Manifold Constrained Variational Problems

B. Dacorogna, I. Fonseca, J. Malý, K. Trivisa

October 20, 1998

Abstract

The integral representation for the relaxation of a class of energy functionals where the admissible fields are constrained to remain on a C^{1} m-dimensional manifold M ⊂ ℝ^{d} is obtained. If f : ℝ^{d×N} → [0, ∞) is a continuous function satisfying 0 ≤ f(ξ) ≤ C(1 + |ξ|^{p}), for C > 0, p ≥ 1, and for all ξ ∈ ℝ^{d×N}, then

F(u, Ω) : = inf_{{u_{n}}} {lim inf_{n→∞} ∫_{Ω} f(∇u_{n}) dx : u_{n} ⇀ u in W^{1,p}, u_{n}(x) ∈ M a.e. x ∈ Ω, n ∈ ℕ}
= ∫_{Ω} Q_{T} f(u, ∇u) dx,

where Ω ⊂ ℝ^{N} is open, bounded, and Q_{T} f(y_{0}, ξ) is the tangential quasiconvexification of f at y_{0} ∈ M, ξ = (ξ^{1}, ..., ξ^{N}), ξ^{i} belong to the tangent space to M at y_{0}, i = 1, ..., N.

1991 Mathematics subject classification (Amer. Math. Soc.): 49J45, 49Q20, 49N60, 73T05, 73V30 Keywords : Quasiconvexification, Radon measures, Radon-Nikodym derivative, partition of unity

1 Introduction

Several interesting questions in materials science require a good understanding of constrained variational problems. Here our motivation to address nonconvex manifold constrained energy relaxation is originated mainly by questions of equilibria for liquid crystals and magnetostrictive materials, a common characteristic being the study of problems of the type

min_{u∈M} I(u), I(u) := ∫_{Ω} f(∇u) dx,

where the class of admissible fields is constrained to remain with values on the C^{∞} (d − 1)- dimensional manifold S^{d−1} := {u ∈ ℝ^{d} : |u| = 1}. Here Ω ⊂ ℝ^{N} is an open, bounded domain, which represents the reference configuration of a certain material body, and the bulk energy density f : 𝕄^{d×N} → [0, +∞) is a continuous function satisfying

0 ≤ f(ξ) ≤ C(1 + |ξ|^{p}),

for some C > 0, p ≥ 1, and all ξ ∈ 𝕄^{d×N}, where 𝕄^{d×N} denotes the set of all d × N matrices.

Often lack of convexity prevents f to meet the requirements ensuring weak lower semicontinuity of I(·) (see [1], [2], [14], [15]), and so we introduce the relaxed energy

F(u) := inf_{{u_{n}}} {lim inf_{n→∞} ∫_{Ω} f(∇u_{n}) dx : u_{n} ⇀ u in W^{1,p}(Ω; ℝ^{d}), u_{n}(x) ∈ M a.e. x ∈ Ω} (1.1)

where M is a C^{1} m-dimensional manifold in ℝ^{d}.

One of the main objectives of relaxation theory is to find the new relaxed (effective) bulk energy density or, equivalently, to give an integral representation for F(·). In the non-constrained case, i.e., when M = ℝ^{d}, it has been shown by Dacorogna [4] that

F(u) = ∫_{Ω} Qf(∇u) dx,

where the quasiconvex envelope Qf of f is defined by

Qf(ξ) := inf {∫_{Q} f(ξ + ∇ϕ(x)) dx : ϕ ∈ W_{0}^{1,∞}(Q; ℝ^{d})} ,  (1.2)

and Q := (0, 1)^{N}. We recall that F(·) is now a W^{1,p}-sequentially weakly lower semicontinuous functional, and (see [4])

inf_{u∈C} ∫_{Ω} f(∇u) dx = min_{u∈C} ∫_{Ω} Qf(∇u) dx,

whenever C is a W^{1,p}(Ω; ℝ^{d}) weakly compact class of admissible fields.

In this paper we identify the relaxed energy F(·) in (1.1) when u ∈ C are constrained to remain with values on a C^{1} m-dimensional manifold M ⊂ ℝ^{d}. Precisely, we show that the relaxed energy in (1.1) can be represented as

F(u) = ∫_{Ω} Q_{T} f(u, ∇u) dx,  (1.3)

where the tangential quasiconvexification Q_{T} f is defined in Section 2 (see Definition 2.1). We derive the formula (see Proposition 2.2)

Q_{T} f(y, ξ) = Qf̄(y, ξ)

with

f̄(y, ξ) := f(P_{y}ξ),  (1.4)

where P_{y} is the orthogonal projection of ℝ^{d} onto the tangent space to M at y, T_{y}(M), P_{y}ξ := (P_{y}ξ^{1}, …, P_{y}ξ^{N}), and ξ^{i} stands for the i^{th} column of the matrix ξ ∈ ℝ^{d×N}. In particular, this concludes (1.3) for globally parametrized constraint manifolds. When M is the unit sphere, M = S^{d−1}, then (1.4) reduces to

f̄(y, ξ) := f((𝕀_{d×d} − y ⊗ y)ξ)

for y ∈ S^{d−1}, ξ = (ξ^{1}, …, ξ^{N}), with the column vectors ξ^{i} being tangent to the unit sphere at the point y, i = 1, …, N (see Example 2.4). This example is relevant in the study of equilibria for liquid crystals, in ferromagnetism, and for magnetostrictive materials (see [6], [16]). For a wide literature concerning existence and regularity of (local) minima of energy functionals where the admissible fields are constrained to have values on the sphere, see [3], [11], [12], [5], [13], and the references therein.

In Proposition 2.5 we prove W^{1,p}(Ω; M)-sequentially weakly lower semicontinuity of

u ↦ ∫_{Ω} Q_{T} f(u, ∇u) dx.

The statement and proof of the relaxation result are presented in Section 3. The main challenge of the analysis is to ensure the locality of the relaxed energy with respect to the domain of integration, and this, in turn, allows us to use the blow-up technique to identify the effective energy density. We will not prove directly the locality property (although it will be an obvious consequence of the representation (1.3)), instead we will introduce an auxilliary functional F_{∞} where the approximating sequences to a macroscopic field u are required to converge uniformly to u on a compact subset of the manifold, outside which they must coincide with u. Locality for F_{∞} is proven in Proposition 3.4, and then we go on to showing that, in fact, F and F_{∞} agree.

2 The Tangential Quasiconvexification

To motivate the introduction of the notion of tangential quasiconvexification, first we derive (1.3) in the simple case where M may be globally parametrized. Suppose that there exists a single C^{1} chart Ψ : M^{∗} → ℝ^{d}, where M^{∗} ⊂ ℝ^{m} is and open set, such that

M = Ψ(M^{∗}).

Further, for simplicity, assume that M^{∗} is the entire space ℝ^{m}. Let u ∈ W^{1,p}(Ω; M). With u^{∗} := Ψ^{−1} ∘ u we have

u = Ψ ∘ u^{∗}, ∇u = ∇Ψ(u^{∗})∇u^{∗},

and so

\begin{aligned}∫_{Ω} f(∇u) dx & = ∫_{Ω} f(∇Ψ(u^{∗})∇u^{∗}) dx \\ & = ∫_{Ω} f̂(u^{∗}, ∇u^{∗}) dx, \end{aligned}

where f̂: ℝ^{m} × 𝕄^{m×N} → [0, +∞) is given by

f̂(y^{∗}, ξ^{∗}) := f(∇Ψ(y^{∗})ξ^{∗}).

Well known relaxation theorems (see [1], [4]) yield that

F(u) := inf_{{u_{n}^{∗}}} {lim inf_{n→∞} ∫_{Ω} f̂(u_{n}^{∗}, ∇u_{n}^{∗}) dx : u_{n}^{∗} ⇀ u^{∗} in W^{1,p}(Ω; ℝ^{m})}
= ∫_{Ω} Qf̂(u^{∗}, ∇u^{∗}) dx,

(2.1)

where (see (1.2)) Qf̂(y^{∗}, ξ^{∗}) := Qf̂(y^{∗}, ·)(ξ^{∗}). If M^{∗} is not the entire space then some additional technical difficulties occur, and if M is not parametrizable by a single chart then previously known results fail to apply. Nevertheless, formula (2.1) indicates how the tangential quasiconvexification Q_{T} could be constructed from local charts.

In what follows, let f : 𝕄^{d×N} → [0, +∞) be a Borel measurable function, and let M ⊂ ℝ^{d} be a C^{1} m-dimensional manifold, 1 ≤ m ≤ d. The tangent space to M at y, where y ∈ M, is denoted T_{y}(M).

Definition 2.1. Let y ∈ M and ξ ∈ [T_{y}(M)]^{N}. The tangential quasiconvexification of f at ξ relative to y is defined by

Q_{T} f(y, ξ) := inf {∫_{Q} f(ξ + ∇φ(x)) : φ ∈ W_{0}^{1,∞}(Q; T_{y}(M))} .

In order to provide alternative characterizations of the tangential quasiconvexification of f, we remark that to each point y ∈ M there corresponds a C^{1} projection Π_{y} of a neighborhood U(y) of y in ℝ^{d} onto M. This projection has the property that for each z ∈ U(y), Π_{y}(z) is the unique point z′ in M ∩ U(z) with

P_{y}(z′ − z) = 0 (in particular, z′ = z if z ∈ M),

where, as in (1.4), P_{y} is the orthogonal projection of ℝ^{d} onto the tangent space T_{y}(M). The projection Π_{y} is constructed via the Implicit Function Theorem. Consider a local chart Ψ : M^{∗} → ℝ^{d} such that M^{∗} ⊂ ℝ^{m} is an open set and Ψ(M^{∗}) = M ∩ Ũ(y), where Ũ(y) is an open neighborhood of y in ℝ^{d}. Set y^{∗} := Ψ^{−1}(y), and denote by ∇Ψ[y^{∗}] the derivative of Ψ at y^{∗} considered as a linear map ∇Ψ[y^{∗}] : ℝ^{m} → T_{y}(M). Since ∇Ψ[y^{∗}] is an isomorphism of ℝ^{m} onto the tangent space T_{y}(M), there is an inverse mapping Λ_{y} : T_{y}(M) → ℝ^{m} such that Λ_{y} ∘ ∇Ψ[y^{∗}] is the identity on ℝ^{m}. Applying the Inplicit Function Theorem to the function

G : M^{∗} × Ũ(y) → ℝ^{m}, (z^{∗}, z) ↦ Λ_{y}(P_{y}(Ψ(z^{∗}) − z)),

we may find a neighborhood U(y) of y, with U(y) ⊂ Ũ(y), and a unique (C^{1}) function Υ : U(y) → M^{∗} such that

G(Υ(z), z) = 0 for all z ∈ U(y).

We set

Π_{y}(z) := Ψ(Υ(z)).  (2.2)

We observe that

P_{y}(Π_{y}(z) − z) = 0 and ∇Π_{y}(y) = P_{y}.

Using the above notation, we provide alternative characterizations for Q_{T} f. Proposition 2.2. For any y ∈ M and ξ ∈ [T_{y}(M)]^{N} we have

(i)

Q_{T} f(y, ξ) = Qf̂(y^{∗}, ξ^{∗}),  (2.3)

where

ξ^{∗} := Λ_{y}ξ

and

f̂(y^{∗}, η^{∗}) := f(∇Ψ(y^{∗})η^{∗})

for all η^{∗} ∈ 𝕄^{m×N}; (ii)

Q_{T} f(y, ξ) = Qf̄(y, ξ),  (2.4)

where

f̄(y, ξ) := f(P_{y}ξ),

with P_{y}ξ := (P_{y}ξ^{1}, …, P_{y}ξ^{N}) and ξ^{i} stands for the i^{th} column of the matrix ξ ∈ ℝ^{d×N}. Proof. (i) We have

Qf̂(y^{∗}, ξ^{∗}) = inf {∫_{Q} f̂(y^{∗}, ξ^{∗} + ∇ϕ(x)) dx : ϕ ∈ W_{0}^{1,∞}(Q; ℝ^{m})}
= inf {∫_{Q} f(∇Ψ[y^{∗}]ξ^{∗} + ∇Ψ[y^{∗}]∇ϕ(x)) dx : ϕ ∈ W_{0}^{1,∞}(Q; ℝ^{m})}
= inf {∫_{Q} f(ξ + ∇Ψ[y^{∗}]∇ϕ(x)) dx : ϕ ∈ W_{0}^{1,∞}(Q; ℝ^{m})} .

(2.5)

Therefore, if ϕ ∈ W_{0}^{1,∞}(Q; ℝ^{m}) then

φ := ∇Ψ[y^{∗}] ∘ ϕ ∈ W_{0}^{1,∞}(Q; T_{y}(M))

and we obtain

Q_{T} f(y, ξ) ≤ Qf̂(y^{∗}, ξ^{∗}).

Conversely, if φ ∈ W_{0}^{1,∞}(Q; T_{y}(M)) then ϕ := Λ_{y} ∘ φ ∈ W_{0}^{1,∞}(Q; ℝ^{m}) and we conclude that

Qf̂(y^{∗}, ξ^{∗}) ≤ Q_{T} f(y, ξ).

(ii) If φ ∈ W_{0}^{1,∞}(Q; T_{y}(M)) then φ ∈ W_{0}^{1,∞}(Q; ℝ^{d}) and P_{y}φ = φ, so that

f̄(y, ξ + φ) = f(y, ξ + φ),

and thus

Qf̄(y, ξ) ≤ Q_{T} f(y, ξ).

Conversely, if φ̄ ∈ W_{0}^{1,∞}(Q; ℝ^{d}), then φ := P_{y} ∘ φ̄ ∈ W_{0}^{1,∞}(Q; T_{y}(M)) and

f(y, ξ + ∇φ) = f̄(y, ξ + ∇φ);

hence

Q_{T} f(y, ξ) ≤ Qf̄(y, ξ).

Remark 2.3 (A formula exploiting the normal field). Let M be a (N−1)-dimensional C^{1} manifold and y ∈ M. If ν(y) is a normal vector to M at y, then the expression y ↦ ν(y) ⊗ ν(y) does not depend on the orientation of M and clearly

P_{y}z = (𝕀_{d×d} − ν(y) ⊗ ν(y)) z for all z ∈ ℝ^{d} and y ∈ M.

Hence in the notation of Proposition 2.2 (ii),

f̄(y, ξ) = f((𝕀_{d×d} − ν(y) ⊗ ν(y)) ξ).

Example 2.4 (The Unit Sphere). Let M = S^{d−1} := {x ∈ ℝ^{d} : |x| = 1}, so here d = N, m = d − 1. We claim that if y ∈ S^{d−1} and ξ ∈ [T_{y}(S^{d−1})]^{N} then

Q_{T} f(y, ξ) = Qf̄(y, ξ),

where

f̄(y, ξ) := f((𝕀_{d×d} − y ⊗ y)ξ).

This may be seen directly from Remark 2.3. Alternatively, we use Proposition 2.2 to derive this formula in terms of parametrization charts for M. Let R be a rotation such that Re_{d} = y, where {e_{1}, e_{2}, …, e_{d}} stands for the canonical basis of ℝ^{d}, and consider the chart

Ψ(x_{1}, …, x_{d−1}) = R (x_{1}, …, x_{d−1}, \sqrt{1 − (x_{1}^{2} + … + x_{d−1}^{2})}) , (x_{1}, …, x_{d−1}) ∈ B_{d−1}(0, 1).

By (2.5) we have that

Q_{T} f(y, ξ) = inf {∫_{Q} f(ξ + ∇Ψ(0)∇ϕ(x)) dx : ϕ ∈ W_{0}^{1,∞}(Q; ℝ^{d−1})} ,  (2.6)

where

∇Ψ(0) = RH, with H := ( \begin{matrix}1 · · · · · · 0 \\ · · · · · · · · \\ · · · · · · · · \\ 0 · · · · · · 1 \\ 0 · · · · · · 0 \end{matrix} )_{d×(d−1)} .

Given ϕ ∈ W_{0}^{1,∞}(Q; ℝ^{d−1}), define ϕ̄ ∈ W_{0}^{1,∞}(Q; ℝ^{d}) by

ϕ̄(x) = RH ϕ(x).

Then

ξ + ∇Ψ(0)∇ϕ(x) = (𝕀 − y ⊗ y)(ξ + ∇ϕ̄(x))

because ξ ∈ [T_{y}(S^{d−1})]^{N}, so (y ⊗ y)ξ = 0, and

\begin{aligned}(𝕀 − y ⊗ y)∇ϕ̄ & = R(𝕀 − e_{d} ⊗ e_{d})R^{T} ∇ϕ̄ \\ & = RHH^{T} R^{T} ∇ϕ̄ \\ & = RH∇ϕ \\ & = ∇Ψ(0)∇ϕ. \end{aligned}  (2.7)

Conversely, given ϕ̄ ∈ W_{0}^{1,∞}(Q; ℝ^{d}) we set

ϕ = H^{T} R^{T} ϕ̄

so that ϕ ∈ W_{0}^{1,∞}(Q; ℝ^{d−1}), and it can be seen easily that (2.7) still holds. This, together with (2.6), implies that

Q_{T} f(y, ξ) = inf {∫_{Q} f((𝕀 − y ⊗ y)(ξ + ∇ϕ̄(x)) dx : ϕ̄ ∈ W_{0}^{1,∞}(Q; ℝ^{d})}
= Qf̄(y, ξ).

Proposition 2.5 (Lower Semicontinuity). Let f : 𝕄^{d×N} → [0, +∞) be a continuous function such that

0 ≤ f(ξ) ≤ C(1 + |ξ|^{p}), p ≥ 1,

for some C > 0 and all ξ ∈ 𝕄^{d×N}. Let

J(u) := ∫_{Ω} Q_{T} f(u, ∇u) dx, u ∈ W^{1,p}(Ω; M).

Then J(·) is W^{1,p}(Ω; M)- sequentially weakly lower semicontinuous.

Proof. For all y ∈ M let Π_{y} : U(y) → U(y) ∩ M be the C^{1} projection onto M introduced in (2.2). Let {η}_{η∈G} be a partition of unity subordinate to {U(y) : y ∈ M} (see [17]), so that

(i) η ≥ 0, η ∈ C_{0}^{∞}(ℝ^{d}; [0, 1]);

(ii) for every η ∈ G there exists y ∈ M such that supp η ⊂ U(y);

(iii) if E ⊂ M is compact then supp η ∩ E ≠ ∅ for only finitely many η ∈ G;

(iv) ∑_{η∈G} η(y) = 1 if y ∈ M.

Consider u, u_{n} ∈ W^{1,p}(Ω; M) such that u_{n} ⇀ u ∈ W^{1,p}, and fix ε > 0. By Lebesgue’s Monotone Convergence Theorem, we may find a compact set E ⊂ M such that

∫_{Ω} Q_{T} f(u, ∇u) dx ≤ ∫_{Ω∩E} Q_{T} f(u, ∇u) dx + ε.  (2.8)

By (iii) and (iv) above, we may choose k ∈ ℕ such that

χ_{E}(y) = ∑_{i=1}^{k} η_{i}(y)χ_{E}(y), where η_{i} ∈ G,

and so, using (ii), with

supp η_{i} ⊂ U(y_{i}), Π_{i} := Π_{y_{i}},

we obtain

∫_{Ω∩E} Q_{T} f(u, ∇u) dx = ∑_{i=1}^{k} ∫_{Ω∩E} η_{i}(u)Q_{T} f(u, ∇u) dx

= ∑_{i=1}^{k} ∫_{Ω∩E} QH_{i}(u, ∇u) dx,

where, according to Definition 2.1 and Remark 2.2 (iii),

H_{i}(y, ξ) := η_{i}(y)f(∇Π_{i}(y)ξ).

Well known lower semicontinuity results for Carathéodory integrands yield (see [1], [4])

∫_{Ω∩E} Q_{T} f(u, ∇u) dx ≤ ∑_{i=1}^{k} lim inf_{n→+∞} ∫_{Ω∩E} QH_{i}(u_{n}, ∇u_{n}) dx

≤ lim inf_{n→+∞} ∑_{i=1}^{k} ∫_{Ω∩E} QH_{i}(u_{n}, ∇u_{n}) dx

= lim inf_{n→+∞} ∑_{i=1}^{k} ∫_{Ω∩E} η_{i}(u_{n})Q_{T} f(u_{n}, ∇u_{n}) dx

≤ lim inf_{n→+∞} ∫_{Ω} Q_{T} f(u_{n}, ∇u_{n}) dx,

which, together with (2.8), concludes the proof.

3 Relaxation

The main result of this section is the integral representation of the relaxed energy, precisely

Theorem 3.1. If f : 𝕄^{d×N} → [0, +∞) is a continuous function satisfying

0 ≤ f(ξ) ≤ C(1 + |ξ|^{p})

for some p ≥ 1, C > 0, and all ξ ∈ 𝕄^{d×N}, and if M is a C^{1} m-dimensional manifold then

F(u) = ∫_{Ω} Q_{T} f(u, ∇u) dx,

where

F(u) := inf_{{u_{n}}} {lim inf_{n→+∞} ∫_{Ω} f(∇u_{n}) dx : u_{n} ⇀ u in W^{1,p}(Ω; M)} .

Proposition 2.5 entails

F(u) ≥ J(u),  (3.1)

and a typical procedure to obtain the converse inequality would involve a “localization” of F(u), i.e., the introduction of the functional

F(u; A) := inf_{{u_{n}}} {lim inf_{n→∞} ∫_{A} f(∇u_{n}) dx : u_{n} ⇀ u in W^{1,p}(Ω; ℝ^{d}), u_{n}(x) ∈ M a.e. x ∈ Ω} ,

where A ∈ A(Ω) and A(Ω) denotes the class of all open subsets of Ω. We would then go on to showing that F(u; ·) is the trace on A(Ω) of a finite Radon measure, absolutely continuous with respect to the N-dimensional Lebesgue measure L^{N} in ℝ^{N}. Finally, establishing the converse of (3.1) would be equivalent to proving that the Radon Nikodym derivative of F(u, ·) with respect to L^{N} is bounded above by the tangential quasiconvexification, i.e.,

\frac{dF(u; ·)}{dL^{N}}(x_{0}) ≤ Q_{T} f(u(x_{0}), ∇u(x_{0})) for L^{N} a.e. x_{0} ∈ Ω.

Now, showing that F(u; ·) is the trace on A(Ω) of a finite Radon measure is equivalent to establishing

(i) (the subadditivity property )

F(u; V) ≤ F(u; V′) + F(u; V \ \overline{V′′})  (3.2)

where V, V′, V′′ ∈ A(Ω), V′′ ⊂⊂ V′ ⊂⊂ V;

(ii) there exists a finite Radon measure ν in Ω such that

F(u, A) ≤ ν(A) for all A ∈ A(Ω).

Using the growth condition imposed on f, (ii) follows immediately by setting

ν := C(1 + |∇u|^{p})L^{N}⌊Ω.

Here we use the notation μ⌊A to denote the restriction of a Borel measure μ to a Borel set A, i.e., μ⌊A(E) := μ(E ∩ A).

The main difficulty lies on part (i), which requires the ability to connect admissible sequences for V \\overline{V′′} and for V′ across a suitable transition layer on V′ \\overline{V′′} and without increasing the total limiting energy. Rather than proving this property directly on F(u, ·), we introduce the auxiliary functional

F_{∞} : W^{1,p}(Ω; M) × A(Ω) → [0, +∞)

defined by

F_{∞}(u; A) := inf_{{u_{n}}} { lim inf_{n→+∞} ∫_{A} f(∇u_{n}) dx : u_{n} ⇀ u in W^{1,p}(Ω; M), u_{n} → u uniformly,

there exists a compact subset E ⊂ M such that u_{n}(x) = u(x) if x ∉ E}.

Our aim is to show that

F(u) = F_{∞}(u; Ω) = ∫_{Ω} Q_{T} f(u, ∇u) dx,

where the last equality for F_{∞}(u; Ω) will be obtained following the scheme outlined above for F(u; Ω). The lemma below will be instrumental in the proof of (3.2) for F_{∞}(u; ·).

Lemma 3.2. Let E ⊂ M be a compact subset. There exist δ > 0, C > 0, and a uniformly continuously differentiable mapping

Φ : D_{δ} × [0, 1] → M,

where

D_{δ} = {(y_{0}, y_{1}) ∈ M × M : dist(y_{0}, E) < δ, dist(y_{1}, E) < δ, |y_{0} − y_{1}| < δ} ,

such that

Φ(y_{0}, y_{1}, 0) = y_{0}, Φ(y_{0}, y_{1}, 1) = y_{1}

and

\frac{∂Φ}{∂t}(y_{0}, y_{1}, t) ≤ C |y_{1} − y_{0}| .

Proof. Cover E with a finite collection of convex open sets U(y_{i}) ⊂ ℝ^{d}, where y_{i} ∈ E∩U(y_{i}), i = 1, …, k, and consider the C^{1} projections onto M introduced in (2.2) (if necessary, the original neighborhoods U(y_{i}) are reduced so as to be rendered convex)

Π_{i} : U(y_{i}) → U(y_{i}) ∩ M.

Define

Φ̃_{i} : (U_{i} ∩ M) × (U_{i} ∩ M) × [0, 1] → U_{i} ∩ M, (y_{0}, y_{1}, t) ↦ Π_{i}((1 − t)y_{0} + ty_{1}).

It is clear that Φ̃_{i}(y_{0}, y_{1}, 0) = y_{0}, and Φ̃_{i}(y_{0}, y_{1}, 1) = y_{1}. Since E is a compact set, we may shrink slightly the open sets U_{i} so as to keep a covering of E, i.e., we choose δ_{1} > 0 such that

E ⊂ ∪_{i=1}^{k}V_{i}, V_{i} := {y ∈ U_{i} : dist(y, ∂U_{i}) > δ_{1}} ,

and, similarly, fix 0 < δ_{2} < 2δ_{1} verifying

E ⊂ ∪_{i=1}^{k}W_{i}, W_{i} := {y ∈ V_{i} : dist(y, ∂V_{i}) > δ_{2}} .

Let

C_{1} := max {1, ∥Π_{i}∥_{W^{1,∞}(Ũ)} : i = 1, …, k}

where

Ũ_{i} := \{y ∈ U_{i} : dist(y, ∂U_{i}) ≥ \frac{δ_{2}}{2} \} .

Set

δ := \frac{δ_{2}}{4C_{1}^{k}} .

We claim that if y_{0}, y_{1} ∈ M are such that dist(y_{0}, E) < δ, dist(y_{1}, E) < δ, and |y_{0} − y_{1}| < δ, then there exists i ∈ {1, …, k} such that

(y_{0}, y_{1}) ∈ Ũ_{i} × Ũ_{i}  (3.3)

and

|Φ̃_{i}(y_{0}, y_{1}, t) − y_{0}| < C_{1}|y_{0} − y_{1}|.  (3.4)

Indeed, if dist(y_{0}, E) < δ then we may find

e ∈ E ⊂ ∪_{i=1}^{k}W_{i}

such that |y_{0} − e| < δ < δ_{2}, |y_{1} − e| < 2δ < δ_{2}. Let i ∈ {1, …, k} be such that e ∈ W_{i}. Since dist(e, ∂V_{i}) > δ_{2}, we have that y_{0}, y_{1} ∈ V_{i}, and since dist(y_{0}, ∂U_{i}), dist(y_{1}, ∂U_{i}) > δ_{1} > \frac{δ_{2}}{2} , we conclude that y_{0}, y_{1} ∈ Ũ_{i}. This proves claim (3.3). Also,

|Φ̃_{i}(y_{0}, y_{1}, t) − y_{0}| = |Π_{i}((1 − t)y_{0} + ty_{1}) − Π_{i}(y_{0})|
≤ C_{1} |y_{1} − y_{0}| ,

and we obtain (3.4).

For i = 1, …, k, let η_{i} ∈ C_{0}^{∞}(U_{i}; [0, 1]) be a cut-off function such that η_{i} = 1 in Ũ_{i} and η_{i}(y) = 0 if dist(y, ∂U_{i}) < δ_{2}/4. Clearly

η_{i}(y) = 0 ⇒ y ∉ Ũ_{i}, η_{i}(y) > 0 ⇒ B (y, δ_{2}/4) ⊂ U_{i}.

Set

η̄_{i}(y_{0}, y_{1}) := η_{i}(y_{0}) η_{i}(y_{1}),

and define

Φ_{1} : (D_{δ} ∩ (Ũ_{1} × Ũ_{1})) × [0, 1] → U_{1} ∩ M, Φ_{1} := Φ̃_{1}.

Suppose that (y_{0}, y_{1}) ∈ (Ũ_{1} × Ũ_{1}) ∩ (M × M), |y_{0} − y_{1}| < δ, and η̄_{2}(y_{0}, y_{1}) > 0. If t ∈ [0, 1] then by (3.4)

|Φ_{1}(y_{0}, y_{1}, t) − y_{0}| ≤ C_{1}δ < \frac{δ_{2}}{4} ,

and, since η_{2}(y_{0}) > 0, B (y_{0}, δ_{2}/4) ⊂ U_{2}. We deduce that

Φ_{1}(y_{0}, y_{1}, t) ∈ U_{2}.

We may then define

Φ_{2} : (D_{δ} ∩ [(Ũ_{1} × Ũ_{1}) ∪ (Ũ_{2} × Ũ_{2})]) × [0, 1] → M,

Φ_{2}(y_{0}, y_{1}, t) := \{ \begin{matrix}Φ_{1}(y_{0}, y_{1}, t) & if η̄_{2}(y_{0}, y_{1}) = 0 \\ Π_{2} (\frac{η̄_{1}(y_{0},y_{1})Φ_{1}(y_{0},y_{1},t)+η̄_{2}(y_{0},y_{1})Φ̃_{2}(y_{0},y_{1},t)}{η̄_{1}(y_{0},y_{1})+η̄_{2}(y_{0},y_{1})}) & if η̄_{2}(y_{0}, y_{1}) > 0. \end{matrix}

Note that Φ_{2}(y_{0}, y_{1}, 0) = y_{0}, Φ_{2}(y_{0}, y_{1}, 1) = y_{1}, and if

η̄_{2}(y_{0}^{n}, y_{1}^{n}) → 0^{+} as y_{0}^{n} → y_{0}, y_{1}^{n} → y_{1}, with η̄_{1}(y_{0}, y_{1}) > 0,

then

Φ_{2}(y_{0}^{n}, y_{1}^{n}, t) → Π_{2}(Φ_{1}(y_{0}, y_{1}, t)) = Φ_{1}(y_{0}, y_{1}, t),

and it follows easily that Φ_{2} is smooth. In addition

|Φ_{2}(y_{0}, y_{1}, t) − y_{0}| < C_{1}^{2}δ

and so, if η_{3}(y_{0}) > 0 then

Φ_{2}(y_{0}, y_{1}, t) ∈ B (y_{0}, C_{1}^{2}δ) ⊂ B (y_{0}, δ_{2}/4) ⊂ U_{3};

hence

Π_{3}(θΦ_{2} + (1 − θ)Φ̃_{3}) is well defined

for all θ ∈ [0, 1]. In this manner, recursively we define

Φ_{i} : (D_{δ} ∩ ⋃_{j≤i}(Ũ_{j} × Ũ_{j})) × [0, 1] → M

by

Φ_{i}(y_{0}, y_{1}, t) := \{ \begin{matrix}Φ_{i−1}(y_{0}, y_{1}, t) & if η̄_{i}(y_{0}, y_{1}) = 0, \\ Π_{i} (\frac{(η̄_{1}+...+η̄_{i−1})Φ_{i−1}+η̄_{i}Φ̃_{i}}{η̄_{1}+...+η̄_{i}} ) & if η̄_{i}(y_{0}, y_{1}) > 0. \end{matrix}

In light of (3.3), Φ := Φ_{k} fulfills the desired requirements.

Remark 3.3. It follows from the construction of Φ in the above lemma that

|Φ(y_{0}, y_{1}, t) − y_{0}| < C |y_{0} − y_{1}|

for some C > 0, and for all (y_{0}, y_{1}, t) ∈ D_{δ} × [0, 1].

Proposition 3.4. If u ∈ W^{1,p}(Ω; M) then F_{∞}(u; ·) is the trace in A(Ω) of a finite Radon measure, absolutely continuous with respect to the N-dimensional Lebesgue measure L^{N}.

Proof. The last part of the statement is trivial, since due to the growth condition on f, we have that

F_{∞}(u; A) ≤ C ∫_{A}(1 + |∇u|^{p}) dx = C (1 + |∇u|^{p})L^{N}⌊Ω(A).  (3.5)

We claim that the subadditivity property holds (see (3.2)), i.e.,

F_{∞}(u; V) ≤ F_{∞}(u; V′) + F_{∞}(u; V \ \overline{V′′})  (3.6)

whenever V, V′, V′′ ∈ A(Ω), V′′ ⊂⊂ V′ ⊂⊂ V. We may assume that V′ has a smooth boundary. Fix ε > 0 and let u_{n} ∈ W^{1,p}(Ω; M) be such that u_{n} → u uniformly, u_{n} ⇀ u in W^{1,p}, u_{n} = u if u(x) ∉ E_{1}, E_{1} ⊂ M compact, and

F_{∞}(u; V \ \overline{V′′}) + ε ≥ lim_{n→+∞} ∫_{V \\overline{V′′}} f(∇u_{n}) dx.

Similarly, let v_{n} ∈ W^{1,p}(Ω; M) be such that v_{n} → u uniformly and weakly in W^{1,p}, v_{n} = u if u(x) ∉ E_{2}, E_{2} ⊂ M compact, with

F_{∞}(u; V′) + ε ≥ lim_{n→+∞} ∫_{V′} f(∇v_{n}) dx.

Without loss of generality, we may assume that (up to a subsequence)

(|∇u_{n}|^{p} + |∇v_{n}|^{p})L^{N}⌊(V′ \ \overline{V′′}) ⇀^{∗} ν,

where ν is a finite, Radon measure, and we choose δ_{0}, 0 < δ_{0} < dist(V′′, ∂V′), such that

ν(S) = 0, S := \{x ∈ V′ \ \overline{V′′} : dist(x, ∂V′) = δ_{0}\} .

Given j ∈ ℕ we consider a smooth cut-off function η_{j} ∈ C_{0}^{∞}(V′; [0, 1]) with ∥∇η_{j}∥_{∞} ≤ Cj and

η_{j} = \{ \begin{matrix}1 & if dist(x, ∂V′) > δ_{0} + 1/j, \\ 0 & if dist(x, ∂V′) < δ_{0} − 1/j. \end{matrix}

Setting E := E_{1}∪E_{2} ⊂ M, E is compact and we consider δ > 0 and a function Φ : D_{δ}×[0, 1] → M satisfying the properties of Lemma 3.2. Choose n large enough so that

∥u_{n} − u∥_{L^{∞}(V \\overline{V′′})}, ∥v_{n} − u∥_{L^{∞}(V′)}, ∥u_{n} − v_{n}∥_{L^{∞}(V′\\overline{V′′})} < δ.

If u(x) ∈ E then dist(u_{n}(x), E), dist(v_{n}(x), E) < δ, and we set

w_{n,j}(x) := Φ(u_{n}(x), v_{n}(x), η_{j}(x)).

If u(x) ∉ E, then define

w_{n,j}(x) := u(x).

By Remark 3.3 it follows that

lim_{j} lim_{n} ∥w_{n,j} − u∥_{L^{∞}(V)} = 0,

and

lim sup_{j} lim sup_{n} ∫_{Ω} |∇w_{n,j}|^{p} dx

≤ C + lim sup_{j} lim sup_{n} ∫_{L_{j}} C(1 + j^{p}|u_{n} − v_{n}|^{p} + |∇u_{n}|^{p} + |∇v_{n}|^{p}) dx < +∞,

where

L_{j} := \{x ∈ V′ \ \overline{V′′} : δ_{0} − \frac{1}{j} < dist(x, ∂V′) < δ_{0} + \frac{1}{j}\} .

Therefore,

F_{∞}(u; V) ≤ lim inf_{j→+∞} lim inf_{n→∞} ∫_{V} f(∇w_{n,j}) dx
≤ F_{∞}(u; V \ \overline{V′′}) + F_{∞}(u; V′) + 2ε
+ lim sup_{j→+∞} lim sup_{n→+∞} ∫_{L_{j}} C(1 + j^{p}|u_{n} − v_{n}|^{p} + |∇u_{n}|^{p} + |∇v_{n}|^{p}) dx
≤ F_{∞}(u; V \ \overline{V′′}) + F_{∞}(u; V′) + 2ε + lim sup_{j→+∞} ν(\overline{L_{j}})
= F_{∞}(u; V \ \overline{V′′}) + F_{∞}(u; V′) + 2ε

because

lim_{j→+∞} ν(\overline{L_{j}}) = ν(S) = 0.

This proves (3.6).

Now we conclude the proof of Proposition 3.4. Up to the extraction of a subsequence, we assume that

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

where

u_{n} ∈ W^{1,p}(Ω, M), ∥u_{n} − u∥_{L^{∞}(Ω)} → 0, u_{n}(x) = u(x) if u(x) ∉ E,

and E ⊂ M is a compact set. Moreover,

f(∇u_{n})L^{N}⌊Ω ⇀^{∗} μ,

where μ is a finite, Radon measure in ℝ^{N}. By (3.5), for all V ∈ A(Ω), ε > 0, we may find C ⊂⊂ V such that

F_{∞}(u; V \ \overline{V′′}) < ε.

First we show that

F_{∞}(u; V) ≤ μ(V) for all V ∈ A(Ω).  (3.7)

In fact, if V′ ∈ A(Ω) then

F_{∞}(u; V′) ≤ lim inf_{n→+∞} ∫_{V′} f(∇u_{n}) dx
≤ μ(\overline{V′}),

thus, using (3.5) and given ε > 0 choose V′′ ⊂⊂ V′ ⊂⊂ V such that

F_{∞}(u; V \ \overline{V′′}) < ε,

so that by (3.6) we have

F_{∞}(u; V) ≤ ε + F_{∞}(u; V′)
≤ ε + μ(\overline{V′})
≤ ε + μ(V)

and (3.7) follows by letting ε → 0^{+}.

Conversely, fixing V ∈ A(Ω), ε > 0, choose V_{ε} ⊂⊂ V open such that

μ(V \ \overline{V}_{ε}) < ε.

Then

μ(V) ≤ ε + μ(\overline{V}_{ε})
= ε + μ(Ω) − μ(Ω \ \overline{V}_{ε})
≤ ε + F_{∞}(u; Ω) − F_{∞}(u; Ω \ \overline{V}_{ε})
≤ ε + F_{∞}(u; V),

where we have used (3.7) and (3.6).

Proof of Theorem 3.1. By Proposition 2.5 we have

F(u) ≥ ∫_{Ω} Q_{T} f(u, ∇u) dx,

and since, clearly, F_{∞}(u; Ω) ≥ F(u), it remains to show that

F_{∞}(u; Ω) ≤ ∫_{Ω} Q_{T} f(u, ∇u) dx,

or, equivalently, and in view of Proposition 3.4,

\frac{dF_{∞}(u; ·)}{dL^{N}}(x_{0}) ≤ Q_{T} f(u(x_{0}), ∇u(x_{0})) for L^{N} a.e. x_{0} ∈ Ω.

Let x_{0} ∈ Ω be a Lebesgue point for u, ∇u, and such that

\frac{dF_{∞}}{dL^{N}} (u; Ω)(x_{0}) exists and is finite.

Fix ε > 0 and let φ ∈ W_{0}^{1,∞}(Q; T_{y_{0}}(M)) be extended periodically to ℝ^{N}, where y_{0} := u(x_{0}) and (see Definiton 2.1)

Q_{T} f(u(x_{0}), ∇u(x_{0})) + ε ≥ ∫_{Q} f(∇u(x_{0}) + ∇φ(x)) dx.  (3.8)

Let Π : U(y_{0}) → U(y_{0})∩M be a C^{1} projection of an open neighborhood U(y_{0}) of y_{0} onto U(y_{0})∩M (see (2.2)). Since f is locally uniformly continuous, given ε > 0 we may find B(y_{0}, δ_{0}) ⊂⊂ U(y_{0}) and 0 < ρ < 1 such that

|ξ_{1}| , |ξ_{2}| ≤ C_{2}, |ξ_{1} − ξ_{2}| < ρ ⟹ |f(ξ_{1}) − f(ξ_{0})| < ε,  (3.9)

where

C_{2} := (∥Π∥_{L^{∞}(B(y_{0},δ_{0}))} + 1) (2 + 2 |∇u(x_{0})| + ∥∇φ∥_{L^{∞}(Q)}) .  (3.10)

Choose δ_{0} > δ > 0 such that

y, y′ ∈ B(y_{0}, δ) ∩ M ⟹ |∇Π(y) − ∇Π(y′)| < θ,  (3.11)

where 0 < θ < 1 has been selected so that

2θ (∥∇Π∥_{L^{∞}(B(y_{0},δ_{0}))} + 1) (1 + |∇u(x_{0})|) (1 + ∥∇φ∥_{L^{∞}(Q)}) < \frac{ρ}{2}.  (3.12)

Let η ∈ C^{∞}(ℝ^{d}; [0, 1]) be a cut-off function such that

η = \{ \begin{matrix}1 & in B(0, δ/4), \\ 0 & outside B(0, δ/2), \end{matrix}

with ∥∇η∥_{∞} ≤ \frac{C}{δ} . We define

u_{n}(x) := \{ \begin{matrix}u(x) if & |u(x) − y_{0}| ≥ \frac{δ}{2}, \\ Π(w_{n}) if & |u(x) − y_{0}| < \frac{δ}{2}, \end{matrix}

where

w_{n}(x) := u(x) + η(u(x) − y_{0})\frac{1}{n}φ(nx).

Note that u_{n} is well defined for

n > 2\frac{∥φ∥_{L^{∞}(Q)}}{δ} ,

because if |u(x) − y_{0}| < \frac{δ}{2} then

|w_{n}(x) − y_{0}| < \frac{δ}{2} + \frac{1}{n}∥φ∥_{L^{∞}(Q)} < δ,

and so w_{n}(x) ∈ U(y_{0}). Notice that

∥u_{n} − u∥_{L^{∞}(Ω)} = ∥u_{n} − u∥_{L^{∞}(\{|u−y_{0}|≤ \frac{δ}{2} \})}
≤ ∥Π∥_{W^{1,∞}(B_{m}(y_{0},δ_{0}))} \frac{1}{n}∥φ∥_{∞}
→ 0

as n → +∞. Therefore,

∥u_{n} − u∥_{L^{∞}(Ω)} → 0 as n → +∞.

In addition, if

|u(x) − y_{0}| ≤ \frac{δ}{2}

then

∇u_{n}(x) = ∇Π(w_{n}) [∇u(x) + η(u(x) − y_{0})∇φ(nx) + \frac{1}{n}φ(nx) ⊗ ∇η(u(x) − y_{0})∇u(x)] ,

so that

\begin{aligned}|∇u_{n}(x)| & ≤ ∥∇Π∥_{L^{∞}(B_{m}(y_{0},δ_{0}))} \\ & × [|∇u(x_{0})| + |∇u(x) − ∇u(x_{0})| \\ & + ∥∇φ∥_{L^{∞}(Q)} + \frac{1}{n}∥φ∥_{L^{∞}(Q)}∥∇η∥_{L^{∞}} |∇u(x)|] \\ & ≤ ∥∇Π∥_{L^{∞}(B_{m}(y_{0},δ_{0}))} [2 |∇u(x) − ∇u(x_{0})| + 2 |∇u(x_{0})| + ∥∇φ∥_{L^{∞}(Q)}] \end{aligned}  (3.13)

provided that

n > max \{2\frac{∥φ∥_{L^{∞}}}{δ} , ∥φ∥_{L^{∞}(Q)}∥∇η∥_{L^{∞}}\} .

We conclude that

|∇u_{n}(x)| ≤ C(1 + |∇u(x) − ∇u(x_{0})| a. e. x ∈ Ω,  (3.14)

and, as a consequence,

{∥∇u_{n}∥_{L^{p}(Ω)}} is bounded.

Setting

z_{n} := ∇u(x_{0})x + \frac{1}{n}φ(nx),

then

|∇z_{n}(x)| ≤ |∇u(x_{0})| + ∥∇φ∥_{L^{∞}(Q)}.

It follows from (3.10) and (3.13) that if |∇u(x) − ∇u(x_{0})| < 1 then

|∇u_{n}(x)| , |∇w_{n}(x)| ≤ C_{2}.  (3.15)

Let

γ := min \{1, (|∇Π(y_{0})| + 1)^{−1} \frac{ρ}{2}\} ,

and assume that

|u(x) − y_{0}| < \frac{δ}{4}, |∇u(x) − ∇u(x_{0})| < γ.

Then η(u(x) − y_{0}) = 1 and

|∇u_{n}(x) − ∇z_{n}(x)| ≤ |∇Π(w_{n})∇u(x) − ∇u(x_{0})|
+ |∇Π(w_{n})∇φ(nx) − ∇φ(nx)|
≤ |∇Π(w_{n}) − ∇Π(y_{0})| |∇u(x)|
+ |∇Π(y_{0})| |∇u(x) − ∇u(x_{0})|
+ |∇Π(w_{n}) − ∇Π(y_{0})| ∥∇φ∥_{L^{∞}(Q)}
=: I_{1} + I_{2} + I_{3}.

By (3.11)

I_{1} ≤ θ(1 + |∇u(x_{0})|),

while, by definition of γ,

I_{2} ≤ \frac{ρ}{2},

and once again by (3.11) we get

I_{3} ≤ θ∥∇φ∥_{L^{∞}(Q)}.

In view of (3.12) we deduce that

|∇u_{n}(x) − ∇z_{n}(x)| < ρ.  (3.16)

We have

\frac{dF_{∞}}{dL^{N}} (u, ·)(x_{0}) = lim_{r→0^{+}} \frac{F_{∞}(u; B(x_{0}, r))}{L^{N}(B(x_{0}, r))}

≤ lim inf_{r→0^{+}} lim inf_{n→+∞} \frac{1}{|B(x_{0}, r)|} ∫_{B(x_{0},r)} f(∇u_{n}) dx

≤ lim inf_{r→0^{+}} lim inf_{n→+∞} \frac{1}{|B(x_{0}, r)|} ∫_{B(x_{0},r)∩\{|u(x)−y_{0}|≥ \frac{δ}{4}\}} f(∇u_{n}) dx  (3.17)

+ lim sup_{r→0^{+}} lim sup_{n→+∞} \frac{1}{|B(x_{0}, r)|} ∫_{B(x_{0},r)∩\{|u(x)−y_{0}|< \frac{δ}{4}\}∩\{|∇u(x)−∇u(x_{0})|<γ\}} f(∇u_{n}) dx

+ lim sup_{r→0^{+}} lim sup_{n→+∞} \frac{1}{|B(x_{0}, r)|} ∫_{B(x_{0},r)∩\{|u(x)−y_{0}|< \frac{δ}{4}\}∩\{|∇u(x)−∇u(x_{0})|≥γ\}} f(∇u_{n}) dx

=: J_{1} + J_{2} + J_{3}.

By virtue of (3.15), (3.16) and (3.9), we obtain

J_{2} ≤ lim sup_{r→0^{+}} lim sup_{n→+∞} \frac{1}{|B(x_{0}, r)|} ∫_{B(x_{0},r)} f(∇z_{n}) dx + ε

= lim sup_{r→0^{+}} lim sup_{n→+∞} \frac{1}{|B(x_{0}, r)|} ∫_{B(x_{0},r)} f(∇u(x_{0}) + ∇φ(nx)) dx + ε

= ∫_{Q} f(∇u(x_{0}) + ∇φ(y)) dy + ε

≤ Q_{T} f(u(x_{0}), ∇u(x_{0})) + 2ε,

(3.18)

where we have used (3.8). By (3.14) we have

J_{1} ≤ lim sup_{r→0^{+}} \frac{1}{|B(x_{0}, r)|} ∫_{B(x_{0},r)∩\{|u(x)−y_{0}|≥ \frac{δ}{4}\}} C(1 + |∇u(x) − ∇u(x_{0})|^{p}) dx
≤ lim sup_{r→0^{+}} \frac{C}{|B(x_{0}, r)|} ∫_{B(x_{0},r)} |∇u(x) − ∇u(x_{0})|^{p} dx  (3.19)

+ lim sup_{r→0^{+}} \frac{4C}{δ} \frac{1}{|B(x_{0}, r)|} ∫_{B(x_{0},r)} |u(x) − y_{0}| dx = 0,

and, finally,

J_{3} ≤ lim sup_{r→0^{+}} \frac{1}{|B(x_{0}, r)|} ∫_{B(x_{0},r)∩\{|∇u(x)−∇u(x_{0})|≥γ\}} C(1 + |∇u(x) − ∇u(x_{0})|^{p}) dx
≤ lim sup_{r→0^{+}} \frac{C}{|B(x_{0}, r)|} ∫_{B(x_{0},r)} |∇u(x) − ∇u(x_{0})|^{p} dx  (3.20)

+ lim sup_{r→0^{+}} \frac{C}{γ} \frac{1}{|B(x_{0}, r)|} ∫_{B(x_{0},r)} |∇u(x) − ∇u(x_{0})| dx = 0.

The conclusion follows from (3.17), (3.18), (3.19), and (3.20).

Acknowledgments: The research of I. Fonseca was partially supported by the National Science Foundation through the Center for Nonlinear Analysis, by the National Science Foundation under grant No. DMS–9500531, and by the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. The research of J. Malý was supported by the grant No. 201/96/0311 of Czech Grant Agency (GAČR) and by the grant No. 189/96 of Charles University (GAUK).

Part of this research was undertaken while K. Trivisa was a postdoctoral associate at the Center for Nonlinear Analysis (CNA) at Carnegie Mellon University, and during the visits of J. Malý to the CNA and to the Max Planck Institute of Mathematics in the Sciences, Leipzig, Germany. The authors are indebted to V. Mizel for many fruitful discussions in the subject of this work.

References

[1] E. Acerbi and N. Fusco, New lower semicontinuity results for polyconvex integrals case, Arch. Rat. Mech. Anal. 86 (1984), 125–145.

[2] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63 (1977), 337–403.

[3] H. Brézis, J. M. Coron and E. H. Lieb, Estimations d’ énergie pour des applications de ℝ^{3} à valeurs dans S^{2}, C. R. Acad. Sc. Paris, t. 303, Série I 5 (1986), 207– 210.

[4] B. Dacorogna, “Direct Methods in Calculus of Variations”, Appl. Math. Sciences 78, Springer–Verlag, Berlin 1989.

[5] A. De Simone, Energy minimizers for large ferromagnetic bodies, Arch. Rat. Mech. Anal. 125 (1993), 99 –143.

[6] J. L. Ericksen, Liquid crystals with variable degree of orientation, Arch. Rational Mech. Anal. 113 (1991), 97–120.

[7] L. C. Evans and R. F. Gariepy, “Measure Theory and Fine Properties of Functions”, CRC Press, 1992.

[8] H. Federer, “Geometric Measure Theory”, Springer, 1969.

[9] I. Fonseca and P. Marcellini, Relaxation of Multiple Integrals in Subcritical Sobolev Spaces., J. Geom. Anal. (to appear).

[10] I. Fonseca, S. Müller and P. Pedregal, Analysis of Concentration and Oscillation Effects Generated by Gradients, SIAM J. Math. Anal 29 (1998), 736-756.

[11] M. Giaquinta, G. Modica and J. Souček, Cartesian Currents and Variational Problems for Mappings into Spheres, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16 (1989), 393–485.

[12] M. Giaquinta, G. Modica and J. Souček, Liquid crystals: relaxed energies, dipoles, singular lines and singular points, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17,3 (1990), 415–437.

[13] R. James and D. Kinderlehrer, Theory of magnetostriction with applications to T b_{x}Dy_{1−x}F e_{2}, Phil. Mag. B 68 (1993), 237–274.

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

[15] C. B. Morrey, “Multiple Integrals in the Calculus of Variations”, Springer, 1996.

[16] E. G. Virga, “Variational Theories for Liquid Crystals”, Chapman & Hall, 1994.

[17] W. P. Ziemer, “Weakly Differentiable Functions”, Springer-Verlag, Berlin 1989.

Bernard Dacorogna Départment de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

E-mail: dacorog@masg1.epfl.ch

Irene Fonseca Department of Mathematical Sciences, Carnegie Mellon University Pittsburgh, PA 15213, USA

E-mail: fonseca@andrew.cmu.edu

Jan Malý KMA Department, Faculty of Mathematics and Physics, Charles University Sokolovská 83, CZ-186 75 Praha, Czech Republic

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

Konstantina Trivisa Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston IL 60208, USA

E-mail: trivisa@math.nwu.edu