = ε_{h} |(z_{h}, u_{h})| . (5.9)

We claim that

|M(∇v_{h}(y))| ≤ C (1 + ε_{h} e^{i_{P} +1} ) |M(∇^{k}w_{h}(y))|. (5.10)

In view of the definition of g this is immediate for x ∈ D_{h} ∪ D_{h}^{+}. Thus it remains to assert (4.10) in D_{h}^{−}. We have

∇v_{h} = (g (|(z_{h}, u_{h})|) I + g′ (|(z_{h}, u_{h})|) \frac{u_{h} ⊗ u_{h}}{|(z_{h}, u_{h})|}) ∇u_{h} + g′ (|(z_{h}, u_{h})|) \frac{u_{h} ⊗ z_{h}}{|(z_{h}, u_{h})|}∇z_{h},

where I is the identity matrix. Since in D_{h}^{−} it holds

|gI + g′ \frac{u_{h} ⊗ u_{h}}{|(z_{h}, u_{h})|}| + |g′ \frac{u_{h} ⊗ z_{h}}{|(z_{h}, u_{h})|}| ≤ C,

we have

|M_{l} ((gI + g′ \frac{u_{h} ⊗ u_{h}}{|(z_{h}, u_{h})|}) ∇u_{h})| ≤ |M_{l} (gI + g′ \frac{u_{h} ⊗ u_{h}}{|(z_{h}, u_{h})|})| |M_{l} (∇u_{h})| ≤ C |M_{l} (∇u_{h})| ,

and

|M_{l} (g′ \frac{u_{h} ⊗ z_{h}}{|(z_{h}, u_{h})|}∇z_{h})| ≤ |M_{l} (g′ \frac{u_{h} ⊗ z_{h}}{|(z_{h}, u_{h})|})| |M_{l} (∇z_{h})| ≤ \{ \begin{matrix}0 & l > 1, \\ C |M_{1} (∇z_{h})| & l = 1, \end{matrix}

where M_{l}(X) is the vector whose components are all the minors of X of order l. Here we have used the facts that

|M_{l}(X + Y)| ≤ C ∑_{i=0}^{l} |M_{i}(X)| |M_{l−i}(Y)|,

that

|M_{l}(X Y)| ≤ |M_{l}(Y)| |M_{l}(Y)|,

and that u_{h} ⊗ z_{h} is a rank one matrix. Then, in view of (4.9),

|M_{l}(∇v_{h}(y))| ≤ C ∑_{i=0}^{l} |M_{i} ((gI + g′ \frac{u_{h} ⊗ u_{h}}{|(z_{h}, u_{h})|}) ∇u_{h})| |M_{l−i} (g′ \frac{u_{h} ⊗ z_{h}}{|(z_{h}, u_{h})|}∇z_{h})|
≤ C (|M_{l} (∇u_{h})| + |M_{l−1} (∇u_{h})| |M_{1} (∇z_{h})|)
≤ C (|M_{l} (∇u_{h})| + ε_{h} |(z_{h}, u_{h})| |M_{l−1} (∇u_{h})|) .

As |(z_{h}, u_{h})| ≤ e^{i_{P} +1} for x ∈ D_{h}^{−}, we conclude that (4.10) holds.

Since D_{h} ⊂ Q_{h}, it follows from (4.6)

(5.11) \frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{h→∞} ∫_{D_{h}} f_{j}(x_{0}, v_{0}, ∇v_{h}) dy − ε − εK.

By (4.4) and (4.8)

∫_{D_{h}^{+}} f_{j}(x_{0}, v_{0}, ∇v_{h}) dy ≤ C_{j} ∫_{D_{h}^{+}} (1 + |M(∇v_{h})|) dy = C_{j} |D_{h}^{+}| → 0, (5.12)

while from (4.4), (4.10) and (4.7), and taking h > h_{P} so that ε_{h}e^{P +1} < 1,

∫_{D_{h}^{−}} f_{j}(x_{0}, v_{0}, ∇v_{h}) dy ≤ C_{j} ∫_{D_{h}^{−}} (1 + |M(∇v_{h})|) dy ≤ CC_{j} ∫_{D_{h}^{−}} (1 + |M(∇^{k}w_{h}(y))|) dy ≤ CC_{j} \frac{1 + K}{P} . (5.13)

Consequently, in view of (4.11), (4.12) and (4.13)

(5.14) \frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{h→∞} ∫_{Q} f_{j}(x_{0}, v_{0}, ∇v_{h}) dy − ε − εK − CC_{j} \frac{1 + K}{P} ,

and by (4.2), (4.7), and in view of the fact that v_{h} ≡ 0 in D_{h}^{+},

sup_{h} ∫_{Q} |M(∇v_{h})| dy ≤ K_{1} < ∞, (5.15)

where K_{1} is independent of h and j. Define

h_{j,ε} (v) := h_{j}(x_{0}, v_{0}, v) + ε |v| , f_{j,ε} (ξ) := h_{j,ε} (M(ξ)) .

Then by (4.14), (4.15), and Proposition 3

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{h→∞} ∫_{Q} h_{j,ε}(M(∇v_{h}(y))) dy − ε − ε (K + K_{1}) − CC_{j} \frac{1 + K}{P}

≥ h_{j,ε}(M(∇^{k}w_{0}(x_{0}))) − ε − ε (K + K_{1}) − CC_{j} \frac{1 + K}{P}

= f_{j}(x_{0}, u(x_{0}), ⋯, ∇^{k}u(x_{0})) + ε |M(∇^{k}w_{0}(x_{0}))| − ε − ε (K + K_{1}) − CC_{j} \frac{1 + K}{P} .

Letting first P → ∞, then taking the supremum in j yields

\frac{dμ}{dL^{N}} (x_{0}) ≥ f(x_{0}, u(x_{0}), ⋯, ∇^{k}u(x_{0})) + ε |M(∇^{k}w_{0}(x_{0}))| − ε − ε (K + K_{1}) ,

by (4.3). To complete the proof it suffices to let ε → 0^{+}.

References

[1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Applied Math. 12 (1959), 623–727.

[2] M. Amar and V. De Cicco, Relaxation of quasi-convex integrals of arbitrary order, Proc. Roy. Soc. Edin. 124 (1994), 927–946.

[3] L. Ambrosio, New lower semicontinuity results for integral functionals, Rend. Accad. Naz. Sci. XL, 11 (1987) 1-42.

[4] L. Ambrosio and G. Dal Maso, On the relaxation in BV (Ω; ℝ^{m}) of quasi-convex integrals, J. Funct. Anal., 109 (1992) 76-97.

[5] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Mathematical Monographs, Oxford University Press, 2000.

[6] J. Ball, J. Currie and P. Olver, Null lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal. 41 (1981), 315–328.

[7] L. Boccardo, D. Giachetti, J.I. Diaz and F. Murat, Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms, J. Diff. Eq. 106 (1993), 215–237.

[8] A. Braides, I. Fonseca and G. Leoni, A-quasiconvexity: relaxation and homogenization, to appear in ESAIM:COCV.

[9] M. Carriero, A. Leaci and F. Tomarelli, Special bounded hessian and elastic-plastic plate, Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 16 (1992), 223–258.

[10] M. Carriero, A. Leaci and F. Tomarelli, Strong minimizers of Blake & Zisserman functional, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 257–285.

[11] M. Carriero, A. Leaci and F. Tomarelli, A second order model in image segmentation: Blake & Zisserman functional, Progr. Nonlinear Differential Equations Appl., 25, Birkhäuser 25 (1996), 57–72.

[12] R. Černý and J. Malý, Counterexample to lower semicontinuity in Calculus of Variations, to appear in Math. Z.

[13] B. Dacorogna, Direct methods in the calculus of variations, Springer-Verlag, New York, 1989.

[14] G. Dal Maso and C. Sbordone, Weak lower semicontinuity of polyconvex integrals: a borderline case, Math. Z. 218 (1995), 603–609.

[15] E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82 (1988), 199–210.

[16] L.C. Evans and R.F. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions, Studies in Advanced Math., CRC Press, 1992.

[17] I. Fonseca and G. Leoni, On lower semicontinuity and relaxation, to appear in the Proc. Royal Soc. Edin..

[18] I. Fonseca and G. Leoni, Some remarks on lower semicontinuity, to appear in Indiana Univ. Math. J.

[19] I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L^{1}, SIAM J. Math. Anal. 23 (1992), 1081–1098.

[20] I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in BV(Ω, ℝ^{p}) for integrands f(x, u, ∇u), Arch. Rat. Mech. Anal. 123 (1993), 1–49.

[21] Fonseca I. and S. Müller, A-quasiconvexity, lower semicontinuity and Young measures, SIAM J. Math. Anal., 30 (1999) 1355-1390.

[22] N. Fusco, Quasiconvessità e semicontinuità per integrali multipli di ordine superiore, Ricerche Mat. 29 (1980), 307–323.

[23] N. Fusco and J. E. Hutchinson, A direct proof for lower semicontinuity of polyconvex functionals, Manus. Math. 85 (1995), 35–50.

[24] M. Guidorzi and L. Poggiolini, Lower semicontinuity for quasiconvex integrals of higher order, NoDEA 6 (1999), 227– 246.

[25] P. Marcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals quasiconvex integrals, Manus. Math. 51 (1985), 1–28.

[26] N. Meyers, Quasi-convexity and lower semi-continuity of multiple variational integrals of any order, Trans. A. M. S. 119 (1965), 125–149.

[27] C. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25–53.

[28] J. Serrin, On the definition and properties of certain variational integrals, Trans. A. M. S. 161 (1961), 139–167.

[29] W.P. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Springer-Verlag, New York, 1989.

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213

E-mail address: fonseca@cmu.edu

Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy 15100

E-mail address: leoni@unipmn.it

Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83,186 75 Praha 8, Czech Republic

E-mail address: maly@karlin.mff.cuni.cz

Dipartimento di Ingegneria Civile, Università degli Studi di Udine, Udine, Italy 33100

E-mail address: roberto.paroni@dic.uniud.it