RELAXATION RESULTS IN MICROMAGNETICS

IRENE FONSECA AND GIOVANNI LEONI

In memory of Ennio De Giorgi

Abstract. In this paper an integral representation formula is obtained for the relaxed energy of a large ferromagnetic body. With Ω ⊂ ℝ^{N} being the reference configuration of the body and m : Ω → S^{N−1} the magnetization, consider the energy

F (m) := ∫_{ℝ^{N}} f(x, χ_{Ω}(x) m(x), u(x), ∇u(x)) dx

where (χ_{Ω} m, ∇u) satisfies Maxwell’s equations, i.e. u ∈ H^{1}(ℝ^{N}) is the unique solution of Δu+div(χ_{Ω} m) = 0 in ℝ^{N}. If f is a Carathéodory function satisfying very mild growth conditions then it is shown that the relaxation of F with respect to L^{∞}-w∗ convergence in L^{∞}(Ω; \overline{B(0, 1)}) is given by

F(m) = ∫_{Ω} Q_{M} f(x, m(x), u(x), ∇u(x)) dx + ∫_{ℝ^{N} \Ω} f(x, 0, u(x), ∇u(x)) dx

where Q_{M} f is the quasiconvex envelope of f relative to the underlying partial differential equations. This class of integrands includes those of the type

f(x, m, u, h) = φ(x, m, u) + ψ(x, u, h)

with ψ(x, u, ·) non convex, thus extending the available relaxation results in micromagnetics.

1. Introduction

In recent years micromagnetics, a continuum model of the macroscopic behavior of a ferromagnetic body, has attracted the attention of researchers in Calculus of Variations (see e.g. the work of [2, 6, 9, 11, 17, 18, 22, 24, 25, 26]). According to this theory, essentially developed by Brown [5], equilibrium states of a body subject to a given external magnetic field h_{e} correspond to absolute (or local) minimizers of the energy functional

F_{e}(m) := \frac{ε}{2} ∫_{Ω} |∇m(x)|^{2} dx + ∫_{Ω} φ(m(x)) dx − ∫_{Ω} h_{e}(x) · m(x) dx + \frac{1}{2} ∫_{ℝ^{3}} |h(x)|^{2} dx. (1.1)

Here Ω ⊂ ℝ^{3} represents the region occupied by the body and the magnetization is a function m : ℝ^{3} → ℝ^{3} such that

|m(x)| = m_{s} χ_{Ω}(x) for L^{N} a.e. x ∈ ℝ^{3}, (1.2)

where m_{s} > 0, the saturation magnetization, is a function of the temperature and of the specific material. We assume that the temperature is fixed, and, without loss of generality, in what follows we take

m_{s} = 1.

Date: November 26, 1999. 1991 Mathematics Subject Classification. Primary 35D99, 35E99, 49J45. Key words and phrases. Micromagnetics, Young measure, relaxation. The research of I. Fonseca was partially supported by the National Science Foundation through the Center for Nonlinear

Analysis 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 GNAFA and the strategic project “Metodi e modelli per la Matematica e l’Ingegneria”.

The authors wish to thank the Center for Nonlinear Analysis, Pittsburgh, USA (NSF Grant No. DMS–9731957) and C.M.A.F., Lisbon, Portugal for their hospitality and support during the preparation of this work.

The induced magnetic field is a function h : ℝ^{3} → ℝ^{3} related to m through the Maxwell’s equations for magnetostatics

A(m, h) := ( \begin{matrix}m + h \\ curl h \end{matrix} ) = 0 in ℝ^{3}. (1.3)

The existence of absolute minimizers for F_{e} has been established by Visintin [26], using the Direct Method of the Calculus of Variations, and their structure strongly depends on the competition between different terms in the energy functional (1.1). Precisely:

• the exchange energy \frac{ε}{2} ∫_{Ω} |∇m(x)|^{2} dx penalizes spatial changes of m and thus models the tendency of the body to exhibit large regions of uniform magnetization (magnetic domains) separated by very thin transitions layers (domain walls). The parameter ε > 0 is obtained by a rescaling argument and is related to size of the body (see [11]);

• the anisotropy energy ∫_{Ω} φ(m(x)) dx drives m to preferred crystallographic directions (easy axes), where the anisotropy energy density φ(m) is a nonnegative, even, continuous function, defined on the unit sphere S^{2} and vanishing only on a finite set of unit vectors which represent the easy axes;

• the external field energy (or interaction energy) ∫_{Ω} h_{e}(x)·m(x) dx fosters the alignment of the magnetization m with the external field h_{e};

• the field energy \frac{1}{2} ∫_{Ω} |h(x)|^{2} dx vanishes only when div(mχ_{Ω}) = 0 and thus models the tendency of the magnetization to be divergence-free.

In [17] James and Kinderlehrer proposed the following simplified version of the energy functional (1.1)

F (m) := ∫_{Ω} φ(m(x)) dx − ∫_{Ω} h_{e}(x) · m(x) dx + \frac{1}{2} ∫_{ℝ^{3}} |h(x)|^{2} dx, (1.4)

which is usually referred as the no-exchange functional. The main new feature of the functional F is that it is no longer semicontinuous with respect to the natural topology for m, the L^{∞}(Ω; ℝ^{3}) weak-* topology. Minimizing sequences may develop oscillations and the existence of minimizers is no longer guaranteed. This is in agreement with the formation of small magnetics domains observed in the experiments. As an example, in the uniaxial case when {φ = 0} = {m_{1}, −m_{1}} for some m_{1} ∈ S^{2}, and for h_{e} = 0, James and Kinderlehrer [17] proved non-existence of minimizers by constructing minimizing sequences which develop finer and finer spatial oscillations, and characterized the Young measures generated by such sequences.

Tartar [24, 25] identified the limiting energy

lim_{ε→0^{+}} F_{ε},

by using a combination of Young measures and H-measures and a result obtained in collaboration with Murat on the relations between Young measures and H-measures [20]. In particular, he showed that the relaxed functional of F, which is defined by

F(m) := inf {lim inf_{k→∞} F (m_{k}) : m_{k} ∈ L^{∞}(Ω; S^{2}), m_{k} ⇀^{∗} m in L^{∞}(Ω; ℝ^{3})}

where |m(x)| ≤ 1, admits the following integral representation formula

F(m) = ∫_{Ω} φ^{∗∗}(m(x)) dx − ∫_{Ω} h_{e}(x) · m(x) dx + \frac{1}{2} ∫_{ℝ^{3}} |h(x)|^{2} dx,

where φ^{∗∗} is the convex envelope of the function φ extended to be +∞ outside the unit sphere S^{2}. The same representation formula for F(m) was later obtained by De Simone [11] using classical methods from the Calculus of Variations. De Simone also showed that when the volume of the body becomes infinite then minimizers and minimizing sequences of F represent the limit behavior of minimizers of the functional F_{ε}, thus asserting that the no-exchange model is a good approximation for large ferromagnetics bodies.

However, as emphasized by Tartar in [24, 25], there are quantitative discrepancies between minimizing sequences and experimental observations of formation of magnetic domains, and it is not clear how robust and accurate this model is (see also [17]). As a first step towards understanding the oscillatory behavior of minimizing sequences, we propose here to address the relaxation of a general class of functionals which includes F.

Let Ω ⊂ ℝ^{N} be an open bounded set with Lipschitz boundary and define

(1.5) M := {m : ℝ^{N} → ℝ^{N} measurable : |m(x)| = χ_{Ω}(x) for L^{N} a.e x ∈ ℝ^{N}}.

Consider a Carathéodory function

f : ℝ^{N} × (S^{N−1} ∪ {0}) × ℝ × ℝ^{N} → [0, ∞)

such that

0 ≤ f(x, m, u, h) ≤ a(x) + C(|u|^{2^{∗}} + |h|^{2}) (1.6)

for L^{N} a.e. x ∈ Ω and all (m, u, h) ∈ (S^{N−1} ∪ {0}) × ℝ × ℝ^{N}, and where a ∈ L^{1}(ℝ^{N}), C > 0 and

2^{∗} := \{ \begin{matrix}\frac{2N}{N−2} & if N > 2, \\ any q ∈ [1, ∞) & if N = 2. \end{matrix}

Given a function m ∈ M there exists a unique function u ∈ H^{1}(ℝ^{N}) such that (see [17])

Δu + div m = 0 in ℝ^{N} (1.7)

in the sense of distributions, and thus we define the energy functional

F (m) := ∫_{ℝ^{N}} f(x, m(x), u(x), ∇u(x)) dx.

Set

(1.8) M^{∗} := {m : ℝ^{N} → ℝ^{N} measurable : |m(x)| ≤ χ_{Ω}(x) for L^{N} a.e x ∈ ℝ^{N}},

and for m ∈ M^{∗} define

F(m) := inf {lim inf_{k→∞} F (m_{k}) : m_{k} ∈ M, m_{k} ⇀^{∗} m in L^{∞}(Ω; ℝ^{N})} .

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

Theorem 1. Under condition (1.6), for all m ∈ M^{∗} we have

F(m) = ∫_{Ω} Q_{M}f(x, m(x), u(x), ∇u(x)) dx + ∫_{ℝ^{N} \Ω} f(x, 0, u(x), ∇u(x)) dx

where the function Q_{M}f : Ω × \overline{B} × ℝ × ℝ^{N} → [0, ∞) coincides with f when |m| = 1, while for |m| < 1 is given by

Q_{M}f(x, m, u, h) := inf \{lim inf_{k→∞} \frac{1}{|B|} ∫_{B} f(x, M_{k}(y), u, h + ∇U_{k}(y)) dy : M_{k} ∈ C^{∞}(\overline{B}; S^{N−1}),

M_{k} ⇀^{∗} m in L^{∞}(B; ℝ^{N}), U_{k}(y) := − ∫_{B} G(ξ, y) div M_{k}(ξ) dξ} ,

where B := B(0, 1) and for ξ, y ∈ \overline{B}, ξ ≠ y,

G(ξ, y) := Γ (\sqrt{|ξ|^{2} + |y|^{2} − 2ξ · y}) − Γ (\sqrt{|ξ|^{2}|y|^{2} + 1 − 2ξ · y}) ,

with

Γ(z) := \{ \begin{matrix}\frac{1}{N(2 − N)|B|} |z|^{2−N} & N > 2, \\ \frac{1}{2π} log |z| & N = 2. \end{matrix}

Remark 1. (i) In the definition of Q_{M}f(x, m, u, h) for |m| < 1 the function U_{k} is the unique solution w ∈ H_{0}^{1} (B) of the equation Δw+div M_{k} = 0 in B.

(ii) In [8] we have recently given an integral representation formula for the relaxed energy F corresponding to general constant rank partial differential operators A which include (1.3), in the case of bounded domains and when no constraints of the type (1.2) are present. As in that paper we work within the general setting of A-quasiconvexity, as introduced by Fonseca and Müller in [14]. The approach is based on the use of Young measures (see also the work of Balder [3] and of Kristensen [19]) together with the blow-up method introduced by Fonseca and Müller in [12].

Corollary 1. Assume that f(x, m, u, h) is of the form

f(x, m, u, h) := φ(x, m, u) + ψ(x, u, h),

where φ : ℝ^{N} × (S^{N−1} ∪ {0}) × ℝ → [0, ∞) and ψ : ℝ^{N} × ℝ × ℝ^{N} → [0, ∞) are Carathéodory functions such that

0 ≤ φ(x, m, u) ≤ a(x) + C|u|^{2^{∗}},
ψ(x, u, ·) is convex in ℝ^{N}, 0 ≤ ψ(x, u, h) ≤ a(x) + C(|u|^{2^{∗}} + |h|^{2})

for L^{N} a.e. x ∈ ℝ^{N} and all (m, u, h) ∈ (S^{N−1} ∪ {0}) × ℝ × ℝ^{N}, and where a ∈ L^{1}(ℝ^{N}) and C > 0. If |m| < 1 then

Q_{M}f(x, m, u, h) = φ^{∗∗}(x, m, u) + ψ(x, u, h),

where, for L^{N} a.e. x ∈ Ω and for each u ∈ ℝ, the function φ^{∗∗}(x, ·, u) is the convex envelope of

φ̂(x, m, u) := \{ \begin{matrix}φ(x, m, u) & if m ∈ S^{N−1}, \\ ∞ & in B(0, 1). \end{matrix}

Remark 2. (i) Corollary 1 is closely related to a recent result of Pedregal [22] where the relaxed functional is defined in terms of Young measures. The analysis in that paper is restricted to the two dimensional case N = 2, where

Δu + div m = 0 if and only if m(x) = ∇u(x) + T ∇v(x)

for some function v and where T is a π/2 rotation in the plane. Clearly this argument cannot be extended to higher dimensions.

In the special case where ψ(h) := \frac{1}{2} |h|^{2} and φ(x, m) := χ_{Ω}(x)g(m) we recover the results of Tartar [24, 25] and De Simone [11].

(ii) To our knowledge Theorem 1.1 is the first relaxation result in micromagnetics which does not make use of the convexity of ψ(x, u, ·) (see e.g [11, 17, 22, 24, 25]). In general, even in the simplest case where

f(x, m, h) := χ_{Ω}(x)φ(m) + ψ(h),

if we start from a function ψ which is not convex in h, then Q_{M}f(x, m, h) will not coincide with χ_{Ω}(x)φ^{∗∗}(m)+ ψ^{∗∗}(h). Indeed let h be such that ψ^{∗∗}(h) < ψ(h). By taking x ∈ Ω and |m| = 1 we get

ψ^{∗∗}(h) + φ(m) < Q_{M}f(x, m, h) = ψ(h) + φ(m).

This remark partially answers a question raised in [22].

2. Preliminaries

Here Ω ⊂ ℝ^{N} is an open, bounded subset with Lipschitz boundary, L^{N} is the N dimensional Lebesgue measure, S^{N−1} := {x ∈ ℝ^{N} : |x| = 1} is the unit sphere and B := B(0, 1) is the unit ball centered at the origin.

We recall briefly some facts about Young measures which will be useful in the sequel (see e.g. [4], [21]). If D is an open set (not necessarily bounded), we denote by C_{c}(D; ℝ^{d}) the set of continuous functions with compact support in D, endowed with the supremum norm. The dual of the closure of C_{c}(D; ℝ^{d}) may be identified with the set of ℝ^{d}-valued Radon measures with finite mass M(D; ℝ^{d}), through the duality

⟨ν, f⟩ := ∫_{D} f(y) dν(y), ν ∈ M(D; ℝ^{d}), f ∈ C_{c}(D; ℝ^{d}).

A map ν : Ω → M(D; ℝ^{d}) is said to be weak-∗ measurable if x ↦ ⟨ν_{x}, f⟩ is measurable for all f ∈ C_{c}(D; ℝ^{d}). We say that a function f : Ω × ℝ^{d} → ℝ is a normal integrand if for L^{N} a.e. x ∈ Ω the function f(x, ·) is lower semicontinuous and there exists a Borel integrand f̃: Ω × ℝ^{d} → [−∞, ∞] such that f (x, ·) = f̃(x, ·) for L^{N} a.e. x ∈ Ω.

A sequence {z_{n}} ⊂ L^{p}(D; ℝ^{d}) is p-equintegrable if {|z_{n}|^{p}} is equintegrable.

The following result is a corollary of the Fundamental Theorem on Young Measures (see [4, 7, 23])

Theorem 2. Let z_{n} : Ω → ℝ^{d} be measurable functions such that

sup_{n∈ℕ} ∫_{Ω} |z_{n}|^{q} dx < ∞,

for some q > 0. Then there exists a subsequence {z_{n_{k}}} of {z_{n}} and a weak-∗ measurable map ν : Ω → M(ℝ^{d}; ℝ^{d}) with

ν_{x} ≥ 0, ||ν_{x}||_{M} = ∫_{ℝ^{d}} dν_{x} = 1 for a.e. x ∈ Ω,

and such that:

(i) if f : Ω × ℝ^{d} → ℝ is a normal integrand bounded from below then

lim inf_{k→∞} ∫_{Ω} f(x, z_{n_{k}}(x)) dx ≥ ∫_{Ω} \overline{f}(x)dx,

where

\overline{f}(x) := ⟨ν_{x}, f(x, ·)⟩ = ∫_{ℝ^{d}} f(x, y) dν_{x}(y);

(ii) if f : Ω × ℝ^{d} → ℝ is a Carathéodory function bounded from below then

lim_{k→∞} ∫_{Ω} f (x, z_{n_{k}}(x)) dx = ∫_{Ω} \overline{f}(x)dx

if and only if {f(·, z_{n_{k}}(·))} is equi-integrable.

The map ν : Ω → M(ℝ^{d}; ℝ^{d}) is called theYoung measure generated by the sequence {z_{n_{k}}}.

Proposition 1. If {z_{n}} generates a Young measure ν and v_{n} → 0 in measure, then {z_{n} +v_{n}} still generates the Young measure ν.

The following regularity results are well known, the former being an immediate consequence of Tiesze

Extension Theorem.

Theorem 3. Let f : C → [0, ∞) be a continuous function, where C is a closed subset of ℝ^{d}. Then there exists a continuous function f̃: ℝ^{d} → [0, ∞) such that f̃ = f on C.

Theorem 4. (Scorza-Dragoni) Let f : A × ℝ^{N} → ℝ be a Carathéodory function, where A is a Borel subset of ℝ^{d}. Then for every ε > 0 and every compact set K ⊂ A there exists a compact set K_{ε} ⊂⊂ K such that |K\K_{ε}| < ε and f : K_{ε} × ℝ^{N} → ℝ is continuous.

Proposition 2. Consider a Carathéodory function

f : ℝ^{N} × (S^{N−1} ∪ {0}) × ℝ × ℝ^{N} → [0, ∞).

Then there exists a Carathéodory function

f̃: ℝ^{2N} × ℝ × ℝ^{N} → [0, ∞)

such that f(x, m, u, h) = f̃(x, m, u, h) for a.e. x ∈ ℝ^{N} and all (m, u, h) ∈ (S^{N−1} ∪ {0}) × ℝ × ℝ^{N}.

Proof of Lemma 2. Since f is Carathéodory, by Theorem 4 for each n, j ∈ there exist compact sets K_{j}^{n} ⊂ B(0, n) with

K_{j}^{n} ⊂ K_{j+1}^{n} ⊂ K_{j+1}^{n+1}, |B(0, n)\K_{j}^{n}| ≤ 1/j, (2.1)

such that f : K_{j}^{n} × (S^{N−1} ∪ {0}) × ℝ × ℝ^{N} → [0, ∞) is continuous. By Theorem 3 we can find a continuous function f_{n,j} : ℝ^{2N} × ℝ × ℝ^{N} → [0, ∞) such that f_{n,j} = f on K_{j}^{n} × (S^{N−1} ∪ {0}) × ℝ × ℝ^{N}. Moreover, by (2.1) we have

f_{n,j} = f_{n,j+1} = f_{n+1,j+1} on K_{j}^{n}.

Set

G := ⋃_{n}^{∞} ⋃_{j}^{∞} K_{j}^{n},  f̃(x, m, u, h) := \{ \begin{matrix}f_{n,j}(x, m, u, h) & if x ∈ K_{j}^{n}, \\ 0 & if x ∈ ℝ^{N} \ G. \end{matrix}

Then f = f̃ on ℝ^{N} \ G and

|ℝ^{N}\G| = lim_{n→∞} |B(0, n)\G| ≤ lim sup_{n→∞} lim sup_{j→∞} |B(0, n)\K_{j}^{n}| = 0.

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

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

The following Sobolev-Gagliardo-Nirenberg Inequality may be found in [16].

Theorem 5. For any v ∈ W^{1,q}(ℝ^{N}), with 1 ≤ q < N,

||v||_{L^{q^{∗}} (ℝ^{N})} ≤ C(q, N)||∇v||_{L^{q}(ℝ^{N})}, (2.2)

where q^{∗} := qN/ (N − q) .

For a proof of the proposition below see Giaquinta [15].

Proposition 3. Let D ⊂ ℝ^{N} be an open bounded set with Lipschitz boundary. If m ∈ L^{∞}(D; ℝ^{N}) and if u ∈ H^{1}(D) is a solution of Δu + div m = 0 in D then ∇u ∈ L_{loc}^{p}(D; ℝ^{N}) and

||∇u||_{L^{p}(D′)} ≤ C(D, D′, N, p) (||∇u||_{L^{2}(D)} + ||u||_{L^{2}(D)} + ||m||_{L^{∞}(D;ℝ^{N})}) (2.3)

for any p ≥ 1 and D′ ⊂⊂ D.

Lemma 1. Let m ∈ M^{∗} and consider a sequence {m_{k}} ⊂ M^{∗} such that

m_{k} ⇀^{∗} m in L^{∞}(Ω; ℝ^{N}). (2.4)

Let u, u_{k} ∈ H^{1}(ℝ^{N}) be such that Δu + div m = Δu_{k} + div m_{k} = 0 in ℝ^{N}. Then

u_{k} ⇀ u in H_{loc}^{1}(ℝ^{N}).

Proof of Lemma 1. By (1.7) we have

∫_{ℝ^{N}} ∇u_{k} · ∇v dx = − ∫_{ℝ^{N}} χ_{Ω}m_{k} · ∇v dx = − ∫_{Ω} m_{k} · ∇v dx (2.5)

for all v ∈ H^{1}(ℝ^{N}). Taking v := u_{k} in (2.5) and applying Hölder’s inequality we obtain

∫_{ℝ^{N}} |∇u_{k}|^{2} dx ≤ (∫_{Ω} |m_{k}|^{2} dx)^{1/2} (∫_{ℝ^{N}} |∇u_{k}|^{2} dx)^{1/2} ,

and thus

∫_{ℝ^{N}} |∇u_{k}|^{2} dx ≤ |Ω| , (2.6)

and by (2.2) we have that

||u_{k}||_{L^{2^{∗}} (ℝ^{N})} ≤ C(N) |Ω| . (2.7)

Thus we may find a subsequence {u_{k_{n}}} of {u_{k}} and a function U ∈ L^{2^{∗}}(ℝ^{N}) with ∇U ∈ L^{2}(ℝ^{N}; ℝ^{N}) such that

u_{k_{n}} ⇀ U in L^{2^{∗}}(ℝ^{N}), ∇u_{k_{n}} ⇀ ∇U in L^{2}(ℝ^{N}; ℝ^{N}). (2.8)

In particular

u_{k_{n}} ⇀ U in H_{loc}^{1}(ℝ^{N}),

and ΔU+div m = 0 in ℝ^{N}. In view of (1.7) we conclude that u (x) = U (x) a.e.

Lemma 2. Let m ∈ M^{∗} and consider a sequence {m_{k}} ⊂ M^{∗} such that

m_{k} ⇀^{∗} m in L^{∞}(Ω; ℝ^{N}).

Let u, u_{k} ∈ H^{1}(ℝ^{N}) be such that Δu + div m = Δu_{k} + div m_{k} = 0 in ℝ^{N}. Then {∇u_{k}} is 2-equintegrable over ℝ^{N}, u_{k} → u in L_{loc}^{∞}(ℝ^{N}; ℝ), and

lim_{k→∞} ||u_{k} − u||_{L^{2^{∗}} (ℝ^{N})} = lim_{k→∞} ||∇u_{k} − ∇u||_{L^{2}(ℝ^{N} \\overline{Ω})} = 0.

Proof of Lemma 2. Consider two balls B_{1} ⊃⊃ B ⊃⊃ \overline{Ω} and fix any p > N. By (2.3)

||∇u_{k}||_{L^{p}(B)} ≤ C(B, B_{1}, N, p) (||u_{k}||_{H^{1}(B_{1})} + 1) , (2.9)

which, together with (2.7) and a standard bootstrap argument, yields that the sequence {u_{k}} is uniformly bounded in W^{1,p}(B; ℝ). By Rellich-Kondrachov Theorem we have u_{k} → u in L^{∞}(B; ℝ), and since B is any fixed ball we conclude that u_{k} → u in L_{loc}^{∞}(ℝ^{N}; ℝ). Moreover, since {∇u_{k}} is bounded in L^{p}(B; ℝ^{N}) it follows that it is 2-equintegrable over B. Hence for any η > 0 there exists δ > 0 such that

∫_{E} |∇(u_{k} − u)|^{2} dx ≤ η, (2.10)

for any measurable set E ⊂ B with |E| ≤ δ. Fix Ω ⊂⊂ Ω′ ⊂⊂ B with |Ω′ \ Ω| ≤ δ and consider a cut-off function φ such that φ ≡ 1 in ℝ^{N} \ \overline{Ω′} and φ ≡ 0 in Ω. Taking as a test function in (2.5) the function

v(x) := φ(x) (u_{k}(x) − u(x)) ,

and using the fact that

∇v = (u_{k} − u)∇φ + φ∇(u_{k} − u),

we get

∫_{ℝ^{N} \\overline{Ω′}} |∇(u_{k} − u)|^{2} dx ≤ ∫_{ℝ^{N} \\overline{Ω}} φ|∇(u_{k} − u)|^{2} dx = − ∫_{ℝ^{N} \\overline{Ω}} (u_{k} − u) ∇φ · ∇(u_{k} − u) dx
≤ ||∇φ||_{L^{∞}} ∫_{Ω′\Ω} |u_{k} − u| |∇(u_{k} − u) | dx → 0

as k → ∞, because u_{k} → u in L^{2}(Ω′) and ∇u_{k} is bounded in L^{2}(ℝ^{N}; ℝ^{N}). Since |Ω′ \ Ω| ≤ δ it follows from (2.10) that

||∇(u_{k} − u)||_{L^{2}(ℝ^{N} \\overline{Ω})} ≤ ||∇(u_{k} − u)||_{L^{2}(ℝ^{N} \\overline{Ω′})} + η,

and thus

lim sup_{k→∞} ||∇(u_{k} − u)||_{L^{2}(ℝ^{N} \\overline{Ω})} ≤ η.

If we let η → 0^{+} we obtain that lim_{k→∞} ||∇(u_{k} − u)||_{L^{2}(ℝ^{N} \\overline{Ω})} = 0. By (2.2)

||u_{k} − u||_{L^{2^{∗}} (ℝ^{N} \\overline{Ω′})} ≤ C||∇ (φ(u_{k} − u)) ||_{L^{2}(ℝ^{N})} ≤ C(||∇φ||_{L^{∞}}||u_{k} − u||_{L^{2}(Ω′\\overline{Ω})} + ||∇(u_{k} − u)||_{L^{2}(ℝ^{N} \\overline{Ω})}),

so letting k → ∞ and using the fact that u_{k} converges to u in L^{∞} (Ω′), it follows that ||u_{k} −u||_{L^{2^{∗}} (ℝ^{N} \\overline{Ω})} → 0. This, together with Lemma 1 and Sobolev Embedding Theorem, implies that ||u_{k} − u||_{L^{2^{∗}} (ℝ^{N})} → 0.

Since \{∇u_{k}\} is 2-equintegrable over B ⊃⊃ \overline{Ω} and ∇u_{k} → ∇u strongly in L^{2}(ℝ^{N} \ \overline{Ω}) this implies that {∇u_{k}} is 2-equintegrable over ℝ^{N}.

Lemma 3. Let D be an open bounded set of ℝ^{N} and consider a sequence

\{(m_{k}, u_{k})\} ⊂ L^{∞}(D; \overline{B}) × (φ_{k} + H_{0}^{1}(D)),

such that Δu_{k} + div m_{k} = 0 in D, where φ_{k} ∈ H^{1}(D). If φ_{k} ⇀ φ in H^{1}(D) and m_{k} ⇀^{∗} m in L^{∞}(Ω; ℝ^{N}) then u_{k} ⇀ u in H^{1}(D), where u ∈ φ + H_{0}^{1}(D) is the unique solution of Δu + div m = 0 in D. Moreover, if {∇φ_{k}} is 2-equi-integrable over D then {∇u_{k}} is also 2-equi-integrable over D. Finally, if in addition we also assume that m_{k} → m strongly in L^{2}(Ω; ℝ^{N}) then u_{k} → strongly in H^{1}(D).

Proof of Lemma 3. Since

∫_{D} ∇u_{k} · ∇v dx = − ∫_{D} m_{k} · ∇v dx (2.11)

for all v ∈ H_{0}^{1}(D), by taking as test functions u_{k} − φ_{k} we obtain

∫_{D} |∇u_{k}|^{2} dx = ∫_{D} ∇u_{k} · ∇φ_{k} dx − ∫_{D} m_{k} · ∇ (u_{k} − φ_{k}) dx.

Hölder’s inequality now yields

||∇u_{k}||_{L^{2}(D)} ≤ C (||∇φ_{k}||_{L^{2}(D)} + ||m_{k}||_{L^{2}(D)}) . (2.12)

Since φ_{k} ⇀ φ in H^{1}(D), by (2.12) we have that

||∇u_{k}||_{L^{2}(D)} ≤ C (sup_{j} ||∇φ_{j}||_{L^{2}(D)} + |D|) < ∞,

which, together with Poincaré Inequality, implies that the sequence {u_{k} − φ_{k}} is uniformly bounded in H_{0}^{1}(D). Hence there exists w ∈ H_{0}^{1}(D) such that (up to a subsequence) u_{k} − φ_{k} ⇀ w in H^{1}(D). Since φ_{k} ⇀ φ in H^{1}(D) and and m_{k} ⇀^{∗} m in L^{∞}(Ω; ℝ^{N}) we have that u_{k} ⇀ u := φ + w in H^{1}(D), and letting k → ∞ in (2.11) we conclude that Δu + div m = 0.

To prove the second part of the lemma assume that {∇φ_{k}} is 2-equiintegrable over D. Then for any η > 0 there exists δ > 0 such that

∫_{E}(|∇φ_{k}|^{2} + 1) dx ≤ η^{2}/4C_{1}^{2}, (2.13)

for any measurable set E ⊂ D with |E| ≤ δ, and where C_{1} := sup_{k} (∫_{D}(1 + |∇u_{k}|^{2}) dx)^{1/2}. Fix S and D′ ⊂⊂ D \ \overline{S} ⊂⊂ D such that |D \ D′| ≤ δ and consider a cut-off function φ such that φ ≡ 1 in S and φ ≡ 0 in D′. Taking as a test function in (2.11) the function

v(x) := φ(x) (u_{k}(x) − φ_{k}(x)) ,

and using the fact that

∇v = (u_{k} − φ_{k})∇φ + φ∇(u_{k} − φ_{k}),

we get

\begin{matrix}∫_{S} |∇u_{k}|^{2} dx ≤ & ∫_{D} ∇u_{k} · φ∇u_{k} dx = − ∫_{D} m_{k} · [(u_{k} − φ_{k})∇φ + φ∇(u_{k} − φ_{k})] dx \\ − ∫_{D} ∇u_{k} · (u_{k} − φ_{k}) ∇φ dx + ∫_{D} ∇u_{k} · φ∇φ_{k} dx \\ ≤ & ||∇φ||_{L^{∞}} ∫_{D\D′} |u_{k} − φ_{k}| (1 + |∇u_{k}|)dx + ∫_{D\D′} (1 + |∇u_{k}|) (1 + |∇φ_{k}|) dx \\ ≤ & 2||∇φ||_{L^{∞}} (∫_{D\D′} |u_{k} − φ_{k}|^{2} dx)^{1/2} (∫_{D}(1 + |∇u_{k}|^{2}) dx)^{1/2} \\ + 2 (∫_{D\D′}(1 + |∇φ_{k}|^{2}) dx)^{1/2} (∫_{D}(1 + |∇u_{k}|^{2}) dx)^{1/2} \\ ≤ & 2C_{1}||∇φ||_{L^{∞}} (∫_{D\D′} |u_{k} − φ_{k}|^{2} dx)^{1/2} + η \end{matrix} (2.14)

by (2.13). Since u_{k} − φ_{k} → 0 in L^{2}(D \ D′) the right hand side of the previous inequality can be made smaller than 2η for all k large. By eventually replacing S with a smaller set we can assume that

∫_{S} |∇u_{k}|^{2} dx ≤ 2η for all k.

To conclude the first part of the proof it remains to show the 2-equi-integrability of {∇u_{k}} in D \ S. Since D \ \overline{S} ⊂⊂ D, by (2.3) for any p > 2

||∇u_{k}||_{L^{p}(D\\overline{S})} ≤ C(D \ \overline{S}, N, p) (||u_{k}||_{H^{1}(D)} + 1) .

The right hand side of the previous inequality is bounded uniformly in k, consequently {∇u_{k}} is uniformly bounded in L^{p}(D \ \overline{S}; ℝ^{N}), and thus 2-equintegrable over D \ \overline{S}, and the claim is proved.

It remains to show that

lim_{k→∞} ||∇u_{k} − ∇u||_{L^{2}(D)} = 0 (2.15)

if m_{k} → m in L^{2}(Ω; ℝ^{N}). From (2.11)

∫_{D} ∇ (u_{k} − u) · ∇v dx = − ∫_{D} (m_{k} − m) · ∇v dx (2.16)

for all v ∈ H_{0}^{1}(D). Since the sequence {∇u_{k}} is 2-equintegrable over D by the first part of the lemma, for any η > 0 there exists δ > 0 such that

∫_{E}(|∇u_{k}|^{2} + |∇u|^{2}) dx ≤ η (2.17)

for any measurable set E ⊂ D with |E| ≤ δ. Let D′ := {x ∈ D : dist (x, ∂D) ≥ ε} where ε > 0 is chosen so small that |D \ D′| ≤ δ. Consider a smooth cut-off function φ ∈ C_{c}^{1} (D) such that φ ≡ 1 in D′. Taking as a test function in (2.16) the function v(x) := φ(x) (u_{k}(x) − u(x)) , and using the fact that

∇v = (u_{k} − u)∇φ + φ∇(u_{k} − u),

we get

∫_{D′} |∇(u_{k} − u)|^{2} dx ≤ ∫_{D} ∇(u_{k} − u) · φ∇(u_{k} − u) dx

= − ∫_{D} (m_{k} − m) · [(u_{k} − u)∇φ + φ∇(u_{k} − u)] dx − ∫_{D} ∇ (u_{k} − u) · (u_{k} − u) ∇φ dx → 0

as k → ∞, since u_{k} ⇀ u in H^{1}(D) and m_{k} → m in L^{2}(Ω; ℝ^{N}). Thus the right hand side of the previous inequality can be made smaller than η for all k large, which, together with (2.17), yields

∫_{D} |∇ (u_{k} − u) |^{2} dx ≤ 3η for all k large,

thus asserting (2.15).

3. Relaxation

To prove Theorem 1 we introduce a localization of F. Let O(Ω) denote the class of all open subsets of Ω and for m ∈ M^{∗} and D ∈ O(Ω) set

F(m; D) := inf {lim inf_{k→∞} ∫_{D} g(x, m_{k}, ∇u_{k}) dx : m_{k} ∈ M^{∗}, |m_{k}| = 1 in D, m_{k} = m outside D,

m_{k} ⇀^{∗} m in L^{∞}(D; ℝ^{N}), u_{k} ∈ H^{1} (ℝ^{N}) , Δu_{k} + div m_{k} = 0 in ℝ^{N}} ,

where g : ℝ^{3N} → [0, ∞] is the Carathéodory function defined as

g(x, m̃, h) := f(x, m̃, u(x), h), (3.1)

and u ∈ H^{1} (ℝ^{N}) is the unique solution of Δu + div m = 0 in ℝ^{N}.

Lemma 4. For all m ∈ M^{∗} we have

F(m) = F(m; Ω) + ∫_{ℝ^{N} \Ω} f(x, 0, u(x), ∇u(x)) dx.

Proof of Lemma 4. Let {m_{k}} ⊂ M be any sequence such that m_{k} ⇀^{∗} m in L^{∞}(Ω; ℝ^{N}), and let u_{k} ∈ H^{1}(ℝ^{N}) be the unique solution of the equation Δu_{k} + div m_{k} = 0 in ℝ^{N}. Assume, in addition, that

lim inf_{k→∞} ∫_{ℝ^{N}} f(x, m_{k}, u_{k}, ∇u_{k}) dx = lim_{k→∞} ∫_{ℝ^{N}} f(x, m_{k}, u_{k}, ∇u_{k}) dx.

By Lemma 2 {∇u_{k}} is 2-equintegrable over ℝ^{N} and

lim_{k→∞} ||u_{k} − u||_{L^{2^{∗}} (ℝ^{N})} = lim_{k→∞} ||∇u_{k} − ∇u||_{L^{2}(ℝ^{N} \\overline{Ω})} = 0.

In view of the growth condition (1.6) we have that {f(·, m_{k}(·), u_{k}(·), ∇u_{k}(·))} is equintegrable over ℝ^{N} and

lim inf_{k→∞} ∫_{ℝ^{N}} f(x, m_{k}, u_{k}, ∇u_{k}) dx = ∫_{ℝ^{N} \\overline{Ω}} f(x, 0, u, ∇u) dx + lim_{k→∞} ∫_{Ω} f(x, m_{k}, u_{k}, ∇u_{k}) dx.

Since ||u_{k}−u||_{L^{∞}(Ω)} → 0, by Proposition 1, and upon extracting a subsequence if necessary the two sequences

{(m_{k}, u_{k}, ∇u_{k})}  and  {(m_{k}, u, ∇u_{k})}

generate the same Young measure {ν_{x}}. By Proposition 2 and Theorem 2(i) we deduce that

lim_{k→∞} ∫_{Ω} f(x, m_{k}, u_{k}, ∇u_{k}) dx = lim_{k→∞} ∫_{Ω} f̃(x, m_{k}, u_{k}, ∇u_{k}) dx

= ∫_{Ω} ∫_{ℝ^{2N} ×ℝ×ℝ^{N}} f̃(x, M, U, H)dν_{x}(M, U, H)dx

= lim_{k→∞} ∫_{Ω} f̃(x, m_{k}, u, ∇u_{k}) dx = lim_{k→∞} ∫_{Ω} f(x, m_{k}, u, ∇u_{k}) dx,

(3.2)

where we have used and the facts that {(m_{k}, u_{k}, ∇u_{k})} and {(m_{k}, u, ∇u_{k})} generate the same Young measure {ν_{x}}, and that {f̃(·, m_{k}(·), u_{k}(·), ∇u_{k}(·))} and {f̃(·, m_{k}(·), u(·), ∇u_{k}(·))} are equi-integrable over Ω by (1.6) and by the 2-equi-integrability of the sequences {(u_{k}, ∇u_{k})} and {(u, ∇u_{k})} over Ω. In conclusion, we have shown that

lim_{k→∞} ∫_{ℝ^{N}} f(x, m_{k}, u_{k}, ∇u_{k}) dx = ∫_{ℝ^{N} \\overline{Ω}} f(x, 0, u, ∇u) dx + lim_{k→∞} ∫_{Ω} f(x, m_{k}, u, ∇u_{k}) dx

and taking the infimum over all sequences {m_{k}} as above we get

F(m) = ∫_{ℝ^{N} \\overline{Ω}} f(x, 0, u, ∇u) dx + F(m; Ω).

Lemma 5. For any D ∈ O(Ω) and m ∈ M^{∗} we have

F(m; D) = F_{0}(m; D),

where

F_{0}(m; D) := inf {lim inf_{k→∞} ∫_{D} g(x, m_{k}, ∇u_{k}) dx : m_{k} ∈ L^{∞}(D; S^{N−1}), m_{k} ⇀^{∗} m in L^{∞}(D; ℝ^{N}),
u_{k} ∈ u + H_{0}^{1}(D), Δu_{k} + div m_{k} = 0 in D}.

Proof of Lemma 5. Let

{(m_{k}, u_{k})} ⊂ L^{∞}(D; S^{N−1}) × (u + H_{0}^{1}(D)),

be such that Δu_{k} + div m_{k} = 0 in D, m_{k} ⇀^{∗} m in L^{∞}(D; ℝ^{N}), and

lim_{k→∞} ∫_{D} g(x, m_{k}(x), ∇u_{k}(x)) dx = lim inf_{k→∞} ∫_{D} g(x, m_{k}(x), ∇u_{k}(x)) dx.

Extend m_{k} to be m outside D and let U_{k} ∈ H^{1}(ℝ^{N}) be the unique solution of ΔU_{k} + div m_{k} = 0 in ℝ^{N}. Define v_{k} := u_{k} − U_{k} and φ_{k} := u − U_{k}. Then

\{ \begin{matrix}Δv_{k} = 0 in D, \\ v_{k} ∈ φ_{k} + H_{0}^{1}(D), \end{matrix}

and by Lemmas 1 and 2 we have φ_{k} ⇀ 0 in H^{1}(D) with {∇φ_{k}} 2-equi-integrable over D. Lemma 3 now yields

v_{k} → 0 in H_{0}^{1}(D), i.e. lim_{k→∞} ||u_{k} − U_{k}||_{H^{1}(D)} = 0. (3.3)

Proceeding as in Lemma 4,

lim inf_{k→∞} ∫_{D} g(x, m_{k}(x), ∇u_{k}(x)) dx = lim inf_{k→∞} ∫_{D} g(x, m_{k}(x), ∇U_{k}(x)) dx ≥ F(m; D)

and thus

F(m; D) ≤ F_{0}(m; D).

To prove the converse inequality let m_{k} ∈ M^{∗}, |m_{k}| = 1 in D, m_{k} = m outside D, m_{k} ⇀^{∗} m in L^{∞}(D; ℝ^{N}). Let U_{k} ∈ H^{1}(ℝ^{N}) be the unique solution of ΔU_{k} + div m_{k} = 0 in ℝ^{N} and let u_{k} ∈ u + H_{0}^{1}(D) be the unique solution of Δu_{k} + div m_{k} = 0 in D. Upon extracting a subsequence, if necessary, we can assume that

lim_{k→∞} ∫_{D} g(x, m_{k}(x), ∇U_{k}(x)) dx = lim inf_{k→∞} ∫_{D} g(x, m_{k}(x), ∇U_{k}(x)) dx

and then proceed as above.

Remark 3. Note that F_{0}(m;D) is well defined. Indeed it is well known that for any D ∈ O(Ω), u ∈ H^{1}(D) and m ∈ L^{2}(D; ℝ^{N}), the generalized Dirichlet problem

\{ \begin{matrix}Δv + div m = 0 in D, \\ v − u ∈ H_{0}^{1}(D) \end{matrix}

is uniquely solvable (see e.g. by Theorem 8.3 in [16]).

Lemma 6. F_{0}(m; ·) is the trace of a Radon measure.

Proof of Lemma 6. As it is usual, it suffices to prove subadditivity (see e.g [1, 14]), i.e.

F_{0}(m; D) ≤ F_{0}(m; D \ \overline{E}) + F_{0}(m; B)

if E ⊂⊂ B ⊂⊂ D ⊂ Ω. Fix η > 0. By the definition of F_{0}(m; ·) there exist two sequences of pairs of functions

\{(m_{k}, u_{k})\} ⊂ L^{∞}(D \ \overline{E}; S^{N−1}) × (u + H_{0}^{1}(D \ \overline{E})) , Δu_{k} + div m_{k} = 0 in D \ \overline{E},
{(w_{k}, z_{k})} ⊂ L^{∞}(B; S^{N−1}) × (u + H_{0}^{1}(B)) , Δz_{k} + div w_{k} = 0 in B,

such that

m_{k} ⇀^{∗} m in L^{∞}(D \ \overline{E}; ℝ^{N}), w_{k} ⇀^{∗} m in L^{∞}(B; ℝ^{N})

and

lim_{k→∞} ∫_{D\\overline{E}} g(x, m_{k}, ∇u_{k}) dx ≤ F_{0}(m; D \ \overline{E}) + η, lim_{k→∞} ∫_{B} g(x, w_{k}, ∇z_{k}) dx ≤ F_{0}(m; B) + η. (3.4)

Since by Lemma 1

∫_{B\\overline{E}}(|∇u_{k}|^{2} + |∇z_{k}|^{2}) dx ≤ K,

where K > 0 is independent of k, we can find an open set S, with E ⊂⊂ S ⊂⊂ B such that for all j sufficiently large

∫_{S_{j}} (a(x) + C(1 + |u|^{2^{∗}} + |∇u_{k}|^{2} + |∇z_{k}|^{2})) dx < η, (3.5)

12 IRENE FONSECA AND GIOVANNI LEONI

where a and C are given in (1.6) and

S_{j} := \{x ∈ S : \frac{1}{j + 1} ≤ dist(x, ∂S) ≤ \frac{1}{j}\} .

Let θ_{j} be smooth cut-off functions with compact support in S, which are equal to 1 when dist(x, ∂S) ≥ \frac{1}{j} and 0 when dist(x, ∂S) ≤ \frac{1}{j+1}. For x ∈ D set

M̂_{j,k} := (1 − θ_{j})m_{k} + θ_{j}w_{k}, M_{j,k} := π(M̂_{j,k}), Û_{j,k} := (1 − θ_{j})u_{k} + θ_{j}z_{k},

where

π(m) := \{ \begin{matrix}m/|m| & if m ≠ 0, \\ e_{1} & if m = 0. \end{matrix}

It is clear that M̂_{j,k} ⇀^{∗} m in L^{∞}(D; ℝ^{N}) as k → ∞ and j → ∞. For j fixed

ΔÛ_{j,k} + div M̂_{j,k} =(1 − θ_{j}) (Δu_{k} + div m_{k}) + θ_{j} (Δz_{k} + div w_{k})
+ ∇θ_{j} · (−2∇u_{k} + 2∇z_{k} − m_{k} + h_{k}) + (−u_{k} + z_{k})Δθ_{j}
=∇θ_{j} · (−2∇u_{k} + 2∇z_{k} − m_{k} + h_{k}) + (−u_{k} + z_{k})Δθ_{j} → 0

(3.6)

in H^{−1}(D) strong, because m_{k}−h_{k} ⇀^{∗} 0 in L^{∞}(B\\overline{E}; ℝ^{N}) and u_{k}−z_{k} ⇀ 0 in H^{1}(B\\overline{E}). Let U_{j,k} ∈ u+H_{0}^{1}(D)

be the solutions of ΔU_{j,k}+div M_{j,k} = 0 in D. For any p ≥ 1

∫_{D} |M̂_{j,k} − M_{j,k}|^{p} dx ≤ 2^{p}|S_{j}| → 0 as j → ∞,

hence, by (3.6),

lim_{j→∞} lim_{k→∞} Δ (Û_{j,k} − U_{j,k}) = lim_{j→∞} lim_{k→∞} div (−M̂_{j,k} + M_{j,k}) = 0 in H^{−1}(D),

and thus, since Û_{j,k} − U_{j,k} ∈ H_{0}^{1}(D),

lim_{j→∞} lim_{k→∞} ||∇ (Û_{j,k} − U_{j,k}) ||_{L^{2}(D;ℝ^{N})} = 0.

Moreover

∫_{D} g(x, M_{j,k}, ∇Û_{j,k}) dx ≤ ∫_{D\\overline{E}} g(x, m_{k}, ∇u_{k}) dx + ∫_{B} g(x, w_{k}, ∇z_{k}) dx
+ ∫_{S_{j}} (a(x) + C(1 + |u|^{2^{∗}} + |∇Û_{j,k}|^{2})) dx
≤ ∫_{D\\overline{E}} g(x, m_{k}, ∇u_{k}) dx + ∫_{B} g(x, w_{k}, ∇z_{k}) dx
+ ∫_{S_{j}} (a(x) + C(1 + |u|^{2^{∗}} + |∇u_{k}|^{2} + |∇z_{k}|^{2})) dx + C ∫_{S_{j}} |∇θ_{j}|^{2}|u_{k} − z_{k}|^{2}) dx

where we have used (1.6). If we now let k → ∞ and then j → ∞ and use (3.4) and (3.5), we get

lim inf_{j→∞} lim inf_{k→∞} ∫_{D} g(x, M_{j,k}, ∇Û_{j,k}) dx ≤ F_{0}(m; D \ \overline{E}) + F_{0}(m; B) + 3η.

Diagonalize to get

(3.7) M_{j} ⇀^{∗} m in L^{∞}(D; ℝ^{N}), ||Û_{j} − U_{j}||_{H^{1}(D)} → 0, ΔU_{j} + div M_{j} = 0 in D,

and

lim inf_{j→∞} ∫_{D} g(x, M_{j}, ∇Û_{j}) dx ≤ F_{0}(m; D \ \overline{E}) + F_{0}(m; B) + 3η. (3.8)

By Proposition 1, the two sequences

(M_{j}, ∇Û_{j})  and  (M_{j}, ∇U_{j})

generate the same Young measure {ν_{x}}, and thus we can proceed as in Lemma 4 to conclude that

F_{0}(m; D) ≤ lim inf_{j→∞} ∫_{D} g(x, M_{j}, ∇U_{j}) dx = lim inf_{j→∞} ∫_{D} g(x, M_{j}, ∇Û_{j}) dx
= F_{0}(m; D \ \overline{E}) + F_{0}(m; B) + 3η.

It now suffices to let η → 0^{+}.

Since

g : Ω × (S^{N−1} ∪ {0}) × ℝ^{N} → [0, ∞)

is Carathéodory, by Theorem 4 for each j ∈ there exists a compact set K_{j} ⊂ Ω, with |Ω\K_{j}| ≤ 1/j, such that g : K_{j} × (S^{N−1} ∪ {0}) × ℝ^{N} → [0, ∞) is continuous. Let K_{j}^{∗} be the set of Lebesgue points of χ_{K_{j}}, and set

ω := ⋃_{j}^{∞}(K_{j} ∩ K_{j}^{∗}). (3.9)

Then

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

Lemma 7. For L^{N} a.e. x_{0} ∈ Ω we have

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) ≥ ĝ(x_{0}, m(x_{0}), ∇u(x_{0}))

where the function ĝ = ĝ(x, m̃, h) coincides with g when |m̃| = 1, while for |m̃| < 1 it is given by

ĝ(x, m̃, h) := inf \{lim inf_{k→∞} \frac{1}{|B|} ∫_{B} g(x, m_{k}(y), h + ∇u_{k}(y)) dy : m_{k} ∈ L^{∞}(B; S^{N−1}),

m_{k} ⇀^{∗} m̃ in L^{∞}(B; ℝ^{N}), u_{k} ∈ H_{0}^{1}(B), Δu_{k} + div m_{k} = 0 in B}.

Proof of Lemma 7. Fix x_{0} ∈ ω, where ω is defined in (3.9), such that

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

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

(3.10)

and

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

where we have chosen the radii r → 0^{+} such that F_{0}(m; ∂B(x_{0}, r)) = 0, and we have used Lemma 6. For each r > 0 fixed let

{(m_{k}, u_{k})} ⊂ L^{∞}(B(x_{0}, r); S^{N−1}) × (u + H_{0}^{1}(B(x_{0}, r)))

be such that Δu_{k} + div m_{k} = 0 in B(x_{0}, r) and

m_{k} ⇀^{∗} m in L^{∞}(B(x_{0}, r); ℝ^{N}), lim_{k→∞} ∫_{B(x_{0},r)} g(x, m_{k}, ∇u_{k}) dx ≤ F_{0}(m; B(x_{0}, r)) + r^{N+1}. (3.11)

We have

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) ≥ lim sup_{r→0^{+}} lim_{k→∞} \frac{1}{|B|} ∫_{B} g(x_{0} + ry, m_{r,k}(y), ∇u_{r,k}(y)) dy,

where

m_{r,k}(y) := m_{k}(x_{0} + ry) and u_{r,k}(y) := \frac{u_{k}(x_{0} + ry) − u(x_{0})}{r}.

We claim that u_{r,k} ⇀ u_{0}(y) := ∇u(x_{0}) y in H^{1}(B) and m_{r,k} ⇀^{∗} m(x_{0}) in L^{∞}(B; ℝ^{N}) if we first let k → ∞ and then r → 0^{+}. Indeed for any fixed φ ∈ L^{∞}(B; ℝ^{N}) belonging to a dense set of L^{1}(B; ℝ^{N})

|∫_{B} φ(y) (m_{r,k}(y) − m(x_{0})) dy| ≤ |\frac{1}{r^{N}} ∫_{B(x_{0},r)} φ (\frac{x − x_{0}}{r} ) (m_{k}(x) − m(x)) dx|

+ C ∥φ∥_{L^{∞}} \frac{1}{r^{N}} ∫_{B(x_{0},r)} |m(x) − m(x_{0})| dx.

If we now let k → ∞ the first integral tends to zero, since m_{k} ⇀^{∗} m in L^{∞}(B(x_{0}, r); ℝ^{N}), while for the second integral we use (3.10)_{1}. The separability property of L^{1}(B; ℝ^{N}) now yields m_{r,k} ⇀^{∗} m(x_{0}) in L^{∞}(B; ℝ^{N}).

To prove the second claim observe that by Lemma 3 u_{k} ⇀ u in H^{1}(B(x_{0}, r)) for any fixed r > 0. Hence, also by (3.10)_{3},

lim_{r→0^{+}} lim_{k→∞} ||u_{r,k} − u_{0}||_{L^{2}(B)} = 0.

Let φ ∈ L^{2}(B; ℝ^{N}). Using Holder’s inequality and then making a change of variables we get

|∫_{B} φ(y) (∇u_{r,k}(y) − ∇u(x_{0})) dy| ≤ |∫_{B} φ(y)(∇u_{k}(x_{0} + ry) − ∇u(x_{0} + ry)) dy|

+ |∫_{B} φ(y)(∇u(x_{0} + ry) − ∇u(x_{0})) dy|

≤ |\frac{1}{r^{N}} ∫_{B(x_{0},r)} φ (\frac{x − x_{0}}{r} ) (∇u_{k}(x) − ∇u(x)) dx|

+ ||φ||_{L^{2}(B)} (\frac{1}{r^{N}} ∫_{B(x_{0},r)} |∇u(x) − ∇u(x_{0})|^{2} dx)^{1/2} .

If we now let k → ∞ the first integral tends to zero, since ∇u_{k} ⇀ ∇u(x_{0}) in L^{2}(B(x_{0}, r); ℝ^{N}). The claim then follows by letting r → 0^{+}, by (3.10)_{2}, and in view of the separability of L^{2}(B; ℝ^{N}).

Moreover the sequence

{(m_{r,k}, u_{r,k})} ⊂ L^{∞}(B; S^{N−1}) × (u_{r} + H_{0}^{1}(B))

is such that

Δu_{r,k} + div m_{r,k} = 0 in B,

where

u_{r}(y) := \frac{u(x_{0} + ry) − u(x_{0})}{r}.

Diagonalize to get subsequences m̂_{j} := m_{r_{j},k_{j}} and û_{j} := u_{r_{j},k_{j}} with

{(m̂_{j}, û_{j})} ⊂ L^{∞}(B; S^{N−1}) × (u_{r_{j}} + H_{0}^{1}(B)), Δû_{j} + div m̂_{j} = 0 in B,

and such that and

\begin{matrix}m̂_{j} ⇀^{∗} m(x_{0}) in L^{∞}(B; ℝ^{N}), û_{j} ⇀ u_{0} in H^{1}(B), \\ \frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) ≥ lim_{j→∞} \frac{1}{|B|} ∫_{B} g(x_{0} + r_{j}y, m̂_{j}(y), ∇û_{j}(y)) dy. \end{matrix} (3.12)

We now distinguish two cases: Case 1: Assume first that |m(x_{0})| = 1. Then for any p > 1

∫_{B} |m̂_{j}(y)|^{p} dy = ∫_{B} |m(x_{0})|^{p} dy = 1

and thus, since L^{p}(B) is uniformly convex, m̂_{j} converges strongly to m(x_{0}) in L^{p}(B; ℝ^{N}). By Lemma 3 and in view of the fact that ∇u_{r_{j}} → ∇u(x_{0}) in L^{2} by (3.10), û_{j} converges strongly to u_{0} in H^{1}(B), and we conclude that

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) = lim_{j→∞} \frac{1}{|B|} ∫_{B} g(x_{0} + r_{j}y, m̂_{j}(y), ∇û_{j}(y)) dy = g(x_{0}, m(x_{0}), ∇u(x_{0})).

This completes the proof in the case |m(x_{0})| = 1. Case 2: If |m(x_{0})| < 1 we claim that

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) ≥ lim_{j→∞} \frac{1}{|B|} ∫_{B} g(x_{0}, m̂_{j}(y), ∇û_{j}(y)) dy. (3.13)

By Lemma 3 and since {∇u_{r_{j}}} is 2-equi-integrable, we deduce that the sequence {∇û_{j}} is also 2-equi-integrable over the set B, and so there exists δ > 0 such that

∫_{E} (a(x_{0} + r_{j}y) + a(x_{0}) + C(1 + |u(x_{0} + r_{j}y)|^{2^{∗}} + |u(x_{0})|^{2^{∗}} + |∇û_{j}(y)|^{2})) dy < η (3.14)

for any measurable set E ⊂ B, with |E| < δ, and where a and C are given in (1.6), and there is an L > 0 such that

|B \ E_{j}| ≤ δ, where E_{j} := {y ∈ B : |u(x_{0} + r_{j}y)| ≤ L, |∇û_{j}(y)| ≤ L}. (3.15)

From the definition of the set ω there exists an integer l such that x_{0} ∈ K_{l} ∩ K_{l}^{∗}. Since

g : K_{l} × (S^{N−1} ∪ \{0\}) × \overline{B(0, L)} → [0, ∞)

is uniformly continuous there exists ρ > 0 such that

|g(x, m, h) − g(x_{1}, m, h)| ≤ η (3.16)

for all x, x_{1} ∈ K_{l} with |x − x_{1}| ≤ ρ and all (m, h) ∈ (S^{N−1} ∪ \{0\}) × \overline{B (0, L)}. By (3.12), (3.15) and (3.16)

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) ≥ lim_{j→∞} \frac{1}{|B|} ∫_{B} g(x_{0} + r_{j}y, m̂_{j}(y), ∇û_{j}(y)) dy

= lim_{j→∞} \frac{1}{|B(x_{0}, r_{j})|} ∫_{B(x_{0},r_{j})} g(x, m̂_{j}((x − x_{0})/r_{j}), ∇û_{j}((x − x_{0})/r_{j})) dx

≥ lim sup_{j→∞} \frac{1}{|B(x_{0}, r_{j})|} ∫_{(x_{0}+r_{j}E_{j})∩K_{l}} g(x, m̂_{j}((x − x_{0})/r_{j}), ∇û_{j}((x − x_{0})/r_{j})) dx

≥ lim sup_{j→∞} \frac{1}{|B(x_{0}, r_{j})|} ∫_{(x_{0}+r_{j}E_{j})∩K_{l}} g(x_{0}, m̂_{j}((x − x_{0})/r_{j}), ∇û_{j}((x − x_{0})/r_{j})) dx − η.

(3.17)

Now, since |∇û_{j}(y)| ≤ L for y ∈ E_{j}, by (1.6) we have that

\frac{1}{|B(x_{0}, r_{j})|} ∫_{(x_{0}+r_{j}E_{j})\K_{l}}g(x_{0}, m̂_{j}((x − x_{0})/r_{j}), ∇û_{j}((x − x_{0})/r_{j})) dx

≤ (a(x_{0}) + C(1 + |u(x_{0})|^{2^{∗}} + L^{2})) \frac{|B(x_{0}, r_{j}) \ K_{l}|}{|B(x_{0}, r_{j})|} → 0

as j → ∞, where we have used the fact that x_{0} is a Lebesgue point of χ_{K_{l}}. Consequently, from (3.17) we get

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) ≥ −η + lim sup_{j→∞} \frac{1}{|B|} ∫_{E_{j}} g(x_{0}, m̂_{j}(y), ∇û_{j}(y)) dy

≥ −2η + lim sup_{j→∞} \frac{1}{|B|} ∫_{B} g(x_{0}, m̂_{j}(y), ∇û_{j}(y)) dy

by (1.6) and (3.14). By letting η → 0^{+} we obtain that

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) ≥ lim sup_{j→∞} \frac{1}{|B|} ∫_{B} g(x_{0}, m̂_{j}(y), ∇û_{j}(y)) dy.

Thus claim (3.13) follows and hence

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) ≥ lim sup_{j→∞} \frac{1}{|B|} ∫_{B} g(x_{0}, m̂_{j}(y), ∇u(x_{0}) + ∇v̂_{j}(y)) dy,

where v̂_{j}(y) := û_{j}(y) − u_{0}(y). Let v_{j} ∈ H_{0}^{1}(B) be the unique solution of Δv_{j}+div m̂_{j} = 0 in B. Since

\{ \begin{matrix}Δ (v̂_{j} − v_{j}) = 0 & in B, \\ v̂_{j} − v_{j} = û_{j} − u_{0} & on ∂B, \end{matrix}

as û_{j} − u_{0} ⇀ 0 in H^{1}(B) and {∇û_{j}} is 2-equi-integrable over the set B, it follows from Lemma 3 that ||v̂_{j} − v_{j}||_{H^{1}(B)} → 0, and so

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) ≥ lim sup_{j→∞} \frac{1}{|B|} ∫_{B} g(x_{0}, m̂_{j}(y), ∇u(x_{0}) + ∇v̂_{j}(y)) dy

= lim sup_{j→∞} \frac{1}{|B|} ∫_{B} g(x_{0}, m̂_{j}(y), ∇u(x_{0}) + ∇v_{j}(y)) dy.

Lemma 8. Let m̃, M ∈ M be piecewise constant functions and λ ∈ (0, 1). Then there exists a sequence

{m_{n}} ⊂ M such that m_{n} ⇀^{∗} λm̃+(1−λ)M in L^{∞}(B; ℝ^{N}), ∇u_{n} → λ∇ũ+(1−λ)∇U strongly in L^{2}(B; ℝ^{N})

and

lim_{n→∞} ∫_{B} g(x, m_{n}(y), h + ∇u_{n}(y)) dy = λ ∫_{B} g(x, m̃(y), h + λ∇ũ(y) + (1 − λ)∇U(y)) dy
+(1 − λ) ∫_{B} g(x, M(y), h + λ∇ũ(y) + (1 − λ)∇U(y)) dy,

where ũ, U, u_{n} ∈ H_{0}^{1}(B) are the unique solutions of Δũ + div m̃ = 0, ΔU + div M = 0, Δu_{n} + div m_{n} = 0, respectively.

Proof of Lemma 8. Consider a partition of the unit ball B into a finite number of disjoint subsets B_{i}, i = 1, …, L, such that m̃ and M are constants on each set B_{i}, say m̃(x) ≡ m_{i} and M(x) ≡ M_{i} in B_{i}. Using an idea of De Simone [11], we let p_{i} ∈ S^{N−1} be such that

p_{i} · (m_{i} − M_{i}) = 0, (3.18)

and consider a 1-periodic function θ such that

θ(t) := \{ \begin{matrix}+1 & t ∈ [0, λ] \\ −1 & t ∈ (λ, 1]. \end{matrix}

Define

B_{i,n}^{±} := \{y ∈ B_{i} : θ(ny · p_{i}) = ±1\}, m_{n}(y) := \{ \begin{matrix}m_{i} & y ∈ B_{i,n}^{+}, \\ M_{i} & y ∈ B_{i,n}^{−}, \end{matrix}

i = 1, …, L. Then m_{n} ⇀^{∗} λm_{i} + (1 − λ)M_{i} in L^{∞}(B_{i}; ℝ^{N}) as n → ∞, for each i = 1, …, L, and, in turn,

m_{n} ⇀^{∗} λm̃ + (1 − λ)M in L^{∞}(B; ℝ^{N}). Let m_{n,i}(y) := m_{n}(y) − (λm_{i} + (1 − λ)M_{i}) in B_{i} and v_{n,i} ∈ H_{0}^{1}(B)

be the unique solution of Δv_{n,i} + div(χ_{B_{i}}m_{n,i}) = 0. We claim that ∇v_{n,i} → 0 in L^{2}(B; ℝ^{N}). Integration by parts gives

\begin{aligned}∫_{B} |∇v_{n,i}(y)|^{2}dy & = − ∫_{B_{i}} ∇v_{n,i}(y) · m_{n,i} dy \\ & = − ∫_{∂B_{i,n}^{−}} v_{n,i}m_{n,i} · ν_{i,n}^{−} dH^{N−1} − ∫_{∂B_{i,n}^{+}} v_{n,i}m_{n,i} · ν_{i,n}^{+} dH^{N−1} \\ & = − ∫_{∂B_{i}} (χ_{∂B_{i,n}^{−}}v_{n,i}m_{n,i} · ν_{i}^{−} + χ_{∂B_{i,n}^{+}}v_{n,i}m_{n,i} · ν_{i}^{+}) dH^{N−1}, \end{aligned} (3.19)

where we have used the fact that, by construction of the sets B_{i,n}^{±}, on (∂B_{i,n}^{−} ∩ ∂B_{i,n}^{−}) \ ∂B_{i} we have ν_{i,n}^{+} = −ν_{i,n}^{−} = p_{i} so that

∫_{∂B_{i,n}^{−}\∂B_{i}} v_{n,i}m_{n,i} · ν_{i,n}^{−} dH^{N−1}+ ∫_{∂B_{i,n}^{+}\∂B_{i}} v_{n,i}m_{n,i} · ν_{i,n}^{+} dH^{N−1}
= ∫_{∂B_{i,n}^{+}\∂B_{i,n}^{−}} v_{n,i}(m_{i} − M_{i}) · p_{i} dH^{N−1} = 0

by (3.18). Since χ_{B_{i}}m_{n,i} ⇀^{∗} 0 in L^{∞}(B; ℝ^{N}) as n → ∞, by Lemma 3 we have that v_{n,i} ⇀ 0 in H_{0}^{1}(B) and thus v_{n,i} → 0 strongly in L^{2}(∂B_{i}). Hence, by (3.19) ∇v_{n,i} → 0 in L^{2}(B; ℝ^{N}) and the claim is asserted.

Let u_{n} ∈ H_{0}^{1}(B) be the unique solution of Δu_{n} + divm_{n} = 0. By linearity we can write

u_{n}(y) = λ ũ(y) + (1 − λ)U(y) + ∑_{i=1}^{L} v_{n,i},

and so ∇u_{n} → λ∇ũ + (1 − λ)∇U strongly in L^{2}(B; ℝ^{N}). Hence, also by (1.6), the sequence {g(x, m_{n}(·), h + ∇u_{n}(·))} is equi-integrable and thus by Theorem 2(ii) and Proposition 1, extracting a subsequence if necessary, we have that

lim_{n→∞} ∫_{B_{i}} g(x, m_{n}(y), h + ∇u_{n}(y)) dy = lim_{n→∞} ∫_{B_{i}} g(x, m_{n}(y), h + λ∇u(y) + (1 − λ)∇U(y)) dy.

To conclude the proof it suffices to observe that

∫_{B} g(x, m_{n}(y), h + λ∇ũ(y) + (1 − λ)∇U(y)) dy = ∫_{B} χ_{∪_{i}B_{i,n}^{+}}(y)g(x, m̃(y), h + λ∇ũ(y) + (1 − λ)∇U(y)) dy
+ ∫_{B} χ_{∪_{i}B_{i,n}^{−}}(y)g(x, M(y), h + λ∇ũ(y) + (1 − λ)∇U(y)) dy,

where

χ_{∪_{i}B_{i,n}^{+}} ⇀^{∗} λ,  χ_{∪_{i}B_{i,n}^{−}} ⇀^{∗} (1 − λ)  in L^{∞}(B; ℝ^{N}).

Lemma 9. ĝ(x, ·, ·) is upper semicontinuos and satisfies the growth condition

0 ≤ ĝ(x, m̃, h) ≤ a(x) + C(|u(x)|^{2^{∗}} + |h|^{2}) (3.20)

for L^{N} a.e. x ∈ ℝ^{N} and all (m̃, h) ∈ ×\overline{B(0, 1)} × ℝ^{N}, where a ∈ L^{1}(ℝ^{N}) and C > 0 are given in (1.6).

Proof of Lemma 9. For L^{N} a.e. x ∈ Ω and every (m̃, h) ∈ \overline{B(0, 1)} × ℝ^{N} and ε > 0, consider a sequence

{(m_{k}, u_{k})} ⊂ L^{∞}(B; S^{N−1}) × H_{0}^{1}(B) such that m_{k} ⇀^{∗} m̃ in L^{∞}(B; ℝ^{N}), Δu_{k}+div m_{k} = 0, and

lim_{k→∞} \frac{1}{|B|} ∫_{B} g(x, m_{k}(y), h + ∇u_{k}(y)) dy ≤ ĝ(x, m̃, h) + ε. (3.21)

By a density argument we may assume that the functions m_{k} are piecewise constant. Note that by Lemma 3

(3.22) u_{k} ⇀ 0 in H^{1} (B) and {∇u_{k}} is 2-equi-integrable over B.

Let \{(m̂_{j}, h_{j})\} ⊂ \overline{B(0, 1)} × ℝ^{N} be any fixed sequence of vectors converging to (m̃, h). Without loss of generality we may assume that

lim sup_{j→∞} ĝ(x, m̂_{j}, h_{j}) = lim_{j→∞} ĝ(x, m̂_{j}, h_{j}).

\underline{Step 1}: Suppose that all the m̂_{j} belong to the segment with endpoints π(m̃) and −π(m̃). Then there exist λ_{j} ∈ [0, 1] such that

λ_{j} → 1 and m̂_{j} = λ_{j} m̃ + (1 − λ_{j})π_{±}(m̃),

where π_{±}(m̃) is either π(m̃) or −π(m̃). If there exists a subsequence \left\{{ \atop m̂_{j_{h}}}\right\} of \left\{{ \atop m̂_{j}}\right\} such that \left|{ \atop m̂_{j_{h}}}\right| = 1 then λ_{j_{h}} = 1, and so \left|{ \atop m̃}\right| = 1 and we have

lim sup_{j→∞} ĝ(x, m̂_{j}, h_{j}) = lim_{h→∞} ĝ(x, m̂_{j_{h}}, h_{j_{h}}) = lim_{h→∞} g(x, m̂_{j_{h}}, h_{j_{h}}) = g(x, m̃, h) = ĝ(x, m̃, h).

If \left|{ \atop m̂_{j}}\right| < 1 for all j (but a finite number) then by Lemma 8 we may find a sequence \{m_{n,k,j}\} ⊂ M such

that m_{n,k,j} ⇀^{∗} λ_{j}m_{k} + (1 − λ_{j})π_{±}(m̃) in L^{∞}(B; ℝ^{N}), ∇u_{n,k,j} → λ_{j}∇u_{k} strongly in L^{2}(B; ℝ^{N}), and

lim_{n→∞} ∫_{B} g(x, m_{n,k,j}(y), h_{j} + ∇u_{n,k,j}(y)) dy =λ_{j} ∫_{B} g(x, m_{k}(y), h_{j} + λ_{j}∇u_{k}(y)) dy
+ (1 − λ_{j}) ∫_{B} g(x, π_{±}(m̃), h_{j} + λ_{j}∇u_{k}(y)) dy,

where u_{n,k,j} ∈ H_{0}^{1}(B) is the unique solution of Δu_{n,k,j} + div m_{n,k,j} = 0. Since m_{n,k,j} ⇀^{∗} λ_{j}m̃ + (1 − λ_{j})π_{±}(m̃) = m̂_{j} in L^{∞}(B; ℝ^{N}) as n → ∞ and k → ∞, in this order, we have

ĝ(x, m̂_{j}, h_{j}) ≤ lim inf_{k→∞} lim_{n→∞} \frac{1}{|B|} ∫_{B} g(x, m_{n,k,j}(y), h_{j} + ∇u_{n,k,j}(y)) dy
≤λ_{j} lim inf_{k→∞} \frac{1}{|B|} ∫_{B} g(x, m_{k}(y), h_{j} + λ_{j}∇u_{k}(y)) dy

+ (1 − λ_{j}) lim sup_{k→∞} \frac{1}{|B|} ∫_{B} g(x, π_{±}(m̃), h_{j} + λ_{j}∇u_{k}(y)) dy,

and, in turn,

(3.23)  lim sup_{j→∞} ĝ(x, m̂_{j}, h_{j}) ≤ lim sup_{j→∞} lim inf_{k→∞} \frac{1}{|B|} ∫_{B} g(x, m_{k}(y), h_{j} + λ_{j}∇u_{k}(y)) dy,

where we have used (1.6), (3.22) and the fact that λ_{j} → 1. By (3.22) and in view of

lim sup_{j→∞} lim inf_{k→∞} ∥h_{j} + λ_{j}∇u_{k}(y) − (h + ∇u_{k}(y))∥_{L^{2}(B;ℝ^{N})} = 0,

we may conclude, as in the proof of Lemma 4 that

lim sup_{j→∞} lim inf_{k→∞} \frac{1}{|B|} ∫_{B} g(x, m_{k}(y), h_{j} + λ_{j}∇u_{k}(y)) dy = lim inf_{k→∞} \frac{1}{|B|} ∫_{B} g(x, m_{k}(y), h + ∇u_{k}(y)) dy.

Hence, by (1.6), (3.21), (3.23),

lim sup_{j→∞} ĝ(x, m̂_{j}, h_{j}) ≤ lim sup_{j→∞} lim inf_{k→∞} \frac{1}{|B|} ∫_{E_{k}} g(x, m_{k}(y), h_{j} + λ_{j}∇u_{k}(y)) dy + ε

≤ lim inf_{k→∞} \frac{1}{|B|} ∫_{E_{k}} g(x, m_{k}(y), h + ∇u_{k}(y)) dy + 2ε

≤ ĝ(x, m̃, h) + 3ε.

By letting ε → 0^{+} we obtain upper semicontinuity.

\underline{Step 2}: In the general case, given a sequence of vectors \{(m_{j}, h_{j})\} ⊂ \overline{B(0, 1)} × ℝ^{N} converging to (m̃, h), we rotate m_{j} of an angle α_{j} until it intersects the segment of end points π(m̃) and −π(m̃) and we denote by m̂_{j} the point of intersection. Then |m_{j} − m̂_{j}| → 0 as j → ∞ (note that when m̃ = 0 this follows from the fact that |m_{j}| = \left|{ \atop m̂_{j}}\right| → 0). Applying Step 1 we can find a sequence \{m_{n,j}\} ⊂ M such that m_{n,j} ⇀^{∗} m̂_{j} in L^{∞}(B; ℝ^{N}) as n → ∞, and

lim sup_{j→∞} lim_{n→∞} \frac{1}{|B|} ∫_{B} g(x, m_{n,j}(y), h_{j} + ∇u_{n,j}(y)) dy ≤ ĝ(x, m̃, h),

where u_{n,j} ∈ H_{0}^{1} (B) is the solution of Δu_{n,j}+div m_{n,j} = 0. By rotating (backwards) m_{n,j} of an angle −α_{j} we

obtain a new sequence {m̃_{n,j}} ⊂ M such that m̃_{n,j} ⇀^{∗} m_{j} in L^{∞}(B; ℝ^{N}) as n → ∞ and ||m̃_{n,j} −m_{n,j}||_{∞} → 0

as j → ∞ uniformly in n. Hence by Lemma 3 ∥∇ũ_{n,j} − ∇u_{n,j}∥_{L^{2}(B)} → 0, where ũ_{n,j} ∈ H_{0}^{1} (B) is the solution of Δũ_{n,j}+div m̃_{n,j} = 0, and so

lim sup_{j→∞} ĝ(x, m_{j}, h_{j}) ≤ lim sup_{j→∞} lim_{n→∞} \frac{1}{|B|} ∫_{B} g(x, m̃_{n,j}(y), h_{j} + ∇ũ_{n,j}(y)) dy
= lim sup_{j→∞} lim_{n→∞} \frac{1}{|B|} ∫_{B} g(x, m_{n,j}(y), h_{j} + ∇u_{n,j}(y)) dy
≤ĝ(x, m̃, h).

\underline{Step 3}: To prove (3.20) observe that by (1.6) and the definition of ĝ we have

ĝ(x, m̃, h) ≤ a(x) + C(|u(x)|^{2^{∗}} + |h|^{2}) + inf \{lim inf_{k→∞} \frac{1}{|B|} ∫_{B} |∇u_{k}(y)|^{2} dy : m_{k} ∈ L^{∞}(B; S^{N−1}),

m_{k} ⇀^{∗} m̃ in L^{∞}(B; ℝ^{N}), u_{k} ∈ H_{0}^{1}(B), Δu_{k} + div m_{k} = 0 in B}.

By Lemma 8 there exists a sequence {(M_{n}, U_{n})} ⊂ M × H_{0}^{1} such that m_{n} ⇀^{∗} λπ(m̃) − (1 − λ)π(m̃) in L^{∞}(B; ℝ^{N}), with λ ∈ (0, 1) such that λπ(m̃) − (1 − λ)π(m̃) = m̃, ∇U_{n} → 0 strongly in L^{2}(B; ℝ^{N}), and thus

ĝ(x, m̃, h) ≤ a(x) + C(|u(x)|^{2^{∗}} + |h|^{2}) + lim_{k→∞} \frac{1}{|B|} ∫_{B} |∇U_{k}(y))|^{2} dy = a(x) + C(|u(x)|^{2^{∗}} + |h|^{2}).

Lemma 10. For L^{N} a.e. x_{0} ∈ Ω we have

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) = ĝ(x_{0}, m(x_{0}), ∇u(x_{0})).

Proof of Lemma 10. Assume first that

m(x) = ∑_{i=1}^{K} ξ_{i}χ_{D_{i}}(x), (3.24)

where |ξ_{i}| ≤ 1 and the sets D_{i} ∈ O(Ω) are pairwise disjoint, |∂D_{i}| = 0, and ∪_{i=1}^{K}\overline{D_{i}} = Ω. Without loss of generality we may assume that x_{0} ∈ ∪_{i=1}^{K}D_{i}, say x_{0} ∈ D_{1}, and that

(3.25) \frac{u (x_{0} + ry) − u (x_{0})}{r} − ∇u (x_{0}) y → 0 in H^{1} (B) , there exists \frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}).

Then

m(x) ≡ ξ_{1} for all x ∈ B(x_{0}, r)

for all r > 0 sufficiently small. Fix η > 0 and consider a sequence {(m_{k}, v_{k})} as in the definition of the function ĝ(x_{0}, m(x_{0}), ∇u(x_{0})) such that

lim_{k→∞} \frac{1}{|B|} ∫_{B} g(x_{0}, m_{k}(y), ∇u(x_{0}) + ∇v_{k}(y)) dy ≤ ĝ(x_{0}, m(x_{0}), ∇u(x_{0})) + η, (3.26)

where m_{k} ≡ m(x_{0}) and v_{k} ≡ 0 if |m(x_{0})| = 1. For any fixed r > 0 sufficiently small define

m̂_{k}(x) := m_{k}((x − x_{0})/r).

Then m̂_{k} ∈ L^{∞}(B(x_{0}, r); S^{N−1}), m̂_{k} ⇀^{∗} m in L^{∞}(B(x_{0}, r); ℝ^{N}), where we used the fact that m restricted to B (x_{0}, r) is identically equal to m(x_{0}). Let ũ_{k} ∈ u+H_{0}^{1}((B(x_{0}, r)) be the unique solution of Δũ_{k} +div m̂_{k} = 0 in B(x_{0}, r). Then

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) = lim_{r→0^{+}} \frac{F_{0}(m; B(x_{0}, r))}{|B(x_{0}, r)|} ≤ lim_{r→0^{+}} lim inf_{k→∞} \frac{1}{|B|} ∫_{B} g(x_{0} + ry, m_{r,k}(y), ∇u_{r,k}(y)) dy

where

m_{r,k}(y) := m̂_{k}(x_{0} + ry) = m_{k}(y) and u_{r,k}(y) := \frac{ũ_{k}(x_{0} + ry) − u(x_{0})}{r}.

Setting û_{k} (y) := ∇u(x_{0})y + v_{k} (y) then

Δ (û_{k} − u_{r,k}) = 0 in B, (û_{k} − u_{r,k})|_{∂B} = ∇u(x_{0})y − \frac{u(x_{0} + ry) − u(x_{0})}{r},

so by Lemma 3 and (3.25)

∥û_{k} − u_{r,k}∥_{H^{1}(B)} → 0

as r → 0^{+}, and, in particular,

lim_{r→0^{+}} lim inf_{k→∞} ∫_{B} |∇u_{r,k} (y) − (∇u(x_{0}) + ∇v_{k} (y))|^{2} dy = 0.

We can now continue exactly as in the proof of Lemma 7 to obtain (3.13), that is

lim_{r→0^{+}} lim inf_{k→∞} ∫_{B} g(x_{0} + ry, m_{r,k}(y), ∇u_{r,k}(y)) dy = lim_{i→∞} ∫_{B} g(x_{0}, m_{k_{i}}(y), ∇u(x_{0}) + ∇v_{k_{i}}(y)) dy

for some subsequence k_{i} → ∞. Hence by (3.26)

\frac{dF_{0}(m; ·)}{dL^{N}}(x_{0}) ≤ ĝ(x_{0}, m(x_{0}), ∇u(x_{0})) + η

and we can now let η → 0^{+}. Together with Lemma 7 this yields that

F_{0}(m; D) = ∫_{D} ĝ(x, m(x), ∇u(x)) dx (3.27)

for all functions m ∈ M^{∗} of the form (3.24).

For a general m ∈ M^{∗} construct a sequence m_{k} ∈ M^{∗} of the form (3.24) such that m_{k} converges to m strongly in L^{∞}(D; ℝ^{N}). A simple diagonalization argument and (3.27) show that

F_{0}(m; D) ≤ lim inf_{k→∞} F_{0}(m_{k}; D) = lim inf_{k→∞} ∫_{D} ĝ(x, m_{k}(x), ∇u_{k}(x)) dx.

Extracting a subsequence, if necessary, we have that

lim inf_{k→∞} ∫_{D} ĝ(x, m_{k}(x), ∇u_{k}(x)) dx = lim_{k→∞} ∫_{D} ĝ(x, m_{k}(x), ∇u_{k}(x)) dx,

and, since by Lemma 3 ∇u_{k} converges to ∇u strongly in L^{2}(B; ℝ^{N}), in view of the growth condition (3.20) we may apply Fatou’s Lemma to a(·) + C(|u(·)|^{2^{∗}} + |∇u_{k}(·)|^{2}) − ĝ(·, m_{k}(·), ∇u_{k}(·)) to obtain

lim sup_{k→∞} ∫_{D} ĝ(x, m_{k}, ∇u_{k}) dx ≤ ∫_{D} lim sup_{k→∞} ĝ(x, m_{k}, ∇u_{k}) dx
≤ ∫_{D} ĝ(x, m, ∇u(x)) dx,

where we have used the upper semicontinuity of ĝ(x, ·, ·) (see Lemma 9). Together with Lemma 7 this yields the desired result.

Proof of Corollary 1. Fix m ∈ M^{∗}, (x, h) ∈ Ω × ℝ^{N}, and define

φ_{1}(m̃) := φ(x, m̃, u(x)), ψ_{1}(h) := ψ(x, u(x), h),

where u ∈ H^{1} (ℝ^{N}) is the unique solution of Δu + div m = 0 in ℝ^{N}. We claim that the function ĝ(x, ·, h) is convex in \overline{B(0, 1)}. Indeed given m_{1}, m_{2} ∈ \overline{B(0, 1)}, λ ∈ (0, 1) and ε > 0, consider two sequences \{(m_{k}, u_{k})\}

and {(M_{k}, U_{k})} in L^{∞}(B; S^{N−1}) × H_{0}^{1}(B), such that m_{k} ⇀^{∗} m_{1}, M_{k} ⇀^{∗} m_{2} in L^{∞}(B; ℝ^{N}), and

lim_{k→∞} \frac{1}{|B|} ∫_{B} g(x, m_{k}(y), h + ∇u_{k}(y)) dy ≤ ĝ(x, m_{1}, h) + ε,
lim_{k→∞} \frac{1}{|B|} ∫_{B} g(x, M_{k}(y), h + ∇U_{k}(y)) dy ≤ ĝ(x, m_{2}, h) + ε.

By a density argument we may assume that the functions m_{k}, M_{k} are piecewise constant, and thus by Lemma 8 we may find a sequence {m_{n,k}} ⊂ M such that m_{n,k} ⇀^{∗} λm_{k} + (1 − λ)M_{k} in L^{∞}(B; ℝ^{N}), ∇u_{n,k} → λ∇u_{k} + (1 − λ)∇U_{k} strongly in L^{2}(B; ℝ^{N}), and

lim_{n→∞} ∫_{B}g(x, m_{n,k}(y), h + ∇u_{n,k}(y)) dy = λ ∫_{B} g(x, m_{k}(y), h + λ∇u_{k}(y) + (1 − λ)∇U_{k}(y)) dy
+ (1 − λ) ∫_{B} g(x, M_{k}(y), h + λ∇u_{k}(y) + (1 − λ)∇U_{k}(y)) dy
=λ ∫_{B} φ_{1}(m_{k}(y)) dy + (1 − λ) ∫_{B} φ_{1}(M_{k}(y)) dy + ∫_{B} ψ_{1}(h + λ∇u_{k}(y) + (1 − λ)∇U_{k}(y)) dy
≤λ ∫_{B} g(x, m_{k}(y), h, ∇u_{k}(y)) dy + (1 − λ) ∫_{B} g(x, M_{k}(y), h + ∇U_{k}(y)) dy

where u_{n,k} ∈ H_{0}^{1}(B) is the unique solution of Δu_{n,k} + div m_{n,k} = 0, and where we have used the convexity of ψ_{1}. Since m_{n,k} ⇀^{∗} λm_{1} + (1 − λ)m_{2} in L^{∞}(B; ℝ^{N}) as n → ∞ and k → ∞, in this order, we have

ĝ(x, λm_{1} + (1 − λ)m_{2}, h) ≤ lim inf_{k→∞} lim_{n→∞} \frac{1}{|B|} ∫_{B} g(x, m_{n,k}(y), h + ∇u_{n,k}(y)) dy

≤λ lim_{k→∞} \frac{1}{|B|} ∫_{B} g(x, m_{k}(y), h, ∇u_{k}(y)) dy

+ (1 − λ) lim_{k→∞} \frac{1}{|B|} ∫_{B} g(x, M_{k}(y), h + ∇U_{k}(y)) dy

≤λĝ(x, m_{1}, h) + (1 − λ)ĝ(x, m_{2}, h) + 2ε.

The claim follows by letting ε → 0^{+}.

Since ĝ(x, ·, h) is convex in \overline{B(0, 1)} we have that

ĝ(x, m, h) ≤ (φ_{1}(·) + ψ_{1}(h))^{∗∗}(m) = φ_{1}^{∗∗}(m) + ψ_{1}(h).

On the other hand, by classical lower semicontinuity results and, in view of the convexity of ψ_{1}, for any sequence {(m_{k}, u_{k})} ⊂ L^{∞}(B; S^{N−1}) × H_{0}^{1}(B) such that m_{k} ⇀^{∗} m in L^{∞}(B; ℝ^{N}) and Δu_{k} + div m_{k} = 0 in B, we have that

lim inf_{k→∞} \frac{1}{|B|} ∫_{B} g(x, m_{k}, h + ∇u_{k}) dy ≥ lim inf_{k→∞} \frac{1}{|B|} ∫_{B}(φ_{1}^{∗∗}(m_{k}) + ψ_{1}(h + ∇u_{k})) dy ≥ φ_{1}^{∗∗}(m) + ψ_{1}(h)

and thus

ĝ(x, m, h) ≥ φ_{1}^{∗∗}(m) + ψ_{1}(h).

This concludes the proof.

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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