A-B QUASICONVEXITY AND IMPLICIT PARTIAL DIFFERENTIAL EQUATIONS

BERNARD DACOROGNA AND IRENE FONSECA

Abstract. The study of existence of solutions of boundary-value problems for differential inclusions

\{ \begin{matrix}Bu(x) ∈ E & a.e. x ∈ Ω, \\ u(x) = φ(x) & for all x ∈ ∂Ω, \end{matrix}

where φ ∈ C_{piec}^{1}(\overline{Ω}; ℝ^{N}), Ω is an open subset of ℝ^{n}, E ⊂ ℝ^{m×n} is a compact set, and B is a m × n-valued first order differential operator, is undertaken. As an application, minima of the energy for large magnetic bodies

E(m) := ∫_{Ω}[φ(m) − ⟨h_{e}; m⟩] dx + \frac{1}{2} ∫_{ℝ^{3}} |h_{m}|^{2} dx

where the magnetization m : Ω → ℝ^{3} is taken with values on the unit sphere S^{2}, h_{m} : ℝ^{3} → ℝ^{3} is the induced magnetic field satisfying curl h_{m} = 0 and div (h_{m} + mχ_{Ω}) = 0, φ is the anisotropic energy density, and the applied external magnetic field is given by h_{e} ∈ ℝ^{3}, are fully characterized. Setting Z := {ξ ∈ S^{2} : ψ(ξ) = min_{m∈S^{2}} ψ(m)} with ψ(m) := φ(m) − ⟨h_{e}; m⟩, it is shown that E admits a minimizer m ∈ L^{∞} with h_{m} ≡ 0 if and only if either 0 is on a face of ∂co Z or 0 ∈ intco Z, where co Z denotes the convex hull of Z.

Keywords: A-B quasiconvexity, differential inclusions, micromagnetism 2000 Mathematics Subject Classification: 35D99, 35E99, 49J45

Contents

1. Introduction 1 2. Preliminaries and statement of the problem 4 3. The approximation lemma 8 4. An abstract existence theorem 14 5. Existence theorems in the applications 19 6. Exact equilibrium solutions in ferromagnetism 22 References 30

1. Introduction

In recent years considerable research activity has been dedicated to the study of Dirichlet problems for first order partial differential equations and systems of the form

\{ \begin{matrix}F_{i} (Du (x)) = 0 i = 1, ..., I, a.e. x ∈ Ω, \\ u = φ on ∂Ω, \end{matrix} (1.1)

where Ω ⊂ ℝ^{n} is a bounded, open set, u : Ω → ℝ^{m}, the functions F_{i} : ℝ^{m×n} → ℝ, i = 1, …, I, are continuous, and the boundary datum φ is given.

The viscosity method was introduced to handle the case where the unknown function u is scalar (i.e. m = 1). In the vectorial case several methods have been developed, notably one using Baire Category Theorem and relaxation theorems, proposed by Cellina, Bressan-Flores and De Blasi-Pianigiani and extended by Dacorogna-Marcellini, and another one called convex integration and introduced by Gromov, further developed in many directions and in the present context by Müller-Šverák and others. We refer to [7] for a detailed bibliography on these matters.

The aim of the present article is to extend some of the results of [7], addressed to the gradient operator D, to the context of compensated compactness of Murat-Tartar, and in the framework of A-B quasiconvexity as introduced by Dacorogna [4] and studied in recent years by many authors following the work of Fonseca-Müller [12]. The property characterizing the operator D,

curl Du = 0,

will be extended here to some more general first order differential operators A (replacing the operator curl ) and B (replacing the operator gradient D) such that AB ≡ 0, namely A could be any combination of div or curl and therefore B would be composed of operators curl^{∗} and grad . Of course our analysis could and should be carried over to a more general class of operators d of the exterior differential calculus. However for the examples of applications we have in mind the present framework suffices.

Instead of summarizing the general results obtained in this paper, and gathered in Section 5, in this introduction we opt to discuss one significant example that has motivated our study. It concerns a problem in micromagnetics, and the findings described below were announced in [6].

Adopting Landau and Lifshitz [16] theory of micromagnetics (see also Brown [3]), we search for minimizers of the energy for a rigid ferromagnetic material occupying a configuration Ω ⊂ ℝ^{3}, where Ω is an open, bounded, Lipschitz domain. The magnetization m : Ω → ℝ^{3} represents a mass density of macroscopic magnetic moment and is subject to the constraint

|m(x)| = M_{T} x ∈ Ω, (1.2)

where T is the temperature, and M_{T} = 0 above the Curie point, i.e. for T ≥ T_{c}. Condition (1.2) ensures that the body is always saturated, and M_{T} is called saturation magnetization. We will assume that the temperature is held fixed, and, as it is usual, without loss of generality we will fix M_{T} = 1.

According to the theory of micromagnetics, observable states of a ferromagnetic body subject to a constant external magnetic field h_{e} ∈ ℝ^{3} correspond to minimizers of the total energy

E_{α}(m) := \frac{α^{2}}{2} ∫_{Ω} |∇m|^{2} dx + ∫_{Ω} φ(m) dx − ∫_{Ω}⟨h_{e}; m⟩ dx + \frac{1}{2} ∫_{ℝ^{3}} |h_{m}|^{2} dx

where φ : S^{2} → ℝ is the nonnegative, even, continuous anisotropic energy density, S^{2} := {ξ ∈ ℝ^{3} : |ξ| = 1}, and h_{m} : ℝ^{3} → ℝ^{3} is the induced magnetic field satisfying, in the sense of distributions,

\{ \begin{matrix}curl h_{m} = 0 & in ℝ^{3}, \\ div (h_{m} + χ_{Ω}m) = 0 & in ℝ^{3}, \end{matrix} (1.3)

where χ_{Ω} is the characteristic function of Ω. The four terms in E_{α} are denoted, respectively, exchange energy, anisotropy energy, interaction energy and magnetostatic energy. A scaling argument shows that for large bodies the exchange energy should become less important, and this leads us to the minimization of (see De Simone [8])

E(m) := E_{0} (m) = ∫_{Ω}[φ(m) − ⟨h_{e}; m⟩] dx + \frac{1}{2} ∫_{ℝ^{3}} |h_{m}|^{2} dx.

Existence of solutions for E_{α} has been obtained by Visintin [24], and a thorough study of the limiting behavior of minimizers for E_{α} and how they relate to minimizers of E may be found in De Simone [8], [9].

Before proceeding further it is convenient to reformulate the problem. We therefore let ψ (η) := φ(η) − ⟨h_{e}; η⟩ and

Z := {ξ ∈ S^{2} : ψ(ξ) = min_{|η|=1} {ψ(η)}} .

It is easy to see that the question of finding minima of

(P) : inf {E (m) : m ∈ L^{∞} (ℝ^{3}; ℝ^{3}) satisfies (1.3)}

is closely linked to the problem of finding m ∈ L^{∞} (ℝ^{3}; ℝ^{3}) (m ≡ 0 in ℝ^{3} \ \overline{Ω}) such that

\{ \begin{matrix}m(x) ∈ Z & a.e. x ∈ Ω, \\ div (χ_{Ω}m) = 0 & in ℝ^{3}. \end{matrix} (1.4)

Precisely, if (1.4) has solutions then (P) attains its minimum and minimizers must satisfy h_{m} ≡ 0 and m ∈ Z for a.e. x ∈ Ω.

When h_{e} = 0 James-Kinderlehrer [15] obtained certain characterizations of minimizers for E, by showing that in the uniaxial case, where Z = {±m_{1}} for some m_{1} ∈ S^{2}, the system (1.4) admits no solution, while if {±m_{1}, ±m_{2}} ⊂ Z for some orthogonal vectors m_{1}, m_{2} ∈ S^{2} then (1.4) does have a solution. The latter case falls within the so-called cubic-symmetry ferromagnetic crystals where Z = {±m_{1}, ±m_{2}, ±m_{3}} with {m_{1}, m_{2}, m_{3}} an orthonormal system in ℝ^{3}. Following the literature on magnetism, the argument is based on the construction of a prototype solution on a prism with cross-sectional shape dictated by the structure of the set Z, subsequently translated, scaled, and pieced together via Vitali Covering Theorem. In addition, James-Kinderlehrer [15] showed that E does not have a minimizer satisfying (1.4) in the presence of certain applied fields and for specific shapes of the domain. Precisely, they proved that if h_{e} = θDm_{1} where φ(m_{1}) = 0, θ ∈ (0, 1], and D = D^{T} > 0 is co-axial with the principal axis of the ellipsoid Ω, then a minimizer of E is given by the uniform magnetization \overline{m} := D^{−1}h_{e}, with corresponding nonzero induced magnetic field.

For further related work we refer to Anzellotti-Baldo-Visintin [1], Gioia-James [14], De Simone-Kohn-Müller-Otto [10], [11], Pedregal [19], and Tartar [22], [23].

In this paper, and using the abstract results developed below, we pursue further the analysis of James-Kinderlehrer [15] so as to give a complete characterization of the minimizers of E satisfying (1.4) and in the presence of a possibly non-vanishing external magnetic field h_{e}. Our analysis does not require that the function φ be even, although this is the natural framework in micromagnetics.

The main result of Section 6 (see Theorem 6.2) establishes that if Z ⊂ S^{2} is compact, then problem (1.4) has or has no solution according to the following cases (in the sequel coZ will denote the convex hull of Z; for the notions of edges and faces of ∂coZ we refer to Section 6):

Case 1 : if 0 ∉ coZ then (1.4) has no solutions;

Case 2 : if 0 ∈ ∂coZ and 0 is on an edge of ∂coZ then (1.4) has no solutions; Case 3 : if 0 ∈ ∂coZ and 0 is on a face de ∂coZ then (1.4) admits solutions; Case 4 : if 0 ∈ intcoZ (the interior of coZ), then (1.4) admits solutions.

Moreover, there exists M ∈ C(\overline{Ω}; ℝ^{3}) satisfying curl M ∈ L^{∞}(Ω; ℝ^{3}) in the sense of distributions, and

\{ \begin{matrix}curl M (x) ∈ Z & a.e. in Ω, \\ M = 0 & on ∂Ω. \end{matrix}

We set m := curl M.

The last case will be solved by the method presented in Sections 3, 4, and 5 of this article, while the other cases are handled by ad hoc methods similar to the ones used by James-Kinderlehrer [15] and strongly influenced by the magnetism literature.

2. Preliminaries and statement of the problem

We start by introducing the first order partial differential operators A and B. Let

U := (V, W) = (V^{1}, ..., V^{k}, W^{k+1}, ..., W^{m}) : ℝ^{n} → ℝ^{k×n} × ℝ^{(m−k)×n} = ℝ^{m×n}

where V^{i} : ℝ^{n} → ℝ^{n}, V^{i} = (V_{r}^{i})_{1≤r≤n}, i = 1, ..., k, and W^{i} : ℝ^{n} → ℝ^{n}, W^{i} = (W_{r}^{i})_{1≤r≤n}, i = k + 1, ..., m.

Consider the first order differential operator

AU := (div V, curl W) ∈ ℝ^{k} × ℝ^{(m−k)× \frac{n(n−1)}{2}}

where

div V := (div V^{1}, ..., div V^{k}) ∈ ℝ^{k},
div V^{i} := ∑_{r=1}^{n} \frac{∂V_{r}^{i}}{∂x_{r}} ∈ ℝ, i = 1, ..., k,

and

curl W := (curl W^{k+1}, ..., curl W^{m}) ∈ ℝ^{(m−k)× \frac{n(n−1)}{2}} ,
curl W^{i} := (\frac{∂W_{r}^{i}}{∂x_{s}} − \frac{∂W_{s}^{i}}{∂x_{r}} )_{1≤r<s≤n} ∈ ℝ^{\frac{n(n−1)}{2}} , i = k + 1, ..., m.

To each function

u = (v, w) ∈ ℝ^{k× \frac{n(n−1)}{2}} × ℝ^{(m−k)}

(in the sequel we will abbreviate N := k \frac{n(n−1)}{2} + m − k) we associate a first order differential operator

Bu := (curl^{∗}v, grad w) ∈ ℝ^{k×n} × ℝ^{(m−k)×n} = ℝ^{m×n}

where

curl^{∗}v := (curl^{∗}v^{1}, ..., curl^{∗}v^{k}) ∈ ℝ^{k×n},
curl^{∗}v^{i} := ((curl^{∗}v^{i})_{1} , ..., (curl^{∗}v^{i})_{n}) ∈ ℝ^{n},
(curl^{∗}v^{i})_{ν} := ∑_{r=1}^{ν−1} (−1)^{ν+r} \frac{∂v_{rν}^{i}}{∂x_{r}} + ∑_{s=ν+1}^{n} (−1)^{ν+s+1} \frac{∂v_{νs}^{i}}{∂x_{s}} ∈ ℝ,

with v^{i} : ℝ^{n} → ℝ^{\frac{n(n−1)}{2}} , v^{i} = (v_{rs}^{i})_{1≤r<s≤n}, i = 1, ..., k. It may be more convenient to write v^{i} as an n × n antisymmetric matrix (that is v_{rs}^{i} = − v_{sr}^{i}) and then we have, equivalently,

(curl^{∗}v^{i})_{ν} = ∑_{r=1}^{n} (−1)^{ν+r} \frac{∂v_{rν}^{i}}{∂x_{r}} .

Similarly

grad w := (grad w^{k+1}, ..., grad w^{m}) ∈ ℝ^{(m−k)×n},
grad w^{i} := (\frac{∂w^{i}}{∂x_{1}} , ..., \frac{∂w^{i}}{∂x_{n}} ) ∈ ℝ^{n},

where w^{i} : ℝ^{n} → ℝ, i = k + 1, ..., m.

Remark 2.1. (i) The important fact linking the operators A and B is that

ABu ≡ 0 for all u ∈ C^{2} (ℝ^{n}; ℝ^{k× \frac{n(n−1)}{2}} × ℝ^{(m−k)}) . (2.1)

(ii) The operators A and B are particular cases of the “d” operators of differential forms. For example div V^{i} (respectively curl W^{i}, curl^{∗}v^{i}, grad w^{i}) is an n (respectively 2, (n − 1), 1) form which, in turn, is the differential of an (n − 1) (respectively 1, (n − 2), 0) form. The identity (2.1) is just a rewriting of

ddω = 0.

Closely related to the operator A is the following set (so-called “characteristic cone” in the language of the theory of compensated compactness) introduced by Murat-Tartar,

Λ := \{ \begin{matrix}λ = ( λ^{1} ; ⋮ ; λ^{m} ) = ( \begin{matrix}λ_{1}^{1} & ⋯ & λ_{n}^{1} \\ ⋮ & ... & ⋮ \\ λ_{1}^{m} & ⋯ & λ_{n}^{m} \end{matrix} ) ∈ ℝ^{m×n} : \\ λ^{i} ⊥ λ^{j} for all i = 1, ..., k, and for all j = k + 1, ..., m, \\ rank ( λ^{1} ; ⋮ ; λ^{k} ) ≤ n − 1, rank ( λ^{k+1} ; ⋮ ; λ^{m} ) ≤ 1 \end{matrix} \} .

We now have the following definitions.

Definition 2.2. (i) A Borel measurable function f : ℝ^{m×n} → ℝ is said to be A-B quasiconvex if

∫_{U} f (ξ + Bφ (x)) dx ≥ f (ξ) meas(U)

for every bounded domain U ⊂ ℝ^{n}, ξ ∈ ℝ^{m×n}, and φ ∈ W_{0}^{1,∞} (U; ℝ^{N}).

(ii) A function f : ℝ^{m×n} → \overline{ℝ} = ℝ ∪ \{+∞\} is said to be Λ-convex if

f (tξ_{1} + (1 − t)ξ_{2}) ≤ t f (ξ_{1}) + (1 − t) f (ξ_{2}) for all ξ_{1}, ξ_{2} such that ξ_{1} − ξ_{2} ∈ Λ.

(iii) Given a set Z ⊂ ℝ^{m×n} we define Λco Z, the Λ convex hull of Z, by

Λco Z := \left\{{λ ∈ ℝ^{m×n} : f (λ) ≤ 0 for all Λ − convex function \atop f : ℝ^{m×n} → \overline{ℝ} = ℝ ∪ \{+∞\} such that f |_{Z}= 0}\right\} .

Remark 2.3. It follows immediately from the above defintion and from the fact that every vector of the canonical orthonormal basis of ℝ^{m×n} may be identified with an element of Λ, that any Λ-convex function is separately convex, and thus locally Lipschitz.

Some basic facts about the preceding notions are (see [4], [12]):

Theorem 2.4. (i) Let f : ℝ^{m×n} → ℝ be a Borel measurable function. (i) if f is convex then f is A-B quasiconvex; (ii) if f is continuous then

f A-B quasiconvex ⇒ f Λ-convex;

(iii) if f is A-B quasiconvex and continuous, then for every sequence such that

u_{ν} → u in L^{∞}(Ω) and Bu_{ν} ⇀^{∗} Bu in L^{∞}(Ω),

the lower semicontinuity property

lim inf_{ν→∞} ∫_{Ω} f (Bu_{ν} (x)) dx ≥ ∫_{Ω} f (Bu (x)) dx

holds for every open, bounded set Ω ⊂ ℝ^{n}.

The following are important examples of A-B quasiconvexity.

Example 2.5. (i) If k = m ≤ n − 1 and AU := (div V^{1}, ..., div V^{k}) then

Λ = ℝ^{m×n},

and therefore Λ convexity reduces to the usual notion of convexity. In particular, the Λ convex hull of a given set is its ordinary convex hull.

If k = m = n and AU := (div V^{1}, ..., div V^{n}) then

Λ = {λ ∈ ℝ^{n×n} : rank (λ) ≤ n − 1} .

An example of non convex function that is A-B quasiconvex is (see Tartar [21])

f (ξ) = (n − 1) trace (ξ^{T} ξ) − (trace ξ)^{2} .

(ii) If k = 0, m = 1, and AU := curl W ∈ ℝ^{\frac{n(n−1)}{2}} then

Λ = ℝ^{n}

and, as in (i) above, Λ convexity is ordinary convexity and the Λ convex hull is the usual convex hull. This case corresponds to scalar variational problems with underlying energy of the type

w ∈ W_{loc}^{1,1}(Ω; ℝ) ↦ ∫_{Ω} f(Dw) dx.

If k = 0, m > 1, and AU := curl W = (curl W^{1}, ..., curl W^{m}) ∈ ℝ^{m× \frac{n(n−1)}{2}} then

Λ = {λ ∈ ℝ^{n×n} : rank (λ) ≤ 1} .

A-B quasiconvexity is then the usual quasiconvexity condition of Morrey [17] for energy densities of the type

w ∈ W_{loc}^{1,1}(Ω; ℝ^{m}) ↦ ∫_{Ω} f(Dw) dx.

In order to compute the Λ convex hull of a given set one can use the following iteration scheme.

Proposition 2.6. Let Z ⊂ ℝ^{m×n}, Λ_{0}co Z := Z, and let for i ∈ ℕ

Λ_{i}co Z := {ξ ∈ ℝ^{m×n} : ξ = tξ_{1} + (1 − t)ξ_{2}, ξ_{1}, ξ_{2} ∈ Λ_{i−1}co Z,
ξ_{1} − ξ_{2} ∈ Λ, t ∈ [0, 1]}.

Then

Λco Z = ∪_{i∈ℕ_{0}}Λ_{i}co Z.

Proof. It can be proved easily by induction (the case i = 0 being trivial) that

Λ_{i}co Z ⊂ Λco Z for all i ∈ ℕ_{0},

from what follows that

∪_{i∈ℕ_{0}}Λ_{i}co Z ⊂ Λco Z.

In order to establish the reverse inclusion, we first need to introduce the notion of Λ-convex envelope of a function f : ℝ^{m×n} → \overline{ℝ} = ℝ ∪ \{+∞\}. We define recursively

Λ_{0}f := f,

Λ_{i+1}f(ξ) := inf{tΛ_{i}f(ξ_{1})+(1−t)Λ_{i}f(ξ_{2}) : t ∈ [0, 1], ξ = tξ_{1} +(1−t)ξ_{2}, ξ_{1} −ξ_{2} ∈ Λ},

and

Λf(ξ) := inf_{i∈ℕ_{0}} Λ_{i}f(ξ). (2.2)

Since Λ_{i+1}f ≤ Λ_{i}f, we deduce that the infimum in (2.2) is actually a limit, and that Λf is the largest Λ-convex function smaller than or equal to f.

Consider the indicator function of the set Z,

I_{Z}(x) := \{ \begin{matrix}0 & x ∈ Z, \\ +∞ & otherwise. \end{matrix}

By induction it can be proved that for all i ∈ ℕ_{0}

Λ_{i}I_{Z} = I_{Λ_{i}co Z},

and thus

ΛI_{Z} = I_{∪_{i∈ℕ_{0}} Λ_{i}co Z}. (2.3)

Since ΛI_{Z} is non-negative and Λ-convex, and ΛI_{Z}|Z = 0, if ξ ∈ Λco Z then ΛI_{Z}(ξ) = 0, which, in view of (2.3), yields ξ ∈ ∪_{i∈ℕ_{0}}Λ_{i}co Z.

We can now state the main problem we will address in the present article (see Section 5).

Problem 2.7. Let Ω ⊂ ℝ^{n} be an open set, let E ⊂ ℝ^{m×n} be compact, and consider the boundary datum φ ∈ C^{1} (\overline{Ω}; ℝ^{N}), where

N := k × \frac{n (n − 1)}{2} + (m − k) .

Set

φ := (ψ, χ)

where

ψ := (ψ^{1}, ..., ψ^{k}) ∈ ℝ^{k× \frac{n(n−1)}{2}} and ψ^{i} = (ψ_{rs}^{i})_{1≤r<s≤n} ∈ ℝ^{\frac{n(n−1)}{2}} ,
χ := (χ^{k+1}, ..., χ^{m}) ∈ ℝ^{(m−k)}.

Assume that

Bφ (x) = (curl^{∗}ψ, grad χ) ∈ E ∪ intΛco E for every x ∈ Ω.

We seek to find a function u ∈ C(\overline{Ω}; ℝ^{N}) such that the distribution Bu belongs to L^{∞}(Ω; ℝ^{m×n}) and

\{ \begin{matrix}Bu (x) ∈ E & a.e. x ∈ Ω, \\ u(x) = φ(x) & for all x ∈ ∂Ω. \end{matrix}

3. The approximation lemma

The proof of our main existence result, Theorem 4.3, is hinged on a density argument together with the approximation lemma below. We adopt the notation introduced in Section 2.

Lemma 3.1 (Approximation lemma). Let Ω ⊂ ℝ^{n} be a bounded open set, let t ∈ (0, 1), let ξ, η ∈ ℝ^{m×n} be such that

ξ − η ∈ Λ,

and let φ ∈ W^{1,∞} (Ω; ℝ^{N}) satisfy

Bφ = tξ + (1 − t) η.

For every ε > 0 there exist u ∈ W^{1,∞} (Ω; ℝ^{N}), and Ω_{ξ}, Ω_{η} ⊂ Ω disjoint open subsets, such that

|meas(Ω_{ξ}) − t meas(Ω)| , |meas(Ω_{η}) − (1 − t) meas(Ω)| ≤ ε, u = φ on a neighborhood of ∂Ω, ∥u − φ∥_{L^{∞}} ≤ ε,

Bu (x) = \{ \begin{matrix}ξ & in Ω_{ξ}, \\ η & in Ω_{η}, \end{matrix}

dist (Bu (x) , [ξ, η]) ≤ ε for all x ∈ Ω.

(3.1)

Proof. We divide the proof into two steps.

Step 1. We start by assuming that

ξ − η =: λ = ( \begin{matrix}0 & λ_{2}^{1} & ⋯ & λ_{n}^{1} \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & λ_{2}^{k} & ⋯ & λ_{n}^{k} \\ λ_{1}^{k+1} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ λ_{1}^{m} & 0 & ⋯ & 0 \end{matrix} ) . (3.2)

In Step 2 we will treat the general case.

We first note that we can assume that Ω is the unit cube. Indeed we can express Ω as a disjoint union of cubes with faces parallel to the coordinate axes and a set of small measure. It then follows that a solution u for (3.1) with respect to Ω may be constructed from solutions of (3.1) when Ω is the cube Q by setting u ≡ φ on the set of small measure and by using homothetics and translations in each of the small subcubes.

Let ε > 0, let Ω_{ε} be a set compactly contained in Ω, and let h ∈ C_{0}^{∞}(Ω) and L = L(Ω) > 0 be such that

meas (Ω \ Ω_{ε}) ≤ ε/2,
0 ≤ h(x) ≤ 1 for all x ∈ Ω,
h(x) = 1 for all x ∈ Ω_{ε},
|Dh(x)| ≤ \frac{L}{ε} for all x ∈ Ω \ Ω_{ε}.

Fix δ ∈ (0, ε). Consider a C^{∞} function g : [0, 1] → ℝ and sets I_{ξ}, I_{η} which are unions of disjoint open subintervals of [0, 1] so that

g′ (x_{1}) = \{ \begin{matrix}1 − t & if x_{1} ∈ I_{ξ}, \\ −t & if x_{1} ∈ I_{η}, \end{matrix}

g′ (x_{1}) ∈ [−t, 1 − t] for all x_{1} ∈ [0, 1] , |meas(I_{ξ}) − t| , |meas(I_{η}) − (1 − t)| ≤ ε/2, |g (x_{1})| ≤ δ for all x_{1} ∈ [0, 1] .

The construction of such function is standard. Let

Ω_{ξ} := {x ∈ Ω_{ε} : x_{1} ∈ I_{ξ}} and Ω_{η} := {x ∈ Ω_{ε} : x_{1} ∈ I_{η}} .

Define α := (β, γ) ∈ C^{∞} (\overline{Ω}; ℝ^{N}) by

α := (β, γ) = (β^{1}, ..., β^{k}, γ^{k+1}, ..., γ^{m}) ∈ ℝ^{k× \frac{n(n−1)}{2}} × ℝ^{(m−k)}

where β^{i} : ℝ^{n} → ℝ^{\frac{n(n−1)}{2}} , β^{i} = (β_{rs}^{i})_{1≤r<s≤n}, i = 1, ..., k, and γ^{i} : ℝ^{n} → ℝ, i = k + 1, ..., m, are given by

β_{rs}^{i}(x) := \{ \begin{matrix}(−1)^{s+1} g (x_{1}) λ_{s}^{i} & if r = 1 \\ 0 & if r ≠ 1 \end{matrix} , i = 1, ..., k,

and

γ^{i} (x) := g (x_{1}) λ_{1}^{i}, i = k + 1, ..., m.

We observe that

Bα = (curl^{∗}β, grad γ) = g′ (x_{1}) (ξ − η) .

Indeed, for i = 1, ..., k, and ν = 1, ...n, we have (recalling that λ_{1}^{i} = 0)

(curl^{∗}β^{i})_{ν} = ∑_{r=1}^{ν−1} (−1)^{ν+r} \frac{∂β_{rν}^{i}}{∂x_{r}} + ∑_{s=ν+1}^{n} (−1)^{ν+s+1} \frac{∂β_{νs}^{i}}{∂x_{s}}

= (−1)^{ν+1} \frac{∂β_{1ν}^{i}}{∂x_{1}} = g′ (x_{1}) λ_{ν}^{i}

which implies that

curl^{∗}β^{i} = g′ (x_{1}) λ^{i}, i = 1, ..., k.

Similarly, for i = k + 1, ..., m, we get

grad γ^{i} = (g′ (x_{1}) λ_{1}^{i}, 0, ..., 0) = g′ (x_{1}) λ^{i}.

We claim that the function

u =: h (α + φ) + (1 − h) φ = h α + φ

satisfies all properties listed in (3.1). The four three statements are immediate in light of the construction of u and since in Ω_{ε} the function h ≡ 1, and so we have

Bu = Bα + Bφ = g′ (x_{1}) (ξ − η) + tξ + (1 − t) η = \{ \begin{matrix}ξ & in Ω_{ξ}, \\ η & in Ω_{η}. \end{matrix}

In order to prove the last property we observe that

Bu = h (Bα + Bφ) + (1 − h) Bφ + R (h, α)
= (t + hg′)ξ + (1 − (t + hg′))η + R (h, α)

where

R (h, α) = ( \begin{matrix}(∑_{r=1}^{ν−1} (−1)^{ν+r} \frac{∂h}{∂x_{r}} β_{rν}^{i} + ∑_{s=ν+1}^{n} (−1)^{ν+s+1} \frac{∂h}{∂x_{s}} β_{νs}^{i}) & {1≤i≤k \atop 1≤ν≤n} \\ (γ^{i}grad h) & k+1≤i≤m \end{matrix} ) .

It is clear that by choosing δ sufficiently small with respect to ε we find that

|R (h, α)| ≤ ε, (3.3)

and since t + hg′, 1 − t − hg′ ≥ 0, by (3.3) we conclude that

dist (Bu (x) , [ξ, η]) ≤ ε, x ∈ Ω.

This achieves the first step of the lemma.

Step 2. We first claim that since ξ − η ∈ Λ we can find Q ∈ ℝ^{n×n} invertible (in fact Q ∈ SO (n)) such that

(ξ − η) Q = ( \begin{matrix}0 & λ_{2}^{1} & ⋯ & λ_{n}^{1} \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & λ_{2}^{k} & ⋯ & λ_{n}^{k} \\ λ_{1}^{k+1} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ λ_{1}^{m} & 0 & ⋯ & 0 \end{matrix} ) . (3.4)

Write

ξ − η = ( μ^{1} ; ⋮ ; μ^{k} ; μ^{k+1} ; ⋮ ; μ^{m} ) .

By definition of Λ we can find q^{1} ∈ ℝ^{n}, |q^{1}| = 1, such that

q^{1} ∈ span {μ^{k+1}, ..., μ^{m}} , q^{1} ⊥ μ^{1}, ..., μ^{k},

(if span {μ^{k+1}, ..., μ^{m}} = {0} then ignore the condition q^{1} ∈ span {μ^{k+1}, ..., μ^{m}}), and choose q^{2}, ..., q^{n} ∈ ℝ^{n}, orthonormal vectors, so that

q^{2}, ..., q^{n} ⊥ μ^{k+1}, ..., μ^{m}.

Let

Q^{T} = ( q^{1} ; ⋮ ; q^{n} ) = ( \begin{matrix}q_{1}^{1} & ⋯ & q_{n}^{1} \\ ⋮ & ... & ⋮ \\ q_{1}^{n} & ⋯ & q_{n}^{n} \end{matrix} ) .

It is clear that

(ξ − η) Q = ( \begin{matrix}⟨μ^{1}; q^{1}⟩ & ⟨μ^{1}; q^{2}⟩ & ⋯ & ⟨μ^{1}; q^{n}⟩ \\ ⋮ & ⋮ & ... & ⋮ \\ ⟨μ^{k}; q^{1}⟩ & ⟨μ^{k}; q^{2}⟩ & ⋯ & ⟨μ^{k}; q^{n}⟩ \\ ⟨μ^{k+1}; q^{1}⟩ & ⟨μ^{k+1}; q^{2}⟩ & ⋯ & ⟨μ^{k+1}; q^{n}⟩ \\ ⋮ & ⋮ & ... & ⋮ \\ ⟨μ^{m}; q^{1}⟩ & ⟨μ^{m}; q^{2}⟩ & ⋯ & ⟨μ^{m}; q^{n}⟩ \end{matrix} )

= ( \begin{matrix}0 & λ_{2}^{1} & ⋯ & λ_{n}^{1} \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & λ_{2}^{k} & ⋯ & λ_{n}^{k} \\ λ_{1}^{k+1} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ λ_{1}^{m} & 0 & ⋯ & 0 \end{matrix} )

and we have established (3.4). The conclusion of the lemma now follows from Step 1 by a change of variables. Precisely, let

Ω̃ := Q^{T} Ω, ξ̃:= ξQ, η̃ = ηQ,

and for i = 1, ..., k, let

φ_{rs}^{i} (y) = (−1)^{r+s} ∑_{a,b=1}^{n} (−1)^{a+b} φ_{ba}^{i} (Q y) q_{b}^{r}q_{a}^{s}, (3.5)

(we consider here φ^{i} as an n × n antisymmetric matrix), while for i = k + 1, ..., m, we set

φ^{i} (y) := φ^{i} (Q y) .

12 BERNARD DACOROGNA AND IRENE FONSECA

The above definition can be written in matrix form as follows

φ^{i} (y) = \{ \begin{matrix}φ^{i} (Q y) adj_{2}Q & if i = 1, ..., k, \\ φ^{i} (Q y) & if i = k + 1, ..., m. \end{matrix} (3.6)

(It is easy to see that when n = 2 then adj_{2}Q = det Q = 1, while when n = 3 then adj_{2}Q = Q). An elementary computation shows that (see Step 3 below) Bφ̃ (y) = Bφ (Q y) Q. (3.7) We therefore get that

Bφ̃ = tξ̃+ (1 − t) η̃

with ξ̃− η̃ as in (3.2). Apply Step 1 to these new data and find ũ, Ω̃ξ̃, Ω̃η̃ satisfying (3.1) with respect to ξ̃ and η̃. It suffices now to set

Ω_{ξ} := Q Ω̃ξ̃, Ω_{η} := Q Ω̃η̃,

and to define for i = 1, ..., k,

u_{ba}^{i} (x) = (−1)^{a+b} ∑_{r,s=1}^{n} (−1)^{r+s} u_{rs}^{i} (Q^{T} x) q_{b}^{r}q_{a}^{s},

while, for i = k + 1, ..., m,

u^{i} (x) := u^{i} (Q^{T} x) .

The result follows using the fact that Bu (x) = Bũ (Q^{T} x) Q^{T}.

Step 3. It only remains to show the elementary result that if

φ^{i} (y) = \{ \begin{matrix}φ^{i} (Q y) adj_{2}Q & if i = 1, ..., k, \\ φ^{i} (Q y) & if i = k + 1, ..., m, \end{matrix} (3.8)

then Bφ̃ (y) = Bφ (Q y) Q. (3.9) This is clear when i = k + 1, ..., m, since

grad φ^{i} (y) = grad φ^{i} (Q y) Q.

It remains to show that for i = 1, ..., k,

curl^{∗}φ^{i} (y) = curl^{∗} [φ^{i} (Q y)] Q

i.e., that for ν = 1, .., n,

(curl^{∗}φ^{i} (y))_{ν} = ∑_{a=1}^{n} (curl^{∗} [φ^{i} (Q y)])_{a} q_{a}^{ν}.

Recall that

(curl^{∗}φ^{i})_{ν} = ∑_{r=1}^{n} (−1)^{ν+r} \frac{∂φ_{rν}^{i}}{∂y_{r}} .

Invoking (3.5), and using the fact that Q ∈ SO (n), thus ∑_{r} q_{s}^{r} q_{b}^{r} = δ_{sb}, we have

(curl^{∗}φ^{i})_{ν} = ∑_{r=1}^{n} (−1)^{ν+r} [(−1)^{ν+r} ∑_{a,b=1}^{n} (−1)^{a+b} \frac{∂}{∂y_{r}} [φ_{ba}^{i} (Q y)] q_{b}^{r}q_{a}^{ν} ]

= ∑_{a,b,r=1}^{n} (−1)^{a+b} [∑_{s=1}^{n} \frac{∂φ_{ba}^{i}}{∂x_{s}} (Q y) q_{s}^{r}] q_{b}^{r}q_{a}^{ν}

= ∑_{a,b,s=1}^{n} (−1)^{a+b} \frac{∂φ_{ba}^{i}}{∂x_{s}} (Q y) δ_{sb} q_{a}^{ν}

= ∑_{a,b=1}^{n} (−1)^{a+b} \frac{∂φ_{ba}^{i}}{∂x_{b}} (Q y) q_{a}^{ν}

= ∑_{a=1}^{n} (curl^{∗} [φ^{i} (Q y)])_{a} q_{a}^{ν}

as claimed. This concludes Step 3 and thus the lemma.

Remark 3.2. (i) We note that, by choosing δ ∈ (0, ε^{2}) in the previous proof, in addition we may require in (3.1) that

∥Du∥_{L^{∞}} ≤ ∥Dφ∥_{L^{∞}} + |ξ − η| + ε.

(ii) It follows immediately from the construction that if φ ∈ C^{1}(\overline{Ω}; ℝ^{N}) (resp. C_{piec}^{1}(\overline{Ω}; ℝ^{N})) then so does u. Here, and in what follows, C_{piec}^{1}(\overline{Ω}; ℝ^{N}) stands for the space of continuous and piecewise C^{1} functions on Ω. (iii) Given ξ ∈ ℝ^{m×n} there exists an affine map u_{ξ} such that Bu_{ξ} = ξ and ||Du_{ξ}||_{L^{∞}(Ω)} ≤ C|ξ|, where C = C(N) . Write first

ξ = ( ξ^{1} ; ⋮ ; ξ^{k} ; ξ^{k+1} ; ⋮ ; ξ^{m} ) .

Define

u_{ξ} := (v, w) = (v^{1}, ..., v^{k}, w^{k+1}, ..., w^{m}) ∈ ℝ^{k× \frac{n(n−1)}{2}} × ℝ^{(m−k)} = ℝ^{N},

where v^{i} : ℝ^{n} → ℝ^{\frac{n(n−1)}{2}} , v^{i} = (v_{rs}^{i})_{1≤r<s≤n}, i = 1, ..., k, and w^{i} : ℝ^{n} → ℝ, i = k + 1, ..., m, are given by

v_{rs}^{i} (x) := \{ \begin{matrix}(−1)^{s+1} [x_{1}ξ_{s}^{i} − \frac{1}{n−1}x_{s}ξ_{1}^{i}] & if r = 1, \\ 0 & if r ≠ 1, \end{matrix} , i = 1, ..., k,

w^{i} (x) := ⟨ξ^{i}; x⟩ , i = k + 1, ..., m.

We claim that Bu_{ξ} = ξ. Indeed, for i = 1, ..., k and ν = 1, we have

(curl^{∗}v^{i})_{1} = ∑_{s=2}^{n} (−1)^{s} \frac{∂v_{1s}^{i}}{∂x_{s}} = ξ_{1}^{i}

while for ν = 2, ...n, we obtain

(curl^{∗}v^{i})_{ν} = ∑_{r=1}^{ν−1} (−1)^{ν+r} \frac{∂v_{rν}^{i}}{∂x_{r}} + ∑_{s=ν+1}^{n} (−1)^{ν+s+1} \frac{∂v_{νs}^{i}}{∂x_{s}}

= (−1)^{ν+1} \frac{∂v_{1ν}^{i}}{∂x_{1}} = ξ_{ν}^{i}.

The case i = k + 1, ..., m, is trivial since

grad w^{i} = ξ^{i}.

The L^{∞} bound on Du_{ξ} is an immediate consequence of the explicit definition of u_{ξ}.

4. An abstract existence theorem

We introduce the notion of sets with the relaxation property with respect to a fixed, underlying set.

Definition 4.1 (Relaxation property). Let E, K ⊂ ℝ^{m×n}. We say that K has the relaxation property with respect to E if for every bounded domain Ω ⊂ ℝ^{n} and for every affine map u_{ξ} with

Bu_{ξ} = ξ ∈ intK

there exists a sequence \{u_{ν}\} ⊂ C_{piec}^{1} (\overline{Ω}; ℝ^{N}) such that

u_{ν} ∈ u_{ξ} + W_{0}^{1,∞} (Ω; ℝ^{N}) ,

u_{ν} → u_{ξ} in L^{∞}(Ω; ℝ^{N}), Bu_{ν} ⇀^{∗} ξ in L^{∞}(Ω; ℝ^{m×n}),

Bu_{ν} (x) ∈ E ∪ int K a.e. in Ω,

∫_{Ω} dist (Bu_{ν} (x) ; E) dx → 0 as ν → ∞.

Remark 4.2. Observe that if K has the relaxation property with respect to E and if f : ℝ^{m×n} → ℝ is A-B quasiconvex and continuous satisfying f |_{E}= 0, then using Remark 3.2 (iii) and Theorem 2.4 (iii) we get

f (ξ) ≤ 0 for all ξ ∈ int K.

Next we state and prove the main abstract existence theorem of this paper.

Theorem 4.3. Let Ω ⊂ ℝ^{n} be open. Let F_{i} : ℝ^{m×n} → ℝ, i = 1, ..., I, be A-B quasiconvex, continuous, and let

E := {ξ ∈ ℝ^{m×n} : F_{i} (ξ) = 0, i = 1, ..., I}

be compact. Let K ⊂ ℝ^{m×n} be a compact set with the relaxation property with respect to E and let φ ∈ C_{piec}^{1} (\overline{Ω}; ℝ^{N}) be such that

Bφ (x) ∈ E ∪ int K a.e. x ∈ Ω.

Then there exists u ∈ C(\overline{Ω}; ℝ^{N}) such that Bu ∈ L^{∞}(Ω; ℝ^{m×n}) and

\{ \begin{matrix}Bu (x) ∈ E & a.e. x ∈ Ω, \\ u(x) = φ(x) & for all x ∈ ∂Ω. \end{matrix} (4.1)

Proof. We divide the proof into three steps and follow the framework of [7].

Step 1. We may assume that Ω ⊂ ℝ^{n} is bounded, since in the general case we decompose Ω as a countable union of open, bounded sets, on each one of which we solve (4.1).

Let V be the C^{0} closure of the set

\{u ∈ φ + W_{0}^{1,∞}(Ω; ℝ^{N}) : u ∈ C_{piec}^{1} (\overline{Ω}; ℝ^{N}) , Bu (x) ∈ E ∪ int K a.e in Ω\} .

Note that φ ∈ V and V is a complete metric space when endowed with the C^{0} norm. The compactness of E and K, the weak lower semicontinuity property of continuous A-B quasiconvex functions (see Theorem 2.4), and Remark 4.2 yield

(4.2) V ⊂ \left\{{u ∈ C(\overline{Ω}; ℝ^{N}) : Bu ∈ L^{∞}(Ω; ℝ^{m×n}), u = φ on ∂Ω \atop F_{i} (Bu (x)) ≤ 0 for i = 1, ..., I, a.e. x ∈ Ω}\right\} .

Step 2. For u ∈ V set

L (u) := ∑^{I} ; i=1 ∫_{Ω} F_{i} (Bu (x)) dx.

Due to the continuity and A-B quasiconvexity of F_{i} (see Theorem 2.4) we have for every u ∈ V

lim inf_{u_{s}→u,u_{s}∈V} L (u_{s}) ≥ L (u) , (4.3)

where we have used the fact that if u_{s} ∈ V are such that u_{s} → u uniformly then Bu_{s} ⇀^{∗} Bu, and also, in view of (4.2),

L(u) ≤ 0, L (u) = 0 ⇔ Bu (x) ∈ E a.e. in Ω. (4.4)

Define

V^{k} := \{u ∈ V : L (u) > −\frac{1}{k} \} .

By (4.3) V^{k} is open in V, and in Step 3 below we prove that V^{k} is dense in V. We deduce from Baire Category Theorem that ∩V^{k} is dense in V. In particular, we may find u ∈ ∩V^{k} and the result now follows from the definition of V and by (4.4).

Step 3. It remains to prove that for fixed k ∈ ℕ, u ∈ V, and ε ∈ (0, 1/k) sufficiently small, we can find u_{ε} ∈ V^{k} so that

∥u_{ε} − u∥_{∞} ≤ ε.

We will prove this property under the further assumptions that u ∈ C_{piec}^{1}(\overline{Ω}; ℝ^{N}) and that

Bu (x) ∈ E ∪ int K a.e in Ω.

The general case will follow by definition of V. Also, by working on each subdomain where u is C^{1}, we can assume, without loss of generality, that u ∈ C^{1} (\overline{Ω}; ℝ^{N}). In particular, setting

Ω_{0} := {x ∈ Ω : Bu (x) ∈ E} , Ω_{1} := Ω \ Ω_{0},

by continuity Ω_{0} is closed and hence Ω_{1} is open.

Before proceeding further we fix the constants. By compactness of E and K we have that

ξ ∈ E ∪ K ⇒ dist (ξ; E) ≤ β

for some β > 0. From the continuity of F_{i} and from the definition of E we can find δ = δ (ε) > 0 such that for any measurable function ξ : ℝ^{n} → E ∪ K it holds

∫_{Ω} dist (ξ (x) ; E) dx ≤ δ ⇒ ∑^{I} ; i=1 ∫_{Ω} F_{i} (ξ (x)) dx ≥ −ε. (4.5)

Using a density argument (see Theorem 10.16 in [7] modified accordingly), we can find ũ ∈ C^{1} (\overline{Ω}_{1}; ℝ^{N}), an integer J = J (ε) , and Ω̃_{j} ⊂ Ω_{1}, 1 ≤ j ≤ J, disjoint open sets, such that

ũ ≡ u on a neighborhood of ∂Ω_{1}, ∥ũ − u∥_{1,∞} ≤ \frac{ε}{2}, Bũ (x) ∈ intK a.e. x ∈ Ω_{1},

meas (Ω_{1} \ ∪_{j=1}^{J}Ω̃_{j}) ≤ \frac{δ}{2β} ,

Bũ ≡ ξ_{j}( constant) on Ω̃_{j}.

Using the relaxation property we may find u_{j} ∈ C_{piec}^{1} (\overline{Ω̃_{j}}; ℝ^{N}) satisfying

u_{j} ∈ ũ + W_{0}^{1,∞} (Ω̃_{j}; ℝ^{N}) ,
∥ũ − u_{j}∥_{∞} ≤ \frac{ε}{2} in Ω̃_{j},
Bu_{j} (x) ∈ E ∪ int K a.e. x ∈ Ω̃_{j},
∫Ω̃_{j} dist (Bu_{j,ν} (x) ; E) dx ≤ \frac{δ}{2} meas\binom{}{Ω̃_{j}}_{meas(Ω_{1})}.

Define

u_{ε} (x) := \{ \begin{matrix}u (x) & if x ∈ Ω_{0}, \\ ũ (x) & if x ∈ Ω_{1} \ ∪_{j=1}^{J}Ω̃_{j}, \\ u_{j} (x) & if x ∈ Ω̃_{j}. \end{matrix}

Observe that u_{ε} ∈ C_{piec}^{1} (\overline{Ω}; ℝ^{N}) ,

u_{ε} ∈ u + W_{0}^{1,∞} (Ω; ℝ^{N}) ,
∥u_{ε} − u∥_{∞} ≤ ε in Ω,
Bu_{ε} (x) ∈ E ∪ intK a.e. x ∈ Ω.

We also have that

∫Ω dist (Bu_{ε} (x) ; E) dx = ∫_{Ω_{0}} dist (Bu_{ε} (x) ; E) dx + ∫_{Ω_{1}} dist (Bu_{ε} (x) ; E) dx
= ∫_{Ω_{1}} dist (Bu_{ε} (x) ; E) dx

= ∫_{Ω_{1}\ ∪_{j=1}^{J}Ω̃_{j}} dist (Bu_{ε} (x) ; E) dx + J ; ∑_{j=1} ∫Ω̃_{j} dist (Bu_{ε} (x) ; E) dx

≤ β meas (Ω_{1} \ ∪_{j=1}^{J}Ω̃_{j}) + \frac{δ}{2}
≤ \frac{δ}{2} + \frac{δ}{2} = δ.

Hence, combining (4.5) with the above inequality, we get that

L (u_{ε}) ≥ −ε > −\frac{1}{k}

which implies that u_{ε} ∈ V^{k}. The claimed density has therefore been established and the proof is complete.

The verification of the relaxation property is, in general, a very delicate and subtle issue. The theorem below will provide a useful tool to establish the relaxation property for certain differential operators B (see Section 5).

Theorem 4.4. Let E, E_{δ} ⊂ ℝ^{m×n}, δ ∈ (0, δ_{0}), be compact sets such that (i) Λco E_{δ} ⊂ intΛco E for every δ ∈ (0, δ_{0}); (ii) for every ε > 0 there exists δ (ε) > 0 such that for any δ ∈ (0, δ (ε))

η ∈ E_{δ} ⇒ dist (η, E) ≤ ε;

(iii) if η ∈ intΛco E then η ∈ Λco E_{δ} for every δ > 0 sufficiently small.

Then Λco E has the relaxation property with respect to E.

Remark 4.5. The above property between the sets E and E_{δ} is called the “approximation property” in [7] and resembles the in-approximation of convex integration (see Müller-Šverák [18]).

Proof. Let Ω ⊂ ℝ^{n} be a bounded, open set, and let u be an affine function with Bu (x) = ξ, ξ ∈ intΛco E. We claim that there exists a sequence u_{ν} ⊂ C_{piec}^{1} (\overline{Ω}; ℝ^{N}) such that

u_{ν} ∈ u + W_{0}^{1,∞} (Ω; ℝ^{N}) ,

u_{ν} → u_{ξ} in L^{∞}(Ω; ℝ^{N}), Bu_{ν} ⇀^{∗} ξ in L^{∞}(Ω; ℝ^{m×n}),

Bu_{ν} (x) ∈ E ∪ int Λco E a.e. in Ω, ∫_{Ω} dist (Bu_{ν} (x) ; E) dx → 0 as ν → ∞.

(4.6)

Fix ε > 0 and let δ = δ(ε) be determined according to (ii). By (iii) we may find δ_{1} < δ such that

ξ ∈ ΛcoE_{δ_{1}}.

In view of Proposition 2.6 we have ξ ∈ Λ_{J}co E_{δ_{1}} for a certain J, and we now proceed by induction on J.

Step 1. Assume that J = 1. We can therefore write

Bu = ξ = tξ_{1} + (1 − t)ξ_{2}, ξ_{1} − ξ_{2} ∈ Λ,

with

ξ_{1}, ξ_{2} ∈ E_{δ_{1}}.

In light of property (i), choose ε′ ∈ (0, ε) such that

(4.7) the ε′ − neighborhood of ΛcoE_{δ_{1}} is contained in intΛcoE.

Using the Approximation Lemma 3.1 and Remark 3.2 (ii), we may find u_{ε} ∈ C^{1} (\overline{Ω}; ℝ^{N}) such that

meas(Ω \ (Ω_{1} ∪ Ω_{2})) = O(ε′), u_{ε} ≡ u near ∂Ω, ∥u_{ε} − u∥_{∞} ≤ ε′,

Bu_{ε}(x) = \{ \begin{matrix}ξ_{1} & in Ω_{1}, \\ ξ_{2} & in Ω_{2}, \end{matrix}

dist (Bu_{ε}(x); Λco E_{δ_{1}}) ≤ ε in Ω,

where we have the fact that (i) implies that

E_{δ_{1}} ⊂ co E ⊂ B(0, M)

for some M > 0 and all δ ∈ (0, δ_{0}), and

co {ξ_{1}, ξ_{2}} ⊂ Λco E_{δ_{1}}.

Now (4.7) ensures that

Bu_{ε} ∈ intΛcoE,

and in view of (ii), taking into account that dist(Bu_{ε}; E) is a bounded function, we conclude that

∫_{Ω} dist(Bu_{ε}; E) = ∫_{Ω_{1}} dist(ξ_{1}; E) + ∫_{Ω_{2}} dist(ξ_{2}; E)
+ ∫_{Ω\(Ω_{1}∪Ω_{2})} dist(Bu_{ε}; E)
= O(ε).

The claim (4.6) follows by letting ε → 0^{+}.

Step 2. Let

ξ ∈ Λ_{J}co E_{δ}

for some J > 1. Then there exist ξ_{1}, ξ_{2} ∈ ℝ^{m×n} such that

ξ = tξ_{1} + (1 − t)ξ_{2}, ξ_{1}, ξ_{2} ∈ Λ_{J−1}co E_{δ}, ξ_{1} − ξ_{2} ∈ Λ.

With ε′ chosen as in Step 1, apply the Approximation Lemma 3.1 and Remark 3.2 (ii) to find a function v_{ε} ∈ C^{1}(\overline{Ω}; ℝ^{N}) and Ω_{1}, Ω_{2} disjoint open sets such that

meas (Ω \ (Ω_{1} ∪ Ω_{2})) ≤ ε′/2, v_{ε} ≡ u on a neighborhood of ∂Ω, ∥v_{ε} − u∥_{∞} ≤ ε′/2,

Bv_{ε}(x) = \{ \begin{matrix}ξ_{1} in Ω_{1}, \\ ξ_{2} in Ω_{2}, \end{matrix}

dist (Bv_{ε}(x); Λco E_{δ}) ≤ ε in Ω.

Note that again by (4.7) we have that

Bv_{ε} ∈ intΛcoE,

although now we are unable to guarantee that

dist(ξ_{i}; E) ≤ ε i = 1, 2.

We use therefore the induction hypothesis on Ω_{1}, Ω_{2} and ξ_{1}, ξ_{2} to obtain v_{ε}^{1} ∈ C_{piec}^{1} in Ω_{1}, v_{ε}^{2} ∈ C_{piec}^{1} in Ω_{2} and satisfying for i = 1, 2,

v_{ε}^{i} ≡ v_{ε} near ∂Ω_{i},
∥v_{ε}^{i} − v_{ε}∥_{∞} ≤ ε′/2 in \overline{Ω}_{i},
Bv_{ε}^{i} ∈ E ∪ intΛcoE a.e. x ∈ Ω_{i},
∫_{Ω_{i}} dist (Bv_{ε}^{i}; E) ≤ ε′.

Setting

u_{ε}(x) := \{ \begin{matrix}v_{ε}(x) & in Ω \ (Ω_{1} ∪ Ω_{2}), \\ v_{ε}^{1}(x) & in Ω_{1}, \\ v_{ε}^{2}(x) & in Ω_{2}, \end{matrix}

we have indeed established (4.6) by choosing ε arbitrarily small.

5. Existence theorems in the applications

In this section we solve Problem 2.7 in the case where Λ = ℝ^{m×n}.

Theorem 5.1. Let Ω ⊂ ℝ^{n} be open. Let Λ = ℝ^{m×n} and let E ⊂ ℝ^{m×n} be compact. Let φ ∈ C_{piec}^{1} (\overline{Ω}; ℝ^{N}) be such that

Bφ (x) ∈ E ∪ intco E a.e. x ∈ Ω.

Then there exists u ∈ C(\overline{Ω}; ℝ^{N}) with Bu ∈ L^{∞}(Ω; ℝ^{m×n}) and satisfying

\{ \begin{matrix}Bu(x) ∈ E & a.e. x ∈ Ω, \\ u(x) = φ(x) & for all x ∈ ∂Ω, \end{matrix}

Remark 5.2. (i) By imposing that Λ = ℝ^{m×n} we are, essentially, restricting to the scalar case.

(ii) Using a more refined version of Vitali Covering Theorem, as in [7] it is possible to handle the case where φ ∈ W^{1,∞} (Ω; ℝ^{N}).

(iii) The result is in fact more precise in that if intco E is non empty then we will find solutions u such that Bu (x) ∈ E_{ext} there where Bφ(x) ∉ E, and where E_{ext} is the set of extreme points of E in the convex sense.

In view of Example 2.5 we obtain the following direct corollary that will be used in the application to ferromagnetism (with m = 1 and n = 3).

Corollary 5.3. Let Ω ⊂ ℝ^{n} be open. Let m ≤ n−1 and let E ⊂ ℝ^{m×n} be compact. Let φ ∈ C_{piec}^{1} (\overline{Ω}; ℝ^{N}), N = m × n (n − 1) /2, be such that

(curl^{∗}φ^{1} (x) , ..., curl^{∗}φ^{m} (x)) ∈ E ∪ intco E a.e. x ∈ Ω.

Then there exists u ∈ C(\overline{Ω}; ℝ^{N}) with curl^{∗}u ∈ L^{∞}(Ω; ℝ^{m×n}) and satisfying

\{ \begin{matrix}(curl^{∗}u^{1} (x) , ..., curl^{∗}u^{m} (x)) ∈ E & a.e. x ∈ Ω, \\ u(x) = φ(x) & for all x ∈ ∂Ω. \end{matrix}

Before proceeding with the proof of the theorem we need this elementary result of convex analysis.

Proposition 5.4. Let E ⊂ ℝ^{N} be compact and such that intco E ≠ ∅. Then there exist convex functions f_{i} : ℝ^{N} → ℝ, i = 1, 2, such that

E_{ext} = {ξ ∈ ℝ^{N} : f_{i} (ξ) = 0, i = 1, 2} ,
co E = co E_{ext} = {ξ ∈ ℝ^{N} : f_{i} (ξ) ≤ 0, i = 1, 2} ,

where E_{ext} denotes the set of extreme points of co E.

Proof. We sketch the proof of Georgy [13] which follows ideas of Bressan-Flores [2] and De Blasi-Pianigiani [20].

The first function f_{1} is the gauge associated to co E, i.e. for any fixed ξ_{0} ∈ int co E we define

f_{1} (ξ) := −1 + inf \{t > 0 : \frac{ξ − ξ_{0}}{t} + ξ_{0} ∈ co E\} ,

which is a convex function that satisfies

f_{1} (ξ) ≤ 0 ⇔ ξ ∈ co E.

The second function f_{2} is the Choquet function, and is defined as follows. Let

φ (ξ) := \{ \begin{matrix}− |ξ|^{2} = − ∑_{i=1}^{N} ξ_{i}^{2} & if ξ ∈ co E, \\ +∞ & otherwise, \end{matrix}

and, denoting by φ^{∗∗} the convex envelope of φ, set

ψ (ξ) := \{ \begin{matrix}φ^{∗∗} (ξ) − φ (ξ) & if ξ ∈ co E, \\ +∞ & otherwise. \end{matrix} .

The function ψ is exactly the Choquet function, which is convex and vanishes only on extreme points of E and otherwise is negative on co E. It is easy to see that ψ is Lipschitz (since φ, and hence φ^{∗∗}, is Lipschitz) all over the compact set co E. It can therefore be extended in a finite and convex way to the whole of ℝ^{N}. It is this extension that we call f_{2}. The conclusion then follows.

We may now return to the proof of Theorem 5.1.

Proof. Since Λ = ℝ^{m×n} we have ΛcoE = coE. If intco E = ∅ then the result is trivial, so we may assume that intco E ≠ ∅. In addition, without loss of generality we suppose that φ ∈ C^{1}(\overline{Ω}; ℝ^{N}).

Let

Ω_{0} := {x ∈ Ω : Bφ(x) ∈ E}, Ω_{1} := Ω \ Ω_{0}.

Then Ω_{1} is open and Bφ(x) ∈ intcoE for all x ∈ Ω_{1}. By Proposition 5.4 we may assume that

E = E_{ext} = {ξ ∈ ℝ^{N} : f_{i} (ξ) = 0, i = 1, 2}

and

co E = co E_{ext} = {ξ ∈ ℝ^{N} : f_{i} (ξ) ≤ 0, i = 1, 2} ,

with f_{i} convex functions, hence A-B quasiconvex and continuous, for i = 1, 2. In view of Theorem 4.3 it suffices to prove that co E has the relaxation property with respect to E to find a function u_{1} ∈ C(\overline{Ω_{1}}; ℝ^{N}) with Bu_{1} ∈ L^{∞}(Ω_{1}; ℝ^{N}) and satisfying

\{ \begin{matrix}Bu_{1}(x) ∈ E_{ext} & a.e. x ∈ Ω_{1}, \\ u_{1}(x) = φ(x) & for all x ∈ ∂Ω, \end{matrix}

and we then set

u(x) := \{ \begin{matrix}φ(x) & if x ∈ Ω_{0}, \\ u_{1}(x) & if x ∈ Ω_{1}. \end{matrix}

In order to establish the relaxation property we use Theorem 4.4. Choose ξ_{0} ∈ intco E and define for δ ∈ (0, 1) the sets

E_{δ} := δξ_{0} + (1 − δ) E.

Observe that

co E_{δ} = δξ_{0} + (1 − δ) co E ⊂ intco E

because if B(ξ_{0}, r) ⊂ coE then it follows that for all η ∈ coE

B(δξ_{0} + (1 − δ)η; δr) ⊂ coE.

Hypothesis (ii) of Theorem 4.4 can be easily verified, and as for (iii) we remark that if η ∈ intcoE then, since

|(\frac{η}{1 − δ} − \frac{δ}{1 − δ} ξ_{0}) − η| → 0 as δ → 0,

we have

\frac{η}{1 − δ} − \frac{δ}{1 − δ} ξ_{0} ∈ coE,

i.e. η ∈ δξ_{0} + (1 − δ)coE = coE_{δ}.

Suppose now that the set E is the set of zeroes of just one A-B quasiconvex function.

Theorem 5.5. Let Ω ⊂ ℝ^{n} be open. Let F : ℝ^{m×n} → ℝ, be A-B quasiconvex, continuous, and coercive (i.e. F (ξ) → +∞ as |ξ| → +∞). Let φ ∈ C_{piec}^{1} (\overline{Ω}; ℝ^{N}) be such that

F (Bφ (x)) ≤ 0 a.e. x ∈ Ω.

Then there exists u ∈ C(\overline{Ω}; ℝ^{N}) with Bu ∈ L^{∞}(Ω; ℝ^{m×n}) and satisfying

\{ \begin{matrix}F (Bu (x)) = 0 & a.e. x ∈ Ω, \\ u(x) = φ(x) & for all x ∈ ∂Ω. \end{matrix}

Remark 5.6. In fact, as it will be clear from the proof, one can weaken the coercivity condition and replace it by coercivity in, at least, one direction λ ∈ Λ, i.e. for every bounded set Ξ ⊂ ℝ^{m×n} there exists a continuous function ρ : ℝ → ℝ, with lim_{t→±∞} ρ (t) = +∞, such that

F (ξ + tλ) ≥ ρ (t) for all t ∈ ℝ and for all ξ ∈ Ξ.

Proof. By working on each subdomain where u ∈ C^{1}, we may assume thar u ∈ C^{1}(\overline{Ω}; ℝ^{N}).

We claim that

Λco E = {F ≤ 0}.

Since F is Λ-convex (see Theorem 2.4 (ii)) we have that

Λco E ⊂ {F ≤ 0}.

Conversely, if F (ξ) = 0 then ξ ∈ E ⊂ Λco E, and if F (ξ) < 0 then choose λ ∈ Λ\{0}. By the coercivity hypothesis there must exist t_{1} < 0 < t_{2} such that F (ξ_{i}) = 0, where ξ_{i} := ξ + t_{i}λ. Now ξ_{1} − ξ_{2} ∈ Λ and ξ = θξ_{1} + (1 − θ)ξ_{2} where θ ∈ [0, 1] is such that 0 = θt_{1} + (1 − θ)t_{2}. In view of Proposition 2.6 we conclude that ξ ∈ Λco E.

As it is usual we may assume that F (Bφ) < 0, by setting u ≡ φ on the closed set where F (Bφ) = 0. By continuity of F we have

Bφ ∈ int Λco E.

Due to the continuity and coercivity of F, the set E := {F = 0} is compact, and in particular Λco E is also compact. Therefore, in order to apply Theorem 4.3 we only have to check that Λco E has the relaxation property with respect to E. Let ξ ∈ int Λco E. If F (ξ) = 0 then, and in view of Remark 3.2, the relaxation property is trivial. Assume now that F (ξ) < 0 and fix λ ∈ Λ\{0}. The coercivity assumption leads to the existence of t_{1} < 0 < t_{2} such that

F (ξ + t_{1}λ) = 0 = F (ξ + t_{2}λ), F (ξ + tλ) < 0 for all t ∈ (t_{1}, t_{2}).

Let now u_{ξ} be an affine map with Bu_{ξ} = ξ, and choose ε > 0 small enough so that θ := \frac{t_{2}−ε}{t_{2}−t_{1}−2ε} > 0. We have

Bu_{ξ} = θξ_{1} + (1 − θ)ξ_{2}

where

ξ_{1} := ξ + (t_{1} + ε)λ, ξ_{2} := ξ + (t_{2} − ε)λ, ξ_{1}, ξ_{2} ∈ int Λco E, ξ_{1} − ξ_{2} ∈ Λ.

Choose 0 < ε′ < ε such that

{dist(·; [ξ_{1}, ξ_{2}]) < ε′} ⊂ int Λco E.

By the Approximation Lemma 3.1 we may find u_{ε} which agrees with u_{ξ} on a neighborhood of ∂Ω, with

||u_{ε} − u_{ξ}||_{L^{∞}} < ε′, dist(Bu_{ε}(x), [ξ_{1}, ξ_{2}]) ≤ ε′.

We conclude that Bu_{ε′} ∈ int Λco E, and it suffices to let ε → 0.

6. Exact equilibrium solutions in ferromagnetism

In the sequel we will adopt the notations of the Introduction, and we will use the notions of edge and face of a convex set. Precisely

Definition 6.1. Given z_{1}, ..., z_{N} ∈ ℝ^{3} we denote by span {z_{1}, ..., z_{N}} the subspace generated by these vectors and by dim span its dimension. If Z ⊂ ℝ^{3} is compact and if ξ ∈ ∂coZ \ Z then we say that ξ is on an edge of ∂co Z if

\begin{matrix}ξ = ∑_{i=1}^{l} t_{i}z_{i} \\ z_{i} ∈ Z, t_{i} > 0, ∑_{i=1}^{l} t_{i} = 1 \end{matrix} \} ⇒ dim span \{z_{1}, ..., z_{l}\} ≤ 1.

If this is not the case then we say that ξ is on a face of ∂co Z.

The main result of this section is the theorem below.

Theorem 6.2. Let Ω ⊂ ℝ^{3} be a bounded, open set with Lipschitz boundary, let Z ⊂ S^{2} be compact and consider the problem

\{ \begin{matrix}m(x) ∈ Z & a.e. x ∈ Ω, \\ div (χ_{Ω}m) = 0 & in ℝ^{3}, \end{matrix} (6.1)

where m ∈ L^{∞} (ℝ^{3}; ℝ^{3}) (m ≡ 0 in ℝ^{3}⧹\overline{Ω}).

Case 1 : if 0 ∉ co Z then (6.1) has no solutions;

Case 2 : if 0 ∈ ∂co Z and 0 is on an edge of ∂co Z then (6.1) has no solutions; Case 3 : if 0 ∈ ∂co Z and 0 is on a face de ∂co Z then (6.1) admits solutions; Case 4 : if 0 ∈ intco Z then (6.1) admits solutions. Moreover, there exists

M ∈ C(\overline{Ω}; ℝ^{3}), with m := curl M ∈ L^{∞}(Ω; ℝ^{3}), satisfying

\{ \begin{matrix}curl M ∈ Z & a.e. in Ω, \\ M = 0 & on ∂Ω. \end{matrix}

Remark 6.3. (i) Note that the condition div (mχ_{Ω}) = 0 implies ∫_{Ω} m = 0.

(ii) In the particular case where the set Z is symmetric with respect to the origin, i.e.

ξ ∈ Z ⇒ −ξ ∈ Z,

and under some further hypotheses, Cases 1 to 3 may be found in the work of James-Kinderlehrer [15] (see also De Simone [8]). Their results were inspired by the magnetism literature and we will also follow these ideas in our proofs for these two cases. However, Case 4 is new and follows from the theory developed earlier in this article. It could happen that in Case 4 problem (6.1) has geometrically constructed solutions of the same type as those we exhibit in Case 3, although we have not followed this avenue in our approach to the proof.

(iii) To contrast with Case 4 we will find in Case 3 a vector valued function M ∈ W^{1,∞} (Ω; ℝ^{3}) satisfying

\{ \begin{matrix}curl M (x) ∈ Z & a.e. in Ω, \\ M ∥ ν & on ∂Ω, \end{matrix}

where ν is a normal to ∂Ω, and we set m := curl M.

Proof. We will study each case separately

Case 1. If 0 ∉ co Z then (6.1) has no solutions. Indeed let ρ : ℝ^{3} → ℝ be the gauge associated to co Z, i.e.

ρ (ξ) ≤ 1 if ξ ∈ co Z and ρ (ξ) > 1 if ξ ∉ co Z.

Since ρ is convex we have by Jensen inequality, and in view of Remark 6.3 (i),

1 < ρ (0) = ρ (\frac{1}{meas(Ω)} ∫_{Ω} m) ≤ \frac{1}{meas(Ω)} ∫_{Ω} ρ (m) .

This implies that m ∉ co Z on a set of positive measure; hence the result.

Case 2. Assume now that 0 ∈ ∂co Z and 0 is on an edge of ∂co Z. As in Case 1 (6.1) has no solutions. To prove this fact we proceed in three steps.

Step 1. Since 0 is on an edge of ∂co Z, we find that there is a unique z_{0} ∈ Z such that −z_{0} ∈ Z and

0 = \frac{1}{2}z_{0} + \frac{1}{2} (−z_{0}) . (6.2)

Indeed assume for the sake of contradiction that there exist z_{0}, z_{1}, z_{2}, z_{3} ∈ Z, all distinct, and s, t ∈ (0, 1) so that

0 = tz_{0} + (1 − t) z_{1} = sz_{2} + (1 − s) z_{3}.

Since the z_{i} ∈ Z ⊂ S^{2}, we deduce that t = s = 1/2, z_{1} = −z_{0} and z_{2} = −z_{3}. From this we immediately get

0 = \frac{1}{4}z_{0} + \frac{1}{4}z_{1} + \frac{1}{4}z_{2} + \frac{1}{4}z_{3}

with

dim span {z_{0}, z_{1}, z_{2}, z_{3}} = dim span {z_{0}, z_{2}} = 2,

contradicting the fact that 0 is on an edge of ∂co Z. The equation (6.2) thus holds.

Step 2. We will now prove that if (6.1) has a solution m ∈ L^{∞} (ℝ^{3}; ℝ^{3}) (m ≡ 0 in ℝ^{3}⧹\overline{Ω}) then necessarily

m (x) ∈ {±z_{0}} a.e. in Ω. (6.3)

Let ε > 0 be arbitrary and let

Z_{0,ε} := {z ∈ Z : min {|z − z_{0}| , |z + z_{0}|} < ε} .

Define next

Ω_{0} : = {x ∈ Ω : m (x) ∈ Z_{0,ε} a.e.}
Ω_{1} : = {x ∈ Ω : m (x) ∈ Z \ Z_{0,ε} a.e.} .

Suppose, for the sake of contradiction, that meas(Ω_{1}) > 0. It follows from Jensen inequality, the definition of Ω_{1}, and from the fact that Z \ Z_{0,ε} is closed, that

\frac{1}{meas(Ω_{1})} ∫_{Ω_{1}} m ∈ co (Z \ Z_{0,ε}) .

By Carathéodory Theorem we can find z_{1}, z_{2}, z_{3}, z_{4} ∈ Z \ Z_{0,ε} and t_{1}, t_{2}, t_{3}, t_{4} > 0 with ∑_{i=1}^{4} t_{i} = 1, such that

\frac{1}{meas(Ω_{1})} ∫_{Ω_{1}} m = ∑_{i=1}^{4} t_{i}z_{i}.

Similarly, there exist z′_{1}, z′_{2}, z′_{3}, z′_{4} ∈ \overline{Z}_{0,ε} and t′_{1}, t′_{2}, t′_{3}, t′_{4} > 0 with ∑_{i=1}^{4} t′_{i} = 1, such that

\frac{1}{meas(Ω_{0})} ∫_{Ω_{0}} m = ∑_{i=1}^{4} t′_{i}z′_{i}.

Combining these two facts we obtain

0 = \frac{1}{meas(Ω)} ∫_{Ω} m = \frac{1}{meas(Ω)} ∫_{Ω_{1}} m + \frac{1}{meas(Ω)} ∫_{Ω_{0}} m

= ∑_{i=1}^{4} \frac{meas(Ω_{1})}{meas(Ω)} t_{i}z_{i} + ∑_{i=1}^{4} \frac{meas(Ω_{0})}{meas(Ω)}t′_{i}z′_{i},

which contradicts the fact that 0 is on an edge of ∂co Z. Hence meas(Ω_{1}) = 0 and thus

m (x) ∈ Z_{0,ε} , a.e. in Ω.

The arbitrariness of ε leads to the desired conclusion (6.3).

Step 3. We are now in a position to conclude. Assume for the sake of contradiction that (6.1) has a solution m ∈ L^{∞} (ℝ^{3}; ℝ^{3}) (m ≡ 0 in ℝ^{3}⧹\overline{Ω}) . Let

Ω_{±} := {x ∈ Ω : m (x) = ±z_{0} a.e.} .

Since div (mχ_{Ω}) = 0 in the sense of distributions, and in view of (6.3), we obtain for every ζ ∈ C_{0}^{∞} (ℝ^{3}),

0 = ∫_{ℝ^{3}}⟨m; ∇ζ⟩ χ_{Ω}dx = ∫_{ℝ^{3}}(χ_{Ω_{+}} − χ_{Ω_{−}}) ⟨z_{0}; ∇ζ⟩ dx = ∫_{ℝ^{3}} (χ_{Ω_{+}} − χ_{Ω_{−}}) \frac{∂ζ}{∂z_{0}} dx

which implies that the function g := χ_{Ω_{+}} − χ_{Ω_{−}} depends only on the variables that are orthogonal to z_{0}, more precisely g is constant on all rays with direction z_{0}, and this is not possible.

Case 3. If 0 ∈ ∂co Z and if 0 is on a face of ∂co Z, then problem (6.1) admits a solution. To assert this fact we address separately two subcases which correspond to the alternatives of Lemma 6.4 below.

Case 3a. There exist z_{1}, z_{2}, z_{3} ∈ Z, all distinct, and t_{1}, t_{2}, t_{3} > 0 with ∑_{i=1}^{3} t_{i} = 1, such that

0 = ∑_{i=1}^{3} t_{i}z_{i}.

We will show that for a given Ω_{0} we can find M ∈ W^{1,∞} (Ω_{0}; ℝ^{3}) such that

\{ \begin{matrix}curl M (x) ∈ \{z_{1}, z_{2}, z_{3}\} & a.e. in Ω_{0}, \\ M ∥ ν & on ∂Ω_{0}, \end{matrix} (6.4)

where ν denotes the exterior normal to ∂Ω_{0}. Since the domain for which we will make such a construction is a sort of a prism, and hence its boundary is not C^{1}, the boundary condition is to be interpreted in the almost everywhere sense with respect to the boundary measure.

The general case follows then by using Vitali Theorem (for more details see James-Kinderlehrer [15]).

Since 0 = ∑_{i=1}^{3} t_{i}z_{i} we have that, for example, {z_{1}, z_{2}} are linearly independent. We will then let α := \frac{t_{1}}{t_{3}} and β := \frac{t_{2}}{t_{3}} (hence z_{3} = −αz_{1} − βz_{2}). We will also define

T^{3} := z_{1} ∧ z_{2}, T^{2} := \frac{z_{2} − ⟨z_{1}; z_{2}⟩ z_{1}}{1 − (⟨z_{1}; z_{2}⟩)^{2}} , T^{1} := \frac{⟨z_{1}; z_{2}⟩ z_{2} − z_{1}}{1 − (⟨z_{1}; z_{2}⟩)^{2}}

and note that (recalling that Z ⊂ S^{2})

T^{2} ∧ T^{3} = z_{1}, T^{1} ∧ T^{3} = z_{2}.

Setting

T := ( \begin{matrix}T^{1} \\ T^{2} \\ T^{3} \end{matrix} ) = ( \begin{matrix}T_{1}^{1} & T_{2}^{1} & T_{3}^{1} \\ T_{1}^{2} & T_{2}^{2} & T_{3}^{2} \\ T_{1}^{3} & T_{2}^{3} & T_{3}^{3} \end{matrix} ) ,

we have det T = T^{1} · (T^{2} ∧ T^{3}) = −1, and so the matrix T is invertible. Consider the triangle

Δ := {(X_{1}, X_{2}) ∈ ℝ^{2} : 0 < X_{1} < 1/β, 0 < X_{2} < 1/α, βX_{1} + αX_{2} < 1}

that we subdivide into a disjoint union of three triangles

Δ_{1} : = \{(X_{1}, X_{2}) ∈ ℝ^{2} : 0 < X_{2} < X_{1} < \frac{1}{β} , βX_{1} + (1 + α) X_{2} < 1\} ,
Δ_{2} : = \{(X_{1}, X_{2}) ∈ ℝ^{2} : 0 < X_{1} < X_{2} < \frac{1}{α}, (1 + β) X_{1} + αX_{2} < 1\} ,
Δ_{3} : = \left\{{(X_{1}, X_{2}) ∈ ℝ^{2} : 1 < (1 + β) X_{1} + αX_{2}, \atop βX_{1} + αX_{2} < 1, 1 < βX_{1} + (1 + α) X_{2}}\right\} .

We next define f : Δ → ℝ as

f (X_{1}, X_{2}) := \{ \begin{matrix}X_{2} & if (X_{1}, X_{2}) ∈ Δ_{1}, \\ X_{1} & if (X_{1}, X_{2}) ∈ Δ_{2}, \\ 1 − (βX_{1} + αX_{2}) & if (X_{1}, X_{2}) ∈ Δ_{3}, \end{matrix}

and observe that f |_{∂Δ}= 0.

Define Ω_{0} and M : ℝ^{3} → ℝ^{3} by

Ω_{0} := {x ∈ ℝ^{3} : T x = (⟨T^{1}; x⟩ , ⟨T^{2}; x⟩ , ⟨T^{3}; x⟩) ∈ Δ × (0, 1)} ,
M (x) := f (⟨T^{1}; x⟩ , ⟨T^{2}; x⟩) T^{3}.

It remains to show that M has all the claimed properties. The fact that curl M ∈ {z_{1}, z_{2}, z_{3}} follows from the definitions of T and M, and from the elementary observation that

curl M = \frac{∂f}{∂X_{1}} (T^{1} ∧ T^{3}) + \frac{∂f}{∂X_{2}} (T^{2} ∧ T^{3}) .

Note that in Δ_{3} the field curl M reduces to −βz_{2} − αz_{1} which is equal to z_{3}. Since

∂Ω_{0}  =  T^{−1} (Δ × {0}) ∪ T^{−1} (Δ × {1}) ∪ T^{−1} (∂Δ × [0, 1])
=: A_{0} ∪ A_{1} ∪ A_{∂},

we deduce that the boundary condition is satisfied on A_{∂} because f |_{∂Δ}= 0, hence M = 0 on A_{∂}, and also on A_{0} ∪ A_{1} where ν ∥ T^{3} and M ∥ T^{3}.

Case 3b. There exist z_{1}, z_{2} ∈ Z, z_{1} ≠ ±z_{2}, with −z_{1}, −z_{2} ∈ Z. We therefore get

0 = \frac{1}{4}z_{1} + \frac{1}{4}z_{2} + \frac{1}{4} (−z_{1}) + \frac{1}{4} (−z_{2}) .

The proof is almost identical to the preceding one except that we need to change somewhat the definitions of Δ and f. Now Δ is the unit square (0, 1)^{2}, that we subdivide into four triangles:

Δ_{1} := {(X_{1}, X_{2}) ∈ ℝ^{2} : 0 < X_{2} < X_{1} < 1 − X_{2}} ,
Δ_{2} := {(X_{1}, X_{2}) ∈ ℝ^{2} : 0 < 1 − X_{1} < X_{2} < X_{1}} ,
Δ_{3} := {(X_{1}, X_{2}) ∈ ℝ^{2} : 0 < 1 − X_{2} < X_{1} < X_{2}} ,
Δ_{4} := {(X_{1}, X_{2}) ∈ ℝ^{2} : 0 < X_{1} < X_{2} < 1 − X_{1}} .

The function f : Δ → ℝ is then given by

f (X_{1}, X_{2}) := \{ \begin{matrix}X_{2} & if (X_{1}, X_{2}) ∈ Δ_{1}, \\ 1 − X_{1} & if (X_{1}, X_{2}) ∈ Δ_{2,} \\ 1 − X_{2} & if (X_{1}, X_{2}) ∈ Δ_{3}, \\ X_{1} & if (X_{1}, X_{2}) ∈ Δ_{4}. \end{matrix}

Case 4. 0 ∈ intco Z. This is a particular case of Corollary 5.3 with m = 1, E := Z, φ ≡ 0, recalling that, in ℝ^{3}, curl M = curl^{∗}M.

We have used in Step 3 of the proof of Theorem 6.2 the following elementary result of convex analysis.

Lemma 6.4. Let Z ⊂ S^{2} be compact and such that 0 is on a face of ∂co Z. Then one of the following two properties holds:

(i) There exist z_{1}, z_{2}, z_{3} ∈ Z, all distinct, and t_{1}, t_{2}, t_{3} > 0 with ∑_{i=1}^{3} t_{i} = 1, such that

0 = ∑_{i=1}^{3} t_{i}z_{i}.

(ii) There exist z_{1}, z_{2} ∈ Z, z_{1} ≠ ±z_{2}, with −z_{1}, −z_{2} ∈ Z. In particular

0 = \frac{1}{4}z_{1} + \frac{1}{4}z_{2} + \frac{1}{4} (−z_{1}) + \frac{1}{4} (−z_{2}) .

Remark 6.5. The two properties are not exclusive. We know by Carathéodory theorem that 0 is always a convex combination of four elements of Z. The lemma asserts that if 0 is not a convex combination of only three elements then necessarily Z contains the four distinct elements ±z_{1}, ±z_{2}.

Proof. We will proceed in two steps.

Step 1. We will start with a preliminary step. Assume that there exist four points z_{1}, z_{2}, z_{3}, z_{4} ∈ Z and t_{1}, t_{2}, t_{3}, t_{4} > 0 with ∑_{i=1}^{4} t_{i} = 1 such that

0 = ∑_{i=1}^{4} t_{i}z_{i} and dim span {z_{1}, z_{2}, z_{3}, z_{4}} = 3.

We will show that necessarily 0 ∈ intco Z (in fact the converse is also true, see [7] Lemma 2.11). Assume, without loss of generality, that {z_{1}, z_{2}, z_{3}} are independent; we therefore have

z_{4} = − (α_{1}z_{1} + α_{2}z_{2} + α_{3}z_{3})

where α_{i} := t_{i}/t_{4} > 0. For ε > 0 define next

C_{ε} := {ξ ∈ ℝ^{3} : ξ = ε_{1}z_{1} + ε_{2}z_{2} + ε_{3}z_{3} with |ε_{i}| < ε, i = 1, 2, 3} .

Clearly C_{ε} is an open set and 0 ∈ C_{ε}. We will show that, for ε > 0 sufficiently small, C_{ε} ⊂ co Z, and this will establish the result. Let ξ ∈ C_{ε}, i.e.

ξ = ε_{1}z_{1} + ε_{2}z_{2} + ε_{3}z_{3}.

Set

s_{4} := \frac{1 − (ε_{1} + ε_{2} + ε_{3})}{1 + (α_{1} + α_{2} + α_{3})}, s_{i} := α_{i}s_{4} + ε_{i}, i = 1, 2, 3.

To ensure that all s_{i} are positive it suffices to choose

ε < min_{i=1,2,3} [\frac{1}{3}, \frac{α_{i}}{1 + (α_{1} + α_{2} + α_{3}) + 3α_{i}} ] .

Since

\{ \begin{matrix}s_{1} + s_{2} + s_{3} + s_{4} = 1, \\ ξ = s_{1}z_{1} + s_{2}z_{2} + s_{3}z_{3} + s_{4}z_{4}, \end{matrix}

we conclude that ξ ∈ co Z.

Step 2. In view of Step 1, since 0 ∈ ∂co Z we deduce that

\begin{matrix}0 = ∑_{i=1}^{4} t_{i}z_{i} \\ z_{i} ∈ Z, t_{i} > 0, ∑_{i=1}^{4} t_{i} = 1 \end{matrix} \} ⇒ dim span \{z_{1}, z_{2}, z_{3}, z_{4}\} ≤ 2.

This, together with the fact that 0 is not on an edge of ∂co Z yields the existence of z_{i} ∈ Z, t_{i} > 0, i = 1, …, 4 such that

(6.5) 0 = ∑_{i=1}^{4} t_{i}z_{i}, ∑_{i=1}^{4} t_{i} = 1, and dim span {z_{1}, z_{2}, z_{3}, z_{4}} = 2.

Assume therefore, without loss of generality, that {z_{1}, z_{2}} in (6.5) are linearly independent and that

z_{3} = − (αz_{1} + βz_{2}) , z_{4} = − (γz_{1} + δz_{2}) ,

for some α, β, γ, δ ∈ ℝ. We thus get

\{ \begin{matrix}t_{1} + t_{2} + t_{3} + t_{4} = 1, \\ t_{1} = αt_{3} + γt_{4}, \\ t_{2} = βt_{3} + δt_{4}. \end{matrix} (6.6)

We will now discuss all the possibilities according to the signs of α, β, γ and δ. We will see that unless α = δ = 0 and β = γ = 1 or α = δ = 1 and β = γ = 0 (i.e. under the conditions of (ii)), the statement (i) of the lemma always holds.

Case 1. α, β > 0. Set

s_{1} := \frac{α}{1 + α + β} , s_{2} := \frac{β}{1 + α + β} , s_{3} := \frac{1}{1 + α + β} , (s_{4} := 0) .

We get

0 = s_{1}z_{1} + s_{2}z_{2} + s_{3}z_{3}

and (i) holds.

Case 2. α ≤ 0 and β ≤ 0. From (6.6) we deduce that γ, δ > 0 and thus choosing

s_{1} := \frac{γ}{1 + γ + δ} , s_{2} := \frac{δ}{1 + γ + δ} , s_{4} := \frac{1}{1 + γ + δ} , (s_{3} := 0) .

we have

0 = s_{1}z_{1} + s_{2}z_{2} + s_{4}z_{4},

which asserts (i).

Case 3. α ≥ 0 and β < 0. From (6.6) we obtain that δ > 0. Two possibilities can then happen: either γ > 0, and we are in a position to apply Case 2, or γ ≤ 0. Using again (6.6) we find that

(αδ − βγ) t_{3} = δt_{1} − γt_{2} > 0

and thus αδ − βγ > 0. Letting

s_{1} := \frac{αδ − βγ}{δ − β + αδ − βγ} , s_{3} := \frac{δ}{δ − β + αδ − βγ} , s_{4} := \frac{−β}{δ − β + αδ − βγ} ,

and s_{2} = 0, the lemma follows from the observation

0 = s_{1}z_{1} + s_{3}z_{3} + s_{4}z_{4}.

Case 4. α < 0 and β ≥ 0. This may be treated as Case 3 with the roles of z_{1} and z_{2} interchanged.

Case 5. α = 0 and β > 0. Since z_{3} = −βz_{2}, β > 0 and z_{2}, z_{3} ∈ Z ⊂ S^{2} we deduce that β = 1. From (6.6) we infer that γ > 0. Three possibilities can then happen: either δ > 0, which was handled in Case 2, or δ < 0, or δ = 0. If δ < 0 then with the choice

s_{1} := \frac{γ}{1 + γ − δ} , s_{3} := \frac{−δ}{1 + γ − δ} , s_{4} := \frac{1}{1 + γ − δ} , (s_{2} := 0)

and in light of the observation

0 = s_{1}z_{1} + s_{3}z_{3} + s_{4}z_{4}

we are led to the assertion (i) of the lemma. Therefore it remains only to discuss the case δ = 0. Since γ > 0, z_{1}, z_{4} ∈ Z ⊂ S^{2} and z_{4} = −γz_{1} we get γ = 1. We find that ±z_{1}, ±z_{2} ∈ Z and hence

0 = \frac{1}{4}z_{1} + \frac{1}{4}z_{2} + \frac{1}{4} (−z_{1}) + \frac{1}{4} (−z_{2}) ,

which is the conclusion (ii) of the lemma.

Case 6. α > 0 and β = 0. This case is almost identical to the above one and we leave out the details.

The proof of the lemma is therefore complete.

We finish this section with some comments on the structure of the set Z in the case of ferromagnetics. We recall that

Z := {ξ ∈ S^{2} : ψ(ξ) = min_{|η|=1} {ψ(η)}} ,

where ψ (η) := φ(η) − ⟨h_{e}; η⟩. Here φ is a non-negative, continuous, even function (i.e. φ(m) = φ(−m) for all m ∈ S^{2}), and h_{e} is a given external magnetic field.

If h_{e} = 0 then Z reduces to the set of minimizers of φ on S^{2}, and since φ is even it follows immediately that 0 ∈ co Z. Next we discuss the case where h_{e} ≠ 0.

Proposition 6.6. If h_{e} ≠ 0 then 0 ∉ int co Z.

Assume, in addition, that φ is an even function separately with respect to each component, i.e.

φ(ξ_{1}, ξ_{2}, ξ_{3}) = φ(−ξ_{1}, ξ_{2}, ξ_{3}) = φ(ξ_{1}, −ξ_{2}, ξ_{3}) = φ(ξ_{1}, ξ_{2}, −ξ_{3}) (6.7)

for all ξ ∈ S^{2}. If either φ ∈ C^{1} or h_{e} = (h_{e}^{1}, h_{e}^{2}, h_{e}^{3}) is such that h_{e}^{i} ≠ 0 for all i ∈ {1, 2, 3}, then 0 ∉ co Z.

Remark 6.7. (i) The conclusions of the above proposition are sharp. Indeed, we may have h_{e} ≠ 0 and still 0 ∈ co Z for some even, continuous function φ. Consider, as an example,

φ(ξ_{1}, ξ_{2}, ξ_{3}) := |ξ_{1} + ξ_{2} + ξ_{3}| , h_{e} := (1, 1, 1).

It can be seen easily that

Z = {ξ ∈ S^{2} : ξ_{1} + ξ_{2} + ξ_{3} ≥ 0} ,

thus, in spite of the fact that 0 ∉ int co Z, we still have

0 ∈ co Z = \{ξ ∈ \overline{B(0, 1)} : ξ_{1} + ξ_{2} + ξ_{3} ≥ 0\} .

Also, the function

φ(ξ_{1}, ξ_{2}, ξ_{3}) := 1 − ξ_{3}^{2} + |ξ_{1}|

satisfies (6.7), and taking h_{e} := (1, 0, 0) then (0, 0, 1), (0, 0, −1) ∈ Z, and we conclude that 0 ∈ co Z. Note, however, that φ ∉ C^{1}.

(ii) The anisotropic energy densities φ considered by Landau and Lifschitz in [16] satisfy property (6.7) with φ ∈ C^{1}. Note that although 0 ∉ co Z, we may still have existence of minima for the energy E, as illustrated by the example treated by James-Kinderlehrer [15] and addressed in the Introduction where h_{e} = θDm_{1} and φ(m_{1}) = 0. Indeed, it is easy to prove that here Z = {m_{1}}, and James-Kinderlehrer [15] provide a solution with induced magnetic field h_{m} ≠ 0.

Proof. If ξ ∈ Z then

φ(ξ) − ⟨h_{e}; ξ⟩ = ψ(ξ) ≤ ψ(−ξ) = φ(−ξ) + ⟨h_{e}; ξ⟩ = φ(ξ) + ⟨h_{e}; ξ⟩ ,

and so

Z ⊂ \{ξ ∈ S^{2} : ⟨h_{e}; ξ⟩ ≥ 0\}, co Z ⊂ \{ξ ∈ \overline{B(0, 1)} : ⟨h_{e}; ξ⟩ ≥ 0\}. (6.8)

If h_{e} ≠ 0 it follows immediately that 0 ∉ int co Z.

Assume now that φ satisfies (6.7) and that h_{e}^{i} ≠ 0 for all i ∈ {1, 2, 3}. If 0 ∈ co Z then

0 = ∑_{i=1}^{4} t_{i}ξ^{i}, ξ^{i} ∈ Z, t_{i} ≥ 0, ∑_{i=1}^{4} t_{i} = 1.

By (6.7) we have that for i ∈ {1, 2, 3, 4}

φ(ξ_{1}^{i}, ξ_{2}^{i}, ξ_{3}^{i}) − ⟨h_{e}; (ξ_{1}^{i}, ξ_{2}^{i}, ξ_{3}^{i})⟩ ≤ φ(−ξ_{1}^{i}, ξ_{2}^{i}, ξ_{3}^{i}) − ⟨h_{e}; (−ξ_{1}^{i}, ξ_{2}^{i}, ξ_{3}^{i})⟩
= φ(ξ_{1}^{i}, ξ_{2}^{i}, ξ_{3}^{i}) − ⟨h_{e}; (−ξ_{1}^{i}, ξ_{2}^{i}, ξ_{3}^{i})⟩ ,

and thus h_{e}^{1} ξ_{1}^{i} ≥ 0; more generally

h_{e}^{j} ξ_{j}^{i} ≥ 0 for all j ∈ {1, 2, 3}, i ∈ {1, 2, 3, 4}. (6.9)

We have

0 = ⟨h_{e}; 0⟩ = ∑_{i=1}^{4} t_{i} ⟨h_{e}; ξ^{i}⟩

which, together with (6.8), yields ⟨h_{e}; ξ^{i}⟩ = 0 for all i ∈ {1, 2, 3, 4}. In view of (6.9) we now have that h_{e}^{j} ξ_{j}^{i} = 0 for all j ∈ {1, 2, 3}, i ∈ {1, 2, 3, 4}, hence ξ^{i} = 0 for all i ∈ {1, 2, 3, 4}, contradicting the fact that ξ ∈ S^{2}. We conclude that 0 ∉ co Z.

Finally, consider the case where h_{e} ≠ 0, and φ ∈ C^{1} satisfies (6.7). Without loss of generality we assume that h_{e}^{1} > 0. We claim that Z ⊂ {x > 0}, from what it will follow that co Z ⊂ {x > 0}, and hence 0 ∉ co Z. Let ξ = (x, y, z) ∈ Z. Since

φ(x, y, z) − ⟨h_{e}; (x, y, z)⟩ ≤ φ(−x, y, z) − ⟨h_{e}; (−x, y, z)⟩ ,

by (6.7) we deduce that 2h_{e}^{1}x ≥ 0, i.e. x ≥ 0. If x = 0 then there exists a Lagrange multiplier λ/2 ∈ ℝ such that

∇ψ(0, y, z) + \frac{λ}{2}∇|id|^{2}(0, y, z) = 0,

and so

\frac{∂φ}{∂x}(0, y, z) − h_{e}^{1} = 0.

But (6.7) implies that \frac{∂φ}{∂x}(0, y, z) = 0, and we conclude that h_{e}^{1} = 0, what is clearly in contradiction with our assumptions.

Acknowledgments. The authors are indebted to David Kinderlehrer and Luc Tartar for stimulating discussions on the subject of this work. The research of I. Fonseca was partially funded by the National Science Foundation under Grant No. DMS-9731957; while B. Dacorogna acknowledges the support of the Center for Nonlinear Analysis (NSF Grant No. DMS-9803791) during his visit to Carnegie-Mellon.

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(Bernard Dacorogna) Département de Mathématiques, EPFL, 1015 Lausanne, Suisse E-mail address, B. Dacorogna: Bernard.Dacorogna@epfl.ch

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

E-mail address, I. Fonseca: fonseca@andrew.cmu.edu