Ana Cristina Barroso
email: abarroso@ptmat.fc.ul.pt

Gisella Croce
email: gisella.croce@univ-lehavre.fr

Ana Margarida

† G Croce

‡ A M Ribeiro

Sufficient Conditions for Existence of Solutions to Vectorial Differential Inclusions and Applications

. We investigate sufficient conditions for existence, more flexible than those available in the literature, so that important applications can be fitted in the theory. We also study some of these applications.

Introduction

In this paper we discuss the existence of W 1,∞ (Ω, R N ) solutions to the vectorial differential inclusion problem Du(x) ∈ E, a.e. x ∈ Ω, u(x) = ϕ(x), x ∈ ∂Ω, (1.1) where Ω is an open subset of R n , E is a given subset of R N ×n and ϕ : Ω → R N . This problem has been intensively studied by Dacorogna and Marcellini [START_REF] Dacorogna | Théorèmes d'existence dans les cas scalaire et vectoriel pour les équations de Hamilton-Jacobi[END_REF], [START_REF] Dacorogna | Implicit partial differential equations[END_REF] through the Baire categories method (see also Müller and Šverák [START_REF] Müller | Convex integration for Lipschitz mappings and counterexamples to regularity[END_REF]). Their result provides a sufficient condition for existence of solutions related with the gradient of the boundary data. It asserts that, if this gradient belongs to a convenient set enjoying the so called relaxation property with respect to the set E (see Definition 3.1, and Theorem 3.2 due to Dacorogna and Pisante [START_REF] Dacorogna | A general existence theorem for differential inclusions in the vector valued case[END_REF]) a dense set of solutions to (1.1) exists.

In the applications, direct verification of the relaxation property is a hard task and sufficient conditions for it were also obtained by Dacorogna and Marcellini [START_REF] Dacorogna | Implicit partial differential equations[END_REF], namely, the approximation property, cf. Definition 3.3 and Theorem 3.4 (see also [START_REF] Dacorogna | Implicit partial differential equations[END_REF]Theorem 6.15] for a more general version). If such a property is satisfied, we can get as a sufficient condition for existence of solutions to 1 (1.1) Dϕ(x) ∈ E ∪ int Rco E, a.e. x ∈ Ω, (1.2) where int Rco E denotes the interior of the rank one convex hull of the set E, that is, the interior of the smallest rank one convex set containing E (see Definition 2.4). However, there are interesting applications for which int Rco E is empty, and thus, condition (1.2) is much too restrictive. This is the case of a well known problem solved by Kirchheim in [START_REF] Kirchheim | Deformations with finitely many gradients and stability of quasiconvex hulls[END_REF] that we will discuss in Section 4. In view of this, our goal in Section 3 is to obtain sufficient conditions for the relaxation property which can be handled in the applications and which are more flexible than (1.2). More precisely, we shall be able to deal with subsets of the hull Rco f E defined as

Rco f E = ξ ∈ R N ×n : f (ξ) ≤ 0, for every rank one convex f ∈ F E ,
where F E = {f : R N ×n → R : f | E ≤ 0}, and which is, in general, a larger hull than Rco E. Recovering results due to Pedregal [START_REF]Laminates and microstructure[END_REF] and to Müller and Šverák [START_REF] Müller | Convex integration for Lipschitz mappings and counterexamples to regularity[END_REF], we obtain in Theorem 2.9 the following characterization of this type of hulls for compact sets E:

Rco f E =          ξ ∈ R N ×n : ∀ ε > 0 ∃ I ∈ N, ∃ (λ i , ξ i ) i=1,...,I with λ i > 0, I i=1 λ i = 1 satisfying (H I (U )), ξ = I i=1 λ i ξ i , I i=1 ξi / ∈Bε(E) λ i < ε          , (1.
3) where U is an open and bounded set containing Rco f E and the property (H I (U )) is introduced in Definition 2.7. Thanks to this characterization we will prove, in particular, the following result (cf. Corollary 3.7).

Theorem 1.1 Let E ⊂ R N ×n be bounded and such that E and Rco f E have the approximation property with K δ = Rco f E δ for some compact sets E δ ⊂ R N ×n . Then int Rco f E has the relaxation property with respect to E.

From this theorem and from the Baire categories method it follows that, to ensure existence of solutions to (1.1) under the approximation property assumption, condition (1.2) can be replaced by Dϕ(x) ∈ E ∪ int Rco f E, a.e. x ∈ Ω.

(1.4)

Based on the characterization of the elements of Rco f E given in (1.3) we will prove in Theorem 3.5 a more general sufficient condition for the relaxation property which allows us to work with subsets of Rco f E. This is very useful in the applications because many times the entire hull Rco f E is not known. Moreover, characterizing Rco f E (or Rco E) may lead to complicated formulas and thus checking condition (1.4) (or (1.2)) becomes very difficult. However, many problems can still be solved provided it is possible to work with convenient subsets of Rco f E. This will be the case in the several applications given in Section 4. We start, in Section 4.1, with the problem already solved by Kirchheim [START_REF] Kirchheim | Deformations with finitely many gradients and stability of quasiconvex hulls[END_REF] on the existence of a non affine map with a finite number of gradients whose values are not rank one connected and with an affine boundary condition. We will show that this example is still in the setting of the Baire categories method thanks to the sufficient conditions for the relaxation property proved in Section 3. In this case the set E is a finite set of matrices with no rank one connections. We observe that we don't need to compute Rco f E and that we get existence of solutions whenever the affine boundary data ϕ satisfies Dϕ ∈ K, for a certain set K ⊂ Rco f E.

Then we will come back to the problem, already considered by Croce [START_REF] Croce | A differential inclusion: the case of an isotropic set[END_REF], of arbitrary compact isotropic subsets E of R 2×2 . Once again, our theory shows here its versatility. For this type of sets the hull Rco f E was characterized by Cardaliaguet and Tahraoui [START_REF] Cardaliaguet | Equivalence between rank one convexity and polyconvexity for isotropic sets of R 2×2 (Part I)[END_REF]. Although it was proved in [START_REF] Croce | A differential inclusion: the case of an isotropic set[END_REF] that this hull coincides with Rco E, we are now able to apply the Baire categories method without using this information.

Finally we consider, in Section 4.3, the case of sets E for which a constraint on the sign of the determinant is imposed on a set of isotropic matrices:

E = ξ ∈ R n×n : (λ 1 (ξ), • • • , λ n (ξ)) ∈ Λ E , det ξ > 0 , (1.5)
where

Λ E is a set contained in {(x 1 , • • • , x n ) ∈ R n : 0 < x 1 ≤ • • • ≤ x n } and 0 ≤ λ 1 (ξ) ≤ • • • ≤ λ n (ξ)
are the singular values of the matrix ξ (cf. Section 4).

Characterizing the hulls of such sets is quite complicated and the only results available were obtained by Cardaliaguet and Tahraoui [START_REF] Cardaliaguet | Equivalence between rank one convexity and polyconvexity for isotropic sets of R 2×2 (Part II)[END_REF] in dimension n = 2. Considering a particular class of sets E we will prove the following result (cf. Theorem 4.10).

Theorem 1.2 Let E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ {(a 1 , b 1 ), (a 2 , b 2 )}, det ξ > 0} with 0 < a 1 < b 1 < a 2 < b 2 . Let Ω ⊂ R 2 be a bounded open set and let ϕ ∈ C 1 piec (Ω, R 2 ) be such that Dϕ ∈ E ∪ int Rco f E a.e. in Ω. Then there exists a map u ∈ ϕ + W 1,∞ 0 (Ω, R 2 ) such that Du(x) ∈ E for a.e. x in Ω.
In addition, we are also able to establish sufficient conditions for sets E of the form (1.5) in dimension n = 2 and n = 3, working with a subset of Rco f E. In particular, in dimension 2, we prove the following result.

Theorem 1.3 Let Λ E be a subset of R 2 containing the line segment joining two distinct points (a 1 , a 2 ) and (b 1 , b 2 ) such that 0 < a 1 ≤ a 2 , 0 < b 1 ≤ b 2 , a 1 < b 1 , a 2 < b 2 , and either a 1 < a 2 or b 1 < b 2 . Let E = ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ Λ E , det ξ > 0 , Ω ⊂ R 2 be a bounded open set and ϕ ∈ C 1 piec (Ω, R 2 ) be such that for a.e. x in Ω, a 1 a 2 < det Dϕ(x) < b 1 b 2 , λ 2 (Dϕ(x)) < λ 1 (Dϕ(x)) b 2 -a 2 b 1 -a 1 + a 2 b 1 -b 2 a 1 b 1 -a 1 .
Then there exists a map u

∈ ϕ + W 1,∞ 0 (Ω, R 2 ) such that Du(x) ∈ E for a.e. x in Ω.
We refer to Theorems 4.11 and 4.12 for more details. We stress the fact that these results are independent of the knowledge of Rco f E, which is not known for n > 2, and that analogous results could be obtained in higher dimensions. In practical applications the conditions stated are easier to verify than the conditions needed to characterize Rco f E.

Review on the generalized notions of convexity

In this section we recall several definitions and properties of some generalized notions of convexity that will be useful throughout this paper. We refer to Dacorogna's monograph [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF] and to Dacorogna and Ribeiro [START_REF] Dacorogna | On some definitions and properties of generalized convex sets arising in the calculus of variations[END_REF] for more details. Several types of hulls in a generalized sense will be recalled here. The main result of this section is the characterization of the hull Rco f E, established in Theorem 2.9.

We start by recalling the notions of polyconvex and rank one convex functions.

Notation 2.1 For ξ ∈ R N ×n we let T (ξ) = (ξ, adj 2 ξ, . . . , adj N ∧n ξ) ∈ R τ (N,n) ,
where adj s ξ stands for the matrix of all s × s subdeterminants of the matrix ξ, 1 ≤ s ≤ N ∧ n = min {N, n} and where

τ = τ (N, n) = N ∧n s=1 N s n s and N s = N ! s! (N -s)! .
In particular, if

N = n = 2, then T (ξ) = (ξ, det ξ) . Definition 2.2 (i) A function f : R N ×n → R ∪ {+∞} is said to be polyconvex if there exists a convex function g : R τ (N,n) → R ∪ {+∞} such that f (ξ) = g(T (ξ)). (ii) A function f : R N ×n → R ∪ {+∞} is said to be rank one convex if f (λξ + (1 -λ)η) ≤ λ f (ξ) + (1 -λ) f (η) for every λ ∈ [0, 1] and every ξ, η ∈ R N ×n with rank(ξ -η) = 1.
It is well known that f polyconvex ⇒ f rank one convex.

Next we recall the corresponding notions of convexity for sets.

Definition 2.3 (i) We say that E ⊂ R N ×n is polyconvex if there exists a convex set K ⊂ R τ (N,n) such that ξ ∈ R N ×n : T (ξ) ∈ K = E. (ii) Let E ⊂ R N ×n .
We say that E is rank one convex if for every λ ∈ [0, 1] and for every ξ, η ∈ E such that rank(ξη) = 1, then λξ + (1λ)η ∈ E.

As shown by Dacorogna and Ribeiro [START_REF] Dacorogna | On some definitions and properties of generalized convex sets arising in the calculus of variations[END_REF] a set E is polyconvex if and only if the following condition is satisfied, for every

I ∈ N I i=1 λ i T (ξ i ) = T I i=1 λ i ξ i ξ i ∈ E, λ i ≥ 0, I i=1 λ i = 1              ⇒ I i=1 λ i ξ i ∈ E.
Moreover, we have the following implication

E polyconvex ⇒ E rank one convex.
As in the classical convex case, for these convexity notions, related convex hulls can be considered.

Definition 2.4

The polyconvex and rank one convex hulls of a set E ⊂ R N ×n are, respectively, the smallest polyconvex and rank one convex sets containing E and are, respectively, denoted by Pco E and Rco E.

Obviously one has the following inclusions

E ⊆ Rco E ⊆ Pco E ⊆ co E,
where co E denotes the convex hull of E.

We recall the usual characterizations for the polyconvex and rank one convex hulls. It was proved by Dacorogna and Marcellini in [START_REF] Dacorogna | Implicit partial differential equations[END_REF] that

Pco E = ξ ∈ R N ×n : T (ξ) = τ +1 i=1 t i T (ξ i ), ξ i ∈ E, t i ≥ 0, τ +1 i=1 t i = 1 (2.1)
and

Rco E = i∈N R i coE, (2.2)
where R 0 coE = E and

R i+1 coE = ξ ∈ R N ×n : ξ = λA + (1 -λ)B, λ ∈ [0, 1], A, B ∈ R i coE, rank(A -B) ≤ 1 , i ≥ 0.
One has (see [START_REF] Dacorogna | On some definitions and properties of generalized convex sets arising in the calculus of variations[END_REF]) that Pco E and Rco E are open if E is open, and Pco E is compact if E is compact. However, in general, it isn't true that Rco E is compact if E is compact (see Kolář [START_REF] Kolář | Non-compact lamination convex hulls Ann[END_REF]).

It is well known that

, for E ⊂ R N ×n , co E = ξ ∈ R N ×n : f (ξ) ≤ 0, for every convex function f ∈ F E ∞ (2.3) co E = ξ ∈ R N ×n : f (ξ) ≤ 0, for every convex function f ∈ F E (2.4)
where co E denotes the closure of the convex hull of E and

F E ∞ = f : R N ×n → R ∪ {+∞} : f | E ≤ 0 F E = f : R N ×n → R : f | E ≤ 0 .
Analogous representations to (2.3) can be obtained in the polyconvex and rank one convex cases:

Pco E = ξ ∈ R N ×n : f (ξ) ≤ 0, for every polyconvex function f ∈ F E ∞ , Rco E = ξ ∈ R N ×n : f (ξ) ≤ 0, for every rank one convex function f ∈ F E ∞ .
However, (2.4) can only be generalized to the polyconvex case if the sets are compact, and, in the rank one convex case, (2.4) is not true, even if compact sets are considered. In view of this, another type of hulls can be defined.

Definition 2.5 For a set E of R N ×n , let

co f E = ξ ∈ R N ×n : f (ξ) ≤ 0, for every convex f ∈ F E Pco f E = ξ ∈ R N ×n : f (ξ) ≤ 0, for every polyconvex f ∈ F E Rco f E = ξ ∈ R N ×n : f (ξ) ≤ 0, for every rank one convex f ∈ F E .
Remark 2.6 1) Notice that these hulls are closed sets. Moreover, they are, respectively, convex, polyconvex and rank one convex.

2) For compact sets E, these are the hulls considered by Müller and Šverák [START_REF] Müller | Convex integration for Lipschitz mappings and counterexamples to regularity[END_REF] to establish an existence result for differential inclusions. We notice that a different definition was introduced for open sets.

Thus, as observed above, co

E = co f E; if E is compact, then Pco E = Pco E = Pco f E, but, in general, Pco E Pco E Pco f E. Moreover, in general, even if E is compact, Rco E Rco E Rco f E.
Next we establish a characterization of the hull Rco f E for a given compact set E. Based on the following result, we will investigate in Section 3 sufficient conditions for the relaxation property (cf. Definition 3.1) which is the key to apply the Baire categories method for vectorial differential inclusions due to Dacorogna and Marcellini [START_REF] Dacorogna | Implicit partial differential equations[END_REF].

Before stating the result we give a definition.

Definition 2.7 Let U be a subset of R N ×n and, for some integer I ≥ 1, let ξ i ∈ R N ×n and λ i > 0, i = 1, ..., I be such that

I i=1 λ i = 1. We say that (λ i , ξ i ) 1≤i≤I satisfy (H I (U )) if (i) in the case I = 1, ξ 1 ∈ U ; (ii) in the case I = 2, ξ 1 , ξ 2 ∈ U and rank(ξ 1 -ξ 2 ) = 1;
(iii) in the case I > 2, up to a permutation, ξ 1 , ξ 2 ∈ U , rank(ξ 1ξ 2 ) = 1 and defining

µ 1 = λ 1 + λ 2 , η 1 = λ1ξ1+λ2ξ2 λ1+λ2 µ i = λ i+1 , η i = ξ i+1 , 2 ≤ i ≤ I -1 then (µ i , η i ) 1≤i≤I-1 satisfy (H I-1 (U )).
Remark 2.8 The property in the above definition was introduced in [7, page 174], but here we have the additional condition that the vertices of the "chain" must be elements of a given set. Moreover, we notice that in the above definition, in particular, all ξ i ∈ U .

Theorem 2.9 Let E ⊂ R N ×n be a compact set and let U be an open and bounded subset of R N ×n containing Rco f E. Then

Rco f E =          ξ ∈ R N ×n : ∀ ε > 0 ∃ I ∈ N, ∃ (λ i , ξ i ) i=1,...,I with λ i > 0, I i=1 λ i = 1 satisfying (H I (U )), ξ = I i=1 λ i ξ i , I i=1 ξi / ∈Bε(E) λ i < ε          , (2.5
) where B ε (E) = {ξ ∈ R N ×n : dist(ξ; E) < ε}.

Remark 2.10 1) The fact that Rco f E is included in the set on the right hand side of (2.5) was obtained by means of Young measures by Müller and Šverák [18, Theorem 2.1] as a refinement of a result due to Pedregal [START_REF]Laminates and microstructure[END_REF]. Below we recall the proof without mentioning Young measures.

2) Since, for compact sets E, Rco f E ⊆ co E, the set U can be chosen to be any convex open set containing E.

3) This result should be compared with the following characterization of Rco E which follows trivially from (2.2): for any set

E ⊂ R N ×n , RcoE =              ξ ∈ R N ×n : ∃ I ∈ N, ∃ (λ i , ξ i ) i=1,...,I with λ i > 0, I i=1 λ i = 1 satisfying (H I (Rco E)), ξ = I i=1 λ i ξ i , ξ i ∈ E, ∀ i = 1, ..., I              .
Proof. Let us call X the set on the right hand side of the identity to be proved. First we show that X ⊆ Rco f E. Let ξ ∈ X and let f : R N ×n → R be any rank one convex function such that f |E ≤ 0. We will show that f (ξ) ≤ 0 by verifying that f (ξ) ≤ δ for any δ > 0.

We start by noticing that, since f is continuous, it is uniformly continuous in U . Thus, fix δ > 0 and let γ > 0 be such that

∀ η 1 , η 2 ∈ U , |η 1 -η 2 | ≤ γ ⇒ |f (η 1 ) -f (η 2 )| ≤ δ. Now, let ε > 0 be such that ε ≤ min{δ, γ} and B ε (E) ⊂ U . By definition of X, let I ε ∈ N, (λ ε i , ξ ε i ) i=1,...,Iε with λ ε i > 0, Iε i=1 λ ε i = 1 satisfying (H Iε (U )), be such that ξ = Iε i=1 λ ε i ξ ε i , Iε i=1 ξ ε i / ∈Bε(E)
λ ε i < ε. Then, using the rank one convexity of f , we have

f (ξ) = f Iε i=1 λ ε i ξ ε i ≤ Iε i=1 λ ε i f (ξ ε i ) = Iε i=1 ξ ε i / ∈Bε(E) λ ε i f (ξ ε i ) + Iε i=1 ξ ε i ∈Bε(E) λ ε i f (ξ ε i ).
We are going to estimate the last two sums. For the first one we have

Iε i=1 ξ ε i / ∈Bε(E) λ ε i f (ξ ε i ) ≤ C Iε i=1 ξ ε i / ∈Bε(E) λ ε i ≤ Cδ,
where C := max U f . For the second sum we use the uniform continuity of f in

U . Since ξ ε i ∈ B ε (E), we can consider η ε i ∈ E such that |η ε i -ξ ε i | < ε. Then, f (ξ ε i ) ≤ f (ξ ε i ) -f (η ε i ) ≤ |f (ξ ε i ) -f (η ε i )| ≤ δ.
We then conclude that f (ξ) ≤ (1 + C)δ. Since δ is arbitrarily small, we obtain f (ξ) ≤ 0, as we wanted.

We will now prove the other inclusion, Rco f E ⊆ X. We suppose by contradiction that ξ ∈ Rco f E and ξ / ∈ X. Then, there exists ε > 0 such that, for every I ∈ N and for every (λ i , ξ i ) satisfying (H I (U )) with ξ = I i=1 λ i ξ i we have

I i=1 ξi / ∈Bε(E) λ i ≥ ε. Defining, for η ∈ U , f (η) := dist(η; E) and g(η) := inf        I i=1 µ i f (η i ) : I ∈ N, (µ i , η i ) i=1,...,I with µ i > 0, I i=1 µ i = 1 satisfying (H I (U ))        ,
it is trivial to see that 0 ≤ g < +∞ and g |E = 0. Moreover, for any (µ i , η i ) i=1,...,I as in the definition of g(ξ), the contradiction assumption gives

I i=1 µ i f (η i ) ≥ I i=1 ηi / ∈Bε(E) µ i f (η i ) ≥ I i=1 ηi / ∈Bε(E) µ i ε ≥ ε 2 > 0
and therefore g(ξ) > 0. Finally, we show that g is rank one convex on U according to the definition in [START_REF] Müller | Convex integration for Lipschitz mappings and counterexamples to regularity[END_REF], that is, if A, B ∈ U with rank(A-B) = 1 and

λA+(1-λ)B ∈ U, ∀ λ ∈ (0, 1) then g(λA+(1-λ)B) ≤ λg(A)+(1-λ)g(B).
To achieve this, it is enough to observe that if (µ i , A i ) i=1,...,I and (γ i , B i ) i=1,...,J are as in the definition of g(A) and g(B), respectively, then (

(λµ i , A i ), ((1-λ)γ i , B i ))
satisfy the conditions in the definition of g(λA

+ (1 -λ)B).
These properties of g allow us to apply the extension result [18, Lemma 2.3] which ensures that there exists a rank one convex function G : R N ×n → R coinciding with g in a neighborhood of Rco f E. This yields the desired contradiction, since we are assuming that ξ ∈ Rco f E. 2

Sufficient conditions for the relaxation property

The Baire categories method, developed by Dacorogna and Marcellini [START_REF] Dacorogna | Implicit partial differential equations[END_REF] for solving vectorial differential inclusions, relies on a fundamental property, called relaxation property, cf. Definition 3.1 below. Due to the difficulty in dealing with this property in the applications, sufficient conditions for it were also obtained in [START_REF] Dacorogna | Implicit partial differential equations[END_REF]. They ensure existence of solutions to the differential inclusion boundary value problem when the gradient of the boundary data is in the interior of the rank one convex hull of the set where the differential inclusion is to be solved. However, in some examples this hull turns out to be too restrictive. Therefore, our goal in this section is to find more flexible sufficient conditions for the relaxation property. This will allow us to handle the problems considered in Section 4. We start by recalling the relaxation property introduced by Dacorogna and Marcellini [START_REF] Dacorogna | Implicit partial differential equations[END_REF] and their related existence theorem for differential inclusions, here in a more general version due to Dacorogna and Pisante [START_REF] Dacorogna | A general existence theorem for differential inclusions in the vector valued case[END_REF]. Definition 3.1 (Relaxation Property) Let E, K ⊂ R N ×n . We say that K has the relaxation property with respect to E if, for every bounded open set Ω ⊂ R n and for every affine function u ξ , such that Du ξ (x) = ξ and Du ξ (x) ∈ K, there exists a sequence

u ν ∈ Af f piec (Ω; R N ) such that u ν ∈ u ξ + W 1,∞ 0 (Ω; R N ), Du ν (x) ∈ E ∪ K, a.e. x in Ω, u ν * ⇀ u ξ in W 1,∞ (Ω; R N ), lim ν→+∞ Ω dist(Du ν (x); E) dx = 0.
Theorem 3.2 Let Ω ⊂ R n be open and bounded. Let E ⊂ R N ×n and K ⊂ R N ×n be compact and bounded, respectively. Assume that K has the relaxation property with respect to E. Let ϕ ∈ Af f piec Ω; R N be such that

Dϕ (x) ∈ E ∪ K, a.e. x in Ω.
Then there exists (a dense set of )

u ∈ ϕ + W 1,∞ 0 Ω; R N such that Du (x) ∈ E, a.e. x in Ω. Moreover, if K is open, ϕ can be taken in C 1 piec (Ω; R N ).
A sufficient condition for the relaxation property is the approximation property [9, Definition 6.12 and Theorem 6.14] (see also [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF]Definition 10.6 and Theorem 10.9]) that we recall next.

Definition 3.3 (Approximation property) Let E ⊂ K ⊂ R N ×n .
The sets E and K are said to have the approximation property if there exists a family of closed sets E δ and K δ , δ > 0, such that (i)

E δ ⊂ K δ ⊂ int K for every δ > 0; (ii) ∀ ε > 0 ∃ δ 0 > 0 : dist(η; E) ≤ ε, ∀ η ∈ E δ , δ ∈ (0, δ 0 ]; (iii) η ∈ int K ⇒ ∃ δ 0 > 0 : η ∈ K δ , ∀ δ ∈ (0, δ 0 ].
We therefore have the following theorem.

Theorem 3.4 Let E ⊂ R N ×n be compact and assume Rco E has the approximation property with K δ = Rco E δ . Then int Rco E has the relaxation property with respect to E.

In the spirit of the approximation property, we establish a sufficient condition for the relaxation property such that larger sets than the rank one convex hull of E can be considered as the set K in Theorem 3.2. More precisely, we will show that hulls like Rco f E are likely to enjoy the relaxation property.

We can now state our main theorem.

Theorem 3.5 Let E, K be two bounded subsets of R N ×n and let, for δ > 0, E δ , K δ be sets verifying the following conditions:

(i) ∀ ε > 0 ∃ δ 0 > 0 : dist(η; E) ≤ ε, ∀ η ∈ E δ , δ ∈ (0, δ 0 ]; (ii) η ∈ int K ⇒ ∃ δ 0 > 0 : η ∈ K δ , ∀ δ ∈ (0, δ 0 ]; (iii) ∀ δ > 0 ∀ ξ ∈ K δ ∃ I ∈ N, ∃ (λ i , ξ i ) 1≤i≤I with λ i > 0, I i=1 λ i = 1, ξ i ∈ R N ×n , satisfying (H I (int K)) and ξ = I i=1 λ i ξ i , I i=1 ξi / ∈E δ λ i < δ.
Then int K has the relaxation property with respect to E.

Remark 3.6 1) In the sense of Proposition 3.13, Theorem 3.5 is a generalization of the usual approximation property for the hulls Rco f .

2) This result is analogous to Theorem 6.15 in [START_REF] Dacorogna | Implicit partial differential equations[END_REF] and allows us to work with subsets of Rco f E.

Before proving the result we establish two corollaries.

Corollary 3.7 Let E ⊂ R N ×n be bounded. Assume that there exist compact sets E δ ⊂ R N ×n such that, defining K δ = Rco f E δ and K = Rco f E, K δ ⊂ int K and conditions (i) and (ii) of Theorem 3.5 are satisfied. Then int Rco f E has the relaxation property with respect to E. Remark 3.8 One can easily see that Theorem 3.5 is also true if in condition (iii) we replace E δ by B δ (E). In this case, Corollary 3.7 follows directly from Theorem 3.5 and Proposition 3.13 below. In the proof that we present here we have to consider artificial approximating sets E δ .

Proof of Corollary 3.7. Let E δ = B δ (E δ ) and let K δ = Rco f E δ . We will show the result as an application of Theorem 3.5 for this choice of approximating sets.

By the hypotheses on E δ and K δ one can easily see that conditions (i) and (ii) of Theorem 3.5 are still satisfied by E δ and K δ . Condition (iii) follows from Theorem 2.9 applied to Rco f E δ , noticing that we are assuming that E δ is compact, and taking U = int Rco f E which contains Rco f E δ by hypothesis. 2

The following result was already proved by Ribeiro [START_REF] Ribeiro | Inclusions différentielles et problèmes variationnels[END_REF].

Corollary 3.9 Let E, K ⊂ R N ×n be such that E is compact and K is bounded. Assume that the following condition holds:

(H) given δ > 0, there exists L = L(δ, E, K) ∈ N such that

∀ ξ ∈ int K \ B δ (E) ∃ η 1 , ..., η J ∈ R N ×n , J ∈ N, J ≤ L, rank(η j ) = 1, j = 1, ..., J [ξ + η 1 + ... + η j-1 -η j , ξ + η 1 + ... + η j-1 + η j ] ⊂ int K, j = 1, ..., J, ξ + η 1 + ... + η J ∈ B δ (E),
where [A, B] represents the segment joining the matrices A and B. Then int K has the relaxation property with respect to E.

Remark 3.10 Using the same ideas of the following proof, it turns out that, under the conditions of Corollary 3.9, the set K is contained in Rco f E.

Proof of Corollary 3.9. We will prove that

E δ = int K ∩ B δ (E) and K δ = (int K ∩ E) ∪ (int K \ B δ (E))
satisfy conditions (i), (ii) and (iii) of Theorem 3.5.

Condition (i) is trivial. To get condition (ii), we observe that if η ∈ int K then either η ∈ E, and thus η ∈ K δ for every δ > 0, or η / ∈ E. In this last case, since E is compact, dist (η; E) > 0 which entails (ii).

It remains to show condition (iii). Let δ > 0, ξ ∈ K δ and consider L = L(δ, E, K) ∈ N as in the hypothesis. If ξ ∈ int K ∩ E, then condition (iii) is satisfied with I = 1 and (λ i , ξ i ) i=1 = (1, ξ) and we are left with the case ξ ∈ int K \ B δ (E). Applying the hypothesis, we have

∃ η 1 , ..., η J ∈ R N ×n , J ∈ N, J ≤ L, rank(η j ) = 1, j = 1, ..., J [ξ + η 1 + ... + η j-1 -η j , ξ + η 1 + ... + η j-1 + η j ] ⊂ int K, j = 1, ..., J, ξ + η 1 + ... + η J ∈ B δ (E).
Thus, by iteratively writing convex combinations using the matrices

η i , i = 1, • • • , J, we obtain ξ = 1 2 (ξ-η 1 )+ 1 2 2 (ξ+η 1 -η 2 )+...+ 1 2 J (ξ+η 1 +...+η J-1 -η J )+ 1 2 J (ξ+η 1 +...+η J ). (3.1)
We notice that if we take

λ j = 1 2 j , if 1 ≤ j ≤ J λ J+1 = 1 2 J and ξ j = ξ + η 1 + ... + η j-1 -η j , if 1 ≤ j ≤ J ξ J+1 = ξ + η 1 + ... + η J ,
then (λ j , ξ j ) 1≤j≤J+1 satisfy (H J+1 (int K)) and (3.1) can be rewritten in the form

ξ = J+1 j=1 λ j ξ j , with J+1 j=1 ξj / ∈B δ (E) λ j ≤ 1 - 1 2 J ≤ 1 - 1 2 L . If all ξ j ∈ B δ (E), then J+1 j=1 ξj / ∈B δ (E)
λ j = 0 and we have achieved condition (iii).

Otherwise, for each ξ j ∈ int K \ B δ (E) we apply again the hypothesis and get, for some

I j ≤ L, ξ j = Ij +1 l=1 λ j l ξ j l such that ( λ j l , ξ j l ) 1≤l≤Ij +1 satisfy (H Ij +1 (int K))
and

Ij +1 l=1 ξ j l / ∈B δ (E) λ j l ≤ 1 - 1 2 L . Therefore ξ = J+1 j=1 ξj ∈B δ (E) λ j ξ j + J+1 j=1 ξj / ∈B δ (E) Ij +1 l=1 λ j λ j l ξ j l ,
where the scalars and matrices in the above expression satisfy (H Ĩ (int K)) for a certain Ĩ ∈ N, and

J+1 j=1 ξj / ∈B δ (E) Ij +1 l=1 ξ j l / ∈B δ (E) λ j λ j l ≤ J+1 j=1 ξj / ∈B δ (E) λ j 1 - 1 2 L ≤ 1 - 1 2 L 2 .
Of course, after a finite number of iterations of this procedure one gets condition (iii). 2

In order to prove Theorem 3.5 we will show the following lemma.

Ω i ε ⊂ Ω such that u ε ∈ u ξ + W 1,∞ 0 (Ω; R N ), Du ε (x) ∈ U ∪ B ε (ξ), a.e. x ∈ Ω, Du ε (x) = ξ i , a.e. x ∈ Ω i ε , i = 1, ..., I, ||u ε -u ξ || L ∞ ≤ ε, | meas(Ω i ε ) -λ i meas(Ω)| ≤ ε, i = 1, ..., I.
The proof of the lemma relies on the following approximation result, due to Müller and Sychev [19, Lemma 3.1], which is a refinement of a classical result. For a ∈ R N and b ∈ R n we will denote by a ⊗ b the N × n matrix whose (i, j) entry is a i b j .

A , Ω B ⊂ Ω, such that                |meas (Ω A ) -t meas (Ω)| ≤ ε, |meas (Ω B ) -(1 -t) meas (Ω)| ≤ ε u(x) = ϕ(x), x ∈ ∂Ω |u(x) -ϕ(x)| ≤ ε, x ∈ Ω Du(x) = A in Ω A B in Ω B Du(x) ∈ ξ + {(1 -t) a ⊗ b, -ta ⊗ b, a ⊗ b 3 , . . . , a ⊗ b k }, a.e. x in Ω.
Proof of Lemma 3.11. We prove the result by induction on I.

If I = 2 it suffices to apply Lemma 3.12 choosing |b 3 |, ..., |b k | sufficiently small so that |a ⊗ b i | ≤ ε for i = 3, ..., k. Now, let I > 2 and consider (λ i , ξ i ) 1≤i≤I as in the hypothesis. Up to a permutation, and defining

µ 1 = λ 1 + λ 2 , η 1 = λ1ξ1+λ2ξ2 λ1+λ2 µ i = λ i+1 , η i = ξ i+1 , 2 ≤ i ≤ I -1, we have ξ 1 , ξ 2 ∈ U , rank(ξ 1 -ξ 2 ) = 1 and (µ i , η i ) 1≤i≤I-1 satisfy (H I-1 (U )).
Then the induction hypothesis ensures the existence of

v ∈ Af f piec (Ω; R N ) and disjoint open sets Ω i ε ⊂ Ω, i = 1, ..., I -1 such that v ∈ u ξ + W 1,∞ 0 (Ω; R N ), Dv(x) ∈ U ∪ B ε (ξ), a.e. x ∈ Ω, Dv(x) = η i , a.e. x ∈ Ω i ε , i = 1, ..., I -1 ||v -u ξ || L ∞ ≤ ε 2 , |meas( Ω i ε ) -µ i meas(Ω)| ≤ ε 2 , i = 1, ..., I -1.
Since (µ i , η i ) 1≤i≤I-1 satisfy (H I-1 (U )), then η 1 ∈ U . Now let 0 < δ < ε be such that the neighborhood of η 1 , B δ (η 1 ), is contained in U and apply again Lemma 3.12 in Ω 1

ε to obtain w ∈ Af f piec ( Ω 1 ε ; R N ) and disjoint open sets Ω i ε ⊂ Ω 1 ε , i = 1, 2 such that w ∈ v + W 1,∞ 0 ( Ω 1 ε ; R N ), Dw(x) ∈ {ξ 1 , ξ 2 } ∪ B δ (η 1 ), a.e. x ∈ Ω 1 ε , Dw(x) = ξ i , a.e. x ∈ Ω i ε , i = 1, 2 ||w -v|| L ∞ ≤ ε 2 , meas(Ω i ε ) - λ i λ 1 + λ 2 meas( Ω 1 ε ) ≤ ε 2 , i = 1, 2. (3.2)
We then obtain the desired result taking Ω 1 ε and Ω 2 ε as above, Ω i ε = Ω i-1 ε for i = 3, ..., I and

u ε = v in Ω \ Ω 1 ε , w in Ω 1
ε . In fact we only need to verify that |meas(Ω i ε )λ i meas(Ω)| ≤ ε for i = 1, 2:

|meas(Ω i ε ) -λ i meas(Ω)| ≤ meas(Ω i ε ) -λi λ1+λ2 meas( Ω 1 ε ) + + λi λ1+λ2 meas( Ω 1 ε ) -λ i meas(Ω) ≤ ε 2 + λi λ1+λ2 meas( Ω 1 ε ) -(λ 1 + λ 2 )meas(Ω) = ε 2 + λi λ1+λ2 meas( Ω 1 ε ) -µ 1 meas(Ω) ≤ ε,
where we have used (3.2). 2

We can now prove Theorem 3.5.

Proof of Theorem 3.5. Let Ω be an open bounded subset of R n , ξ ∈ int K and let us denote by u ξ an affine map such that Du ξ = ξ. We want to construct a sequence u ε ∈ Af f piec (Ω; R N ) such that

u ε ∈ u ξ + W 1,∞ 0 (Ω; R N ), Du ε (x) ∈ E ∪ int K, a.e. x ∈ Ω, u ε * ⇀ u ξ in W 1,∞ (Ω; R N ), lim ε→0 + Ω dist(Du ε (x); E) dx = 0.
Fix ε > 0. From condition (ii), and since ξ ∈ int K, we have

ξ ∈ K δ for δ ≤ δ 0 . Choose 0 < δ ≤ δ 0 such that δ < ε and dist(η; E) ≤ ε, ∀ η ∈ E δ . (3.3)
This is possible from condition (i). We then apply condition (iii) to obtain

I = I(δ) ∈ N, (λ i , ξ i ) 1≤i≤I with λ i > 0, I i=1 λ i = 1, ξ i ∈ R N ×n satisfying (H I (int K)) and such that ξ = I i=1 λ i ξ i and I i=1 ξi / ∈E δ λ i < δ. (3.4)
By Lemma 3.11 we now get u ε ∈ Af f piec (Ω; R N ) and disjoint open sets Ω i ε ⊂ Ω such that, for ε sufficiently small,

u ε ∈ u ξ + W 1,∞ 0 (Ω; R N ), Du ε (x) ∈ int K, a.e. x ∈ Ω, Du ε (x) = ξ i , a.e. x ∈ Ω i ε , i = 1, ..., I, ||u ε -u ξ || L ∞ ≤ ε, |meas(Ω i ε ) -λ i meas(Ω)| ≤ ε I , i = 1, ..., I. (3.5)
Since K is bounded, up to a subsequence, we have u ε * ⇀ u ξ in W 1,∞ (Ω; R N ). We will finish the proof by verifying that lim

ε→0 + Ω dist(Du ε (x); E) dx = 0.
Since E and K are bounded there exists a positive constant c such that dist(η; E) ≤ c, ∀ η ∈ int K.

(3.6)

Then, using (3.3), (3.6), (3.5) and (3.4), we obtain the following estimates,

Ω dist(Du ε (x); E) dx = = I i=1 ξi∈E δ Ω i ε dist(ξ i ; E) dx + I i=1 ξi / ∈E δ Ω i ε dist(ξ i ; E) dx+ + Ω\(∪ I i=1 Ω i ε ) dist(Du ε (x); E) dx ≤ ε meas(Ω) + c I i=1 ξi / ∈E δ meas(Ω i ε ) + c meas Ω \ ∪ I i=1 Ω i ε ≤ ε meas(Ω) + c I i=1 ξi / ∈E δ ε I + λ i meas(Ω) + c ε ≤ ε meas(Ω) + c ε + c ε meas(Ω) + c ε.
This completes the proof. 2

As already mentioned, the characterization of Rco f E obtained in Section 2 entails a similar condition to condition (iii) of Theorem 3.5 under the approximation property assumption. This is stated in the next proposition. Proposition 3.13 Let E ⊂ R N ×n be a bounded set and for δ > 0 let E δ be compact sets such that Rco

f E δ ⊂ int Rco f E and (i) ∀ ε > 0 ∃ δ 0 > 0 : dist(η; E) ≤ ε, ∀ η ∈ E δ , δ ∈ (0, δ 0 ]; (ii) η ∈ int Rco f E ⇒ ∃ δ 0 > 0 : η ∈ Rco f E δ , ∀ δ ∈ (0, δ 0 ].
Then the following condition is satisfied:

∀ δ > 0 ∀ ξ ∈ Rco f E δ ∃ I ∈ N, ∃ (λ i , ξ i ) 1≤i≤I with λ i > 0, I i=1 λ i = 1, ξ i ∈ R N ×n , satisfying (H I (int Rco f E)) and ξ = I i=1 λ i ξ i , I i=1 ξi / ∈B δ (E) λ i < δ. Proof. Let δ > 0 and ξ ∈ Rco f E δ . Since Rco f E δ ⊂ int Rco f E, ξ ∈ int Rco f E so, using (ii), we conclude that ξ ∈ Rco f E µ for all µ ≤ µ 1 .
Thus, by the characterization of Rco f E µ stated in Theorem 2.9 with U = int Rco f E, for every µ ≤ µ 1 and for every ε > 0, there exist I ∈ N and (λ i , η i ) i=1,...,I with λ i > 0,

I i=1 λ i = 1 satisfying (H I (U )) and such that ξ = I i=1 λ i η i , I i=1 ηi / ∈Bε(Eµ) λ i < ε. (3.7)
On the other hand, by (i), for every γ > 0 there exists µ 2 > 0 such that

dist(η; E) ≤ γ, ∀ η ∈ E µ with µ ≤ µ 2 . (3.8)
Choosing in the previous conditions γ = δ 3 , µ 0 ≤ min{µ 1 , µ 2 }, ε = δ 3 we conclude, by (3.7), that there exist I ∈ N and (λ i , η i ) i=1,...,I with λ i > 0, I i=1 λ i = 1 satisfying (H I (U )) and such that

ξ = I i=1 λ i η i , I i=1 ηi / ∈B δ/3 (Eµ 0 ) λ i < δ 3 .
To obtain the desired condition, we observe that

I i=1 ηi / ∈B δ (E) λ i ≤ I i=1 ηi / ∈B 2δ/3 (Eµ 0 ) λ i < 2δ 3 < δ. Indeed, if η i / ∈ B δ (E) then dist(η i ; E) ≥ δ. From (3.8), dist(η; E) ≤ δ 3 , ∀ η ∈ E µ0 . This means that E µ0 ⊂ B δ/3 (E) and thus dist(η i ; E µ0 ) ≥ 2δ 3 . 2

Applications

We will now recall some properties of isotropic sets and investigate similar properties when a restriction on the sign of the determinant is considered. These results will be useful in the study of some differential inclusions related with this type of sets which we present in subsections 4.2 and 4.3. We start by giving the precise definition of isotropic set.

Definition 4.1 Let E be a subset of R n×n . We say E is isotropic if RES ⊆ E for every R, S in the orthogonal group O(n).

Isotropic sets can be easily described by means of the singular values of its matrices. Indeed, let 0 ≤ λ 1 (ξ) ≤ • • • ≤ λ n (ξ) denote the singular values of the matrix ξ, that is, the eigenvalues of the matrix ξξ t , then the isotropic sets E of R n×n are those which can be written in the form

E = {ξ ∈ R n×n : (λ 1 (ξ), • • • , λ n (ξ)) ∈ Λ E } , (4.1)
where

Λ E is a set contained in {(x 1 , • • • , x n ) ∈ R n : 0 ≤ x 1 ≤ • • • ≤ x n }.
This is a consequence of some properties of the singular values that we recall next.

The following decomposition holds (see [START_REF] Horn | Topics in matrix analysis[END_REF]): for every matrix ξ ∈ R n×n there exist R, S ∈ O(n) such that

ξ = R diag(λ 1 (ξ), • • • , λ n (ξ))S = R    λ 1 (ξ) . . . λ n (ξ)    S (4.2)
and, for every ξ ∈ R n×n , R, S ∈ O(n)

λ i (ξ) = λ i (RξS).
Moreover, one has

n i=1 λ i (ξ) = | det ξ| and n i=1 (λ i (ξ)) 2 = |ξ| 2 .
In particular, in the 2 × 2 case, λ 1 and λ 2 are given by

λ 1 (ξ) = 1 2 |ξ| 2 + 2| det ξ| -|ξ| 2 -2| det ξ| λ 2 (ξ) = 1 2 |ξ| 2 + 2| det ξ| + |ξ| 2 -2| det ξ| .
The In this section we will also be interested in sets of the form

E = ξ ∈ R n×n : (λ 1 (ξ), • • • , λ n (ξ)) ∈ Λ E , det ξ ≥ 0 , (4.3)
where, as before, Λ E is a set contained in

{(x 1 , • • • , x n ) ∈ R n : 0 ≤ x 1 ≤ • • • ≤ x n }.
We observe that these are not isotropic sets, but just a class of SO(n) invariant sets, where SO(n) denotes the special orthogonal group.

Theorem 4.2 If E ⊆ R n×n has the form (4.3) for some compact set Λ E , then Rco f E has the same form, with Λ Rco f E also compact.

Proof. Let E ⊆ R n×n be a set of the form (4.3) for some compact set Λ E . Then it is trivial to conclude that Rco f E is compact and Rco f E ⊆ {ξ ∈ R n×n : det ξ ≥ 0}, this follows from the fact that ξ →det ξ is rank one convex. We will show that

Rco f E = ξ ∈ R n×n : (λ 1 (ξ), • • • , λ n (ξ)) ∈ Λ Rco f E , det ξ ≥ 0 where Λ Rco f E = {x ∈ R n : x = (λ 1 (ξ), • • • , λ n (ξ)) for some ξ ∈ Rco f E} .
Notice that, in particular, Λ Rco f E is compact. To achieve the desired representation of Rco f E we only need to show that if ξ / ∈ Rco f E then for every R, S ∈ SO(n) one has RξS / ∈ Rco f E. Let ξ / ∈ Rco f E, then there exists a rank one convex function f : R n×n → R such that f | E ≤ 0 and f (ξ) > 0. Let R, S ∈ SO(n) and define f 1 (η) := f (R -1 ηS -1 ). Then f 1 is rank one convex and for all η ∈ E, f 1 (η) = f (R -1 ηS -1 ) ≤ 0, as R -1 ηS -1 ∈ E. However f 1 (RξS) = f (ξ) > 0 and so RξS doesn't belong to Rco f E. 4.1 Non-affine map with a finite number of gradients without rank one connections

In [START_REF] Kirchheim | Deformations with finitely many gradients and stability of quasiconvex hulls[END_REF], Kirchheim proved the existence of non-affine maps with a finite number of values for the gradient but without rank one connections between them (see also the result obtained by Kirchheim and Preiss, cf. [START_REF] Kirchheim | Rigidity and geometry of microstructures[END_REF]Corollary 4.40], where a non-affine map whose gradient takes five possible values not rank one connected was constructed). Kirchheim's result is the following.

Theorem 4.3 Let N, n ≥ 2, m ∈ N and Ω ⊂ R n be a bounded open set. Then there is a set E = {ξ 1 , ..., ξ m } ⊂ R N ×n such that rank(ξ i -ξ j ) = min{N, n}, if i = j
and there are ξ / ∈ E and u

∈ u ξ + W 1,∞ 0 (Ω; R N ) such that Du(x) ∈ E, a.e. x ∈ Ω,
where u ξ represents a map such that Du ξ = ξ.

This theorem was obtained thanks to an abstract result also due to Kirchheim (cf. [START_REF] Kirchheim | Deformations with finitely many gradients and stability of quasiconvex hulls[END_REF]Theorem 5]). What we want to show in this section is that the same result can also be achieved by the Baire categories method, cf. Theorem 3.2. Evidently, since, as described in the statement of the theorem, the elements of the set E are not rank one connected, the gradient of the affine boundary data ξ does not belong to Rco E = E. Therefore, to prove the relaxation property required to apply Theorem 3.2, one cannot use the usual approximation property (cf. Definition 3.3). However, as we will see, this difficulty can be overcome by means of Corollary 3.9. Indeed, the set E constructed by Kirchheim is such that the gradient of the boundary data ξ belongs to the interior of Rco f E (cf. Remark 3.10).

We recall in the following lemma the properties of the set E constructed by Kirchheim. For the construction of the set we refer again to [START_REF] Kirchheim | Deformations with finitely many gradients and stability of quasiconvex hulls[END_REF].

., ξ m } ⊂ R N ×n , m ∈ N, such that rank(ξ i -ξ j ) = min{N, n}, if i = j
and dist(ξ; B 1 2 (0)) > 0, for every ξ ∈ E. Moreover, for every ξ ∈ E there exists

M ξ ⊂ R N ×n such that i) M ξ ⊂ ξ + {µ ∈ R N ×n : rank µ = 1} ii) M ξ ⊂ B 1 2 (0), #M ξ < 4N n iii) ∂B 1 2 (0) ⊂ ξ∈E int(co({ξ} ∪ M ξ )).
We now prove Theorem 4.3, for the set E considered in the previous lemma, using the Baire categories method. This proof was already obtained by Ribeiro in [START_REF] Ribeiro | Inclusions différentielles et problèmes variationnels[END_REF] but we include it here for the sake of completeness.

Proof of Theorem 4.3. We consider a set E with the properties of the above lemma and, with the same notations of the lemma, we define

K = B 1 2 (0) ∪ ξ∈E int(co({ξ} ∪ M ξ )).
Notice that K is an open and bounded set. Since K \ E is non empty (for instance, it contains 0) let ξ ∈ K \ E. We will show that K has the relaxation property with respect to E by applying Corollary 3.9. Once this is proved, we conclude by Theorem 3.2 that there exists

u ∈ u ξ + W 1,∞ 0 (Ω; R N ) such that Du ∈ E with ξ / ∈ E.
We only need to ensure condition (H) of Corollary 3.9. Let δ > 0 and

η ∈ K \ B δ (E). If η ∈ B 1 2 (0)
, by definition of K and condition iii) of Lemma 4.4 one can easily reduce to the case η ∈ int(co({ξ} ∪ M ξ )) \ B 1 2 (0), with ξ ∈ E, moving along any rank one direction.

Consider now the case where η ∈ int(co({ξ} ∪ M ξ )) \ B 1 2 (0), for some ξ ∈ E. In this case we can write

η = k j=1 λ j µ j +   1 - k j=1 λ j   ξ,
for some λ j ∈ (0, 1) such that k j=1 λ j < 1 and for some µ j ∈ M ξ . By condition ii) of Lemma 4.4 we have k < 4N n.

Let j * ∈ {1, ..., k} be such that λ j * |ξµ j * | = max 1≤j≤k λ j |ξµ j | and consider the rank one direction ξµ j * . We will show that it is possible to find c > 0, independent of η, such that

η 1 = c(ξ-µ j * ) satisfies [η-η 1 , η+η 1 ] ⊂ int K.
In particular, since dist(ξ; B 1 2 (0)) > 0, for every ξ ∈ E, and µ j ∈ B 1 2 (0) by condition ii) of Lemma 4.4, it follows that, for some C > 0 independent of η and ξ,

|η 1 | ≥ C. (4.4)
To find the constant c we proceed in the following way. We notice that, for

|t| < min    λ j * , 1 - k j=1 λ j    , η+t(ξ-µ j * ) = k j=1 j =j * λ j µ j +(λ j * -t)µ j * +   1 - k j=1 λ j + t   ξ ∈ int(co({ξ}∪M ξ )).
Thus we only need to show that min

   λ j * , 1 - k j=1 λ j    ≥ c > 0.
The estimate for λ j * follows from

δ < |η -ξ| ≤ k j=1 λ j |µ j -ξ| ≤ 4N nλ j * |µ j * -ξ| ≤ 4N nλ j * max ξ∈E µ∈M ξ |µ -ξ|.
On the other hand, since

1 2 ≤ |η| ≤ k j=1 λ j |µ j | +   1 - k j=1 λ j   |ξ| ≤ ≤   max ξ∈E µ∈M ξ |µ|   k j=1 λ j +   1 - k j=1 λ j   max ξ∈E |ξ|, one gets 1 - k j=1 λ j ≥ 1 2 -max ξ∈E µ∈M ξ |µ| max ξ∈E |ξ| -max ξ∈E µ∈M ξ |µ| > 0 as wished.
We argue that repeating the same reasoning with the matrix η + η 1 and so on, after i iterations of this procedure, we obtain a sequence of rank one matrices η 1 , ...,

η i satisfying [η + η 1 + • • • + η j-1 -η j , η + η 1 + • • • + η j-1 + η j ] ⊆ int K, j = 1, • • • , i and η + η 1 + ... + η i ∈ B δ (E) where i ≤ L(δ, E, K) is independent of η.
Indeed, without loss of generality, assume that |η

+ η 1 | ≥ |η -η 1 |. Then it follows that |η + η 1 | ≥ C since, by (4.4), 2 |η + η 1 | 2 ≥ |η + η 1 | 2 + |η -η 1 | 2 = 2 |η| 2 + 2 |η 1 | 2 ≥ 2C 2 . If η + η 1 / ∈ B δ (E) we obtain, as before, η 2 such that |η 2 | ≥ C and [η + η 1 - η 2 , η +η 1 +η 2 ] ⊂ int K = K. Again, assuming that |η + η 1 + η 2 | ≥ |η + η 1 -η 2 |, one has 2 |η + η 1 + η 2 | 2 ≥ |η + η 1 + η 2 | 2 + |η + η 1 -η 2 | 2 = 2 |η + η 1 | 2 + 2 |η 2 | 2 ≥ 4C 2 .
After i iterations of this procedure one gets η + η 1 + ...

+ η i ∈ int K = K with |η + η 1 + ... + η i | ≥ √ iC.
Thus, |η + η 1 + ... + η i | → +∞, as i → +∞, contradicting the fact that K is bounded. Therefore, for some i bounded by a constant L = L(δ, E, K), we must have η + η 1 + ...

+ η i ∈ K ∩ B δ (E).
This concludes the proof of condition (H) of Corollary 3.9 and thus the proof. 2

Isotropic differential inclusion

In this section we discuss the differential inclusion problem

Du(x) ∈ E, a.e. x ∈ Ω, u(x) = ϕ(x), x ∈ ∂Ω, (4.5)
where Ω is an open bounded subset of R n and E is a compact subset of R n×n which is isotropic, that is to say, invariant under orthogonal transformations. We observe that a result due to Dacorogna and Marcellini [9, Theorem 7.28] provides a sufficient condition for existence of solutions to this problem. Indeed, denoting by

λ 1 (ξ) ≤ λ 2 (ξ) ≤ • • • ≤ λ n (ξ) the singular values of ξ ∈ R n×n , if there exists η ∈ E with λ i (η) = γ i > 0 and ϕ ∈ C 1 piec (Ω; R n ) is such that Dϕ ∈ E ∪ ξ ∈ R n×n : n i=τ λ i (ξ) < n i=τ γ i , τ = 1, ..., n then (4.5) has W 1,∞ (Ω; R n ) solutions.
In the 2 dimensional case (n = 2), a less restrictive condition can be obtained, although it is more difficult to check in concrete examples. This was studied by Croce [START_REF] Croce | A differential inclusion: the case of an isotropic set[END_REF] (see also [START_REF] Croce | Sur quelques inclusions différentielles de premier et de deuxième ordre[END_REF]), using the Baire categories method that we discussed in Section 3, and by Barroso, Croce and Ribeiro [START_REF] Barroso | Existence for a nonlinear problem involving isotropic deformations[END_REF] using the convex integration method due to Müller and Šverák [START_REF] Müller | Attainment results for the two-well problem by convex integration Geometric analysis and the calculus of variations[END_REF][START_REF] Müller | Convex integration for Lipschitz mappings and counterexamples to regularity[END_REF]. With both methods, the result obtained was the following.

Theorem 4.5 Let E := {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ Λ E }, where Λ E ⊂ {(x, y) ∈ R 2 : 0 < x ≤ y} is a compact set and let K = ξ ∈ R 2×2 : f θ (λ 1 (ξ), λ 2 (ξ)) < max (a,b)∈ΛE f θ (a, b), ∀ θ ∈ [0, max (a,b)∈ΛE b] ,
where f θ (x, y)

:= xy + θ(y -x). Then, if Ω ⊂ R 2 is a bounded open set and if ϕ ∈ C 1 piec (Ω, R 2 ) is such that Dϕ ∈ E ∪ K a.e. in Ω, there exists a map u ∈ ϕ + W 1,∞ 0 (Ω, R 2 ) such that Du ∈ E a.e. in Ω.
We notice that it turns out that K = int Rco E = int Rco f E. To achieve the previous theorem two fundamental results due to Cardaliaguet and Tahraoui [START_REF] Cardaliaguet | Equivalence between rank one convexity and polyconvexity for isotropic sets of R 2×2 (Part I)[END_REF] were used. On one hand, they characterized the polyconvex hull of any set E as in the theorem; based on their description, Croce [START_REF] Croce | A differential inclusion: the case of an isotropic set[END_REF] then showed that

Pco E = ξ ∈ R 2×2 : f θ (λ 1 (ξ), λ 2 (ξ)) ≤ max (a,b)∈ΛE f θ (a, b), ∀ θ ∈ [0, max (a,b)∈ΛE b] .
On the other hand, Cardaliaguet and Tahraoui [START_REF] Cardaliaguet | Equivalence between rank one convexity and polyconvexity for isotropic sets of R 2×2 (Part I)[END_REF] showed that, in dimension 2, compact isotropic sets which are rank one convex are also polyconvex (see also [START_REF] Conti | Polyconvexity equals rank-one convexity for connected isotropic sets in M 2×2[END_REF]). Since the hull Rco f E of a compact isotropic set E is also compact and isotropic (cf. Theorem 4.2) and since it is also rank one convex, one immediately obtains that Rco f E = Pco E and thus a characterization for Rco f E. This was fundamental to study the differential inclusion by means of the convex integration method. Indeed, to prove the required in-approximation it is necessary to know the hull Rco f E. In the prior work of Croce [START_REF] Croce | A differential inclusion: the case of an isotropic set[END_REF], where the Baire categories method was used via the approximation property, the appropriate hull to consider was Rco E. Contrary to the case of Rco f E, a characterization of Rco E does not follow immediately from Cardaliaguet and Tahraoui's results since, in general, Rco E may not be compact. For this reason, in [START_REF] Croce | A differential inclusion: the case of an isotropic set[END_REF], Rco E had to be computed (and the conclusion was that it coincides with Rco f E). However, thanks to the theory presented in Section 3, the results of Cardaliaguet and Tahraoui are sufficient to obtain Theorem 4.5 using the Baire categories method. We proceed with a brief sketch of this proof which is essentially the one given in [START_REF] Croce | A differential inclusion: the case of an isotropic set[END_REF], but with no need of computing Rco E.

Proof of Theorem 4.5. We recall that by the results in [START_REF] Cardaliaguet | Equivalence between rank one convexity and polyconvexity for isotropic sets of R 2×2 (Part I)[END_REF] and [START_REF] Croce | A differential inclusion: the case of an isotropic set[END_REF],

Rco f E = ξ ∈ R 2×2 : f θ (λ 1 (ξ), λ 2 (ξ)) ≤ max (a,b)∈ΛE f θ (a, b), ∀ θ ∈ [0, max (a,b)∈ΛE b] and int Rco f E = ξ ∈ R 2×2 :f θ (λ 1 (ξ), λ 2 (ξ)) < max (a,b)∈ΛE f θ (a, b), ∀θ ∈ [0, max (a,b)∈ΛE b] .
Thus, by Corollary 3.7 and Theorem 3.2, it is enough to construct compact sets E δ such that Rco f E δ ⊂ int Rco f E and satisfying conditions (i) and (ii) of Theorem 3.5 with K δ = Rco f E δ . In fact, this is the case if we consider

E δ = (a,b)∈ΛE ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) = (a -δ, b -δ) , for 0 ≤ δ ≤ min (a,b)∈ΛE a 2
. We refer to [START_REF] Croce | A differential inclusion: the case of an isotropic set[END_REF] for the details of the proof. 2

Differential inclusions for some SO(n) invariant sets

We consider in this section the differential inclusion problem

Du(x) ∈ E, a.e. x ∈ Ω, u(x) = ϕ(x), x ∈ ∂Ω, (4.6)
in the case E has the form

E = ξ ∈ R n×n : (λ 1 (ξ), • • • , λ n (ξ)) ∈ Λ E , det ξ > 0 , with Λ E ⊆ {(x 1 , • • • , x n ) ∈ R n : 0 < x 1 ≤ • • • ≤ x n }.
As already observed, these are not isotropic sets, but just a class of SO(n) invariant sets.

For n = 2, Cardaliaguet and Tahraoui [START_REF] Cardaliaguet | Equivalence between rank one convexity and polyconvexity for isotropic sets of R 2×2 (Part II)[END_REF] defined the set R(Λ E ) for any compact set

Λ E ⊂ {(x 1 , x 2 ) ∈ R 2 : 0 ≤ x 1 ≤ x 2 } as the smallest compact subset of {(x 1 , x 2 ) ∈ R 2 : 0 ≤ x 1 ≤ x 2 } containing Λ E such that {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ R(Λ E ), det ξ ≥ 0}
is rank one convex. This hull can be used to describe Rco f E (see Lemma 4.8). The representation of this envelop is quite complicated and leads to some difficulty in dealing with it in order to show existence of solutions to problem (4.6). In Theorem 4.10 we consider a particular set E composed by matrices with two possible singular values and, using Cardaliaguet and Tahraoui's results, we give a sufficient condition for existence, relating the gradient of the boundary data and the hull Rco f E.

For n > 2, a representation of Rco f E is not available. Of course, if one wants to ensure existence of solutions to (4.6), one may not need to know the entire hull. Moreover, we notice that in the applications it is more convenient to have simpler conditions to check than those describing the hull Rco f E obtained by Cardaliaguet and Tahraoui [START_REF] Cardaliaguet | Equivalence between rank one convexity and polyconvexity for isotropic sets of R 2×2 (Part II)[END_REF] for the 2 dimensional case. In this sense, in Theorems 4.11 and 4.12 we will establish sufficient conditions for existence of solutions to problem (4.6) for certain sets E in dimension 2 and 3. Analogous results could be obtained in higher dimensions however, due to the heavy notation already present in the 3 dimensional case, we have only considered these two settings.

Set of singular values consisting of two points

In this section we are going to consider the case where

E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ Λ E , det ξ > 0}
with Λ E = {(a 1 , a 2 ), (b 1 , b 2 )} and 0 < a 1 < b 1 < a 2 < b 2 . We start by studying the set Rco f E. To this effect we will use the following characterization of R(Λ E ) obtained in [3, Proposition 8.6, Theorem 7.1 and Definition 1.1].

Theorem 4.6 Let Λ be a compact subset of {(x 1 , x 2 ) ∈ R 2 : 0 ≤ x 1 ≤ x 2 } such that R(Λ) is connected. Then R(Λ) = {(x 1 , x 2 ) ∈ R 2 :0 ≤ x 1 ≤ x 2 , x 1 ≥ α, σ 3 (x 1 ) ≤ x 2 ≤ inf{σ 1 (x 1 ), σ 2 (x 1 )}} where α = inf (x1,x2)∈Λ x 1 , σ 1 (x 1 ) = inf (θ,γ)∈Σ1 f 1 θ,γ (x 1 ), f 1 θ,γ (x 1 ) = θ + γ-θ 2 θ-x1 , x 1 < θ, +∞, otherwise, σ 2 (x 1 ) = inf (θ,γ)∈Σ2 f 2 θ,γ (x 1 ), f 2 θ,γ (x 1 ) = θ + γ -θ 2 θ + x 1 , σ 3 (x 1 ) = sup (θ,γ)∈Σ3 f 3 θ,γ (x 1 ), f 3 θ,γ (x 1 ) = -θ + γ-θ 2 x1-θ , x 1 > θ , +∞, otherwise , Σ 0 = {(θ, γ) ∈ R 2 : θ ≥ 0, γ ≥ θ 2 }, Σ 1 = {(θ, γ) ∈ Σ 0 : x 2 ≤ f 1 θ,γ (x 1 ), ∀ (x 1 , x 2 ) ∈ Λ}, Σ 2 = {(θ, γ) ∈ Σ 0 : x 2 ≤ f 2 θ,γ (x 1 ), ∀ (x 1 , x 2 ) ∈ Λ}, Σ 3 = {(θ, γ) ∈ Σ 0 : x 2 ≥ f 3 θ,γ (x 1 ), ∀ (x 1 , x 2 ) ∈ Λ}. Moreover, there exists a convex function h : R 2×2 × R → R such that ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ R(Λ), det ξ ≥ 0 = ξ ∈ R 2×2 : h(ξ, det ξ) ≤ 0, det ξ ≥ 0 .
The following sufficient condition for R(Λ) to be connected was also proven in [START_REF] Cardaliaguet | Equivalence between rank one convexity and polyconvexity for isotropic sets of R 2×2 (Part II)[END_REF]Proposition 8.4].

Proposition 4.7 Let Λ be a compact subset of {(x 1 , x 2 ) ∈ R 2 : 0 ≤ x 1 ≤ x 2 }
and assume that Λ satisfies the following property: if there exist C 1 and C 2 , compact subsets of Λ, such that

C 1 ∩ C 2 = ∅, C 1 ∪ C 2 = Λ and sup (x1,x2)∈C1 x 2 < inf (x1,x2)∈C2
x 1 , then either

C 1 = ∅ or C 2 = ∅. Then R(Λ) is connected. If E = ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ Λ E , det ξ ≥ 0 , where Λ E is a compact subset of {(x 1 , x 2 ) ∈ R 2 : 0 ≤ x 1 ≤ x 2 }, then R(Λ E )
describes the hull Rco f E, as we prove in the following lemma.

Lemma 4.8 Let E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ Λ E , det ξ ≥ 0} where Λ E is a compact subset of {(x 1 , x 2 ) ∈ R 2 : 0 ≤ x 1 ≤ x 2 } such that R(Λ E ) is connected. Then Rco f E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ R(Λ E ), det ξ ≥ 0}.
Proof. We set

E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ R(Λ E ), det ξ ≥ 0}.
Since R(Λ E ) is connected, by Theorem 4.6 there exists a convex function h :

R 2×2 × R → R such that E = {ξ ∈ R 2×2 : h(ξ, det ξ) ≤ 0, det ξ ≥ 0}. Note that h(ξ, det ξ) ≤ 0 for every ξ ∈ E, since E ⊂ E. According to Definition 2.5 of Pco f E, one has Pco f E ⊆ E. This implies that Rco f E ⊆ E.
On the other hand, Rco f E is a compact and rank one convex set. By Theorem 4.2,

Rco f E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ Λ, det ξ ≥ 0} for some compact subset Λ of {(x 1 , x 2 ) ∈ R 2 : 0 ≤ x 1 ≤ x 2 }. By definition of R(Λ E ), Λ ⊇ R(Λ E ) and thus Rco f E ⊇ E.
2 Using the two previous results we can show the following formula for Rco f E, for the set E considered in this section.

Proposition 4.9 Let E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ Λ E , det ξ > 0}, where Λ E = {(a 1 , a 2 ), (b 1 , b 2 )}, 0 < a 1 < b 1 < a 2 < b 2 , and define θ = -a 1 a 2 + b 1 b 2 b 1 + b 2 -a 1 -a 2 . Then Rco f E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ R(Λ E ), det ξ > 0} (4.7) where R(Λ E ) is the set of points (x 1 , x 2 ) ∈ R 2 such that                      a 1 ≤ x 1 ≤ x 2 ≤ b 2 , x 2 ≥ a 1 a 2 x 1 , x 2 ≤ -a 1 a 2 + θ(a 1 + a 2 ) -θx 1 -x 1 + θ , if a 1 ≤ x 1 ≤ b 1 x 2 ≤ b 1 b 2 x 1 . (4.8) Moreover, int Rco f E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ rel intR(Λ E ), det ξ > 0},
where rel intR(Λ E ) is the relative interior of R(Λ E ) with respect to the set

{(x 1 , x 2 ) ∈ R 2 : 0 < x 1 ≤ x 2 } and is the set of points (x 1 , x 2 ) ∈ R 2 such that                      a 1 < x 1 ≤ x 2 < b 2 , x 2 > a 1 a 2 x 1 , x 2 < -a 1 a 2 + θ(a 1 + a 2 ) -θx 1 -x 1 + θ , if a 1 < x 1 < b 1 x 2 < b 1 b 2 x 1 .
Proof. Notice that, since a 1 > 0 and λ 1 (ξ)λ 2 (ξ) = | det ξ|, our set E satisfies the hypotheses of Lemma 4.8 since, by Proposition 4.7, the set R(Λ E ) is connected. Thus (4.7) holds and to establish (4.8) we will write the inequalities σ 3 (x 1 ) ≤ x 2 ≤ inf{σ 1 (x 1 ), σ 2 (x 1 )}, given by Theorem 4.6, for x 1 ≥ a 1 , in a more explicit way. Let us start by studying x 2 ≥ σ 3 (x 1 ). It is easy to see that

Σ 3 = (θ, γ) ∈ R 2 : 0 ≤ θ < a 1 , θ 2 ≤ γ ≤ a 1 a 2 -θ(a 2 -a 1 ) , as 0 < a 1 < b 1 < a 2 < b 2 . Therefore x 2 ≥ σ 3 (x 1 ) is equivalent to x 1 x 2 -θ(x 2 -x 1 ) ≥ a 1 a 2 -θ(a 2 -a 1 ) , θ ∈ [0, a 1 ) since x 1 > θ, that is, x 2 ≥ sup θ∈[0,a1) a 1 a 2 -θ(a 2 -a 1 ) -θx 1 x 1 -θ = a 1 a 2 x 1 . (4.9)
In the same way, to study x 2 ≤ σ 2 (x 1 ), we remark that

Σ 2 = (θ, γ) ∈ R 2 : θ ≥ 0, γ ≥ max{θ 2 , b 1 b 2 + θ(b 2 -b 1 )} . Therefore x 2 ≤ σ 2 (x 1 ) is equivalent to x 1 x 2 + θ(x 2 -x 1 ) ≤ max{θ 2 , b 1 b 2 + θ(b 2 -b 1 )} , θ ≥ 0 , that is, x 2 ≤ min b 1 b 2 x 1 , b 2 . (4.10)
We now analize the inequality x 2 ≤ σ 1 (x 1 ). It is easy to see that

Σ 1 = {(θ, γ) ∈ R 2 : θ ≥ 0, γ ≥ max{θ 2 , -a 1 a 2 + θ(a 1 + a 2 ), -b 1 b 2 + θ(b 1 + b 2 )}} . Therefore x 2 ≤ σ 1 (x 1 ) is equivalent to x 2 ≤ γ -θx 1 θ -x 1 , ∀ (θ, γ) ∈ Σ 1 : θ > x 1 , that is, -x 1 x 2 +θ(x 1 +x 2 ) ≤ max{θ 2 , -a 1 a 2 +θ(a 1 +a 2 ), -b 1 b 2 +θ(b 1 +b 2 )}, ∀ θ > x 1 .
By (4.10), x 2 ≤ b 2 and so

-x 1 x 2 +θ(x 1 +x 2 ) ≤ max{θ 2 , -a 1 a 2 +θ(a 1 +a 2 ), -b 1 b 2 +θ(b 1 +b 2 )}, θ ∈ (x 1 , b 2 ]. Since θ > x 1 ≥ a 1 and 0 < a 1 < b 1 < a 2 < b 2 , one has x 2 ≤ inf θ∈(x1,b2] max{-a 1 a 2 + θ(a 1 + a 2 ), -b 1 b 2 + θ(b 1 + b 2 )} -θx 1 -x 1 + θ . (4.11)
We observe that b 1 < θ < a 2 (4.12) and max{-a 1 a 2 +θ(a

1 +a 2 ), -b 1 b 2 +θ(b 1 +b 2 )} = -a 1 a 2 + θ(a 1 + a 2 ), a 1 ≤ θ ≤ θ -b 1 b 2 + θ(b 1 + b 2 ), θ ≤ θ ≤ b 2 .
To study (4.11) we distinguish the cases x 1 ≥ θ and x 1 < θ. In the first case, (4.11) gives

x 2 ≤ inf θ∈(x1,b2] -b 1 b 2 + θ(b 1 + b 2 ) -θx 1 -x 1 + θ .
Notice that the sign of the derivative of the function

g(θ) := -b 1 b 2 + θ(b 1 + b 2 ) -θx 1 -x 1 + θ
does not depend on θ. Therefore we have

x 2 ≤ min g(b 2 ), lim θ→x + 1 g(θ) = g(b 2 ) = b 2 , if x 1 ≥ θ . (4.13)
In the second case, (4.11) yields

x 2 ≤ min inf θ∈(x1,θ] -a 1 a 2 + θ(a 1 + a 2 ) -θx 1 -x 1 + θ , min θ∈[θ,b2] -b 1 b 2 + θ(b 1 + b 2 ) -θx 1 -x 1 + θ .
As above, the sign of the derivatives of g(θ) and

f (θ) := -a 1 a 2 + θ(a 1 + a 2 ) -θx 1 -x 1 + θ
does not depend on θ, so we obtain

x 2 ≤ min f (θ), lim θ→x + 1 f (θ), g(θ), g(b 2 ) = min f (θ), b 2 ,
that is,

x 2 ≤ min -a 1 a 2 + θ(a 1 + a 2 ) -θx 1 -x 1 + θ , b 2 , if x 1 < θ . Therefore if x 1 < θ x 2 ≤    -a 1 a 2 + θ(a 1 + a 2 ) -θx 1 -x 1 + θ , x 1 ≤ b 1 b 2 , b 1 < x 1 < θ . (4.14)
In conclusion, from (4.9), (4.10), (4.13) and (4.14) we get

a 1 a 2 x 1 ≤ x 2 ≤ min b 2 , b 1 b 2 x 1 , if x 1 > b 1
and

a 1 a 2 x 1 ≤ x 2 ≤ min b 2 , b 1 b 2 x 1 , -a 1 a 2 + θ(a 1 + a 2 ) -θx 1 -x 1 + θ , if x 1 ≤ b 1 .
By (4.12) and the fact that Proof. We will prove the result using Theorem 3.2 and Corollary 3.7. Let δ > 0 be sufficiently small such that a 1 + δ < b 1 < a 2 < b 2δ. We define

x 1 → -a 1 a 2 + θ(a 1 + a 2 ) -θx 1 -x 1 + θ passes through (a 1 , a
E δ = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ Λ E δ , det ξ > 0},
with Λ E δ = {(a 1 + δ, a 2 ), (b 1 , b 2δ)}. Observe that E δ is a compact set. By Proposition 4.9, Rco f E δ is given by the matrices ξ ∈ R 2×2 with positive determinant such that (x 1 , x 2 ) = (λ 1 (ξ), λ 2 (ξ)) satisfies

a 1 + δ ≤ x 1 ≤ x 2 ≤ b 2 -δ (4.15)
x 2 ≥ (a 1 + δ)a 2 x 1 (4.16)

x 2 ≤ -(a 1 + δ)a 2 + θ δ (a 1 + δ + a 2 ) -θ δ x 1 -x 1 + θ δ , if a 1 + δ ≤ x 1 ≤ b 1 (4.17) x 2 ≤ b 1 (b 2 -δ) x 1 (4.18)
where

θ δ = -(a 1 + δ)a 2 + b 1 (b 2 -δ) b 1 + b 2 -a 1 -a 2 -2δ .
We are going to verify the hypotheses of Corollary 3.7. We start by proving that Rco f E δ ⊂ int Rco f E. Let ξ ∈ Rco f E δ and denote (λ 1 (ξ), λ 2 (ξ)) by (x 1 , x 2 ). Since (x 1 , x 2 ) satisfies inequalities (4.15), (4.16) and (4.18), it is clear that . Since x 1 ≥ a 1 + δ, it suffices to show that if

a 1 < x 1 ≤ x 2 < b 2 ,
x 1 ∈ [a 1 + δ, b 1 ), then -(a 1 + δ)a 2 + θ δ (a 1 + δ + a 2 ) -θ δ x 1 -x 1 + θ δ < -a 1 a 2 + θ(a 1 + a 2 ) -θx 1 -x 1 + θ .
As b 1 < θ < a 2 and b 1 < θ δ < a 2 , the above inequality is equivalent to

(θ δ -θ)(x 1 -a 1 )(x 1 -a 2 ) < δ(-x 1 + θ)(a 2 -θ δ ) , (4.20)
for x 1 ∈ [a 1 + δ, b 1 ). This inequality holds whenever θ δθ > 0, since the left hand side of (4.20) is negative and the right hand side is positive. To show it also holds in the case θ δθ < 0, we notice that the graph of x 1 → (θ δθ)(x 1a 1 )(x 1a 2 ) is a concave parabola passing through (a 1 , 0), (a 2 , 0), whereas the graph of x 1 → δ(-x 1 + θ)(a 2θ δ ) is a straight line with negative

E δ := ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ Λ E δ , det ξ > 0 , K δ := ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ Λ K δ , det ξ > 0 ,
where

Λ E δ :=   (α1,α2)∈rel int Λ E α 1 + c(δ, α 1 , α 2 ), α 1 α 2 α 1 + c(δ, α 1 , α 2 )   (x 1 , x 2 ) ∈ R 2 : 0 < x 1 ≤ x 2 , a 1 a 2 + δ ≤ x 1 x 2 ≤ b 1 b 2 -δ , Λ K δ := (α1,α2)∈Λ E δ {(x 1 , x 2 ) ∈ R 2 : x 1 x 2 = α 1 α 2 , 0 < x 1 ≤ x 2 ≤ α 2 }.
We proceed with the proof of conditions (i), (ii) and (iii) of Theorem 3.5. To prove condition (i), by the matrix decomposition (4.2), it is enough to show that, for any given ε > 0, there exists δ 0 > 0 such that dist((x 1 , x 2 ); Λ E ) ≤ ε, ∀ (x 1 , x 2 ) ∈ Λ E δ , δ ∈ (0, δ 0 ].

The elements of Λ E δ are of the form α 1 + c(δ, α 1 , α 2 ), Defining (β 1 , β 2 ) = α 1 + c(δ, α 1 , α 2 ), α1α2 α1+c(δ,α1,α2) , we observe that (β 1 , β 2 ) ∈ Λ E δ . Finally, λ 2 (η) < β 2 is equivalent to c > c(δ, α 1 , α 2 ) and this is true for sufficiently small δ, since lim δ→0 c(δ, α 1 , α 2 ) = 0 and c > 0.

It remains to prove condition (iii). We notice that, using (4.21), we conclude that any ξ ∈ K δ belongs to Rco ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) = (α 1 , α 2 ), det ξ > 0 , for some (α 1 , α 2 ) ∈ Λ E δ . Since Λ E δ ⊂ Λ K , this hull is a subset of K = int K and condition (iii) follows from (2.2) (see also Remark 2.10 -3)).

2

We consider next the 3 dimensional version of the previous result.

Lemma 3 .

 3 [START_REF] Dacorogna | On some definitions and properties of generalized convex sets arising in the calculus of variations[END_REF] Let I > 1 be an integer and let U ⊂ R N ×n be an open set. For 1 ≤ i ≤ I, let ξ i ∈ R N ×n and λ i > 0 be such thatI i=1 λ i = 1 and (λ i , ξ i ) 1≤i≤Isatisfy (H I (U )). Denote by ξ the sum I i=1 λ i ξ i and let u ξ be an affine map such that Du ξ = ξ. Then, for any given ε > 0 and any bounded open set Ω ⊂ R n , there exist u ε ∈ Af f piec (Ω; R N ) and disjoint open sets

Lemma 3 . 12 (

 312 Approximation lemma) Let Ω ⊂ R n be a bounded open set. Let A, B ∈ R N ×n be such that A -B = a ⊗ b, with a ∈ R N and b ∈ R n . Let b 3 , . . . , b k ∈ R n , k ≥ n, be such that 0 ∈ int co{b, -b, b 3 , . . . , b k } and, for t ∈ [0, 1], let ϕ be an affine map such that Dϕ(x) = ξ = tA + (1t)B, x ∈ Ω (i.e. A = ξ + (1t) a ⊗ b and B = ξta ⊗ b). Then, for every ε > 0, there exists u ∈ Af f piec (Ω; R N ) and there exist disjoint open sets Ω

2

 2

Lemma 4 . 4

 44 Let N, n ≥ 2 and denote by B 1 2 (0) the open ball of R N ×n centered at 0 and with radius 1 2 . Then there exists a set E = {ξ 1 , ..

  2 ) and (b 1 , b 2 ), we get(4.8). The formula of int Rco f E is easy to obtain from the above representation. 2We are now in position to prove an existence result for problem (4.6).Theorem 4.10Let E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ Λ E , det ξ > 0} where Λ E = {(a 1 , a 2 ), (b 1 , b 2 )} and 0 < a 1 < b 1 < a 2 < b 2 . Let Ω ⊂ R 2 be a bounded open set and let ϕ ∈ C 1 piec (Ω, R 2 ) be such that Dϕ ∈ E ∪ int Rco f E a.e.in Ω. Then there exists a map u ∈ ϕ + W 1,∞ 0 (Ω, R 2 ) such that Du(x) ∈ E for a.e. x in Ω.

1 , α 2 )α 1 +/a 2 1 ,

 1211 ∈ rel int Λ E ⊂ Λ E . Thus, choosing δ 0 = c(δ, α 1 , α 2 ), α 1 α 2 α 1 + c(δ, α 1 , α 2 ) -(α 1 , α 2 ) ≤ δ 1 + b 2 2where we have used the fact that any (α 1 , α 2 ) ∈ rel int Λ E satisfies α 1 ≥ a 1 and α 2 ≤ b 2 (this follows from the hypotheses on a 1 , a 2 , b 1 , b 2 ). Now we prove condition (ii). Let η ∈ int K = K. It suffices to show that, for sufficiently small δ, λ 1 (η)λ 2 (η) = β 1 β 2 , for some (β 1 , β 2 ) ∈ Λ E δ with λ 2 (η) ≤ β 2 . By (4.22), (λ 1 (η), λ 2 (η)) = α 1 + c, α1α2 α1+c for some (α 1 , α 2 ) ∈ rel int Λ E and 0 < c ≤ √ α 1 α 2α 1 .The monotonicity of the function ψ introduced above, yields α 1 α 2 ∈ (a 1 a 2 , b 1 b 2 ). Thus, for small δ, one has α 1 α 2 ∈ [a 1 a 2 + δ, b 1 b 2δ].

c(δ, α) = min δ, α 1 α 2 α 3 α 1 + δ - α 1 α 2 α 1

 1 + δ , α = (α 1 , α 2 , α 3 ) andΛ K δ := (α1,α2,α3)∈Λ E δ (x 1 , x 2 , x 3 ) ∈ R 3 : x 1 x 2 x 3 = α 1 α 2 α 3 , 0 < x 1 ≤ x 2 ≤ x 3 ≤ α 3 , x 2 x 3 ≤ α 2 α 3 } .

2

 2

  x 2 > a 1 a 2 x 1 and x 2 < b 1 b 2 x 1 . It remains to show that x 2 < -a 1 a 2 + θ(a 1 + a 2 )θx 1 -x 1 + θ , if a 1 < x 1 < b 1 (4.19) where θ = -a 1 a 2 + b 1 b 2 b 1 + b 2a 1a 2

Acknowledgements

The research of Ana Cristina Barroso was partially supported by Fundação para a Ciência e Tecnologia, through Financiamento Base 2010-ISFL/1/209, PTDC/MAT/109973/2009 and UTA-CMU/MAT/0005/2009. The research of Ana Margarida Ribeiro was partially supported by Fundação para a Ciência e Tecnologia, through Financiamento Base 2010-ISFL/1/297, PTDC/MAT/-109973/2009 and UTA-CMU/MAT/0005/2009. This work was undertaken during a visit of Ana Margarida Ribeiro to the Mathematics Department of the École Polytechnique Fédérale de Lausanne whose kind hospitality and support is gratefully acknowledged.

slope passing through (θ, 0). Therefore it is sufficient to prove (4.20) for x 1 = b 1 , that is,

It is not difficult to see that the above inequality holds if and only if δ < b 1a 1 which is satisfied by the hypotheses on δ.

The other conditions of Corollary 3.7 are easy to check. Indeed, any η ∈ E δ can be written as R diag(a 1 + δ, a 2 ) S or R diag(b 1 , b 2δ) S, for some R, S ∈ SO [START_REF] Cardaliaguet | Equivalence between rank one convexity and polyconvexity for isotropic sets of R 2×2 (Part I)[END_REF]. In both cases dist(η; E) ≤ δ. This proves that for every ε > 0 dist(η; E) ≤ ε for every η ∈ E δ , with δ ≤ ε.

To prove the last condition of Corollary 3.7, let η [START_REF] Kolář | Non-compact lamination convex hulls Ann[END_REF]), (4.17) and (4.18) for sufficiently small δ, that is, η ∈ Rco f E δ for sufficiently small δ. 2

Set of singular values containing a line segment

In this section we establish sufficient conditions for existence of solutions to problem (4.6) when n = 2 and n = 3. Our results rely on the hypothesis that the set of singular values of the matrices in E contains a line segment. We start by considering the 2 dimensional case.

, and assume that

with a 1 < b 1 , a 2 < b 2 , and either a 1 < a 2 or b 1 < b 2 . Let

where

and rel int Γ is the relative interior of Γ with respect to the line joining (a 1 , a 2 ) and (b 1 , b 2 ), that is,

Proof. We start by noticing that, by the regularity of the boundary condition ϕ, one can assume that E is compact. Let E be the subset of E whose singular values lie in the segment joining (a 1 , a 2 ) and (b 1 , b 2 ):

Using Theorem 3.5, we will show that K has the relaxation property with respect to E and thus with respect to E, since E ⊆ E. The result will then follow as an application of Theorem 3.2. Notice first that E is a bounded set since Λ E is compact and λ 2 is a norm.

We will prove that the set

and assume, by contradiction, that, for each δ > 0, there exists

and thus, the existence of the bijection referred to above implies that

In particular, again by the continuity of the bijection between [a 1 a 2 , b 1 b 2 ] and Λ E , lim

. Since, by hypothesis, x n / ∈ Λ K , then x n 2 ≥ β n 2 and passing to the limit, as n → +∞, we get y 2 ≥ α 2 , which is a contradiction. So we conclude that Λ K is open.

Since the singular values are continuous functions and λ 1 ≤ λ 2 , it follows that K is an open set. It remains thus to show that K has the relaxation property with respect to E, which will be achieved through Theorem 3.5.

Before proceeding, we observe that K ⊂ int Rco E. Indeed, it follows from a result due to Dacorogna and Tanteri [START_REF] Dacorogna | Implicit partial differential equations and the constraints of nonlinear elasticity[END_REF] (see also [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF]Theorem 7.43

Therefore, it is clear that K ⊆ Rco E and since it is open, the desired inclusion follows. Moreover this inclusion implies that K is bounded since E is bounded. We also note that, since

Now we will prove the relaxation property introducing convenient approximating sets E δ and K δ . For sufficiently small δ, let

, and assume that

where

and rel int Γ is the relative interior of Γ with respect to the line joining (a 1 , a 2 , a 3 ) and (b 1 , b 2 , b 3 ), that is,

Let Ω ⊂ R 3 be a bounded open set and let ϕ ∈ C 1 piec (Ω, R 3 ) be such that Dϕ(x) ∈ E ∪ K for a.e. x in Ω. Then there exists a map u

Proof. The proof of this result follows the lines of that of Theorem 4.11. Due to the heavy notation we won't present it here in full detail. The reader can follow the proof of Theorem 4.11 taking into account that in this case the set Λ K can be written in the form

∈ R 3 :

The approximating sets E δ and K δ can be defined, for sufficiently small δ, by

where Λ E δ is the set (α1,α2,α3)∈rel int Γ