RELAXATION OF CONVEX FUNCTIONALS: THE GAP PROBLEM

E. ACERBI, G. BOUCHITTÉ, AND I. FONSECA

Abstract. It is shown that the relaxed energy

F(u, A) := inf{lim inf_{n→+∞} ∫_{A} f(x, ∇u_{n})dx : {u_{n}} ⊂ W_{loc}^{1,β}(A) , u_{n} → u in L^{1}(A)},

admits the representation

F(u, A) = ∫_{A} f(x, ∇u)dx + μ_{s}(A) ,

where f is a convex, Carathéodory integrand satisfying a non standard “α-β”growth hypothesis, β ∈ [α, Nα/(N − 1)). Sufficient conditions guaranteeing that μ_{s} = 0 are discussed. An example asserting that this representation may fail in the quasiconvex case is provided.

2000 Mathematics Subject Classification 49J45, 74B20 Keywords: convexity, Besicovich Covering Theorem, Radon Nikodym derivative, Lavrentiev phenomenon

Contents

1. Introduction 1 2. Preliminaries 4 3. A Representation Theorem 7 4. Remarks on the Lavrentiev Phenomenon 17 5. A Counterexample for Carathéodory Quasiconvex Integrands 20 References 21

1. Introduction

The Lavrentiev phenomenon, or gap problem, has stirred renewed interest in recent years as it challenges traditional theories in the Calculus of Variations. A prototype model, relevant to the study of cavitation in rubber-like materials (see [3], [20], [27], [28], among others), assigns to each deformation u ∈ W^{1,N}(Ω; ℝ^{N}) the total energy

F (u, Ω) := ∫_{Ω} |∇u|^{N−1} + |det∇u| dx, (1.1)

where Ω is an open, bounded domain in ℝ^{N}. Clearly sequences of deformations in W^{1,N}(Ω; ℝ^{N}) with bounded energy will be weakly compact in W^{1,N−1}(Ω; ℝ^{N}) but not necessarily in W^{1,N}, so it may be possible to approach energetically functions u ∈ W^{1,N−1}(Ω; ℝ^{N}) \ W^{1,N}(Ω; ℝ^{N}). We then seek to characterize the limiting, effective energy associated to u. This example has been studied at length, and in particular we refer to [1], [9], [16]. More generally, consider a bulk energy density f : Ω × ℝ^{d×N} → [0, +∞) satisfying the following standing hypotheses:

(H1) f is Carathéodory; (H3) [“α-β”growth condition] |z|^{α} ≤ f(x, z) ≤ C(1 + |z|^{β}) for all z ∈ ℝ^{d×N}, L^{N} a.e. x ∈ Ω, and for some

C > 0, where 1 < α ≤ β < \frac{Nα}{N−1}.

The research of E. Acerbi was partially supported by GNAFA-CNR under the Research Project “Modelli variazionali sotto ipotesi non standard – 2000.”

The research of I. Fonseca was partially supported by the National Science Foundation under Grant No. DMS–9731957 and by the Center for Nonlinear Analysis (NSF Grant No. DMS–9803791). I. Fonseca also thanks the Universities of Parma and of Toulon et Var for their hospitality during the final stage of the paper.

The authors are indebted to the Max-Planck Institute in Leipzig for its support during the preparation of this work.

For every Borel set E ⊂ Ω and every u ∈ W_{loc}^{1,1}(Ω; ℝ^{d}) we set

F (u, E) := ∫_{E} f(x, ∇u(x)) dx,

and we introduce the relaxed functional

F(u, A) := inf{lim inf_{n→+∞} F (u_{n}, A) : {u_{n}} ⊂ W_{loc}^{1,β}(A), u_{n} → u in L^{1}(A)}.

We search for an integral representation for F(u, A).

When α = β and (H1), (H3) hold, then it is well known that (see [AF], [B], [D], [M])

F(u, A) = ∫_{A} Qf(x, ∇u) dx,

where the quasiconvex envelope Qf of f is defined by

Qf(ξ) := inf {∫_{(0,1)^{N}} f(ξ + ∇φ(x)) dx : φ ∈ W_{0}^{1,∞}(Q; ℝ^{d})} .

If α < β − β/N then one may have F(u, Ω) = 0 (see [4]), and in the case where α = β − β/N it may happen that F(u, ·) is not even subadditive (see [11]). When f does not depend on the position vector x ∈ Ω these degeneracies cannot occur if β ∈ [α, αN/(N − 1)). Within this range, and using the global method for relaxation introduced by Bouchitté, Fonseca and Mascarenhas (see [7]), together with an extension operator from W^{1,p} into W^{1,q} obtained by Fonseca and Malý (see [15], Lemma 2.2), it was proven in [6], Theorem 3.1, that if f = f(∇u) satisfies (H3) and if F(u, Ω) < +∞ then

F(u, A) = ∫_{A} Qf(∇u) dx + μ_{s}(A) (1.2)

for all Borel sets A ⊂ Ω and for some finite, Radon measure μ_{s}, singular with respect to L^{N} Ω, the N-dimensional Lebesgue measure in Ω. Earlier results on lower semicontinuity for certain ranges α < β and with quasiconvex integrands were obtained by [24], [25], and for polyconvex energy densities with α ≥ N −1, β = N, we refer to [1], [8], [9], [11], [12], [17], [18], [21], [22], [23].

In this paper we address the effect of considering an inhomogeneous density, i.e. f = f(x, ξ). In the main theorem of this paper we show that the analogue to (1.2),

F(u, A) = ∫_{A} Qf(x, ∇u) dx + μ_{s}(A) (1.3)

still holds provided f satisfies (H1), (H3) and

(H2) f(x, ·) is convex for L^{N} a.e. x ∈ Ω.

Precisely, we prove

Theorem 1.1. If (H1)–(H3) hold, if A is an open subset of Ω, if u ∈ W^{1,α}(Ω), and if F(u, A) < +∞, then for every open set B ⊂ A

F(u, B) = ∫_{B} f(x, ∇u) dx + μ_{s}(B)

where μ_{s} is a nonnegative Radon measure singular with respect to L^{N}.

Again the proof of this result is strongly hinged on the global method of relaxation introduced in [7] (see also [5]), although some of the arguments may now be greatly simplified by exploiting the convexity assumption. In particular, the uplifting operator P introduced in [15] (see also [6]) is no longer needed. The proof presented in Section 3 concerns the scalar case where d = 1 and the argument is entirely similar in the vector valued-scalar case. Therefore and for simplicity, we leave the obvious adaptations to the reader.

Section 4 is devoted to the study of several sufficient conditions ensuring that the Lavrentiev phenomenon does not occur. In view of Ioffe’s Theorem (see Theorem 2.4), the functional F is lower semicontinuous and thus F ≤ F. We then say that the Lavrentiev phenomenon occurs when at some point we have the strict inequality F < F.

The first result asserts that the Lavrentiev phenomenon is local, precisely, Proposition 1.2. If for every x ∈ Ω there is a ball B = B_{ρ(x)}(x) ⊂ Ω such that

F (u, B_{ρ(x)}(x)) = F (u, B_{ρ(x)}(x)) (1.4)

then F(u, ·) = F (u, ·).

An interesting model example of convex functionals depending on x and with a growth-coercivity gap as in (H3) is provided by ∫ |Du|^{p(x)} dx for some function p : Ω → ℝ (see [29], [30]), or, more generally,

u ↦ ∫_{Ω} f(x, ∇u(x)) dx

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

(1.5) |z|^{p(x)} ≤ f(x, z) ≤ C(1 + |z|^{p(x)}) for all z ∈ ℝ^{N}, a.e. x ∈ Ω,

and for some C > 0. The proof of Theorem 1.3 below follows closely the original argument of Zhikov and Fan (see [29]), although some of the technical difficulties encountered by those authors may now be avoided using the theory developed in this paper.

Let p : Ω ↦ ℝ be a continuous function such that

1 < α ≤ p(x) ≤ β

and

(1.6) |p(x) − p(y)| ≤ \frac{γ}{|log |x − y||} whenever 0 < |x − y| ≤ \frac{1}{2},

for some α, β, γ > 0.

Theorem 1.3. Let f : Ω × ℝ^{N} → [0, +∞) satisfy (H1), (H2), (1.5), and assume that (1.6) holds. If u ∈ W_{loc}^{1,α}(Ω) is such that f(·, ∇u) ∈ L_{loc}^{1}(Ω) then

F(u, ·) = F (u, ·).

The last section of this work, Section 5, is dedicated in its entirety to the treatment of an example falling into the class of prototype energies (1.1) which satisfy (H1), (H3), are polyconvex rather than convex, and for which a genuine measurable dependence on x prevents (1.3) to hold. This fact provides one more evidence that quasiconvex energies and convex energies do share quite different properties, and that care must be taken when generalizing results from the convex to the non-convex setting. With Proposition 1.4 we show that Theorem 1.1 may be false if stated for integrands f : Ω × ℝ^{N} × ℝ^{d×N} → [0, +∞), d, N > 1, satisfying (H1), (H3), and

(H2′) f(x, ·) is quasiconvex for L^{N} a.e. x ∈ Ω.

We recall that a Borel function g : ℝ^{d×N} → ℝ is said to be quasiconvex (see [Morrey]) if

∫_{(0,1)^{N}} g(ξ + ∇φ(y)) dy ≥ g(ξ)

for all ξ ∈ ℝ^{d×N} and all φ ∈ W_{0}^{1,∞}((0, 1)^{N}; ℝ^{d}). Clearly convex functions are quasiconvex, and a simple example of a quasiconvex function which is not convex is given by ξ ↦ | det ξ| (for a detailed study of quasiconvexity we refer the reader to [1], [2], [9], [11], [12], [16], [17], [23], [24], [25], [26]). In the absence of growth-coercivity gap, i.e. when α = β in (H3), it was proven by Acerbi and Fusco (see [2]), that F (·, Ω) is sequentially weakly lower semicontinuous in W^{1,α}(Ω; ℝ^{d}), and, in addition, it can be shown that (see also [10])

F(u, Ω) = ∫_{Ω} f(x, ∇u(x)) dx.

If there is a gap and under (H1), (H3), then Theorem 3.1 in [15] still holds, and so for u ∈ W^{1,α}(Ω; ℝ^{d}) such that F(u, Ω) < +∞ we have

F(u, A) = ∫_{A} f^{u}(x) dx + μ_{s}(A) for all A open subset of Ω ,

where μ_{s} is a non-negative finite Radon measure on Ω, singular with respect to L^{N} Ω, and for some density f^{u}.

The next natural question is: under which conditions can we guarantee that

f^{u}(x) = f(x, ∇u(x)) for a.e. x ∈ Ω ? (1.7)

Theorem 1.1 tells us that (H2), i.e. the convexity of f(x, ·) for a.e. x ∈ Ω, implies the identity in (1.7), and Theorem 3.2 in [6] ensures (1.7) if (H2′) holds and if f does not depend on x.

Here we show that this last statement is close to being optimal, in that if f has a genuine measurable dependence on x, and even though (H1), (H2′), (H3) still hold, then it may happen that

f^{u}(x) < f(x, ∇u(x)) for all x ∈ E,

where E is some set with |E| > 0. The construction of such function f is motivated by the result of Gangbo (see [18], Theorem 3.1) asserting that

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

is lower semicontinuous in W^{1,N}(Ω; ℝ^{N}) for the weak topology of W^{1,p}(Ω; ℝ^{N}), with N − 1 < p < N, if and only if |∂K| = 0.

Proposition 1.4. Fix θ ∈ (0, 1), let p ∈ (N − 1, N), let Ω ⊂ ℝ^{N} be an open, bounded domain, let K be a compact subset of Ω with |∂K| > 0, and let f : Ω × ℝ^{N×N} → [0, +∞) be given by

f(x, ξ) := θ|ξ|^{p} + χ_{K} | det ξ| for a.e. x ∈ Ω, and all ξ ∈ ℝ^{N×N}.

If id(·) is the identity function and 𝕀 is the identity matrix, then there exists θ_{0} ∈ (0, 1) such that if θ < θ_{0} then for all x ∈ ∂K

f^{id}(x) < f(x, 𝕀).

It is clear that f satisfies (H1), (H3) with α := p, β := N, and f(x, ·) is quasiconvex. In light of Proposition 1.4, we have

F(id, A) = ∫_{A} f^{id}(x) dx + μ_{s}(A) ,

where there holds, for θ sufficiently small, f^{id}(x) < f(x, 𝕀) for all x ∈ ∂K.

2. Preliminaries

Let Ω be an open, bounded subset of ℝ^{N}, and consider a function f : Ω × ℝ^{N} → [0, +∞) satisfying the following standing hypotheses:

(H1) f is Carathéodory; (H2) f(x, ·) is convex for L^{N} a.e. x ∈ Ω; (H3) |z|^{α} ≤ f(x, z) ≤ C(1 + |z|^{β}) for all z ∈ ℝ^{N}, L^{N} a.e. x ∈ Ω, and for some C > 0, where

1 < α ≤ β < \frac{Nα}{N − 1}. (2.1)

We recall that f is said to be Carathéodory if f(·, z) is L^{N}-measurable for all z and f(x, ·) is continuous for L^{N} a.e. x ∈ Ω, where L^{N} stands for the N-dimensional Lebesgue measure on ℝ^{N}. Often we will write |E| in place of L^{N}(E). Also, throughout this paper the letter C will denote a generic positive constant which may vary from line to line and within the same formula. Note that in view of (2.1), if A ⊂ ℝ^{N} is an open set with Lipschitz boundary then the inclusion W^{1,α}(∂A) ⊂ W^{1−1/β,β}(∂A) is continuous and compact, where W^{1−1/β,β}(∂A) is the space of traces on ∂A of functions in W^{1,β}(A). More generally standard results in the theory of Sobolev spaces yield the lifting property presented below.

Lemma 2.1. If A ⊂ ℝ^{N} is an open, bounded, Lipschitz domain and if (2.1) holds, then there exists a continuous, linear, compact mapping

E_{A} : W^{1,α}(∂A) → W^{1,β}(A)  such that  tr E_{A}(g) = g on ∂A.

Remark 2.2. Since the “uplift” operator E_{A} of Lemma 2.1 is constructed with symmetric convolution kernels, it is readily verified that affine functions remain unchanged, i.e., if u(x) = a + b · x then E_{A} u = u.

Remark 2.3. (i) The growth hypothesis (H3) together with the convexity of f imply that f is β-locally Lipschitz continuous with respect to z, i.e., for a.e. x ∈ Ω and all z, w ∈ ℝ^{N} (see [10], [24], [25])

|f(x, z) − f(x, w)| ≤ C(1 + |z|^{β−1} + |w|^{β−1})|z − w|. (2.2)

(ii) Let {z_{i}} be a countable dense subset of ℝ^{N}, and for all i let K_{i} be the set of Lebesgue points of the function f(·, z_{i}). Setting 𝕂(f) := ∩K_{i}, then |Ω \ 𝕂(f)| = 0, and if x_{0} ∈ 𝕂(f) then it is a Lebesgue point for f(·, z) for all z ∈ ℝ^{N}. Indeed, fix 0 < ε < 1, let M := 1 + |z|, and choose z_{i} with |z − z_{i}| < ε. By (2.2)

|f(x, z) − f(x_{0}, z)| ≤ CM^{β−1}|z − z_{i}| + |f(x, z_{i}) − f(x_{0}, z_{i})|,

thus for ρ > 0,

− ∫_{B_{ρ}(x_{0})} |f(x, z) − f(x_{0}, z)| dx ≤ CM^{β−1}ε + − ∫_{B_{ρ}(x_{0})} |f(x, z_{i}) − f(x_{0}, z_{i})| dx

and since x_{0} ∈ ∩K_{i}, we now have

lim sup_{ρ→0} − ∫_{B_{ρ}(x_{0})} |f(x, z) − f(x_{0}, z)| dx ≤ CM^{β−1}ε.

The assertion follows by letting ε → 0.

For every Borel set E ⊂ Ω and every u ∈ W_{loc}^{1,1}(Ω), we set

F (u, E) := ∫_{E} f(x, ∇u(x)) dx.

Given an open set A ⊂ Ω and u ∈ W_{loc}^{1,1}(Ω), the set of admissible sequences for u in A is defined by

A(u; A) := {{u_{n}} ⊂ W_{loc}^{1,β}(A) : u_{n} → u in L^{1}(A)} ,

and we introduce the relaxed functional

(2.3) F(u, A) := inf{lim inf_{n→+∞} F (u_{n}, A) : {u_{n}} ∈ A(u, A)}.

We define the set of good sequences for u in A to be

G(u, A) := {{u_{n}} ∈ A(u, A) : F(u, A) = lim_{n→+∞} F (u_{n}, A)} . (2.4)

It is clear that G(u, A) is nonempty. If F (u, A) < +∞ then by (H3) we have that ∇u ∈ L^{α}(A), and so for almost all balls B_{ρ} ⊂ A the trace of u on ∂B_{ρ} belongs to W^{1,α}(∂B_{ρ}). In view of Lemma 2.1, we may define for every such ball

m(u, B_{ρ}) := inf {F (v, B_{ρ}) : v ∈ W^{1,β}(B_{ρ}), v − u ∈ W_{0}^{1,α}(B_{ρ})} ,

and we also set

F_{av}(u, A) := \frac{1}{|A|}F(u, A) , m_{av}(u, B_{ρ}) = \frac{1}{|B_{ρ}|}m(u, B_{ρ}),

where B_{ρ} stands for a ball B(x, ρ) of center x ∈ Ω and radius ρ > 0.

Next we recall a semicontinuity result by Ioffe (see [19]).

Theorem 2.4. Let g : Ω × ℝ^{d} × ℝ^{d×N} → [0, +∞) be a Carathéodory function such that g(x, u, ·) is convex for every u ∈ ℝ^{d} and for L^{N} a.e. x ∈ Ω. Then the functional

G(u) := ∫_{Ω} g(x, u, ∇u) dx

is lower semicontinuous on W^{1,1}(Ω; ℝ^{d}) with respect to the weak convergence in W^{1,1}(Ω; ℝ^{d}).

By Ioffe’s Theorem, the functional F is lower semicontinuous, thus

F ≤ F .

On the other hand, if u ∈ W_{loc}^{1,β}(A) ∩ W^{1,α}(A) then the constant sequence {u} is admissible and thus F(u, A) ≤ F (u, A). This, together with the previous inequality, yields

(2.5) F(u, A) = F (u, A) whenever u ∈ W_{loc}^{1,β}(A) ∩ W^{1,α}(A) .

Using a good sequence in G(u, A) to approach F(u, A), and in view of (2.5), it now follows that

F = sup \{H : H ≤ \overline{F} , H is L^{1} lower semicontinuous\}

where

\overline{F} (u, A) := \{ \begin{matrix}F (u, A) & if u ∈ W_{loc}^{1,β}(A), \\ +∞ & otherwise. \end{matrix}

The following lemma will be instrumental to show that F(u, ·) is the trace in 𝔸(Ω) of a positive Radon measure, where 𝔸(Ω) denotes the class of open subsets of Ω.

Lemma 2.5. Let λ : 𝔸(Ω) → [0, +∞) and μ be such that

(i) μ is a finite Radon measure on Ω; (ii) λ(Ω) ≥ μ(Ω); (iii) λ(A) ≤ μ(\overline{A}) for all A ∈ 𝔸(Ω); (iv) (subadditivity) λ(A) ≤ λ(A \ \overline{C}) + λ(B) for all A, B, C ∈ 𝔸(Ω) such that C ⊂⊂ B ⊂ A;

(v) (inner regularity) for all A ∈ 𝔸(Ω), ε > 0, there exists C ∈ 𝔸(Ω) such that C ⊂⊂ A and λ(A \ \overline{C}) < ε.

Then λ = μ on 𝔸(Ω).

Proof. We start by showing that

λ(A) ≤ μ(A) for all A ∈ 𝔸(Ω). (2.6)

Fix A ∈ 𝔸(Ω), ε > 0. By (iv) and (v) we may find open sets C ⊂⊂ B ⊂⊂ A such that λ(A \ \overline{C}) < ε and

λ(A) ≤ λ(A \ \overline{C}) + λ(B) ≤ ε + μ(\overline{B}) ≤ ε + μ(A),

where we used (iii). Clearly (2.6) now follows by letting ε → 0^{+}.

In order to prove the converse inequality, since μ is inner regular for ε > 0 we may find an open set A′ ⊂⊂ A such that μ(A) ≤ ε + μ(\overline{A′}), and by virtue of (ii), (2.6), and (iv),

μ(A) ≤ ε + μ(\overline{A′}) = ε + μ(Ω) − μ(Ω \ \overline{A′}) ≤ ε + λ(Ω) − λ(Ω \ \overline{A′}) ≤ ε + λ(A).

By the arbitrariness of ε > 0, we now conclude that

μ(A) ≤ λ(A).

Remark 2.6. If (i) and (iii) in the statement of Lemma 2.5 hold, then the inner regularity property (v) is satisfied provided we can show that there exists C > 0 such that if U ∈ 𝔸(Ω), U = ∪_{i=1}^{∞}U_{i}, U_{i} ∈ 𝔸(Ω), U_{i} ⊂⊂ U_{i+1} ⊂⊂ U, then

λ (U \ \overline{U_{1}}) ≤ C ∑_{i=1}^{∞} λ (U_{i+3} \ \overline{U_{i}}) . (2.7)

Indeed, if U ∈ 𝔸(Ω), take U_{i} := {x ∈ U : dist(x, ∂U) > 1/i}, and choose k large enough so that

μ(U \ U_{k}) < \frac{ε}{4C} .

We have

λ(U \ \overline{U_{k}}) ≤ C ∑_{i=k}^{∞} λ (U_{i+3} \ \overline{U_{i}}) ≤ C ∑_{i=k}^{∞} μ (\overline{U_{i+3} \ \overline{U_{i}}})

≤ C ∑_{i=k}^{∞} μ(U_{i+4} \ U_{i}) ≤ 4C μ(U \ U_{k}) < ε.

3. A Representation Theorem

In this section we prove Theorem 1.1. We start by showing that F(u, ·) is the trace in 𝔸(Ω) of a Radon measure. For every A ∈ 𝔸(Ω) set

λ(A) := F(u, A) .

Consider a sequence {u_{n}} ∈ G(u, A) such that

F(u, Ω) = lim_{n→+∞} F (u_{n}, Ω) = lim_{n→+∞} ∫_{Ω} f(x, ∇u_{n}) dx,

and define the Radon measures on ℝ^{N}

λ_{n} := f(x, ∇u_{n})L^{N} Ω .

Since these measures are equibounded, and up to the extraction of a subsequence, we have

λ_{n} ⇀^{∗} μ in the sense of measures

for some finite Radon measure μ, i.e.,

∫_{Ω} φ dλ_{n} → ∫_{Ω} φ dμ for all φ ∈ C_{c}(Ω).

Note that

(3.1) μ(Ω) ≤ lim inf_{n→+∞} λ_{n}(Ω) = lim_{n→+∞} ∫_{Ω} f(x, ∇u_{n}) dx = λ(Ω),

and if A ∈ 𝔸(Ω) then

(3.2) λ(A) ≤ lim inf_{n→+∞} ∫_{A} f(x, ∇u_{n}) dx = lim inf_{n→+∞} λ_{n}(A) ≤ μ(\overline{A}).

Therefore, by virtue of Lemma 2.5 we have

λ = μ on 𝔸(Ω)

provided λ satisfies the subadditivity and inner regularity properties (iv) and (v). These depend mostly on the growth condition (H3) and on (2.1), and were asserted in [15] (see Lemma 2.4) by means of an uplifting operator P more refined and complex than the operator E_{A} introduced in Lemma 2.1. Below we give an alternative proof which relies heavily on the convexity assumption (H2), and it bypasses the need to exploit the operator P.

Proposition 3.1. If A, B, C ∈ 𝔸(Ω), C ⊂⊂ B ⊂ A, then

F(u, A) ≤ F(u, B) + F(u, A \ \overline{C}) .

Proof. Fix \{v_{n}\} ∈ G(u, B) and \{w_{n}\} ∈ G(u, A\\overline{C}), let φ ∈ C_{c}^{∞}(B) be such that 0 ≤ φ ≤ 1, \{φ < 1\} ⊂⊂ A\\overline{C}, and set

u_{n} := φv_{n} + (1 − φ)w_{n} ,

where φv_{n} is taken to be zero there where φ = 0, even though v_{n} may be not defined on that set, and analogously (1 − φ)w_{n} is to be equal to zero on {φ = 1}. Clearly {u_{n}} ∈ A(u, A), and for all 0 < t < 1 also {tu_{n}} ∈ A(tu, A), thus

F(tu, A) ≤ lim inf_{n→+∞} F (tu_{n}, A) . (3.3)

By the convexity of f, and since 0 < t < 1, 0 ≤ φ ≤ 1, we may find some positive constant M such that

F (tu_{n}, A) = ∫_{A} f(x, t[φ∇v_{n} + (1 − φ)∇w_{n}] + (1 − t)\frac{v_{n} − w_{n}}{1 − t} t∇φ) dx
≤ t ∫_{A} f(x, φ∇v_{n} + (1 − φ)∇w_{n}) dx + (1 − t) ∫_{A} M (1 + |\frac{v_{n} − w_{n}}{1 − t} t∇φ|^{β}) dx
≤ ∫_{A}[φ(x)f(x, ∇v_{n}) + (1 − φ(x))f(x, ∇w_{n})] dx + M(1 − t)|A|

+M \frac{∥∇φ∥_{L^{∞}}^{β}}{(1 − t)^{β−1}} ∫_{B\\overline{C}} |v_{n} − w_{n}|^{β} dx

≤ F (v_{n}, B) + F (w_{n}, A \ \overline{C}) + M(1 − t)|A| + M \frac{∥∇φ∥_{L^{∞}}^{β}}{(1 − t)^{β−1}} ||v_{n} − w_{n}||_{L^{β}(B\\overline{C})}^{β},

where we have used the growth condition in (H3). Also, by the coercivity assumption in (H3), we deduce that \{∥∇v_{n}∥_{L^{α}(B\\overline{C})} + ∥∇w_{n}∥_{L^{α}(B\\overline{C})}\} is bounded, and since v_{n} − w_{n} → u − u = 0 in L^{1}(B \ \overline{C}), we have that v_{n} − w_{n} ⇀ 0 weakly in W^{1,α}(B \ \overline{C}), and therefore strongly in L^{β}(B \ \overline{C}) by (2.1) and Rellich Theorem. This, together with (3.3), yields

F(tu, A) ≤ F(u, B) + F(u, A \ \overline{C}) + C(1 − t)|A|,

and the result now follows by letting t ↗ 1 and using the L^{1} lower semicontinuity property of F(·, A).

Finally, and in light of Remark 2.6, in order to assert inner regularity for λ it suffices to establish (2.7).

Proposition 3.2. If U ∈ 𝔸(Ω), U = ∪_{i=1}^{∞}U_{i}, U_{i} ∈ 𝔸(Ω), U_{i} ⊂⊂ U_{i+1} ⊂⊂ U, then

F (u, U \ \overline{U_{1}}) ≤ 2 ∑_{i=1}^{∞} F (u, U_{i+3} \ \overline{U_{i}}) .

Proof. Fix U ∈ 𝔸(Ω) and suppose that U = ∪_{i=1}^{∞}U_{i}, U_{i} ∈ 𝔸(Ω), U_{i} ⊂⊂ U_{i+1} ⊂⊂ U. Choose {u_{n}^{(i)}} ∈ G(u, U_{i+2} \ \overline{U_{i}}) and \{u_{n}^{∞}\} ∈ G(u, U), so that, recalling that λ = F(u, ·),

λ(U_{i+2} \ \overline{U_{i}}) = lim_{n→+∞} ∫_{U_{i+2}\\overline{U_{i}}} f(x, ∇u_{n}^{(i)}) dx, λ(U) = lim_{n→+∞} ∫_{U} f(x, ∇u_{n}^{∞}(x)) dx.

Let φ_{i} be a smooth cut-off function such that φ_{i} = 1 in U_{i}, φ_{i} = 0 outside U_{i+1}, and {0 < φ_{i} < 1} ⊂⊂ U_{i+1} \ \overline{U_{i}}.

Up to the extraction of a subsequence, we may assume that f(·, ∇u_{n}^{∞}) U ⇀^{∗} η, where η is a finite Radon measure on U. Fix ε > 0 and choose M ∈ ℕ large enough so that η(U \ \overline{U_{M}}) < ε . Define

ū_{n} : = χ_{U_{2}\\overline{U_{1}}} u_{n}^{(1)} + ∑_{i=2}^{M} χ_{U_{i+1}\U_{i}} [φ_{i}u_{n}^{(i−1)} + (1 − φ_{i})u_{n}^{(i)}]
+χ_{U\U_{M+1}} [φ_{M+1}u_{n}^{(M)} + (1 − φ_{M+1})u_{n}^{∞}] .

Clearly ū_{n} ∈ A(u, U \ \overline{U_{1}}), and we have

λ(U \ \overline{U_{1}}) ≤ lim inf_{t→1} lim inf_{n→+∞} ∫_{U\\overline{U_{1}}} f(x, t∇ū_{n}) dx
≤ lim sup_{t→1} lim sup_{n→+∞} t ∫_{U_{2}\\overline{U_{1}}} f(x, ∇u_{n}^{(1)}) dx + lim sup_{t→1} (1 − t) ∫_{U_{2}\\overline{U_{1}}} f(x, 0) dx

+ lim sup_{t→1} lim sup_{n→+∞} ∑_{i=2}^{M} ∫_{U_{i+1}\U_{i}} f(x, t∇ū_{n}) dx

(3.4) + lim sup_{t→1} lim sup_{n→+∞} ∫_{U\U_{M+1}} f(x, t∇ū_{n}).

Now

(3.5)  lim sup_{t→1} lim sup_{n→+∞} t ∫_{U_{2}\\overline{U_{1}}} f(x, ∇u_{n}^{(1)}) dx + lim sup_{t→1} (1 − t) ∫_{U_{2}\\overline{U_{1}}} f(x, 0) dx ≤ λ(U_{3} \ \overline{U_{1}}),

and

lim sup_{t→1} lim sup_{n→+∞} ∑_{i=2}^{M} ∫_{U_{i+1}\U_{i}} f(x, t∇ū_{n}) dx

= lim sup_{t→1} lim sup_{n→+∞} ∑_{i=2}^{M} ∫_{U_{i+1}\U_{i}} f(x, t [φ_{i}∇u_{n}^{(i−1)} + (1 − φ_{i})∇u_{n}^{(i)}]

+(1 − t)\frac{u_{n}^{(i−1)} − u_{n}^{(i)}}{1 − t} t ∇φ_{i}) dx

≤ lim sup_{t→1} lim sup_{n→+∞} t ∑_{i=2}^{M} \{∫_{U_{i+1}\\overline{U_{i−1}}} φ_{i}f(x, ∇u_{n}^{(i−1)}) dx

+ ∫_{U_{i+2}\\overline{U_{i}}}(1 − φ_{i})f(x, ∇u_{n}^{(i)}) dx\}

+ lim sup_{t→1} lim sup_{n→+∞} (1 − t) ∑_{i=2}^{M} ∫_{U_{i+1}\\overline{U_{i}}} C (1 + \frac{|u_{n}^{(i−1)} − u_{n}^{(i)}|^{β}}{(1 − t)^{β}} t^{β} |∇φ_{i}|^{β}) dx

≤ ∑_{i=2}^{M} [λ(U_{i+1} \ \overline{U_{i−1}}) + λ(U_{i+2} \ \overline{U_{i}})] , (3.6)

where we have used the fact that u_{n}^{(i−1)} − u_{n}^{(i)} → 0 in L^{β}(U_{i+1} \ \overline{U_{i}}).

In a similar way, we can show that

lim sup_{t→1} lim sup_{n→+∞} ∫_{U\U_{M+1}} f(x, t∇ū_{n}) ≤ lim sup_{n→+∞} ∫_{U_{M+2}\U_{M+1}} f(x, ∇u_{n}^{(M)})

+ lim sup_{n→+∞} ∫_{U\U_{M+1}} f(x, ∇u_{n}^{∞}) dx

≤ λ(U_{M+3} \ \overline{U_{M}}) + η(U \ \overline{U_{M}})
(3.7)  ≤ λ(U_{M+3} \ \overline{U_{M}}) + ε.

In view of (3.4), (3.5), (3.6), and (3.7), we obtain

λ (U \ \overline{U_{1}}) ≤ 2 ∑_{i=1}^{∞} λ (U_{i+3} \ \overline{U_{i}}) + ε.

Letting ε → 0 yields property (2.7).

Proposition 3.3. If (H1)–(H3) hold and if F(u, Ω) < +∞ then there exists a finite Radon measure μ on Ω such that F(u, ·) is the trace of μ on 𝔸(Ω).

Proof. The proof follows immediately from Lemma 2.5, Remark 2.6, (3.1), (3.2), Propositions 3.1 and 3.2.

In the sequel we will often write μ to designate the measure F(u, ·).

Lemma 3.4. If (H1)–(H3) hold and if F(u, Ω) < +∞ then there exists a sequence {u_{n}} with the following properties:

(a) u_{n} ∈ W_{loc}^{1,β}(Ω);

(b) u_{n} ⇀ u in W^{1,α}(Ω);

(c) F(u, Ω) = lim_{n→+∞} F (u_{n}, Ω);

(d) F(u, A) = lim_{n→+∞} F (u_{n}, A) for all open sets A ⊂ Ω such that μ(∂A) = 0;

(e) for all x_{0} ∈ Ω and for almost all 0 < t < R := dist(x_{0}, ∂Ω), there exists a subsequence {u_{k_{n}}} such that F(u, B_{t}(x_{0})) = lim_{n→+∞} F (u_{k_{n}}, B_{t}(x_{0})) and u_{k_{n}} ⇀ u in W^{1,α}(∂B_{t}(x_{0})).

Proof. By the definition of F there exists a sequence {u_{n}} satisfying properties (a), (b), (c). If A ∈ 𝔸(Ω) is such that μ(∂A) = 0 then

lim sup_{n→+∞} F (u_{n}, A) = lim_{n→+∞} F (u_{n}, Ω) − lim inf_{n→+∞} F (u_{n}, Ω \ \overline{A})
≤ F(u, Ω) − F(u, Ω \ \overline{A})
= F(u, A)
≤ lim inf_{n→+∞} F (u_{n}, A)

and property (d) follows. To prove (e), fix x_{0} ∈ Ω and remark that due to the finiteness of the measure μ, property (d) holds for B_{t}(x_{0}) for L^{1} a.e. t ∈ (0, R). With

M := sup_{n} ∫_{B_{R}(x_{0})} |∇u_{n}|^{α} dx < +∞,

define for K > 0

E_{K,n} = ⋂_{m≥n}{t ∈ (0, R) : μ_{s}(∂B_{t}(x_{0})) = 0, ∫_{∂B_{t}(x_{0})} |∇u_{m}|^{α} dH^{N−1} ≥ K}.

Clearly |E_{K,n}| ≤ M/K, and since E_{K,n+1} ⊃ E_{K,n}, we still have |E_{K}| ≤ M/K where E_{K} := ∪_{n}E_{K,n}. As a consequence, if E := ∩_{K}E_{K} then |E| = 0, and if t ∉ E then

lim inf_{n→+∞} ∫_{∂B_{t}(x_{0})} |∇u_{n}|^{α} dH^{N−1} < +∞,

which, together with (d), asserts the existence of a subsequence fulfilling (e).

Definition 3.5. If F(u, Ω) < +∞ then we say that B_{ρ}(x_{0}) is an uplift ball for u if (i) μ(∂B_{ρ}(x_{0})) = 0; (ii) there exists a sequence {u_{n}} satisfying properties (a), (b), and (c) of Lemma 3.4 and u_{n} ⇀ u in

W^{1,α}(∂B_{ρ}(x_{0})).

Remark 3.6. By Lemma 3.4, if F(u, Ω) < +∞ then for all x_{0} ∈ Ω almost all balls centered at x_{0} are uplift balls for u. Also, by means of a diagonal argument we may assume that the sequence {u_{n}} satisfying (ii) in Definition 3.5 is the same for a countable set of uplift balls.

Lemma 3.7. Assume that F(u, Ω) < +∞ and let B be an uplift ball for u. Let v ∈ W^{1,β}(B) satisfy v = u on ∂B, set

ū(x) := \{ \begin{matrix}v(x) & if x ∈ B, \\ u(x) & otherwise, \end{matrix}

and let μ̄ be the measure which extends F(ū, ·) to all Borel sets of Ω. Then

μ̄(∂B) = 0.

Proof. Let {u_{n}} be a sequence as in Definition 3.5 (ii), and let B′ ⊂⊂ B ⊂⊂ B′′ be uplift balls concentric with B, so close together that if we denote by A the annulus B′′ \ \overline{B′}, and using the fact that F(u, ∂B) = 0, then

F(u, A) < ε , F (v, A ∩ B) < ε.

Since the trace on ∂B of u_{n} − u converges to zero weakly on W^{1,α}(∂B), by Lemma 2.1 if we take v_{n} := E_{B}(u_{n} − u) then

v_{n} ∈ W^{1,β}(B) ,  v_{n} = u_{n} − u on ∂B ,  v_{n} → 0 in W^{1,β}(B).

We define in A

ū_{n}(x) := \{ \begin{matrix}u_{n}(x) & if x ∈ A \ B, \\ v_{n}(x) + v(x) & if x ∈ A ∩ B. \end{matrix}

Clearly ū_{n} ∈ W^{1,β}(A), and

F (ū_{n}, A) = F (u_{n}, A \ B) + F (ū_{n}, A ∩ B) ≤ F (u_{n}, A) + F (ū_{n}, A ∩ B).

Since ū_{n} → ū in L^{1}(A) and ū_{n} → v in W^{1,β}(A ∩ B), by the W^{1,β}-continuity of F and the property (d) satisfied by {u_{n}}, in view of the fact that μ(∂A) = 0, we deduce that

F(ū, A) ≤ lim inf_{n→+∞} F (ū_{n}, A)
≤ lim sup_{n→+∞} F (ū_{n}, A \ B) + lim sup_{n→+∞} F (ū_{n}, A ∩ B)
≤ lim_{n→+∞} F (u_{n}, A) + F (v, A ∩ B)
= F(u, A) + F (v, A ∩ B) < 2ε

and the assertion follows.

Proposition 3.8. If F(u, Ω) < +∞ then for L^{N}-a.e. x_{0} ∈ Ω

lim inf_{ρ→0} m_{av}(u, B_{ρ}(x_{0})) ≤ f(x_{0}, ∇u(x_{0})).

Proof. Using the notation introduced in Remark 2.3, let x_{0} ∈ 𝕂(f) be a Lebesgue point for u and ∇u, and let ρ_{j} ↘ 0 be any sequence such that B_{ρ_{1}}(x_{0}) ⊂ Ω. We recall that, since u ∈ W^{1,α}(B_{ρ_{1}}), for L^{N}-a.e. such point x_{0} we have (see [13], Theorem 2, p.230)

− ∫_{B_{ρ_{j}} (x_{0})}[|∇u(x) − ∇u(x_{0})|^{α} + \frac{1}{ρ_{j}^{α}} |u(x) − u(x_{0}) − ∇u(x_{0})(x − x_{0})|^{α}] dx → 0.

On B := B_{1}(0) we define

u_{j}(y) := \frac{1}{ρ_{j}} [u(x_{0} + ρ_{j}y) − u(x_{0})] , u_{0}(y) := ∇u(x_{0})y. (3.8)

Then

− ∫_{B}[|∇u_{j} − ∇u_{0}|^{α} + |u_{j} − u_{0}|^{α}] dy → 0,

i.e.,

∫_{0}^{1} dt ∫_{∂B_{t}} [|∇u(x_{0} + ρ_{j}y) − ∇u(x_{0})|^{α} + |u_{j}(y) − ∇u(x_{0})y|^{α}] dH^{N−1} → 0,

where B_{t} := B_{t}(0), and so, up to a subsequence,

∫_{∂B_{t}} [|∇u(x_{0} + ρ_{j}y) − ∇u(x_{0})|^{α} + |u_{j}(y) − ∇u(x_{0})y|^{α}] dH^{N−1} → 0 (3.9)

for L^{1} a.e. t ∈ (0, 1). As for almost all t

(3.10)  μ(∂B_{tρ_{j}}) = 0 and u_{j} ∈ W^{1,α}(∂B_{t})  for all j,

we may find t̄∈ (0, 1) satisfying (3.9), (3.10), and we relabel as {ρ_{j}} the sequence {t̄ρ_{j}}. We then have that the sequence defined in (3.8) with the new ρ_{j} satisfies

∫_{∂B} [|∇u(x_{0} + ρ_{j}y) − ∇u(x_{0})|^{α} + |u_{j}(y) − ∇u(x_{0})y|^{α}] dH^{N−1} → 0,

μ(∂B_{ρ_{j}}) = 0, u_{j} ∈ W^{1,α}(∂B) for all j, (3.11)

and u_{j} still converges to u_{0} in W^{1,α}(B). Setting v_{j} := E_{B} u_{j}, and as E_{B} u_{0} = u_{0} (see Remark 2.2), from Lemma 2.1 and (3.11) we deduce that

v_{j} → u_{0} in W^{1,β}(B),

and

m_{av}(u, B_{ρ_{j}}(x_{0})) ≤ − ∫_{B_{ρ_{j}}} f (x, ∇ (ρ_{j}v_{j} (\frac{x − x_{0}}{ρ_{j}} ) + u(x_{0}))) dy

= − ∫_{B} f(x_{0} + ρ_{j}y, ∇v_{j}(y)) dy.

Since f is β-locally Lipschitz (see (2.2)), we obtain

∫_{B}|f(x_{0} + ρ_{j}y, ∇v_{j}(y)) − f(x_{0} + ρ_{j}y, ∇u(x_{0}))| dy
≤ C ∫_{B}(1 + |∇v_{j}(y)|^{β−1} + |∇u(x_{0})|^{β−1})|∇v_{j}(y) − ∇u(x_{0})| dy
≤ C∥∇v_{j} − ∇u_{0}∥_{L^{β}(B)}

which vanishes as j → +∞. Therefore

lim inf_{ρ→0} m_{av}(u, B_{ρ}(x_{0})) ≤ lim sup_{j→+∞} m_{av}(u, B_{ρ_{j}}(x_{0}))
≤ lim sup_{j→+∞} − ∫_{B} f(x_{0} + ρ_{j}y, ∇u(x_{0})) dy
(3.12)  = lim sup_{j→+∞} − ∫_{B_{ρ_{j}} (x_{0})} f(x, ∇u(x_{0})) dx.

By (3.12) and the definition of the set 𝕂(f) (see Remark 2.3), we conclude that

lim inf_{ρ→0} m_{av}(u, B_{ρ}(x_{0})) ≤ f(x_{0}, ∇u(x_{0})).

Remark 3.9. Recall that Ioffe’s Theorem (Theorem 2.4) ensures that F ≤ F. In particular, if F(u, Ω) < +∞ then f(·, ∇u) ∈ L^{1}(Ω). Moreover, if the point x_{0} in the previous proof is also a Lebesgue point for f(·, ∇), which holds L^{N}-a.e. in Ω, then the sequences {ρ_{j}} and {u_{j}} we found above satisfy

μ_{s}(∂B_{ρ_{j}}) = 0,

u ∈ W^{1,α}(∂B_{ρ_{j}}),

u_{j} → u in W^{1,α}(B_{ρ_{j}}) and in W^{1,α}(∂B_{ρ_{j}}),

lim sup_{j→+∞} m_{av}(u, B_{ρ_{j}}(x_{0})) ≤ lim_{j→+∞} − ∫_{B_{ρ_{j}} (x_{0})} f(x, ∇u(x)) dx ,

and from the last inequality we deduce, in particular, that for all t > 0 there exists j̄ such that for all j ≥ j̄

m(u, B_{ρ_{j}}(x_{0})) ≤ ∫_{B_{ρ_{j}} (x_{0})} f(x, ∇u(x)) dx + t|B_{ρ_{j}}(x_{0})|.

Proposition 3.10. For L^{N}-a.e. x_{0} ∈ Ω

lim inf_{ρ→0} F_{av}(u, B_{ρ}(x_{0})) ≤ f(x_{0}, ∇u(x_{0})).

Proof. Let x_{0} be a Lebesgue point for f(·, ∇u(·)) such that

\frac{dμ^{s}}{dL^{N}} (x_{0}) = 0 (3.13)

where μ_{s} is the singular part of μ with respect to L^{N} Ω. Fix an open set Ω′ ⊂ Ω and define for every t ∈ (0, 1)

E_{t} := {x ∈ Ω′ : ∃ρ_{j} ↘ 0 such that m(u, B_{ρ_{j}}(x_{0})) ≤ ∫_{B_{ρ_{j}} (x_{0})} f(x, ∇u(x)) dx + t|B_{ρ_{j}}(x_{0})|}.

By Remark 3.9 we have that |Ω′ \ E_{t}| = 0, thus, in particular, E_{t} is L^{N}-measurable. Fix ε > 0 and choose an open set ω and a compact set K such that

K ⊂ E_{t} ⊂ ω ⊂ Ω′ ,

with

μ_{s}(ω) ≤ ε + μ_{s}(E_{t}) , μ_{s}(ω \ K) ≤ ε + μ^{s}(E_{t}). (3.14)

Fix δ > 0 and set

X^{δ} := \{B_{ρ}(x) : x ∈ E_{t}, ρ < δ, \overline{B_{ρ}(x)} ⊂ ω, μ_{s}(∂B_{ρ}(x)) = 0, u ∈ W^{1,α}(∂B_{ρ}(x)),
m(u, B_{ρ}(x)) ≤ ∫_{B_{ρ}(x)} f(y, ∇u(y)) dy + t|B_{ρ}(x)|}
Y^{δ} := \{B_{ρ}(x) : ρ < δ, \overline{B_{ρ}(x)} ⊂ ω \ K, μ_{s}(∂B_{ρ}(x)) = 0\} .

The balls of X^{δ} and Y^{δ} form a fine cover of ω, and so Besicovitch Covering Theorem yields a countable subcover

ω = N ∪ ( ⋃_{i=1}^{+∞} B_{i}^{Y}) ∪ ( ⋃_{i=1}^{+∞} B_{i}^{X} ) (3.15)

with μ(N) = 0, so that the balls B_{i}^{Y} ∈ Y^{δ} and B_{i}^{X} ∈ X^{δ} are all disjoint. By (3.14) we may take n large enough such that

μ[N ∪ (∪_{i}B_{i}^{Y}) ∪ (∪_{i>n}B_{i}^{X})] < ε + μ^{s}(E_{t}), (3.16)

and for all i ≤ n choose v_{i} ∈ W^{1,β}(B_{i}^{X}) such that v_{i} − u ∈ W_{0}^{1,α}(B_{i}^{X}) and

F (v_{i}, B_{i}^{X}) ≤ m(u, B_{i}^{X}) + δ|B_{i}^{X} |. (3.17)

Due to the choice of X^{δ} and by (H3) we have

∫_{B_{i}^{X}} |Dv_{i}|^{α} dx ≤ ∫_{B_{i}^{X}} f(x, ∇v_{i}(x)) dx ≤ m(u, B_{i}^{X}) + δ|B_{i}^{X} |
≤ ∫_{B_{i}^{X}} f(x, ∇u(x)) dx + (δ + t)|B_{i}^{X} |,

and in view of the boundary assumption on v_{i} and Poincaré inequality, using again (H3),

∥v_{i} − u∥_{L^{α}(B_{i}^{X})}^{α} ≤ Cδ^{α}(∫_{B_{i}^{X}} f(x, ∇u(x)) dx + (δ + t)|B_{i}^{X} |),

where we have invoked the fact that the radius of B_{i}^{X} is less than δ. Set

u_{δ}(x) := \{ \begin{matrix}v_{i}(x) & if x ∈ B_{i}^{X}, i ≤ n, \\ u(x) & otherwise. \end{matrix}

The previous inequality yields

∥u_{δ} − u∥_{L^{α}(ω)}^{α} ≤ Cδ^{α}(∫_{ω} f(x, ∇u(x)) dx + |ω|) ,

thus u_{δ} → u in L^{α}(ω).

By Lemma 3.7, (3.15), and (3.16), we have

F(u_{δ}, ω) = F(u, ω \ ∪_{i≤n}\overline{B_{i}^{X}}) + ∑_{i≤n} F(u_{δ}, B_{i}^{X}) ≤ ε + μ^{s}(E_{t}) + ∑_{i≤n} F(v_{i}, B_{i}^{X}).

Since v_{i} ∈ W^{1,β}(B_{i}^{X}), by (2.5), (3.17), and in light of the choice of X^{δ},

F(u_{δ}, ω) ≤ ε + μ^{s}(E_{t}) + ∑_{i≤n} m(u, B_{i}^{X}) + δ| ∪_{i≤n} B_{i}^{X} |

≤ ε + μ^{s}(E_{t}) + ∫_{ω} f(x, ∇u(x)) dx + (δ + t)|ω|.

The L^{1} lower semicontinuity of F(·, ω) now yields

F(u, ω) ≤ lim inf_{δ→0} F(u_{δ}, ω) ≤ ε + μ^{s}(E_{t}) + ∫_{ω} f(x, ∇u(x)) dx + t|ω|,

so letting ω ↘ E_{t} and then ε → 0 we get

μ(E_{t}) ≤ μ^{s}(E_{t}) + t|E_{t}| + ∫_{E_{t}} f(x, ∇u(x)) dx.

Since |Ω′ \ E_{t}| = 0 this implies

F(u, Ω′) ≤ μ(E_{t}) + μ^{s}(Ω′) ≤ 2μ^{s}(Ω′) + t|Ω′| + ∫_{Ω′} f(x, ∇u(x)) dx.

Taking Ω′ := B_{ρ}(x_{0}) in the previous inequality, dividing through by |B_{ρ}(x_{0})|, and letting ρ → 0 we obtain

lim sup_{ρ→0} F_{av}(u, B_{ρ}(x_{0})) ≤ t + f(x_{0}, ∇u(x_{0})),

and by (3.13) the result now follows by the arbitrariness of t.

We are now in position to prove Theorem 1.1.

Proof of Theorem 1.1. The statement of Theorem 1.1 is now easily asserted by appealing to Proposition 3.3, Proposition 3.10, and to the fact that (see Theorem 2.4)

∫_{A} f(x, ∇u) dx ≤ F(u, A)

for all A ∈ Ω.

In the remaining of this section we show that, near x_{0} ∈ Ω, F(u·) may be identified with a Dirichlet problem. Precisely,

Theorem 3.11. If F(u, Ω) < +∞ then for μ-a.e. x_{0} ∈ Ω

lim_{ρ→0} \frac{m(u, B_{ρ}(x_{0}))}{μ(B_{ρ}(x_{0}))} = 1.

We first prove an auxiliary result.

Proposition 3.12. If F(u, Ω) < +∞ and if B is an uplift ball for u then

F(u, B) ≥ m(u, B).

Proof. By the definition of uplift ball we have that u ∈ W^{1,α}(∂B), and consider a sequence {u_{n}} satisfying the conditions of Lemma 3.5 (ii). Set

ū := E_{B} u , w_{n} := E_{B}(u_{n} − u),

so that ū, w_{n} ∈ W^{1,β}(B), ū = u on ∂B. Since u_{n} ⇀ u in W^{1,α}(∂B), by Lemma 2.1

w_{n} → 0 in W^{1,β}(B). (3.18)

Define for all w ∈ W^{1,β}(B)

h(w) := inf{F (v, B) : v ∈ W^{1,β}(B), v = ū + w on ∂B} ,

and remark that

h(0) = m(u, B) , h(w_{n}) = m(u_{n}, B). (3.19)

In order to show that h is lower semicontinuous at 0 with respect to the strong topology of W^{1,β}(B), we prove that it is convex and that it is locally bounded at 0. To establish convexity we fix w_{1}, w_{2} ∈ W^{1,β}(B), 0 < t < 1, ε > 0, and choose v_{1}, v_{2} ∈ W^{1,β}(B) such that

v_{i} = ū + w_{i} on ∂B , F (v_{i}, B) ≤ h(w_{i}) + ε.

Then tv_{1} + (1 − t)v_{2} = ū + tw_{1} + (1 − t)w_{2} on ∂B, and by the convexity of f

h(tw_{1} + (1 − t)w_{2}) ≤ F (tv_{1} + (1 − t)v_{2}, B) ≤ tF (v_{1}, B) + (1 − t)F (v_{2}, B)
≤ t h(w_{1}) + (1 − t) h(w_{2}) + 2ε.

It suffices now to let ε → 0.

On the other hand, for all w ∈ W^{1,β}(B)

h(w) ≤ F (ū + w, B) ≤ C (|B| + ∥∇(ū + w)∥_{L^{β}(B)}^{β}) ≤ C (|B| + ∥ū + w∥_{W^{1,β}(B)}^{β}) ,

from which we conclude that h is bounded in the unit ball of W^{1,β}(B).

In view of Lemma 3.4 (c), (3.18), and (3.19) we now have

F(u, B) = lim_{n→+∞} F (u_{n}, B) ≥ lim inf_{n→+∞} m(u_{n}, B) = lim inf_{n→+∞} h(w_{n}) ≥ h(0) = m(u, B).

Proof of Theorem 3.11. By Proposition 3.12 we have

lim sup_{ρ→0} \frac{m(u, B_{ρ}(x_{0}))}{μ(B_{ρ}(x_{0}))} ≤ 1, (3.20)

where we have used the fact that the limit on the left hand side of the previous inequality exists μ-a.e. For all t > 0 set

G_{t} := {B_{ρ} ⊂ Ω : B_{ρ} uplift ball, F(u, B_{ρ}) > m(u, B_{ρ}) + tμ(B_{ρ})}

and

E_{t} := {x ∈ Ω : ∃ρ_{h} → 0 with B_{ρ_{h}}(x) ∈ G_{t}}.

We claim that μ(E_{t}) = 0, therefore establishing the converse of (3.20). Indeed, if μ(E_{t}) = 0 then for μ-a.e. x_{0} ∈ Ω there exists δ_{0} > 0 such that

B_{δ}(x_{0}) ∉ G_{t} for all 0 < δ < δ_{0}. (3.21)

Consider any sequence {r_{n}} of positive numbers with r_{n} → 0, fix t > 0, and choose 0 < δ_{n} < r_{n} such that B_{δ_{n}} is an uplift ball and F(u, B_{r_{n}}\B_{δ_{n}}) < t, where here we have used the fact that μ(∂B_{δ_{n}}) = 0 (see Remark 3.6). By (3.21) we have

F(u, B_{r_{n}}) = F(u, B_{δ_{n}}) + F(u, B_{r_{n}} \ B_{δ_{n}}) ≤ F(u, B_{δ_{n}}) + t
≤ m(u, B_{δ_{n}}) + tμ(B_{δ_{n}}) + t ≤ m(u, B_{r_{n}}) + tμ(B_{r_{n}}) + t.

The arbitrariness of t yields

F(u, B_{r_{n}}) ≤ m(u, B_{r_{n}}),

and, in turn, the arbitrariness of {r_{n}} allows us to conclude that

lim inf_{ρ→0} \frac{m(u, B_{ρ}(x_{0}))}{μ(B_{ρ}(x_{0}))} ≥ 1.

We now establish the claim. Note that at this point there is no guarantee that E_{t} is even measurable, so we must proceed with care. For all δ > 0 define

U_{δ} := ⋃_{B_{ρ}(x)∈G_{t}, x∈E_{t}, ρ<δ} B_{ρ}(x) , U_{0} := ⋂_{δ>0} U_{δ}.

The sets U_{δ} are open, thus U_{0} is a Borel set, and U_{0} ⊃ E_{t}. It suffices, therefore, to prove that μ(U_{0}) = 0. Suppose that, on the contrary, μ(U_{0}) = γ > 0. Let ε < γ/2, fix δ_{0} > 0, and choose a compact set K ⊂ U_{0} such that

μ(U_{δ_{0}} \ K) < ε. (3.22)

We now define for all δ < δ_{0}

X^{δ} := {B_{ρ}(x) : x ∈ E_{t}, ρ < δ, B_{ρ}(x) ∈ G_{t}}

and

Y^{δ} := \{B_{ρ}(x) : ρ < δ, \overline{B_{ρ}(x)} ⊂ U_{δ} \ K, μ(∂B_{ρ}(x)) = 0\}.

These sets form a fine covering of U_{0}, so by Besicovitch Covering Theorem we may find a countable subcover such that

U_{0} ⊂ N ∪ ( ⋃_{i=1}^{+∞} B_{i}^{Y}) ∪ ( ⋃_{i=1}^{+∞} B_{i}^{X} )

with μ(N) = 0 and for suitable mutually disjoint balls B_{i}^{Y} ∈ Y^{δ} and B_{i}^{X} ∈ X^{δ}. By (3.22) we may find n large enough such that

μ[N ∪ (∪_{i}B_{i}^{Y}) ∪ (∪_{i>n}B_{i}^{X})] < ε,

and thus

μ (∪_{i≤n}B_{i}^{X} ) ≥ μ(U_{0}) − ε ≥ \frac{γ}{2} . (3.23)

For all i ≤ n we choose v_{i} ∈ W^{1,β}(B_{i}^{X}) such that v_{i} − u ∈ W_{0}^{1,α}(B_{i}^{X}) and

F (v_{i}, B_{i}^{X}) ≤ m(u, B_{i}^{X}) + δμ(B_{i}^{X}).

By (H3) and Proposition 3.12 we then get

∫_{B_{i}^{X}} |∇v_{i}|^{α} dx ≤ F (v_{i}, B_{i}^{X}) ≤ m(u, B_{i}^{X}) + δμ(B_{i}^{X}) ≤ F(u, B_{i}^{X}) + δμ(B_{i}^{X}) ,

hence

∑_{i≤n} ∫_{B_{i}^{X}} |Dv_{i}|^{α} dx ≤ F(u, Ω) + δμ(Ω) < +∞.

Poincaré inequality now implies that

∑_{i≤n} ∥v_{i} − u∥_{L^{α}(B_{i}^{X})} ≤ C δ^{α} ,

where we used the fact that the radii of the balls B_{i}^{X} are less than δ. We define

u_{δ}(x) := \{ \begin{matrix}v_{i}(x) & if x ∈ B_{i}^{X}, i ≤ n, \\ u(x) & otherwise. \end{matrix}

Clearly u_{δ} → u in L^{α}(U_{δ_{0}}), and by our choice of the functions v_{i} and by definition of the set E_{t} we deduce that

F(u_{δ}, U_{δ_{0}}) = F(u, U_{δ_{0}} \ ∪_{i≤n}\overline{B_{i}^{X}}) + ∑_{i≤n} F (v_{i}, B_{i}^{X})
≤ F(u, U_{δ_{0}} \ ∪_{i≤n}B_{i}^{X}) + ∑_{i≤n}[m(u, B_{i}^{X}) + δμ(B_{i}^{X})]
≤ F(u, U_{δ_{0}} \ ∪_{i≤n}B_{i}^{X}) + ∑_{i≤n}[F(u, B_{i}^{X}) + (δ − t)μ(B_{i}^{X})]
≤ F(u, U_{δ_{0}}) + (δ − t)μ(∪_{i≤n}B_{i}^{X}).

By (3.23) and in view of the L^{1} lower semicontinuity of F, the latter inequality implies that

\frac{tγ}{2} ≤ lim inf_{δ→0}[F(u, U_{δ_{0}}) − F(u_{δ}, U_{δ_{0}})] ≤ 0

and we have reached a contradiction. We conclude that μ(U_{0}) = 0.

Remark 3.13. If F(u, Ω) < +∞ then

f(x, ∇u(x)) = lim_{ρ→0} \frac{F(u, B_{r}(x))}{|B_{r}(x)|} = lim_{ρ→0} \frac{m(u, B_{r}(x))}{|B_{r}(x)|} for L^{N}-a.e. x.

Indeed, if f(x, ∇u(x)) > 0 then this follows from Theorems 1.1 and 3.11, and if f(x, ∇u(x)) = 0 then it suffices to use Proposition 3.12 to deduce that

lim sup_{ρ→0} \frac{m(u, B_{r}(x))}{|B_{r}(x)|} ≤ lim_{ρ→0} \frac{F(u, B_{r}(x))}{|B_{r}(x)|} = f(x, ∇u(x))

where the last equality is asserted by Theorem 1.1. Moreover

μ_{s}({x}) = lim_{ρ→0} m(u, B_{r}(x)).

Indeed, if μ_{s}({x}) = 0 then this equality is a direct consequence of Proposition 3.12, while if μ_{s}({x}) > 0 then Theorem 3.11 yields

1 = lim_{r→0} \frac{m(u, B_{r}(x))}{μ(B(x))} = \frac{lim_{r→0} m(u, B_{r}(x))}{μ_{s}(\{x\})}.

4. Remarks on the Lavrentiev Phenomenon

In this section we prove Proposition 1.2 and Theorem 1.3.

Proof of Proposition 1.2. In view of Theorem 1.1 it suffices to prove that μ_{s}(Ω) = 0. For every x ∈ Ω let B_{ρ(x)}(x) be a ball satisfying (1.4). By Theorem 1.1 we have that μ_{s} (B_{ρ(x)}(x)) = 0, and so μ_{s} (\overline{B_{r}(x)}) = 0 for all 0 < r < ρ(x). Since

Ω = ⋃_{x∈Ω, 0<r<ρ(x)} B_{r}(x),

using Besicovitch Covering Theorem we may find a countable family of disjoint closed balls and a μ_{s}-null set N such that

Ω = ⋃_{i=1}^{∞} \overline{B_{r_{i}}(x_{i})} ∪ N.

We conclude that

μ_{s}(Ω) = ∑_{i=1}^{∞} μ_{s} (\overline{B_{r_{i}}(x_{i})}) = 0.

Remark 4.1. It can be easily verified that condition (1.4) is equivalent to assuming that there exists a sequence {u_{n}} ⊂ W^{1,β}(B) such that

u_{n} ⇀ u in W^{1,α}(B), F (u_{n}, B) → F (u, B). (4.1)

That (1.4) implies (4.1) follows immediately from the definition of F(u, B): it suffices to choose any good sequence for u in B (see (2.4)). Conversely, by Ioffe’s Theorem (see Theorem 2.4) we have that F (u, B_{ρ(x)}(x)) ≥ F (u, B_{ρ(x)}(x)), while (4.1) yields the converse inequality; hence

F (u, B_{ρ(x)}(x)) = F (u, B_{ρ(x)}(x)) for all x ∈ Ω.

Theorem 1.3 follows immediately from Proposition 1.2, Remark 4.1, and the lemma below.

Lemma 4.2. If f is a Carathéodory function satisfying (1.5) and (1.6) then there exists a sequence {u_{n}} such that

u_{n} ∈ C_{0}^{∞}(Ω), u_{n} → u in L_{loc}^{1}(Ω), f(x, ∇u_{n}) → f(x, ∇u) in L_{loc}^{1}(Ω).

Moreover for every x_{0} ∈ Ω and for almost all ρ < dist(x_{0}, ∂Ω)

u_{n} → u strongly in W^{1,α}(B_{ρ}(x_{0})) and weakly in W^{1,α}(∂B_{ρ}(x_{0})).

Proof. Let ρ(x) be a standard symmetric mollifier with support in B_{1}(0), for every n ∈ ℕ set ρ_{n}(x) := n^{N}ρ(nx), and set

u_{n} := ū_{n} ∗ ρ_{n}, ū_{n}(x) := \{ \begin{matrix}u(x) & if dist (x, ∂Ω) > 1/n, \\ 0 & otherwise. \end{matrix}

Clearly u_{n} ∈ C_{0}^{∞}(Ω), u_{n} → u in W_{loc}^{1,α}(Ω), thus u_{n} → u strongly in W^{1,α}(B_{r}(x_{0})) for any ball B_{r}(x_{0}) with x_{0} ∈ Ω and r < dist(x_{0}, ∂Ω). In particular, for almost all such r we have u_{n} ⇀ u in W^{1,α}(∂B_{r}(x_{0})). We only have to show that

f(x, ∇u_{n}) → f(x, ∇u) in L_{loc}^{1}(Ω). (4.2)

Fix A ⊂⊂ Ω, and let dist(A, ∂Ω) ≥ 2ε_{0}; without loss of generality we may always take n > 1/ε_{0}, thus in A we have Du_{n} = Du ∗ ρ_{n}. Set A′ := {x ∈ Ω : dist(x, ∂Ω) ≥ ε_{0}}, and define

g(x) := f(x, Du(x)), p_{n}(x) = min{p(y) : |x − y| ≤ 1/n}, φ_{n}(x, z) = |z|^{p_{n}(x)}.

18 E. ACERBI, G. BOUCHITTÉ, AND I. FONSECA

Clearly φ_{n} is a Carathéodory function, φ(x, ·) is convex for all x ∈ Ω, and if |x − y| < 1/n then p_{n}(x) ≤ p(y) and so

φ_{n}(x, z) = |z|^{p_{n}(x)} ≤ 1 + |z|^{p(y)} ≤ 1 + f(y, z).

In view of this inequality and by Jensen’s inequality, if x ∈ A then

φ_{n}(x, ∇u_{n}(x)) = φ_{n}(x, ∫_{B(x,1/n)} ∇u(y))ρ_{n}(x − y) dy)
≤ ∫_{B(x,1/n)} φ_{n}(x, ∇u(y))ρ_{n}(x − y) dy
≤ ∫_{B(x,1/n)} [1 + f(y, ∇u(y))] ρ_{n}(x − y) dy
(4.3)  = 1 + (g ∗ ρ_{n})(x).

Now remark that if x ∈ A then, with α′ := 1 − 1/α, and using the fact that u ∈ W_{loc}^{1,α} (see (1.5)), we have

|∇u_{n}(x)| = |∫_{A′} ∇u(y)ρ_{n}(x − y) dy|
≤ C(α)∥∇u∥_{L^{α}(A′)} n^{N} |∫_{A′} ρ^{α′}(n(x − y)) dy|^{1/α′}
≤ C(α)∥∇u∥_{L^{α}(A′)} n^{N}n^{−N/α′} |∫_{ℝ^{N}} ρ^{α′}(y) dy|^{1/α′}
= C′∥∇u∥_{L^{α}(A′)} n^{N/α},

where C′ = C′(α) is a positive constant. By (1.5) we deduce that

\frac{1}{C} f(x, ∇u_{n}(x)) − 1 ≤ |∇u_{n}(x)|^{p(x)} = |∇u_{n}(x)|^{p(x)−p_{n}(x)}φ_{n}(x, ∇u_{n}(x))
≤ (1 + C′^{β−α} ∥∇u∥_{L^{α}(A′)}^{β−α}) (n^{p(x)−p_{n}(x)})^{N/α}φ_{n}(x, ∇u_{n}(x))
≤ C e^{Nγ/α} φ_{n}(x, ∇u_{n}(x)),

where we have used the fact that

0 ≤ p(x) − p_{n}(x) ≤ γ/ log n ,

so that

(n^{p(x)−p_{n}(x)})^{N/α} ≤ e^{Nγ/α}.

In view of (4.3) we now have for all x ∈ A

0 ≤ f(x, ∇u_{n}(x)) ≤ C(1 + (g ∗ ρ_{n})(x)).

Since ∇u_{n} → ∇u a.e. in A, by the continuity of f(x, ·) and Fatou’s Lemma we have

lim inf_{n→+∞} ∫_{A} f(x, ∇u_{n}(x)) dx ≥ ∫_{A} f(x, ∇u(x)) dx. (4.4)

Moreover, once more by Fatou’s Lemma and using the fact that g ∗ ρ_{n} → g in L^{1}(A),

∫_{A} C(1 + g(x)) − lim sup_{n→+∞} ∫_{A} f(x, ∇u_{n}(x)) dx = lim inf_{n→+∞} ∫_{A} [C(1 + (g ∗ ρ_{n})(x)) − f(x, ∇u_{n}(x))] dx
≥ ∫_{A} [C(1 + g(x)) − f(x, ∇u(x))] dx,

and we conclude that

lim sup_{n→+∞} ∫_{A} f(x, ∇u_{n}(x)) dx ≤ ∫_{A} f(x, ∇u(x)) dx.

This, together with (4.4), yields the convergence of f(·, ∇u_{n}(·)) to f(·, ∇u(·)) in L^{1}(A).

The last result of this section shows that under some additional assumptions on the convergence of the traces of approximating sequences, it is possible to impose Dirichlet boundary conditions on the admissible sequences without increasing the overall energy. Since a priori the infimum taken over the class of functions restricted under the Dirichlet boundary condition is greater than the infimum over the unconstrained class, we may interpret this result as a situation where the Lavrentiev phenomenon is avoided.

Proposition 4.3. Let F(u, Ω) < +∞, let A ⊂ Ω be an open set with Lipschitz boundary, and assume that u_{n} ∈ W^{1,β}(A) satisfies

u_{n} ⇀ u weakly in W^{1,α}(A) and weakly in W^{1,α}(∂A),

and

F(u, A) = lim_{n→+∞} F (u_{n}, A).

Then there exists a sequence v_{n} ∈ W^{1,β}(A) such that

v_{n} = u on ∂A, v_{n} ⇀ u weakly in W^{1,α}(A), F(u, A) = lim_{n→+∞} F (v_{n}, A).

Moreover if the sequence {u_{n}} converges to u strongly in W^{1,α}(A) then so does {v_{n}}.

Proof of Proposition 4.3. Set w_{n} = E_{A}(u_{n} − u), so that by Lemma 2.1 w_{n} → 0 strongly in W^{1,β}(A) and w_{n} = u_{n} − u on ∂A. For k ∈ ℕ let ε = 1/k and define h_{ε} : W^{1,β}(A) → [0, +∞] by

h_{ε}(w) := inf{F (v, A) : v ∈ W^{1,β}(A), v = u + w on ∂A, ∥v − u∥_{L^{1}(A)} < ε}.

Since for n ∈ ℕ sufficiently large

u_{n} = u + w_{n} on ∂A and ∥u_{n} − u∥_{L^{1}(A)} < ε,

it follows that

h_{ε}(w_{n}) ≤ F (u_{n}, A). (4.5)

A similar proof to that of Proposition 3.12 yields the convexity of h_{ε}. Also, let C(ε) > 0 be such that

∥w∥_{W^{1,β}(A)} < C(ε) ⇒ ∥w∥_{L^{1}(A)} < ε/2.

We claim that h_{ε} is bounded in the ball in W^{1,β}(A) of center zero and radius C(ε). Indeed, as u ∈ W^{1,α}(∂A) there exists ū ∈ W^{1,β}(A) such that u − ū ∈ W_{0}^{1,α}(A), but then there exists φ_{ε} ∈ C_{0}^{∞}(A) such that ∥(u−ū)−φ_{ε}∥_{W^{1,α}(A)} < ε/2. Setting u_{ε} := ū+φ_{ε} we have u_{ε} ∈ W^{1,β}(A), u_{ε} = u on ∂A, ∥u_{ε}−u∥_{W^{1,α}(A)} < ε/2, and thus for all w ∈ W^{1,β}(A) with ∥w∥_{W^{1,β}(A)} < C(ε), and in view of (H3)

h_{ε}(w) ≤ F (u_{ε} + w, A) ≤ C (|A| + ∥(u_{ε} + w)∥_{W^{1,β}(A)}^{β}) ≤ C′(ε)

for some constant C′(ε) > 0. Therefore the convex function h_{ε} is lower semicontinuous at the origin, and by (4.5) we have

F(u, A) = lim_{n→+∞} F (u_{n}, A) ≥ lim_{n→+∞} h_{ε}(w_{n}) ≥ h_{ε}(0),

that is

F(u, A) ≥ inf{F (v, A) : v ∈ W^{1,β}(A), v = u on ∂A, ∥v − u∥_{L^{1}(A)} < ε}.

In particular, there exists v_{ε} ∈ W^{1,β}(A) such that v_{ε} = u on ∂A and

∥v_{ε} − u∥_{L^{1}(A)} < ε , F (v_{ε}, A) ≤ F(u, A) + ε (4.6)

and the result now follows by letting ε → 0.

If u_{n} → u strongly in W^{1,α}(A) then one may replace the L^{1} norm in the definition of h_{ε} by the norm in W^{1,α}, thus obtaining in (4.6) that ∥v_{ε} − u∥_{W^{1,α}(A)} < ε.

5. A Counterexample for Carathéodory Quasiconvex Integrands

Here we present the proof of Proposition 1.4.

Recall that under (H1), (H3), by Theorem 3.1 in [15] if u ∈ W^{1,α}(Ω; ℝ^{d}) is such that F(u, Ω) < +∞ then

F(u, A) = ∫_{A} f^{u}(x) dx + μ_{s}(A) for all A ∈ 𝔸(Ω),

where μ_{s} is a non-negative finite Radon measure on Ω, singular with respect to L^{N} Ω, and (see Theorem 3.2 in [6] for a.e. x_{0} ∈ Ω

f^{u}(x_{0}) = lim_{ρ→0, ρ∈R(u,x_{0})} \frac{m(u, B_{ρ}(x_{0}))}{|B_{ρ}(x_{0})|}. (5.1)

Here R(u, x_{0}) stands for the set of good radii of the function u at the point x_{0} ∈ Ω (see [6]), Definition 3.4), and it can be shown that almost all r ∈ (0, dist(x_{0}, ∂Ω)) belong to R(u, x_{0}).

The following lemma may be found in [6], Lemma 3.5.

Lemma 5.1. Under assumptions (H1) and (H3), if R ∈ R(u, x_{0}) and if {u_{n}} ⊂ W^{1,p}(Ω; ℝ^{d}) is such that u_{n} ⇀ u weakly in W^{1,p}(∂B_{R}(x_{0}); ℝ^{d}) then

m(u, B_{R}(x_{0})) = lim_{n→+∞} m(u_{n}, B_{R}(x)).

Next we simplify the representation formula for f^{u} in the general quasiconvex context and in the case where u is affine.

Lemma 5.2. If (H1) and (H3) hold, and if u is affine, then for L^{N}-a.e. x_{0} ∈ Ω

f^{u}(x_{0}) = lim_{n→+∞} {inf − ∫_{B} f(x_{0} + ε_{n}y, ∇w(y)) dy : w ∈ W^{1,β}(B; ℝ^{d}), w − u ∈ W_{0}^{1,α}(B; ℝ^{d})} ,

where B is the unit ball centered at the origin and {ε_{n}} ⊂ R(u, x_{0}) is any sequence of positive numbers converging to 0.

Proof. Consider {ε_{n}} ⊂ R(u, x_{0}) with ε_{n} → 0^{+}. By (5.1) we have

f^{u}(x_{0}) = lim_{n→+∞}{inf − ∫_{B_{εn} (x_{0})} f(x, ∇v(x)) dx : v ∈ W^{1,β}(B_{ε_{n}}(x_{0}); ℝ^{d}), v − u ∈ W_{0}^{1,α}(B_{ε_{n}}(x_{0}); ℝ^{d})}

= lim_{n→+∞}{inf − ∫_{B} f(x_{0} + ε_{n}y, ∇v(x_{0} + ε_{n}y)) dy : v ∈ W^{1,β}(B_{ε_{n}}(x_{0}); ℝ^{d}), v − u ∈ W_{0}^{1,α}(B_{ε_{n}}(x_{0}); ℝ^{d})}
= lim_{n→+∞}{inf − ∫_{B} f(x_{0} + ε_{n}y, ∇w(y)) dy : w ∈ W^{1,β}(B; ℝ^{d}), w − u ∈ W_{0}^{1,α}(B; ℝ^{d})},

where we used the substitution

w(y) := \frac{v(x_{0} + ε_{n}y) − u(x_{0}) + ε_{n}u(0)}{ε_{n}}

for y ∈ B, and we invoked the fact that u is affine.

Proof of Proposition 1.4. Let K, p and f be as stated. We claim that for all δ ∈ (0, 1) there exists θ_{0} ∈ (0, 1) such that for all θ < θ_{0} and for all x_{0} ∈ ∂K

f^{id}(x_{0}) < f(x_{0}, 𝕀) − 1 + δ. (5.2)

Fix x_{0} ∈ ∂K and let ε ∈ (0, 1). Since 0 ∈ ∂(K − x_{0}) we may find y_{ε} ∈ B_{ε^{2}}(0) such that y_{ε} ∉ K − x_{0}. Define x_{ε} := y_{ε}/ε. Then |x_{ε}| < ε and x_{ε} ∉ (K − x_{0})/ε. Denoting by B the unit ball B_{1}(0), we clearly have

B(x_{ε}, 1 − |x_{ε}|) ⊂ B, |B \ B(x_{ε}, 1 − |x_{ε}|)| → 0 as ε → 0.

Since (K − x_{0})/ε is closed, there exists r_{ε} ∈ (0, 1) such that

B(x_{ε}, r_{ε}) ⊂ B(x_{ε}, 1 − |x_{ε}|) \ \frac{K − x_{0}}{ε} , (5.3)

and we define in B the function

v_{ε}(x) := \{ \begin{matrix}x & if x ∉ B(x_{ε}, 1 − |x_{ε}|), \\ x_{ε} + (1 − |x_{ε}|) \frac{x−x_{ε}}{|x−x_{ε}|} & if r_{ε} < |x − x_{ε}| < 1 − |x_{ε}|, \\ x_{ε} + \frac{1−|x_{ε}|}{r_{ε}} (x − x_{ε}) & if |x − x_{ε}| < r_{ε}. \end{matrix}

Clearly v_{ε} ∈ W^{1,∞}(B; ℝ^{N}), v_{ε} = id on ∂B, and

− ∫_{B} |∇v_{ε}|^{p} dx ≤ 1 + ∫_{B(x_{ε},1−|x_{ε}|)\B(x_{ε},r_{ε})}|\frac{𝕀}{|x − x_{ε}|} − \frac{(x − x_{ε}) ⊗ (x − x_{ε})}{|x − x_{ε}|^{3}} |^{p} dx

+ ∫_{B_{rε} (x_{ε})} |\frac{1 − |x_{ε}|}{r_{ε}} |^{p} dx

(5.4)  ≤ 1 + C ∫_{r_{ε}}^{1−|x_{ε}|} r^{N−1−p} dr + C r_{ε}^{N−p} ≤ C_{0}

where C_{0} is a constant independent of x_{0}, and where we have used the fact that p < N . In view of Lemma 5.2, and choosing a sequence {ε_{n}} ⊂ R(id, x_{0}) of positive numbers converging to 0, we have

f^{id}(x_{0}) = lim_{n→+∞}{inf − ∫_{B}[χ_{K}(x_{0} + ε_{n}x) | det ∇w(x)| + θ|∇w(x)|^{p}] dx : w ∈ W^{1,N}(B; ℝ^{N}), w = id on ∂B}

≤ lim inf_{n→+∞} − ∫_{B} [χ_{\frac{K−x_{0}}{εn}} (x) | det ∇v_{ε_{n}}(x)| + θ|∇v_{ε_{n}}(x)|^{p}] dx

≤ lim inf_{n→+∞} \frac{|B \ B(x_{ε_{n}}, 1 − |x_{ε_{n}}|)|}{|B|} + θ C_{0} = θ C_{0},

where we have used (5.3), (5.4), and the fact that det ∇v_{ε_{n}}(x) = 0 if r_{ε_{n}} < |x − x_{ε_{n}}| < 1 − |x_{ε_{n}}|. Since f(x_{0}, 𝕀) ≥ 1, we have now established the claim with θ_{0} = δ/C_{0}.

References

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Dipartimento di Matematica, Via Massimo D’Azeglio 85/A, 43100 Parma, Italy

   E-mail address: acerbi@prmat.math.unipr.it

Département de Mathématiques, Université de Toulon et du Var - BP 132, 83957 La Garde Cedex, France

   E-mail address: bouchitte@univ-tln.fr

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

   E-mail address: fonseca@andrew.cmu.edu