Gisella Croce

A DIFFERENTIAL INCLUSION: THE CASE OF AN ISOTROPIC SET

Keywords: 1991 Mathematics Subject Classification. 35F30, 34A60, 52A30 Rank one convex hull, polyconvex hull, differential inclusion, isotropic set

In this article we are interested in the following problem: to find a map u

where Ω is an open set of R 2 and E is a compact isotropic set of R 2×2 . We will show an existence theorem under suitable hypotheses on ϕ.

Introduction

In this article we study the following problem: let Ω be an open set of R 2 ; we investigate the existence of maps u : Ω → R 2 (weakly differentiable) that satisfy (1.1) Du ∈ E a.e. in Ω u(x) = ϕ(x) x ∈ ∂Ω where ϕ : Ω → R 2 is sufficiently regular and E is a compact isotropic set of R 2×2 (that is AEB ⊆ E for every A, B ∈ O(2)). In an equivalent way E can be written as (1.2) E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ K} , for some compact set K ⊂ T = {(x, y) ∈ R 2 : 0 ≤ x ≤ y}, where we have denoted by λ 1 (ξ) ≤ λ 2 (ξ) the singular values of the matrix ξ, that is the eigenvalues of the matrix ξξ t , which are (1.3)

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)| .
We will assume throughout the article that (1.4) min{x, (x, y) ∈ K} > 0 .

Thanks to the properties of the singular values (see [START_REF] Horn | Topics in matrix analysis[END_REF]), the problem (1.1) can be rewritten in the following equivalent way: x ∈ ∂Ω .

We will show the following existence theorem for the problem (1.1):

Theorem 1.1. Let E be defined by (1.2) where K ⊂ T is a compact set such that min{x, (x, y) ∈ K} > 0 . Let Ω ⊂ R 2 be an open set. Let ϕ ∈ C 1 piec (Ω; R 2 ) such that Dϕ(x) ∈ E ∪ int RcoE in Ω (where Rco E denotes the rank one convex hull of E and int RcoE its interior). Then there exists a map u ∈ ϕ + W 1,∞ 0 (Ω; R 2 ) such that Du ∈ E a.e. in Ω.

Our result will be a generalization of an existence theorem by Dacorogna and Marcellini (see [START_REF] Dacorogna | Implicit partial differential equations[END_REF]), where they investigated the case in which K contains only one point, but K ⊆ R n , n ≥ 2.

To establish our theorem we will use an abstract existence result proved by Dacorogna and Marcellini [START_REF] Dacorogna | Implicit partial differential equations[END_REF] and then refined by Dacorogna and Pisante [START_REF] Dacorogna | A general existence theorem for differential inclusions in the vector valued case[END_REF], which is based on a functional analytic method, that uses the Baire category theorem. However we recall that the kind of problem like (1.1) can also be solved by another method, established by Gromov [START_REF] Gromov | Partial differential relations[END_REF] and then presented by Müller and Šverák (see [START_REF] Müller | Convex integration for Lipschitz mappings and counterexamples to regularity[END_REF] for example) in a more analytic manner.

Moreover it will be useful the representation of the rank one convex hull of the set E (we will show that the rank one convex hull coincides with the polyconvex one): for this, we will apply some results established by Cardaliaguet and Tahraoui in [START_REF] Cardaliaguet | Equivalence between rank-one convexity and polyconvexity for isotropic sets of R 2×2[END_REF].

Definitions and preliminary results

This section is devoted to the study of the polyconvex hull of an isotropic compact set of R 2×2 : this is useful to study the rank one convex hull, as we will see in the next section.

We first give the definitions of polyconvex hull and rank one convex hull of a set E ⊆ R 2×2 : we will follow the definitions of Dacorogna and Marcellini [START_REF] Dacorogna | Implicit partial differential equations[END_REF].

Definition 2.1. Let E ⊆ R 2×2 ; if R = R ∪ {+∞}, we define Pco E = {ξ ∈ R 2×2 : f (ξ) ≤ 0, ∀ f : R 2×2 → R polyconvex , f | E ≤ 0}; Rco E = {ξ ∈ R 2×2 : f (ξ) ≤ 0, ∀ f : R 2×2 → R rank one convex , f | E ≤ 0}.
Remark 2.2. i) We observe that, according to this definition, the rank one convex hull of a compact set is not necessarily closed (some examples can be found in [START_REF] Kolář | Non-compact lamination convex hulls[END_REF]). According to our definition Rco E is the smallest rank one convex set which contains E (see [START_REF] Dacorogna | Implicit partial differential equations[END_REF]). Some authors call our envelop the laminate convex hull of E. ii) We will use also the following representation for Rco E (see [START_REF] Dacorogna | Implicit partial differential equations[END_REF]

): RcoE = i∈N R i coE, where R 0 coE = E, R i+1 coE = {tA+(1-t)B : A, B ∈ R i coE, rk(A-B) ≤ 1, t ∈ [0, 1]} .
iii) If E is bounded, then (see [START_REF] Dacorogna | Implicit partial differential equations[END_REF])

Pco E = {ξ ∈ R 2×2 : f (ξ) ≤ 0, ∀ f : R 2×2 → R polyconvex , f | E ≤ 0} = Pco E .
This means that for a compact set, our notion of polyconvexity coincides with that one of Cardaliaguet and Tahraoui in [START_REF] Cardaliaguet | Equivalence between rank-one convexity and polyconvexity for isotropic sets of R 2×2[END_REF]. We observe that the polyconvex hull of a compact set E is compact. iv) We recall that Rco E ⊆ Pco E for every set E.

We now pass to the study of the polyconvex hull of our set E. We recall the following result established in [START_REF] Cardaliaguet | Equivalence between rank-one convexity and polyconvexity for isotropic sets of R 2×2[END_REF]:

Proposition 2.3. Let K ⊂ T be a compact set and E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ K} . Let (2.1) Σ = (θ, γ) ∈ R 2 + : γ ≥ θ 2 et ∀ (x, y) ∈ K y ≤ θ + γ -θ 2 x + θ σ(x) = inf (θ,γ)∈Σ θ + γ -θ 2 x + θ , ∀ x ≥ 0. Then PcoE = {ξ ∈ R 2×2 : λ 2 (ξ) ≤ σ(λ 1 (ξ))} .
It will be useful for our purposes to write in a different way the function σ defined by (2.1). We will use the notation m(θ) = max Proposition 2.4. Let K ⊂ T be a compact set satisfying (1.4) and σ be the function defined by (2.1). Then

(2.2) σ(x) = min θ∈[0, max (a,b)∈K b] θx + m(θ) x + θ .
Proof. We divide the proof into two steps: in the first one we study the set Σ defined in proposition 2.3, and in the second one we state the formula for the function σ.

Step 1: Study of the set Σ: By definition

Σ = (θ, γ) ∈ R 2 + : γ ≥ θ 2 et ∀ (a, b) ∈ K b ≤ θ + γ -θ 2 a + θ .
As for every (a, b) ∈ K we have a > 0 , thanks to (1.4)

Σ = (θ, γ) ∈ R 2 + : γ ≥ sup{θ 2 , max (a,b)∈K {ab + θ(b -a)}} .
We observe that if θ ≥ max and so max

(a,b)∈K {ab + θ(b -a)} ≤ θ 2 . If θ < max (a,b)∈K b then, considering (a, max (a,b)∈K b) ∈ K we have max (a,b)∈K {ab + θ(b -a)} ≥ a( max (a,b)∈K b -θ) + θ max (a,b)∈K b > θ max (a,b)∈K b > θ 2 . If θ = max (a,b)∈K b we have (2.3) max (a,b)∈K {ab + [ max (a,b)∈K b](b -a)} = [ max (a,b)∈K b] 2 .
From this study we can infer that

Σ = Σ 1 ∪ Σ 2 ,
where

Σ 1 = {(θ, γ) ∈ R 2 : θ ∈ [0, max (a,b)∈K b], γ ≥ max (a,b)∈K {ab + θ(b -a)}} Σ 2 = {(θ, γ) ∈ R 2 : θ ≥ max (a,b)∈K b, γ ≥ θ 2 } .
Step 2: Study of the function σ: We define

g x (θ, γ) = θx + γ x + θ for x ≥ 0,
and

Γ 1 = {(θ, max (a,b)∈K {ab + θ(b -a)}), θ ∈ [0, max (a,b)∈K b]} Γ 2 = {(θ, θ 2 ), θ ≥ max (a,b)∈K b} .
We observe that Γ 1 ∪ Γ 2 ⊂ Σ 1 ∪ Σ 2 . We are going to show that

σ(x) = inf Γ1∪Γ2 g x (θ, γ).
We know from proposition 2.3 that σ(x) = inf

(θ,γ)∈Σ θ + γ -θ 2 x + θ . Now, if x = 0 σ(0) = inf (θ,γ)∈Σ θ + γ -θ 2 θ = inf (θ,γ)∈Σ γ θ .
As (θ 0 fixed) the function γ θ 0 is increasing,

σ(0) = inf (θ,γ)∈Σ γ θ = inf Γ1∪Γ2 g 0 (θ, γ). If x > 0, σ(x) = inf (θ,γ)∈Σ θ + γ -θ 2 x + θ = inf (θ,γ)∈Σ g x (θ, γ) . We observe that ∂g x ∂γ > 0:
this implies that, if θ 0 ≥ 0 is fixed, g x (θ 0 , γ) is increasing (in γ), and so for x > 0

σ(x) = inf Σ g x (θ, γ) = inf Γ1∪Γ2 g x (θ, γ) ,
as wished. We observe that x + θ .

g x | Γ1 = θx + max
Moreover

g x | Γ2 = θx + θ 2 x + θ = θ, θ ≥ max (a,b)∈K b : consequently inf Γ2 g x = max (a,b)∈K b.
This study implies that

σ(x) = inf Γ1∪Γ2 g x (θ, γ) = inf    inf θ∈[0, max (a,b)∈K b] θx + max (a,b)∈K {ab + θ(b -a)} x + θ , max (a,b)∈K b    .
Due to (2.3)

x max θx + m(θ) x + θ .

We are going to show that

∀ x ≥ 0 inf θ∈[0, max (a,b)∈K b] θx + m(θ) x + θ = min θ∈[0, max (a,b)∈K b] θx + m(θ) x + θ ,
and so the formula of the statement. For x > 0 this is trivial. For x = 0 we have to study the function of θ

m(θ) θ = max (a,b)∈K {ab + θ(b -a)} θ : we observe that, max (a,b)∈K {ab + θ(b -a)} θ ≥ max (a,b)∈K ab θ → ∞, θ → 0.
This implies that there exists ε > 0 such that

(2.4) inf θ∈[0, max (a,b)∈K b] m(θ) θ = inf θ∈[ε, max (a,b)∈K b] m(θ) θ = min θ∈[ε, max (a,b)∈K b] m(θ) θ ,
and so we have the result.

In the next proposition we study the interior of Pco E .

Proposition 2.5. Let K ⊂ T be a compact set satisfying (1.4).

Let E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ K} . Then, if σ is the function defined by (2.2), int PcoE = {ξ ∈ R 2×2 : λ 2 (ξ) < σ(λ 1 (ξ))} .
Proof. The proof is divided into two steps.

Step 1) To show that int PcoE ⊇ {ξ ∈ R 2×2 : λ 2 (ξ) < σ(λ 1 (ξ))} , as λ i (ξ) are continuous functions, i = 1, 2, it is sufficient to show that the function σ is continuous for x ≥ 0. This is easy for x > 0. To study the point x = 0, we are going to show that if x n → 0 + , then σ(x n ) → σ(0). We can say from (2.4) that there exists θ 0 ∈ (0, max

(a,b)∈K b] such that σ(0) = m(θ0) θ0 . Then, by definition of σ(x n ) σ(x n ) -σ(0) ≤ θ 0 x n + m(θ 0 ) x n + θ 0 - m(θ 0 ) θ 0 → 0, n → ∞ .
We are going to show that σ(

x n ) -σ(0) ≥ h(x n ), with h(x n ) → 0, if n → ∞: this will imply that σ(x n ) → σ(0)
, and so the continuity of σ in 0.

For every x n there exists θ n ∈ [0, max

(a,b)∈K b] such that σ(x n ) = θ n x n + m(θ n ) x n + θ n .
Then, by definition of σ(0)

σ(x n ) -σ(0) = θ n x n + m(θ n ) x n + θ n - min θ∈[0, max (a,b)∈K b] m(θ) θ ≥ θ n x n + m(θ n ) x n + θ n - m(θ n ) θ n = x n θ n (x n + θ n ) [θ 2 n -m(θ n )] .
We will show that h(

x n ) = xn θn(xn+θn) [θ 2 n -m(θ n )] → 0, n → ∞ . As θ 2 n -m(θ n ) is bounded, it is now sufficient to prove that x n θ n (x n + θ n ) → 0, n → ∞ .
We observe that lim inf

n→∞ θ n > 0 . In fact, if lim inf n→∞ θ n = 0 , then there exists a sub- sequence n k such that lim k→∞ θ n k = 0; consequently σ(x n k ) = θ n k x n k + m(θ n k ) x n k + θ n k ≥ max (a,b)∈K ab x n k + θ n k → ∞, k → ∞ .
The matrix

ξ n k = x n k 0 0 σ(x n k ) belongs to Pco E and λ 2 (ξ n k ) → ∞ : recalling that λ 2 is a norm over R 2×2 (see [6]
), we got a contradiction because PcoE is bounded, as E is bounded. Then lim inf n→∞ θ n = a > 0 : this implies that, for n sufficiently large θ n ≥ a 2 and so

x n θ n (x n + θ n ) ≤ x n a 2 (x n + a 2 ) → 0, n → ∞ ,
that is the result.

Step 2) We now show that int Pco E ⊆ {ξ ∈ R 2×2 : λ 2 (ξ) < σ(λ 1 (ξ))} . Suppose that there exists a matrix η ∈ int Pco E such that λ 2 (η) = σ(λ 1 (η)); therefore the ball B ε (η) ⊆ Pco E, for some ε > 0. Let A, B ∈ O(2) be such that

AηB = λ 1 (η) 0 0 λ 2 (η) ;
we define

D = A -1 0 0 0 d B -1 , with 0 < d < ε. Then we have λ 1 (η + D) = λ 1 (AηB + ADB) = λ 1 (η) λ 2 (η + D) = λ 2 (AηB + ADB) = λ 2 (η) + d . The matrix η + D ∈ B ε (η) ⊆ Pco E, as d < ε: this implies that λ 2 (η + D) ≤ σ(λ 1 (η + D)). From other hand, λ 2 (η + D) = λ 2 (η) + d = σ(λ 1 (η)) + d = σ(λ 1 (η + D)) + d > σ(λ 1 (η + D)) ,
and this is a contradiction: therefore λ 2 (η) < σ(λ 1 (η)) .

Remark 2.6. The previous results imply that if ξ ∈ ∂Pco E, then there exists θ ∈ [0, max

(a,b)∈K b] such that λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) = m( θ) .

The rank one convex hull

In this section we are going to prove the representation theorem of the rank one convex hull of E. It will be useful to show our existence result. We recall that we use the notation m(θ) = max Theorem 3.1. Let K ⊂ T be a compact set satisfying (1.4). Let

E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ K} . Then RcoE = ξ ∈ R 2×2 : λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) ≤ m(θ), ∀ θ ∈ [0, max (a,b)∈K b] , int RcoE = ξ ∈ R 2×2 : λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) < m(θ), ∀ θ ∈ [0, max (a,b)∈K b] .
We will first prove the following lemma:

Lemma 3.2. Let K = {(a 1 , b 1 ), (a 2 , b 2 )}, 0 < a 1 < a 2 , a 1 b 1 ≤ a 2 b 2 , b 2 ≤ b 1 , a 1 ≤ b 1 , a 2 ≤ b 2 and E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ K}. Then Rco E = {ξ ∈ R 2×2 : λ 2 (ξ) ≤ b 1 , λ 1 (ξ)λ 2 (ξ) ≤ a 2 b 2 λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) ≤ a 1 b 1 + θ(b 1 -a 1 )} where θ = a 2 b 2 -a 1 b 1 b 1 -a 1 -b 2 + a 2 .
Remark 3.3. We remember that Dacorogna and Marcellini [START_REF] Dacorogna | Implicit partial differential equations[END_REF] proved that if K is composed by one point (a, b) we have

Rco E = {ξ ∈ R 2×2 : λ 2 (ξ) ≤ b, λ 1 (ξ)λ 2 (ξ) ≤ ab} .
Proof. In [START_REF] Cardaliaguet | Equivalence between rank-one convexity and polyconvexity for isotropic sets of R 2×2[END_REF] it is showed that the function σ (2.2) defined for E is

σ(x) = inf b 1 , a 2 b 2 x , θx + a 1 b 1 + θ(b 1 -a 1 ) θ + x .
Thanks to proposition 2.3, this implies that Pco E is the set of matrices ξ such that

λ 2 (ξ) ≤ b 1 , (3.1) λ 1 (ξ)λ 2 (ξ) ≤ a 2 b 2 , (3.2) λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) ≤ a 1 b 1 + θ(b 1 -a 1 ) . (3.3)
Therefore to prove the formula of RcoE it is sufficient to prove that PcoE = RcoE. For this we will show that ∂PcoE ⊆ RcoE : this will imply the not trivial inclusion PcoE ⊆ RcoE and so the result. In fact, let ξ ∈ intPcoE; as PcoE is compact, then for any rank one matrix λ ∈ R 2×2 , there exist

t 1 = t 1 (λ) < 0 < t 2 = t 2 (λ) such that ξ + t i λ ∈ ∂PcoE ⊆ RcoE, i = 1, 2. Defining ξ i = ξ + t i λ, i = 1, 2, we have ξ = t 2 t 2 -t 1 ξ 1 - t 1 t 2 -t 1 ξ 2 ∈ RcoE , because rk(ξ 1 -ξ 2 ) = 1.
Now, let ξ ∈ ∂PcoE: necessarily λ 2 (ξ) = σ(λ 1 (ξ)) and so ξ satisfies either (3.1) or (3.2) or (3.3) with equality. We are going to treat these cases separately (steps 1, 2, 3 respectively) to show that ξ ∈ RcoE. We can assume without loss of generality that ξ = diag(λ 1 (ξ), λ 2 (ξ)), as Rco E is isotropic: in fact, using the same notations as in the second point of remark 2.2, we have by induction on i that R i coE is isotropic and so RcoE is isotropic.

step 1) If ξ satisfies (3.1) with equality, (3.3) implies that λ 1 (ξ) ≤ a 1 ; then ξ ∈ RcoE, as

ξ = λ 1 (ξ) 0 0 b 1 = t a 1 0 0 b 1 + (1 -t) -a 1 0 0 b 1 , t ∈ (0, 1) .
In the next steps we can assume that ξ satisfies (3.1) with strict inequality. step 2) We suppose that ξ satisfies (3.2) with equality. Moreover we can assume that ξ satisfies (3.3) with strict inequality, otherwise these two equalities imply

ξ = a 2 0 0 b 2 ∈ E.
If we define

V = {ξ ∈ R 2×2 : λ 1 (ξ)λ 2 (ξ) = a 2 b 2 }, Y = V ∩ ∂PcoE
we have that ξ ∈ rel intY 1 . Let Z be the rank one matrix defined by

Z = 1 -λ2(ξ) λ1(ξ) 1 -λ2(ξ) λ1 (ξ)
:

then λ 1 (ξ + tZ)λ 2 (ξ + tZ) = λ 1 (ξ)λ 2 (ξ) = a 2 b 2 ∀ t ∈ R.
This implies, as Y is compact, that there exist t 1 < 0 < t 2 : ξ + t i Z ∈ ∂Y, i = 1, 2. Consequently ξ + t i Z satisfies either (3.1) and (3.2) as equalities or (3.3) and (3.2) as equalities: from the previous studies we obtain that ξ + t i Z ∈ RcoE, i = 1, 2 and so ξ ∈ RcoE.

In the next step we can assume that (3.1) and (3.2) are satisfied as strict inequalities.

step 3) We assume that ξ satisfies (3.3) with equality. Using the explicit expressions of λ 1 , λ 2 (see (1.3)), it is easy to prove that if (λ 1 (ξ), λ 2 (ξ)) = (x, y) the matrix defined by

A =   1 y- θ x+ θ -y- θ x+ θ -y- θ x+ θ  
has the following properties: it is well defined (as y ≥ θ because θ2 ≤ max

(a,b)∈K ab + θ(b -a) = xy + θ(y -x)) and (3.4) λ 1 (ξ + tA)λ 2 (ξ + tA) + θ[λ 2 (ξ + tA) -λ 1 (ξ + tA)] = xy + θ(y -x) ∀ t ∈ [t -, t + ], t -= - xy(x + θ) θ(x + y) , t + = (y -x)(x + θ) x + y .
In fact

λ 1 (ξ + tA)λ 2 (ξ + tA) = | det(ξ + tA)| = xy -xt y - θ x + θ + ty ;
One can easily check that this equality is verified for every t and so (3.4) is verified.

We prove now that there exists t 1 ∈ [t -, 0] such that λ 2 (ξ + t 1 A) = b 1 : this implies that ξ + t 1 A satisfies (3.1) and (3.3) as equalities: as we saw in the first step, ξ + t 1 A ∈ RcoE. Moreover we prove also that there exists

t 2 ∈ [0, t + ] such that λ 1 (ξ + t 2 A)λ 2 (ξ + t 2 A) = a 2 b 2 : this implies that ξ + t 2 A satisfies (3.
2) and (3.3) as equalities: as we saw in the second step, ξ + t 2 A ∈ RcoE. Consequently ξ ∈ RcoE, as it can be written as rank one combination of ξ + t 1 A and ξ + t 2 A. Existence of t 1 ) We consider F (t) = λ 2 (ξ + tA)b 1 . The existence of t 1 follows from the fact that this function is continuous and F (0) < 0 < F (t -) : in fact

F (t -) = ξ + t -A -b 1 > 0 ⇐⇒ b 1 < ξ + t -A = = x 2 ( θ -y) θ(x + y) 2 + y 2 ( θ + x) θ(x + y) 2 + 2 x 2 y 2 (x + θ)(y -θ) θ2 (x + y) 2 = x 2 (y -θ) + y 2 ( θ + x) θ(x + y) = xy + θ(y -x) θ .
The last inequality is equivalent to

θb 1 ≤ xy + θ(y -x) = a 1 b 1 + θ(b 1 -a 1 ) ⇐⇒ b 1 ≥ θ which is true. Existence of t 2 ) We consider G(t) = λ 1 (ξ + tA)λ 2 (ξ + tA) -a 2 b 2 .
The existence of t 2 follows from the fact that G is continuous and G(0

) < 0 < G(t + ) : in fact G(t + ) > 0 if and only if xy -x (y -θ)(y -x) x + y + y (x + θ)(y -x) x + y ≥ a 2 b 2 . Using that xy + θ(y -x) = a 2 b 2 + θ(b 2 -a 2 ) we get xy(x + y) + θ(y -x)(y + x) ≥ a 2 b 2 (x + y) ⇐⇒ a 2 b 2 + θ(b 2 -a 2 ) ≥ a 2 b 2
which is true.

To prove theorem 3.1 it will be useful to recall the following properties about convex functions and their sub-differential (we will follow the definition of [START_REF] Rockafellar | Convex analysis[END_REF]). Definition 3.4. Let f : R → R be a convex function and θ ∈ R. The subdifferential of f in θ is the set

∂f ( θ) = {θ * ∈ R : f (θ) ≥ f ( θ) + θ * (θ -θ) ∀ θ ∈ R} . Proposition 3.5. Let f : R → R be a convex function. Then i) ∂f (θ) is non empty, compact and convex for every θ ∈ R. ii) If θ is a point of differentiability of f then ∂f (θ) = {f ′ (θ)} .
iii) The set of points of differentiability of f is dense in R and

∂f (x) = co S(x) , S(x) = { lim n→∞ f ′ (x n ) f differentiable in x n , x n → x} .
Remark 3.6. The proof of i) can be found at page 218 of [START_REF] Rockafellar | Convex analysis[END_REF]; ii) is theorem 26.1 of [START_REF] Rockafellar | Convex analysis[END_REF]; the proof of iii) is a direct combination of theorems 25.6 and 17.2 of [START_REF] Rockafellar | Convex analysis[END_REF].

We pass now to the proof of theorem 3.1.

Proof. Thanks to propositions 2.3 and 2.4 it is sufficient to prove that Pco E = Rco E. For this, as in lemma 3.2, we will show the inclusion ∂Pco E ⊆ Rco E .

Let ξ ∈ ∂Pco E. We have seen in remark 2.6 that there exists θ ∈ [0, max

(a,b)∈K b] such that λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) = m( θ)
, and for every θ ∈ [0, max

(a,b)∈K b] λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) ≤ m(θ) .
We define

F (θ) =              max (a,b)∈K ab θ ≤ 0 m(θ) θ ∈ [0, max (a,b)∈K b] θ 2 θ ≥ max (a,b)∈K b .
We observe that ξ satisfies

λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) ≤ F (θ), ∀ θ ∈ R ,
and there exists θ ∈ [0, max

(a,b)∈K b] such that λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) = F ( θ) .
The following remarks will be useful:

• λ 2 (ξ) -λ 1 (ξ) ∈ ∂F ( θ) .
• One can easily check that F is convex: the previous proposition implies that ∂F ( θ) = [α( θ), β( θ)] for some α( θ), β( θ) ∈ R.

• If θ 0 is a point of differentiability for the function

F then ∂F (θ 0 ) = {F ′ (θ 0 )} = b 0 -a 0 for some (a 0 , b 0 ) ∈ K, such that F (θ 0 ) = max (a,b)∈K {ab + θ 0 (b -a)} = a 0 b 0 + θ 0 (b 0 -a 0 ) .
We are going to show that ξ ∈ Rco E: for this we will find a set a ⊂ K composed by one or two points such that letting A = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ a} we have ξ ∈ RcoA ⊆ Rco E. We will distinguish the cases θ = 0, θ ∈ (0, max step 1) We analyse the case θ = 0, for which max (a,b)∈K ab = λ 1 (ξ)λ 2 (ξ). We study the set S(0) defined in proposition 3.5 (the points θ n will be points of differentiability of F throughout this proof): 0) is 0 or ba for some (a, b) ∈ K, because of the compactness of K. The fact that K is compact implies also that p M = bā, for some (ā, b) ∈ K and so

S(0) = { lim n→∞ F ′ (θ n ), θ n → 0} = { lim n→∞ F ′ (θ n ), θ n → 0 + } ∪ {0} as for every θ < 0 F is constant. Let p M = sup{ lim n→∞ F ′ (θ n ), θ n → 0 + } ≥ 0 . Let θ n → 0 + be points of differentiability for F : then F ′ (θ n ) = b n -a n , for some (a n , b n ) ∈ K; therefore every point of S(
∂F (0) = coS(0) = [0, b -ā] ∋ λ 2 (ξ) -λ 1 (ξ) .
It is easy to see that there exists (ã, b) ∈ K such that b-ā = b-ã and max In fact by definition of bā, ∀ ε > 0 there exists θ ε n which goes to 0 for n → ∞ and a sequence (

a ε n , b ε n ) ∈ K such that (3.5) b -ā -ε ≤ lim n→∞ F ′ (θ ε n ) = lim n→∞ b ε n -a ε n ≤ b -ā.
We observe that, as θ ε n is a point of differentiability of F max

(a,b)∈K ab + θ ε n (b -a) = a ε n b ε n + θ ε n (b ε n -a ε n ) . Now, if we consider the points (a ε n , b ε n ) ∈ K, as K is compact, we can say, up to a sub-sequence that (a ε n , b ε n ) → (a ε , b ε ) ∈ K . For the same reason, if ε → 0 (a ε , b ε ) → (ã, b) ∈ K .
Passing to the limit for n → ∞ in the last relation, we obtain from the continuity in θ of the function max

(a,b)∈K ab + θ(b -a) lim n→∞ max (a,b)∈K ab + θ ε n (b -a) = max (a,b)∈K ab = lim n→∞ a ε n b ε n + θ ε n (b ε n -a ε n ) = a ε b ε ,
and so max

(a,b)∈K ab = λ 1 (ξ)λ 2 (ξ) = lim ε→0 a ε b ε = ãb . From the relation (3.5) we get, b -ā ≤ lim ε→0 lim n→∞ b ε n -a ε n = b -ã ≤ b -ā ⇐⇒ b -ā = b -ã .
Then we have that λ 1 (ξ)λ 2 (ξ) = ãb and λ 2 (ξ)λ 1 (ξ) ≤ bã, that is λ 2 (ξ) ≤ b . This is equivalent, thanks to remark 3.3, to say that ξ ∈ RcoA ⊆ RcoE, where

A = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) = (ã, b)} . step 2) We study the case θ ∈ (0, max (a,b)∈K b). As in step 1, if p m = inf{ lim n→∞ F ′ (θ n ), θ n → θ} p M = sup{ lim n→∞ F ′ (θ n ), θ n → θ} , we have for some (a i , b i ) ∈ K, i = 1, 2 ∂F ( θ) = [b 2 -a 2 , b 1 -a 1 ] .
Following the same kind of study as in step 1, one can show that there exist (ã i , bi ) ∈ K, i = 1, 2 such that

λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) = F ( θ) = max (a,b)∈K {ab + θ(b -a)} = = ãi bi + θ( bi -ãi ), bi -ãi = b i -a i .
We now show that ξ ∈ Rco A, where

A = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) = ( ãi , bi ), i = 1, 2} :
for this, thanks to lemma 3.2, it is sufficient to show that b2 ≤ λ 2 (ξ) ≤ b1 ,

ã1 b1 ≤ λ 1 (ξ)λ 2 (ξ) ≤ ã2 b2 , ã1 < ã2 . For every θ ∈ [0, max (a,b)
ã2 b2 + θ( b2 -ã2 ) < λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) < ã1 b1 + θ( b1 -ã1 )
for every θ ≥ θ and in particular for θ = λ 2 (ξ) we have that b2 ≤ λ 2 (ξ) ≤ b1 , that is the first condition is verified. As b2 -ã2 ≤ λ 2 (ξ)λ 1 (ξ) ≤ b1 -ã1 , then ã1 b1 ≤ λ 1 (ξ)λ 2 (ξ) ≤ ã2 b2 and ã1 < ã2 .

step 3) We study the case θ = max

p m = inf{ lim n→∞ F ′ (θ n ), θ n → max (a,b)∈K
b} and we have, as in the previous steps

∂F ( θ) = [ b -ā, β( θ)] , (ā, b) ∈ K .
It is easy to show that there exists (ã, b) ∈ K such that b-ā = b-ã and max

(a,b)∈K ab+ θ(b -a) = [ max (a,b)∈K b] 2 = ãb + θ( b -ã). Therefore b = max (a,b)∈K b . Defining A = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) = (ã, b)} we have that ξ ∈ Rco A ⊆ Rco E, thank to remark 3.3: in fact λ 2 (ξ) -λ 1 (ξ) ≥ b -ã = max (a,b)∈K b -ã , that is λ 1 (ξ) ≤ ã , and λ 2 (ξ) = b .
The formula for the interior of Rco E follows from proposition 2.5 and from the fact that Pco E = Rco E , as we have just showed.

The existence theorem

In this section we are going to show theorem 1.1. The proof will be a direct combination of theorem 4.5 and of proposition 4.2. To do this it will be useful to define the so called approximation property (this definition is given in [START_REF] Dacorogna | Implicit partial differential equations[END_REF]). Definition 4.1. Let E ⊂ K(E) ⊂ R 2×2 . We say that E and K(E) have the approximation property if there exists a family of closed sets E δ and K(E δ ), δ > 0, such that 1) E δ ⊂ K(E δ ) ⊂ intK(E) for every δ > 0; 2) for every ε > 0 there exists δ 0 = δ 0 (ε) > 0 such that dist(η, E) ≤ ε for every η ∈ E δ and δ ∈ [0, δ 0 ]; 3) if η ∈ intK(E) then η ∈ K(E δ ) for every δ > 0 sufficiently small.

We can now show the following result. Proposition 4.2. Let E be defined by (1.4). Then E and RcoE have the approximation property with

E = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) ∈ K} with K compact satisfying
K(E δ ) = Rco E δ , if E δ = (a,b)∈K E (a,b) δ , E (a,b) δ = {ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) = (a -δ, b -δ)}, for 0 ≤ δ ≤ min (a,b)∈K a/2.
It will be useful the following result due to Cardaliaguet and Tahraoui [START_REF] Cardaliaguet | Equivalence between rank-one convexity and polyconvexity for isotropic sets of R 2×2[END_REF]: Proposition 4.3. For every θ ≥ 0 the function H θ : R 2×2 → R defined by

H θ (ξ) = max{λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) -θ 2 , 0}
is rank one convex. Remark 4.4. In [START_REF] Cardaliaguet | Equivalence between rank-one convexity and polyconvexity for isotropic sets of R 2×2[END_REF] Cardaliaguet and Tahraoui show that the function H θ is polyconvex.

We can now prove proposition 4.2.

Proof. We remark that E δ is compact: this will let us use the representation theorem 3.1. In the following three steps, we show the three conditions of the approximation property respectively. 1)Rco(E δ ) ⊂ intRcoE, ∀ δ > 0 :

Let (a, b) ∈ K be fixed. Then we have

E (a,b) δ ⊆ {ξ ∈ R 2×2 : λ 1 (ξ)λ 2 (ξ) < ab, λ 2 (ξ) < b} = int Rco{ξ ∈ R 2×2 : (λ 1 (ξ), λ 2 (ξ)) = (a, b)} ⊆ int RcoE;
this implies, if we pass to the union over K, that

E δ ⊆ int RcoE .
From this inclusion we can infer that

RcoE δ ⊆ Rco(int RcoE) = int RcoE ,
as the interior of Rco E is rank one convex. In fact let ξ, ξ + A ∈ int RcoE, with rk(A) = 1, that is, thank to theorem 3.1, for every θ ∈ [0, max

(a,b)∈K b] (4.1) λ 1 (ξ)λ 2 (ξ) + θ(λ 2 (ξ) -λ 1 (ξ)) < max (a,b)∈K ab + θ(b -a) λ 1 (ξ + A)λ 2 (ξ + A) + θ(λ 2 (ξ + A) -λ 1 (ξ + A)) < max (a,b)∈K ab + θ(b -a) .
We want to show that ξ + sA ∈ int RcoE, s ∈ [0, 1].

Surely ξ + sA ∈ RcoE, because ξ, ξ + A ∈ RcoE. Now, let us suppose that there exists θ ∈ [0, max

(a,b)∈K b] such that (4.2) max (a,b)∈K ab + θ(b -a) = λ 1 (ξ + sA)λ 2 (ξ + sA) + θ(λ 2 (ξ + sA) -λ 1 (ξ + sA)) .
We can assume that θ = max b . This is a contradiction as λ 2 is a norm over

R 2×2 . Therefore we can write θ2 < max (a,b)∈K ab + θ(b -a) = λ 1 (ξ + sA)λ 2 (ξ + sA) + θ(λ 2 (ξ + sA) -λ 1 (ξ + sA)) .
Using the expression of the function Hθ defined in proposition 4.3, we have

Hθ(ξ + sA) = λ 1 (ξ + sA)λ 2 (ξ + sA) + θ(λ 2 (ξ + sA) -λ 1 (ξ + sA)) -θ2 = max (a,b)∈K ab + θ(b -a) -θ2 > 0.
Thanks to the fact that Hθ(ξ) is rank one convex, from proposition 4.3

0 < Hθ(ξ + sA) ≤ sHθ(ξ + A) + (1 -s)Hθ(ξ) ≤ max{Hθ(ξ), Hθ(ξ + A)} .
Without loss of generality we can assume that max{Hθ(ξ), Hθ(ξ +A)} = Hθ(ξ +A) .

If Hθ(ξ + A) = 0 we have a contradiction. If Hθ(ξ + A) > 0, we have, as ξ + A ∈ int RcoE

Hθ(ξ + A) = λ 1 (ξ + A)λ 2 (ξ + A) + θ(λ 2 (ξ + A) -λ 1 (ξ + A)) -θ2 < max (a,b)∈K ab + θ(b -a) -θ2
and so we have obtained

Hθ(ξ + sA) = max (a,b)∈K ab + θ(b -a) -θ2 ≤ Hθ(ξ + A) < max (a,b)∈K ab + θ(b -a) -θ2 which is a contradiction. 2)∀ ε > 0 ∃ δ 0 = δ 0 (ε) > 0 : dist(η, E) ≤ ε ∀ η ∈ E δ , δ ∈ [0, δ 0 ] : Let η ∈ E δ ; then there exists (a, b) ∈ K such that η ∈ E (a,b) δ . We define X = a 0 0 b ∈ E . Let A, B ∈ O(2) be such that AηB = λ 1 (η) 0 0 λ 2 (η) . Then η -A -1 XB -1 = AηB -AA -1 XB -1 B = AηB -X = √ 2δ 2 . This implies that dist(η, E) ≤ η -A -1 XB -1 = √ 2δ 2 → 0, δ → 0;
moreover this limit is uniform with respect to η.

3)If η ∈ intRcoE then η ∈ RcoE δ ∀ δ > 0 sufficiently small: Let η ∈ intRcoE; if (λ 1 (η), λ 2 (η)) = (x, y) thanks to theorem 3.1 we have to show the following implication:

xy + θ(y -x) < max (a,b)∈K ab + θ(b -a) ∀ θ ∈ [0, max (a,b)∈K b] ⇓ xy + θ(y -x) < max (a,b)∈K (a -δ)(b -δ) + θ(b -a)
uniformly with respect to θ ∈ [0, max 4.1. Proof of the existence theorem. We are going to recall an abstract existence theorem (established by Dacorogna and Pisante [START_REF] Dacorogna | A general existence theorem for differential inclusions in the vector valued case[END_REF]) that we will apply. Then there exists u ∈ ϕ + W 1,∞ 0 Ω; R 2 such that Du (x) ∈ E, a.e. in Ω.

Our theorem then follows immediately. In fact, E is compact and we verified that E and RcoE have the approximation property in proposition 4.2. Then we obtain the existence theorem 1.1 thanks to theorem 4.5.

Representation of Rco E

In this section we are going to give an explicit formula of Rco E for a set E (1.2) defined by a set K composed by a finite number of points. We will treat the cases in which K is composed by one, two elements, and finally a particular K composed by three elements; for the general formula and for its proof we refer to [2]. We recall that the representation of Rco E, for E defined by K composed by one element was already obtained by Dacorogna and Marcellini (see [START_REF] Dacorogna | Implicit partial differential equations[END_REF]); Cardaliaguet and Tahraoui in [START_REF] Cardaliaguet | Equivalence between rank-one convexity and polyconvexity for isotropic sets of R 2×2[END_REF] showed the formula for the case of K composed by two elements.

To give the representation of Rco E it is sufficient to give the formula of the function σ defined by (2.1): indeed Rco E = {ξ ∈ R 2×2 : λ 2 (ξ) ≤ σ(λ 1 (ξ))} . x , b 3 , xθ(j, j + 1) + a j b j + θ(j, j + 1)(b ja j )

x + θ(j, j + 1) , j = 1, 2 .

2 = a 2 +

 22 b 2 a.e. in Ω, (a, b) ∈ K, |det Du(x)| = ab a.e. in Ω, (a, b) ∈ K, u(x) = ϕ(x)

  (a,b)∈K {ab + θ(ba)} throughout all this article.

b

  then for every (a, b) ∈ K one has ab + θ(ba) ≤ θa + θ(ba) = θb ≤ θ 2

  {ab + θ(ba)}

  (a,b)∈K {ab + θ(ba)} .

b

  respectively in the steps 1, 2, 3.

  ∈K b] we have that θ 2 ≤ max (a,b)∈K ab + θ(ba), as seen in proposition 2.4: writing this inequality for θ = θ we get λ 2 (ξ) > θ. As

4 λ 2

 42 we have, as we saw in proposition 2.(ξ) = max (a,b)∈K b. We define

  (a,b)∈K b. In fact, due to (2.3) [ max (a,b)∈K b] 2 = max (a,b)∈K {ab + [ max (a,b)∈K b](ba)} ; therefore if we choose θ = max (a,b)∈K b in (4.1) and in (4.2) we have λ 2 (ξ), λ 2 (ξ + A) < max (a,b)∈K b and λ 2 (ξ + sA) = max (a,b)∈K

  (a,b)∈K bδ]. For this it is to show that lim δ→0 max (a,b)∈K(aδ)(bδ) + θ(ba) = max (a,b)∈K ab + θ(ba) uniformly with respect to θ ∈ [0, max (a,b)∈K b-δ]. We have, as (a-δ)(b-δ)+θ(b-a) ≥ 0 ∀ δ ∈ [0, min (a,b)∈K a/2], | max (a,b)∈K (aδ)(bδ) + θ(ba)max(a,b)∈K ab + θ(ba)| ≤ max (a,b)∈K |(aδ)(bδ) + θ(ba)abθ(ba)| ≤ max (a,b)∈K δ(a + b + δ) → 0, δ → 0 , uniformly with respect to θ ∈ [0, max (a,b)∈K bδ]. Consequently we showed the third condition of the approximation property too.

Theorem 4 . 5 .

 45 Let Ω ⊂ R 2 be an open set. Let E ⊂ R 2×2 be a compact set.Assume that E and Rco E have the approximation property withK(E δ ) = Rco E δ . Let ϕ ∈ C 1piec Ω; R 2 be such that Dϕ (x) ∈ E ∪ int Rco E, a.e. in Ω.

Proposition 5 . 1 . 2 )b 1 x , b 1 . 3 )b 2 x , b 2 . 5 . 3 .

 51213253 Let E be defined by(1.2) with K = {(a, b), 0 < a ≤ b}. Thenσ(x) = inf ab x , b . Proposition 5.2. Let E be defined by (1.2) with K = {(a 1 , b 1 ), (a 2 , b 2 ), 0 < a i ≤ b i , i = 1, 2}. Without loss of generality we can assume that a 1 ≥ a 2 . Then 1) If a 1 b 1 > a 2 b 2 and b 2 > b 1 then σ(x) = inf a 1 b 1 x , b 2 , θ(1, 2)x + a 1 b 1 + θ(1, 2)(b 1a 1 ) x + θ(1, 2) where θ(1, 2) = a 1 b 1a 2 b 2 b 2a 2 -(b 1a 1 ) . If b 1 ≥ b 2 then σ(x) = inf a 1 If b 1 < b 2 and a 1 b 1 ≤ a 2 b 2 , then σ(x) = inf a 2Proposition Let E be defined by(1.2) with K = {(a i , b i ), 0 < a i ≤ b i , i = 1, 2, 3}. We define θ(i, j) = a i b ia j b j b ja j -(b ia i ) . Let us suppose that a 1 b 1 > a 2 b 2 > a 3 b 3 , b 3 > max{b 1 , b 2 }, b 2a 2 >b 1a 1 and θ(1, 2) < θ(1, 3) . Then σ(x) = inf a 1 b 1

Acknowledgements. I would like to thank Bernard Dacorogna for having proposed me the subject of this article and for his scientific help; I'm very grateful to the referees for their precious remarks and suggestions. As well, I beneficiated of some discussions with Ana Margarida Fernandes Ribeiro and Giovanni Pisante.

Research supported by Fonds National Suisse (21-61390-00).


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

THE REGULARIZING EFFECTS OF SOME LOWER ORDER TERMS IN AN ELLIPTIC EQUATION WITH DEGENERATE COERCIVITY

Keywords: 1991 Mathematics Subject Classification. 35B45, 35D05, 35J65, 35J70 nonlinear elliptic problem, boundary value problem, irregular data, degenerate ellipticity, distributional solution

In this article we study an elliptic problem with degenerate coercivity. We will show that the presence of some lower order terms has a regularizing effect on the solutions, if we assume that the datum f ∈ L m (Ω) for some m ≥ 1.

Introduction

Consider the problem (1.1)

-∆u

+ |u| p-1 u = f in Ω u = 0 on ∂Ω .
It is known that the lower order term |u| p-1 u has a regularizing effect on the solutions: indeed the solution u belongs to the Lebesgue space L p (Ω) under the weak assumption that the datum f ∈ L 1 (Ω) (see [START_REF] Brézis | Semi-linear second-order elliptic equations in L 1[END_REF]); moreover ∇u ∈ L q (Ω), q < 2p p+1 (see [START_REF] Boccardo | Nonlinear elliptic equations in R N without growth restrictions on the data[END_REF]). A stronger effect can be observed if we consider a lower order term h(u) where h : [0, σ) → R + is a continuous, increasing function with a vertical asymptote in σ (σ > 0). Indeed in [START_REF] Boccardo | Quasilinear elliptic equations with natural growth terms: the regularizing effects of the lower order terms[END_REF] it is shown that if f is an L 1 (Ω) positive function, then there exists a bounded H 1 0 (Ω) solution to

(1.2) -∆u + h(u) = f in Ω u = 0 on ∂Ω .

All these results have been proved for general nonlinear coercive elliptic problems too.

In this article we want to analyse the regularizing effects of the same kind of lower order terms as in problems (1.1) and (1.2) in the case of an elliptic operator with degenerate coercivity. More in details, let us consider the differential operator (1.3) A(v) = -div (a(x, v)∇v) , v ∈ H 1 0 (Ω) under the following assumptions: Ω is an open bounded subset of R N , N ≥ 3, and a : Ω × R → R is a Carathéodory function such that for a.e. x ∈ Ω and for every s ∈ R for some real positive constants α, β, γ. Assumption (1.4) implies that the differential operator A is well defined on H 1 0 (Ω), but it fails to be coercive on the same space when v is large (see [START_REF] Porretta | Uniqueness and homogenization for a class of noncoercive operators in divergence form. Special issue in honor of Calogero Vinti[END_REF]). Due to the lack of coercivity, the classical theory for elliptic operators acting between spaces in duality (see [START_REF] Lions | Quelques méthodes de resolution de problèmes aux limites non linéaires[END_REF]) cannot be applied. Despite this difficulty, some papers (see [START_REF] Alvino | Existence results for nonlinear elliptic equations with degenerate coercivity[END_REF], [START_REF] Boccardo | Some elliptic problems with degenerate coercivity[END_REF], [START_REF] Boccardo | Some remarks on a class of elliptic equations with degenerate coercivity[END_REF], [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity[END_REF], [START_REF] Cirmi | On the existence of solutions to non-linear degenerate elliptic equations with measures data[END_REF], [START_REF] Giachetti | Existence results for some nonuniformly elliptic equations with irregular data[END_REF] and [START_REF] Giachetti | Elliptic equations with degenerate coercivity: gradient regularity[END_REF]) have been written about Dirichlet problems with the differential operator A. In particular in [START_REF] Alvino | Existence results for nonlinear elliptic equations with degenerate coercivity[END_REF], [START_REF] Boccardo | Some elliptic problems with degenerate coercivity[END_REF] and [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity[END_REF] it has been shown the existence of solutions to the differential problem

(1.6) -div (a(x, u)∇u) = f in Ω u = 0 on ∂Ω in the case where f ∈ L m (Ω) with m ≥ 1 and γ ∈ (0, 1]. Indeed, let (1.7) r = N m(1 -γ) N -2m and (1.8) q = N m(1 -γ) N -m(1 + γ) . If f ∈ L m (Ω), with 1 ≤ m ≤ max N N +1-γ(N -1) , 1 problem (1.6) admits a solution u such that |u| s ∈ L 1 (Ω), s < r and |∇u| s ∈ L 1 (Ω), s < q . If N N +1-γ(N -1) < m ≤ 2N N (1-γ)+2(γ+1) , then u ∈ W 1,q 0 (Ω). If 2N N (1-γ)+2(γ+1) ≤ m < N 2 , then u ∈ H 1 0 (Ω) ∩ L r (Ω) . If m > N 2 , then u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) .
In the case where γ > 1, the effect of the degenerate coercivity is even worst: indeed problem (1.6) may have no solution at all, even if the datum f is a constant function (see [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity[END_REF] and [START_REF] Alvino | Existence results for nonlinear elliptic equations with degenerate coercivity[END_REF]). In any case the degenerate coercivity of the operator A gives solutions less regular than the solutions to uniformly elliptic problems.

In the spirit of problem (1.1), the first problem that we will consider in this article is

(1.9) -div (a(x, u)∇u) + |u| p-1 u = f in Ω u = 0 on ∂Ω .
We will study the existence of distributional solutions, that is the existence of some function u in W 1,1 0 (Ω) (at least) such that |u| p ∈ L 1 (Ω) and

Ω a(x, u)∇u • ∇ϕ + Ω |u| p-1 u ϕ = Ω f ϕ
for every ϕ ∈ C ∞ 0 (Ω). We will also use a weaker notion of solution, the entropy solution. We recall that this type of solution has been introduced in [START_REF] Benilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF]. To give this definition we need the following lemma (see [START_REF] Benilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF] for the proof). We define

T k (s) = max{-k, min{k, s}} Lemma 1.1. Let u : Ω → R be measurable function such that T k (u) ∈ H 1 0 (Ω) for every k > 0. Then there exists an unique measurable v : Ω → R N such that ∇T k (u) = v χ {|u|<k} q.o. in Ω ∀ k > 0 ;
the map v is called the weak gradient of u. If u ∈ W 1,1 0 (Ω), then the weak gradient of u coincides with its standard distributional gradient.

Definition 1.2. Let f be an L m (Ω) function, with m ≥ 1. A measurable function u : Ω → R such that |u| p ∈ L 1 (Ω) and T k (u) ∈ H 1 0 (Ω) for every k > 0 is an entropy solution to problem (1.9) if Ω a(x, u)∇u • ∇T k (u -ϕ) + Ω |u| p-1 u T k (u -ϕ) ≤ Ω f T k (u -ϕ)
for every k > 0 and for every ϕ ∈ H 1 0 (Ω) ∩ L ∞ (Ω). Note that this definition is useful in the case where the solution of problem (1.9) does not necessary belong to a Sobolev space. Indeed, about the gradient of the solution, it has a sense under the weak hypothesis that ∇T k (u) ∈ L 2 (Ω), we don't need that ∇u ∈ L 1 (Ω), as for distributional solutions.

Definition 1.3. The Marcinkiewicz space M s (Ω), s > 0, is made up of all measurable functions v : Ω → R such that |{|v| ≥ k}| ≤ C k s for all k > 0, for a positive constant C > 0.

One can show that, if |Ω| is finite and if 0 < ε < s -1

(1.10) L s (Ω) ⊂ M s (Ω) ⊂ L s-ε (Ω) .
We can now state our existence results for problem (1.9).

Theorem 1.4. Let f ∈ L 1 (Ω).

(1) If p > γ + 1, then there exists a distributional solution u to problem (1.9) such that

u ∈ W 1,s 0 (Ω) ∩ L p (Ω) , s < 2p γ + 1 + p .
(2) If 0 < p ≤ γ + 1, then there exists an entropy solution u to problem (1.9) such that

|u| p ∈ L 1 (Ω) and |∇u| ∈ M 2p γ+1+p (Ω).
Observe that the presence of the lower order term |u| p-1 u guarantees the existence of a distributional solution if f is a L 1 (Ω) function. On the contrary, problem (1.6) may have no distributional solution, because the summability of the gradient of the solutions may be lower than 1.

In the case where f ∈ L m (Ω) with m > 1, we will prove the following existence result.

Theorem 1.5. Let f ∈ L m (Ω), with m > 1. (1) If p ≥ γ + 1 m -1
, then there exists a distributional solution u to problem (1.9)

such that u ∈ H 1 0 (Ω) ∩ L pm (Ω) . (2) If γ m -1 < p < γ + 1 m -1
, then there exists a distributional solution u to problem (1.9) such that

|u| pm ∈ L 1 (Ω) and u ∈ W 1, 2pm γ+1+p 0
(Ω) .

(

) If 0 < p ≤ γ m -1 3
, then there exists an entropy solution u to problem (1.9) such that

|u| pm ∈ L 1 (Ω) and |∇u| ∈ M 2pm γ+1+p (
Ω). In this case too the lower order term |u| p-1 u has a regularizing effect. Indeed, if p > γ m-1 , we have a distributional solution for any m > 1. This is not always the case for problem (1.6). If p is larger, that is p > γ+1 m-1 we even find a finite energy (i.e., H 1 0 (Ω)) solution, for any m > 1. Notice that in the case where γ = 0, we get the same results obtained in [START_REF] Cirmi | Regularity of the solutions to nonlinear elliptic equations with a lower-order term[END_REF], for elliptic coercive problems; moreover these results are a generalization of [START_REF] Boccardo | Some remarks on a class of elliptic equations with degenerate coercivity[END_REF].

In the spirit of problem (1.2), the second problem that we will study in this paper is

(1.11) -div (a(x, u)∇u) + h(u) = f in Ω u = 0 on Ω
under the following assumptions on the lower order term h. Let σ be a real positive constant; we assume that the function h : [0, σ) → R is increasing, continuous, h(0) = 0 and lim s→σ -h(s) = +∞. We will show the existence of a solution u such that h(u) ∈ L 1 (Ω) and (1.11). This result shows that the presence of the lower order term h(u) has a strong regularizing effect: it is sufficient to compare the summability that we have found to summabilities (1.7) and (1.8) for the solution to the same problem, without the lower order term h(u).

Ω a(x, u)∇u • ∇ϕ + Ω h(u)ϕ = Ω f ϕ for every ϕ ∈ H 1 0 (Ω) ∩ L ∞ (Ω). Theorem 1.6. Let f be a positive L 1 (Ω) function. Then there exists a solution u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) to problem
Notice that the same results of Theorems 1.4, 1.5 and 1.6 can be obtained for more general lower order terms, depending also on the variable x.

Preliminaries

We will use the following approximating problems to prove Theorems 1.4 and 1.5:

(2.1) -div (a(x, T n (v))∇v) + |v| p-1 v = T n (f ) in Ω v = 0 on ∂Ω .
The following lemma will be very useful, as it gives us an a priori estimate on the summability of the solutions to problems (2.1).

Lemma 2.1. Let f ∈ L m (Ω), m ≥ 1.
Then, for every n ∈ N, there exists a solution u n ∈ H 1 0 (Ω) to problems (2.1) such that

(2.2) Ω |u n | pm ≤ Ω |f | m ; u n satisfies (2.3) Ω a(x, T n (u n ))∇u n • ∇ϕ + Ω |u n | p-1 u n ϕ = Ω T n (f ) ϕ for every ϕ ∈ H 1 0 (Ω) . Proof. The operator v → -div (a(x, T n (v))∇v) + |v| p-1 v , v ∈ H 1 0
(Ω) satisfies the hypothesis of standard theorems for elliptic operators (see [START_REF] Lions | Quelques méthodes de resolution de problèmes aux limites non linéaires[END_REF]); therefore there exists a solution u n ∈ H 1 0 (Ω) to problems (2.1) which satisfies (2.3). To show estimate (2.2), we will consider the cases m > 1 e m = 1 separately.

Case m > 1:

Choosing ϕ = |u n | p(m-1) sgn(u n ) in (2.3) one gets Ω |u n | pm ≤ Ω |f ||u n | p(m-1) ;
the Hölder inequality on the right hand side yields

Ω |u n | pm ≤ f m Ω |u n | pm 1-1 m which implies (2.2). Case m = 1: Choosing ϕ = T k (u n ) k in (2.3) we get Ω |u n | p-1 u n T k (u n ) k ≤ Ω T n (f ) T k (u n ) k ≤ Ω |f | .
Fatou's lemma implies, for k → 0, estimate (2.2).

In the study of entropy solutions to problem (1.9) we will need the following lemmas.

Lemma 2.2. Let u be a measurable function in M s (Ω), s > 0, and suppose that there exists a positive constant ρ > 0 such that

Ω |∇T k (u)| 2 ≤ C k ρ ∀ k > 0 , ∀ n ∈ N . Then |∇u| ∈ M 2s ρ+s (Ω). Remark 2.3. Lemma 2.2 is true for sequences too. That is, if u n is a sequence of measurable functions such that |{|u n | ≥ k}| ≤ C k s ∀ n ∈ N s > 0,
and there exists a positive constant ρ > 0 such that

Ω |∇T k (u n )| 2 ≤ C k ρ ∀ k > 0 , then |{|∇u n | ≥ k}| ≤ C k 2s ρ+s ∀ n ∈ N .
Proof. See [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity[END_REF].

Lemma 2.4. Let u n be a sequence of measurable functions such that T k (u n ) is bounded in H 1 0 (Ω) for every k > 0. Then there exists a measurable function u such that T k (u) ∈ H 1 0 (Ω) and, up to a subsequence, u n → u a.e. in Ω and T k (u n ) → T k (u) weakly in H 1 0 (Ω). Proof. See [START_REF] Benilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF].

Proofs

To show Theorems 1.4 and 1.5, we will pass to the limit in (2.3). All along the proofs C will denote a constant depending only on p, m, α, f m and γ.

Let us begin with the proof of 1.4.

Proof. We divide our proof into two parts: in the first one we will study the case where p > γ + 1, showing the existence of a distributional solution to problem (1.9), and in the second one we will show the existence of an entropy solution for 0 < p ≤ γ + 1.

PART I: Let p > γ + 1. If we choose ϕ = [(1 + |u n |) 1-λ -1]sgn(u n ), with λ > 1, in (2.
3) we get, using assumption (1.4) on a

(3.1) C Ω |∇u n | 2 (1 + |u n |) γ+λ ≤ Ω |f | . Now, let q < 2; writing Ω |∇u n | q = Ω |∇u n | q (1 + |u n |) q 2 (γ+λ) (1 + |u n |) q 2 (γ+λ)
and using the Hölder inequality with exponent 2 q , we get from (3.1)

Ω |∇u n | q ≤ C Ω (1 + |u n |) q 2-q (γ+λ) 1-q 2 .
Thanks to Lemma 2.1, the right hand side is uniformly bounded if q 2-q (γ + λ) = p, that is q = 2p γ+p+λ . Since λ > 1, then q > 1 and so we get that u n is uniformly bounded in W 1,β 0 (Ω), for β < 2p γ+p+1 . As a consequence there exists a function u ∈ W 1,β 0 (Ω), β < 2p γ+p+1 such that u n → u weakly in W 1,β 0 (Ω), up to a subsequence. Moreover u n → u a.e. in Ω and this implies that u ∈ L p (Ω).

We are going to show that u is a distributional solution to problem (1.9) passing to the limit in (2.3). We will suppose that ϕ ∈ C ∞ 0 (Ω). For the first term of the left-hand side, it is sufficient to observe that ∇u n → ∇u weakly in L β (Ω) and a(x, T n (u n ))∇ϕ → a(x, u)∇ϕ in L m (Ω) for every m, due to assumption (1.5) on a. The limit of the second term is a little more delicate. Even if the principal part of our differential operator is degenerate, on can show the following estimate, in the spirit of [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF]:

(3.2) {|un|>t} |u n | p ≤ {|un|>t} |f | .
Indeed, let ψ i be a sequence of increasing, positive, uniformly bounded C ∞ (Ω) functions, such that

ψ i (s) →    1, s ≥ t 0, |s| < t -1, s ≤ -t . Choosing ψ i (u n ) in (2.3), we get Ω |u n | p-1 u n ψ i (u n ) ≤ Ω T n (f ) ψ i (u n ) .
The limit on i implies (3.2). We are going to use this inequality to show that if E is any measurable subset of Ω, then lim

|E|→0 E
|u n | p = 0 uniformly with respect to n .

Using assumption (3.2), for any t > 0 we have

E |u n | p ≤ t p |E| + E∩{|un|>t} |u n | p ≤ t p |E| + {|un|>t} |f | .
The previous lemma and the fact that f ∈ L 1 (Ω) allow us to say that for any given ε > 0, there exists t ε such that

{|un|>tε} |f | ≤ ε .
In this way

E |u n | p ≤ t p ε |E| + ε and so lim |E|→0 E |u n | p ≤ ε ∀ ε > 0 .
We thus proved that lim

|E|→0 E
|u n | p = 0 uniformly with respect to n. Vitali's theorem implies that |u n | p-1 u n → |u| p-1 u in L 1 (Ω) and so we can pass to the limit in the second term of the left-hand side of (2.3). We get a distributional solution to problem (1.9) in this way.

Part II: Let p > 0. Let us choose T k (u n ) as test function in (2.3):

Ω a(x, T n (u n ))∇u n • ∇T k (u n ) + Ω |u n | p-1 u n T k (u n ) = Ω T n (f ) T k (u n ) .
In this way we have

Ω a(x, T n (u n ))∇u n • ∇T k (u n ) ≤ C k ,
and so

Ω |∇T k (u n )| 2 ≤ Ck(1 + k) γ .
Lemma 2.4 implies the existence of a measurable function u such that ∀ k > 0, T k (u) ∈ H 1 0 (Ω), and, up to a subsequence, T k (u n ) → T k (u) weakly in H 1 0 (Ω), and u n → u a.e. in Ω. Fatou's lemma implies that |u| p ∈ L 1 (Ω). To prove that |∇u| ∈ M 2p γ+p+1 (Ω), one can pass to the limit, as n → ∞, in the previous inequality to get

Ω |∇T k (u)| 2 ≤ C k (1 + k) γ . Using Lemma 2.2, |∇u| ∈ M 2p γ+1+p (Ω).
We claim that u is an entropy solution. Indeed, let us choose

T k (u n -ϕ), ϕ ∈ H 1 0 (Ω) ∩ L ∞ (Ω) as test function in (2.3): Ω a(x, T n (u n ))∇u n •∇T k (u n -ϕ)+ Ω |u n | p-1 u n T k (u n -ϕ) = Ω T n (f ) T k (u n -ϕ) .
For the second term of the left hand side and for the right hand side, one can use the Fatou's lemma to pass to the limit. For the first term of the left hand side, we can use the same technique as in [START_REF] Boccardo | Existence and regularity results for some elliptic equations with degenerate coercivity[END_REF] (Theorem 1.17). Indeed, let us write it as

Ω a(x, T n (u n ))|∇T k (u n -ϕ)| 2 + Ω a(x, T n (u n ))∇ϕ • ∇T k (u n -ϕ) .
Passing to the limit in the first term, we have, since

T k (u n ) → T k (u) weakly in H 1 0 (Ω) and u n → u a.e. in Ω, lim inf n→∞ Ω a(x, T n (u n ))|∇T k (u n -ϕ)| 2 ≥ Ω a(x, u)|∇T k (u -ϕ)| 2 ;
the second term tends to

Ω a(x, u)∇ϕ • ∇T k (u -ϕ) . Therefore lim inf n→∞ Ω a(x, T n (u n ))∇u n • ∇T k (u n -ϕ) ≥ Ω a(x, u)∇u • ∇T k (u -ϕ) .
Let us show Theorem 1.5, that is the existence of solutions to problem (1.9) in the case where f ∈ L m (Ω), m > 1.

Proof. We will divide our proof into three parts, according to the different values of p.

PART I: Suppose

p ≥ γ + 1 m -1 . If we choose ϕ = [(1 + |u n |) γ+1 -1]sgn(u n ) in (2.
3) we get, using assumption (1.4) on a

C Ω |∇u n | 2 (1 + |T n (u n )|) γ (1 + |T n (u n )|) γ ≤ Ω |T n (f )| [(1 + |u n |) γ+1 -1] . As a consequence Ω |∇u n | 2 ≤ C Ω |f ||u n | γ+1 .
The Hölder inequality on the right hand side and Lemma 2.1 imply that

Ω |f ||u n | γ+1 ≤ f m   Ω |u n | (γ+1) m m-1   m-1 m < ∞ if (γ + 1) m m-1 ≤ pm. Under this assumption Ω |∇u n | 2 ≤ C , ∀ n ∈ N .
Up to a subsequence, there exists a function u ∈ H 1 0 (Ω) such that u n → u weakly in H 1 0 (Ω) and a.e. in Ω. Moreover u ∈ L pm (Ω). We are going to show that u is a solution to problem (1.9), passing to the limit in (2.3). For the first term of the left-hand side, it is sufficient to observe that a(x, T n (u n ))∇ϕ → a(x, u)∇ϕ in L r (Ω), for any r ≥ 1, due to assumption (1.5) on a. For the second term, since |u n | p-1 u n is uniformly bounded in L m (Ω) with m > 1 and u n → u a.e., we can deduce that

|u n | p-1 u n → |u| p-1 u in L 1 (Ω). PART II: Suppose γ m -1 < p < γ + 1 m -1 . If we choose ϕ = [(1 + |u n |) p(m-1) - 1]sgn(u n ) in (2.
3), we get, using assumption (1.4) on a 1) . Now, using the Hölder inequality in the right hand side of the previous inequality and Lemma 2.1, we get

Ω |∇u n | 2 (1 + |u n |) γ-p(m-1)+1 ≤ C Ω |f | |u n | p(m-
(3.3) Ω |∇u n | 2 (1 + |u n |) γ-p(m-1)+1 ≤ C   Ω |u n | pm   1-1 m ≤ C ∀ n ∈ N .
From the other hand, let σ < 2; writing

Ω |∇u n | σ = Ω |∇u n | σ (1 + |u n |) σ 2 [γ-p(m-1)+1] (1 + |u n |) σ 2 [γ-p(m-1)+1]
and using the Hölder inequality with exponent 2 σ , we get from (3.3)

Ω |∇u n | σ ≤ C Ω (1 + |u n |) σ 2-σ [γ-p(m-1)+1] 1-σ 2 . Thanks to Lemma 2.1, if σ 2 -σ [γ -p(m -1) + 1] = pm, that is σ = 2pm γ + p + 1
, the last quantity is uniformly bounded. Notice that σ < 2, since we are assuming that

p < γ + 1 m -1 . If 2pm γ + p + 1 > 1, the fact that Ω |∇u n | 2pm γ+p+1 ≤ C ∀ n ∈ N implies the existence of a function u ∈ W 1, 2pm γ+p+1 0
(Ω) such that, up to a subsequence,

u n → u weakly in W 1, 2pm γ+p+1 0
(Ω) and a.e. in Ω; moreover |u| pm ∈ L 1 (Ω). One can show that u is a distributional solution to problem (1.9), as in PART I.

PART III: Suppose that p ≤ γ m -1 . We show that there exists an entropy solution to problem (1.9). Estimate (3.3) (obtained indipendently from the value of p) implies that

{|un|>k} |∇u n | 2 (1 + |u n |) γ-p(m-1)+1 ≤ C and consequently {|un|<k} |∇T k (u n )| 2 ≤ C(1 + k) γ-p(m-1)+1 .
Thanks to Lemma 2.4, there exists a function u such that

T k (u) ∈ H 1 0 (Ω), ∀ k, T k (u n ) → T k (u) weakly in H 1 0
(Ω) and u n → u a.e. in Ω. Therefore we can pass to the limit as n → ∞ in the previous inequality to get

Ω |∇T k (u)| 2 ≤ C(1 + k) γ-p(m-1)+1 . Lemma 2.2 gives us, if p < γ+1 m-1 |∇u| ∈ M 2pm γ+1+p (Ω) . Since |u n | pm is uniformly bounded in L 1 (Ω)
, we can say that |u| pm ∈ L 1 (Ω). One can show that u is an entropy solution, using the same method as in PART II of the proof of Theorem 1.4.

We are going to show Theorem 1.6. We will follow the technique used in [START_REF] Boccardo | On the regularizing effect of strongly increasing lower order terms[END_REF]. In the lemma below we prove the existence of a solution to problem (1.11) in the case where the datum f is a bounded function.

Lemma 3.1. Let f be a positive L ∞ (Ω) function. Then there exists a distributional solution u ∈ L ∞ (Ω) ∩ H 1 0 (Ω) to problem (1.11). Moreover 0 ≤ u(x) ≤ σ a.e. in Ω. Proof. The standard theorems on elliptic operators (see [START_REF] Lions | Quelques méthodes de resolution de problèmes aux limites non linéaires[END_REF]) imply the existence of a solution u n ∈ H 1 0 (Ω) to the approximating problems -div (a(x,

T n (v))∇v) + h n (v) = f in Ω v = 0 on Ω where h n (s) =    h(s), h(s) < n and s < σ n, h(s) ≥ n and s < σ n, s ≥ σ . The use of (u n -h -1 ( f ∞ )) + as test function yields Ω (h n (u n ) -f )(u n -h -1 ( f ∞ )) + ≤ 0.
From the previous inequality we have

0 ≥ {h(un)≥ f ∞ } (T n (h(u n )) -f )(u n -h -1 ( f ∞ )) = {n>h(un)≥ f ∞ } (h(u n ) -f )(u n -h -1 ( f ∞ )) + {h(un)≥n≥ f ∞ } (n -f )(u n -h -1 ( f ∞ )) ≥ 0. This implies that 0 ≤ h(u n ) ≤ f ∞ ,
and so

(3.4) 0 ≤ u n ≤ h -1 ( f ∞ ) ≤ σ -ε .
Choosing ϕ = u n we get an uniform bound for the H 1 0 (Ω) norm of u n . Indeed using assumption (1.4) on a and the fact that h(u n ) ≥ 0, (3.4)

implies that α Ω |∇u n | 2 (1 + σ) γ ≤ f 1 σ .
This yields the existence of a function u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) such that, up to a subsequence, u n → u weakly in H 1 0 (Ω) and a.e. in Ω. Remark that u n = T n (u n ) for n large enough. Consequently u n satisfies

Ω a(x, u n )∇u n • ∇ϕ + Ω h n (u n )ϕ = Ω f ϕ ∀ ϕ ∈ H 1 0 (Ω) ∩ L ∞ (Ω) .
Let us show that u is a solution to (1.11), passing to the limit in the previous equality. About the first term it is sufficient to observe that a(x, u n )∇u n → a(x, u)∇u weakly in L 2 (Ω), because a(x, u n ) → a(x, u) in L 2 (Ω) and ∇u n → ∇u weakly in L 2 (Ω). One can use the same technique as in [START_REF] Boccardo | On the regularizing effect of strongly increasing lower order terms[END_REF] to pass to the limit in the second term using (3.4) and to show that h(u) ∈ L 1 (Ω).

We remark that the previous lemma gives us an upper bound for the solution u (that is u ≤ σε) independently on the datum f .

We are now able to show Theorem 1.6.

Proof. The following sequence of approximating problems -div (a(x, v)∇v) + h(v) = T n (f ) in Ω v = 0 on ∂Ω has a solution u n such that 0 ≤ u n ≤ σε for every n ∈ N, according to the previous lemma; u n satisfies (3.5)

Ω a(x, u n )∇u n • ∇ϕ + Ω h n (u n )ϕ = Ω T n (f ) ϕ ∀ ϕ ∈ H 1 0 (Ω) ∩ L ∞ (Ω) .
As in the previous lemma one has

α Ω |∇u n | 2 (1 + σ) γ ≤ ||f || 1 σ .
Therefore there exists a function u ∈ H 1 0 (Ω) such that, up to a subsequence, u n → u weakly in H 1 0 (Ω) and a.e. in Ω; moreover 0 ≤ u ≤ σ. To show that u is a solution to (1.11), we pass to the limit in (3.5). As in the previous Lemma, it is easy to prove that a(x, u n )∇u n → a(x, u)∇u weakly in L 2 (Ω). One can use the same technique as in [START_REF] Boccardo | On the regularizing effect of strongly increasing lower order terms[END_REF] to pass to the limit in the second term. Indeed showing the estimate Since lim s→σ |{s ≤ u n < σ}| = 0 we get that {h(u n )} is equi-integrable. This implies that h(u n ) → h(u) in L 1 (Ω), as required.

  |a(x, s)| ≤ β

  {s≤un<σ} h(u n ) ≤ {s≤un<σ} f one can prove that for any measurable set E ⊂ Ω E h(u n ) ≤ {s≤un<σ} f + h(s)|E| .


A Sarthou
email: sarthou@ipb.fr

JP S Vincent

Caltagirone

A second-order curvilinear to Cartesian transformation of immersed interfaces and boundaries. Application to ctitious domains and multiphase ows

Keywords: Curvilinear grids, structured Cartesian grids, ctitious domains, grid transformation, Front Tracking, VOF, Level-Set, multiphase ows

A global methodology dealing with ctitious domains of all kinds on orthogonal curvilinear grids is presented. The main idea is to transform the curvilinear workframe and its associated elements (velocity, immersed interfaces...) into a Cartesian grid. On such a grid, many operations can be performed much faster than on curvilinear grids. The method is coupled with a Thread Ray-casting algorithm which work on Cartesian grids only. This algorithm computes quickly the Heaviside function related to the interior of an object on an Eulerian grid. The approach is also coupled with an immersed boundary method (L 2 -penalty method) or with phase advection with VOF-PLIC, VOF-TVD, Front-tracking or Level-set methods. Applications, convergence and speed tests are performed for shape initializations, immersed boundary methods, and interface tracking.

Introduction

Structured Cartesian grids are convenient to use in CFD simulations. However, this kind of mesh structure involves many limitations on the shape of the numerical domain. A rst solution is to use ctitious domain methods (e.g. immersed boundary methods) which enable the immersion of complex boundaries on Cartesian grids. An other solution is to use curvilinear structured grids which couple some advantages of the structured approach (trivial connectivities) with the possibility to deal accurately with complex numerical boundaries and rened boundary layers.

In [START_REF] Thompson | General curvilinear coordinate systems[END_REF], Thompson proposes a general method to work with curvilinear grids. The main idea is to consider a xed square grid in a rectangular transformed region where the numerical calculations are performed with modied operators. This approach can be extended to moving boundaries or interfaces and does not require the eective construction of the transformed domain. Nevertheless, the question of the motion and reconstruction of the mesh (especially the implementation eort and the computational cost), as for the ALE method, is asked. This point is critical for multiphase ows and particularly for dispersed phases or for cases with many moving objects, such as uidized beds. Even if less accurate, the ctitious domain methods seem to be more convenient and fast as the calculation grid can remain xed.

One can found some examples of ctitious domains on curvilinear grids in the literature. An adaptation of the VOF-PLIC method on curvilinear grids can be found in [START_REF] Jang | A study on the extension of a vof/plic based method to a curvilinear co-ordinate system[END_REF]. In [START_REF] Muradoglu | An auxiliary grid method for computations of multiphase ows in complex geometries[END_REF], Muradoglu and Kayaalp use an auxiliary Cartesian grid superimposed with the curvilinear mesh to manage the front tracking operations. Kernel functions are used to interpolate the velocity from one grid to an other. In [START_REF] Huang | Coupled ghost uid/two-pahse level set method for curvilinear body-tter grids[END_REF], Huang et al. extend the Ghost-uid method [START_REF] Fedkiw | A non-oscillatory eulerian approach to interfaces in multimaterial ows (the ghost uid method)[END_REF] to curvilinear grids. In [START_REF] Borazjani | Curvilinear immersed boundary method for simulating uid structure interaction with complex 3d rigid bodies[END_REF][START_REF] Roman | An improved immersed boundary method for curvilinear grids[END_REF], authors perform a uid-structure coupling with an immersed boundary method on curvilinear grids. In these works, the transformation dened by [START_REF] Thompson | General curvilinear coordinate systems[END_REF] is not used.

The present work proposes an optimized approach using the transformation of [START_REF] Thompson | General curvilinear coordinate systems[END_REF] to deal with immobile boundaries while moving interfaces and boundaries are treated with ctitious domain methods. A transformation of the original immersed interface is proposed and its projection onto a volume function is accelerated thanks to the use of the transformed Cartesian space. Contrary to the aforementioned references, immersed boundary methods and phase advection are performed considering only the transformed Cartesian grids with the original ad hoc Cartesian algorithms.

Methodology

The method is presented in 2D. The extension to 3D problems is straightforward. Let us consider a curvilinear orthogonal structured mesh T h covering a numerical domain Ω. Its structured vertices are denoted x c (c denotes the doublet of indexes (i, j)), with 1 i L, 1 j M (and 1 k N in 3D). The mesh cells are denoted K c with x c the south-west vertex of each cell. An embedded domain of complex interface Σ divides Ω in two subdomains Ω 0 and Ω 1 . If uid ows are considered, each subdomain can represent a phase or a solid object. The base principle is to unfold the curvilinear triangulation T h to obtain a Cartesian grid Th of unit space step dening a new computational domain Ω. For this purpose, we dene a bi-continuous bijective operator F : Ω → Ω such as the transformed position of a grid node

x c is F (x c ) = xc = (i, j) T .
Once the Cartesian grid has been built, the immersed interface Σ has to be unfolded too. Numerically, Σ is piecewise linear and can be dened by its vertices denoted by σ l of position x l . In the Cartesian frame, Σ becomes a deformed interface Σ composed of vertices σl . The transformation F is built cell by cell. The restriction of F to K c is the operator

F c = (F 1 c , F 2 c
) T , and we have

F 1 c (x c ) = xc , F 2 c (x c ) = ŷc . Each operator F n is built with a Q 1 polynomial. In 2D, Q 1 (x, y) = a 1 xy + a 2 x + a 3 y + a 4 and in 3D, Q 1 (x, y, z) = a 1 xyz + a 2 xy + a 3 yz + a 4 zx + a 5 x + a 6 y + a 7 z + a 8 .
The coecients of each Q 1 are dened by solving two systems composed of the equations F 1 c (x c,n ) = xc,n , n = 1, 2, 3, 4 and the equations F 2 c (x c,n ) = ŷc,n , n = 1, 2, 3, 4 where x c,n are the positions of the 4 vertices of the cell K c (see Fig. 1). Consequently, 3 systems of 8 equations are solved in 3D for all cells. The analytical solution can be used in 2D while a numerical solver is required in 3D.

The new position of each Lagrangian vertex σ l can be obtained with xl = F (x l ). The cell containing the vertex is found thanks to a search algorithm. The distance between the vertex and a rst Eulerian node is calculated. A new node which is a neighbor of the previous one is chosen such as the distance to the vertex is decreasing. Such a "point-in-cell" operation has to be performed only at the start of the whole calculation as the Cartesian frame is used for the further localizations. As demonstrated below, the computational cost of The discrete binary Heaviside function χ is dened as:

χ(x) =    1 if x ∈ Ω 1 0.5 if x ∈ Σ 0 otherwise (1)
This function is the binary indicator of the presence of an Eulerian point in Ω 1 and is built with a point-in-solid method presented below. An rst-order accurate localization of the interface can be retrieved with χ. This function is obtained with a Ray-casting (RC) method. The principle is to cast a ray from each Eulerian point to innity and to test the number of intersections between the ray and the Lagrangian mesh. If the number of intersections is odd, the Eulerian point is inside the object, and outside otherwise. The RC method can be enhanced by classifying the elements of the Lagrangian mesh with an octree sub-structure [START_REF] Puech | Quadtrees, octrees, hyperoctrees: a unied analytical approach to tree data structures used in graphics, geometric modeling and image processing[END_REF] which recursively subdivides the space in boxes. If a ray does not intersect a box, it does not intersect the triangles inside the box. Some details on the implementation and a short review of point-in-solid strategies can be found in [START_REF] Ogayar | Point in solid strategies[END_REF].

Concerning the standard RC method, its computational cost is generally very expensive as L × M × N × S intersections tests have to be performed, with S the number of elements of the Lagrangian mesh. Acceptable times are reached (with L × M × N × log(S) intersections) with an octree but the implementation eort is non-negligible. A simple optimization, called here the Thread Raycasting (TRC) which is known in the computer graphics community, is used. For the standard RC method, the direction of a ray is indierent. If all rays are launched in the same direction, Ox for instance, many intersection tests are done more than once for a set of points in a same Eulerian mesh row in the Ox direction. Hence, only one ray can be cast per row. If rays are cast in a given direction (the best choice is the one with the greatest number of cells), the computational cost is divided by the number of cells in this direction and we have (L × M × N × S)/ max(L, M, N ) intersections to test. Ideally, this method is coupled with the octree.

It is important to notice that this method works only on Cartesian meshes. The computational cost of the curvilinear to Cartesian transformation is justied by the gain of times between the RC and the TRC.

As F is bi-continuous and bijective, x i,j,k ∈ Ω 1 ⇔ xi,j,k ∈ Ω1 and χ(x i,j,k ) = χ(x i,j,k ). As a consequence, the discrete Heaviside function in structured storage veries χ (i,j,k) = χ(i,j,k) and χ can be computed in the Cartesian frame. This property is however true for the continuous transformation. From a discrete point of view, this property is true for a sucient number S of Lagrangian elements σ l .

The Level-set function

The level-set (LS) function φ is

φ(x) = -dist Σ (x) if x ∈ Ω 1 dist Σ (x) otherwise (2)
with dist Σ (x) = inf p∈Σ x -p . The unsigned distance is computed geometrically and the sign is obtained from χ. If the distance is calculated for all Eulerian nodes (a simple optimisation is to compute the distance only near the interface), this operation is generally expensive as L × M × N × S distance tests have to be performed. Details on implementation and optimisations can be found in [START_REF] Jones | 3d distance elds: A survey of techniques and applications[END_REF]. In 1D, the LS function gives the exact location of the interface. For higher dimensions, a second-order localization of the interface can be obtained by taking φ = 0. The normal n and the curvature κ of the LS function are obtained with

n = ∇φ ∇φ , κ = ∇ • ∇φ ∇φ ( 3
)
Theses quantities are wrong in the curvilinear domain if the Level-set function is computed in the Cartesian domain. However, the interface normal n in the curvilinear frame can be retrieved from φ computed in the Cartesian frame with

n = 1 ∆ξ , 1 ∆η , 1 ∆ζ T ∇ • φ 1 ∆ξ , 1 ∆η , 1 ∆ζ T ∇ • φ (4)
with ξ, η, ζ the curvilinear coordinates,∆ξ, ∆η, ∆ζ the local mesh steps and ∇

the Cartesian gradient operator. Using n, the curvature can be computed with the curvilinear divergence operator.

The VOF function

The volume-of-uid (VOF) function C, also called the color or phase function, is the volume ratio of a given phase in an elementary volume (generally the nite volume cell). This function is typically used to localize a uid phase in multiphase ows and is the base of the 1-uid model [START_REF] Kataoka | Local instant formulation of two-phase ow[END_REF]. This function can be computed from the values of the Level-set function near the interface with the formula of [START_REF] Sussman | A level set approach for computing solutions to incompressible two-phase ow[END_REF]:

C(x) ≈        0 if φ < - 1 2 1 + φ - 1 π sin(πφ/ ) if |φ| 1 if φ > (5)
with ε the interface width. An alternative method is proposed in [START_REF] Shin | Modelling three-dimensional multiphase ow using a level contour reconstruction method for front tracking without connectivity[END_REF] and does not required χ. The following Poisson equation is solved:

∇ 2 C = ∇ • Σ nδ(x -x i )dΓ ( 6
)
where n is the unit normal to the interface, δ a smoothed Dirac function, x a position on the Eulerian grid and x l a marker (vertex) position on Σ. The original interface is recovered in C = 0.

Immersed boundary method

A L 2 -penalty method is used to impose an immersed Dirichlet boundary condition on Σ. A penalty term χ ε (u -u l ) is added to the initial conservation equation. As ε 1, the initial equation is negligible where χ = 1 (i.e. in Ω 1 ). To reach a second order of accuracy, the sub-mesh penalty method (SMPM) [START_REF] Sarthou | Eulerian-Lagrangian grid coupling and penalty methods for the simulation of multiphase ows interacting with complex objects[END_REF] is used. The penalty term is discretized as χ ε (P i u -u l ) where P i is a local interpolator of the solution from both sides of the interface. Polynomials are used to construct P i . As the interface location is required to build the penalty term, the intersections between Σ and the mesh lines are used. These points can be found with a geometric intersection test, or using the Level-set and VOF functions. Both gives an implicit representation of the interfacer through their isosurfaces (φ = 0 and C = 0.5). As demonstrated later, choosing one or an other impacts on the accuracy of the SMPM.

Phase advection

The advection of phases can be performed using the Cartesian frame. The initial advection equation is written

∂C ∂t + u • ∇C = 0 (7)
with ∇ the curvilinear gradient operator. One can write

∇ = ( ∂ ∂ξ , ∂ ∂η , ∂ ∂ζ ) T = ( ∂x ∂ξ , ∂y ∂η , ∂z ∂ζ ) T ( ∂ ∂x , ∂ ∂y , ∂ ∂z ) T (8)
Consequently,

∂C ∂t + u • ∇C = ∂C ∂t + û • ∇ Ĉ = 0 (9)
with the transformed velocity in the Cartesian frame û = If we consider in 1D the initial position x 0 of a characteristic curve, its evolution in the curvilinear frame is given by x(t) = x 0 + ut. If the characteristic curve covers a cell in a time T with a velocity u in the original frame, one can see that a velocity u∆x/∆ξ is required to cover a transformed cell in the same time.

Other applications

As the Cartesian frame has a unit space step, any point (x l , y l , z l ) of the Cartesian space belongs to the cell K x l , y l , z l . This property saves computational time for point-in-cell operations, e.g. when a quantity has to be extrapolated from the Eulerian grid to a Lagrangian point. In [START_REF] Vincent | Augmented Lagrangian and penalty methods for the simulation of two-phase ows interacting with moving solids. Application to hydroplaning ows interacting with real tire tread patterns[END_REF], the 3D simulation of the hydroplaning of a tire is performed on a irregular mesh and the present algorithms are used to deal with multiphase ows and the calculation of the forces exerted by the uid on the tire. The pressure and stress tensor are extrapolated on the Lagrangian surface of the tire in order to determine the drag and lift forces. For each surface element, the extrapolation requires to pick 3 points in the uid and to interpolate a quantity from the Eulerian nodes to these Lagrangian nodes. The same occurs for the Euler-Lagrange advection methods, such as the VOF-SM method [START_REF] Vincent | Eulerian-lagrangian multiscale methods for solving scalar equations -application to incompressible two-phase ows[END_REF], which requires many coupling between Lagrangian and Eulerian informations. For each interpolations from the Cartesian grid to a Lagrangian point, a unique simple Q 1 interpolation is used.

Parallelism

As these operations are generally independently performed with respect to Eulerian or Lagrangian elements, the present algorithms are easy to parallelize with OpenMP or GPGPU approaches. Concerning domain decomposition, for MPI approaches, the interface Σ is not always entirely inside or outside the computational domain treated by a processor. The present algorithms generally works in this case, except for the LCR VOF construction (6) (the resolved equation requires the surface to be entirely inside the domain), and the curvilinear to Cartesian transformation. If a Lagrangian vertex σ l is not inside an Eulerian cell, its transformation cannot be determined (eventually approximated for some curvilinear grids with irregular mesh steps and straight boundaries). For both cases, the solution is to cut Σ and to close it with parts of the segments which denes the boundaries of the Eulerian grid.

3 Results

Computational time

Speed tests have been performed on a P4 2.4 GHz for several meshes with and without the octree. Three Lagrangian meshes are used, a sphere (18000 triangles, regular), the Stanford bunny (Fig. 3, 10122 triangles, irregular) and the Lascaux cave ([9], 271136 triangles, irregular).

Figure 3: The Stanford bunny

The Tab. 1 shows computational times for these three Lagrangian meshes coupled with a 100 3 Eulerian mesh, for three operations: Transf. is the curvilinear to Cartesian transformation, TRC the Thread Ray-casting method and LS the computation of the Level-set function. The computational time of the standard Ray-casting method is not measured here but is supposed to be 100 times higher than for the TRC. Hence, even if the transformation is longer to perform than the TRC, the total cost is negligible compared to the RC. As expected, all the routines are faster with the octree. The gain for the curvilinear to Cartesian projection is between 1.35 and 1.73 only. For this case, a derivation of the octree for point-in-cell operations, a kD-tree [START_REF] Louis | Multidimensional binary search trees used for associative searching[END_REF], is used. For the Ray-casting, the gain is smaller than 2 on small meshes, and more than 10 for the Lascaux cave, where an octree with more subdivisions can be used. For the computation of the Level-set, the gain ratio is from 35 to 75.

The most important result is that the maximum computational time for the whole method is shorter than a minute on a P4 with and without the octree negligible against the computational time required to solve the linear system resulting from the discretization of a conservation equation with a 100 3 mesh.

Concerning the computational time of the SMPM, the construction of the penalty term is negligible when volume functions are used. When the Lagrangian elements are used, the method takes roughly the same time as the TRC.

Immersed boundary method

The SMPM coupled with diverse surface representations is tested here. The accuracy of the surface representations can be deduced from the accuracy of the SMPM for each representation. The Laplace equation is solved with two curvilinear grids (see Fig. 4). A circular interface is considered and a Dirichlet boundary condition is imposed on the circle with the SMPM [START_REF] Sarthou | Eulerian-Lagrangian grid coupling and penalty methods for the simulation of multiphase ows interacting with complex objects[END_REF]. This method has a second order of space accuracy and requires the accurate location of the boundary. The computational mesh is a curvilinear converging pipe. Fig. 5 left shows a spatial convergence study for the L 2 and L ∞ error norms. The interface is located either with Σ (GI), the curvilinear Level-set function (LS-CURVI), the Cartesian Level-set function (LS), the Sussman Heaviside reconstructed function (SUS) 5 and the VOF function from the Front-Tracking method (FT). As can be seen, the GI, LS-CURVI and LS methods allow the SMP method to almost keep its second order. The expected loss of accuracy between the LS-CURVI and the LS methods is too slight to disqualify the use of the Cartesian grid. The SUS method shows lower performances due to its construction using a sinus function (with a linear reconstruction, the same accuracy as with the LS is found). The FT shows a good convergence for the rst meshes (with a second order) but is globally the less accurate function and does not converge for the nest meshes. An implementation error or a saturation eect could be involved.

Multiphase ows

The methodology is used with VOF-PLIC [START_REF] Youngs | Time-dependent multimaterial ow with large uid distortion[END_REF], Level-Set [START_REF] Sussman | A level set approach for computing solutions to incompressible two-phase ow[END_REF], VOF-TVD [START_REF] Vincent | Ecient solving method for unsteady incompressible interfacial ow problems[END_REF] and LCR Front-Tracking [START_REF] Shin | Modelling three-dimensional multiphase ow using a level contour reconstruction method for front tracking without connectivity[END_REF] advection methods. The Fig. 5 right shows a cross-shaped phase advected at a front positive speed on a curvilinear grid with a Front-Tracking method. The shape and mass of the initial phase are well conserved if only the Lagrangian markers are considered (which is not the case for the 1-uid model where the VOF function is used). Fig. 7 shows the convergence of the error on the mass conservation. As expected, a regular second-order of convergence is found for the LS and FT methods. Concerning the VOF-PLIC and VOF-TVD methods, the transformation removes their ability to exactly conserve the phase volume and the conservation depends on the accuracy of the interface advection. It is known for the VOF-PLIC method that the convergence order of the interface position decreases with the mesh [START_REF] Rider | Reconstructing volume tracking[END_REF][START_REF] Aulisa | Interface reconstruction with least-squares t and split advection in three-dimensional cartesian geometry[END_REF]. Whatsoever, the VOF-PLIC and VOF-TVD methods remain globally more accurate than the two other methods. For the FT method, the error on the advection of the Lagrangian markers is negligible against the error on the LCR operation 6. Contrary to the test in section 3.2, the saturation is not reached and a similar second order is obtained. However, the reconstruction of the VOF function using the method of [START_REF] Shin | Modelling three-dimensional multiphase ow using a level contour reconstruction method for front tracking without connectivity[END_REF] could be replaced by a more accurate method. Concerning the TVD method, good results are obtained but the cause of its irregular convergence has to be investigated. A new second-order methodology to work with ctitious domains on curvilinear grids has been drawn here. It allows methods implemented on Cartesian grids to be easily extended to curvilinear grids saving implementation eorts and computational cost. The Thread Ray-casting is used on curvilinear grids and greatly improves the performances of the Heaviside construction. Used with many implicit and explicit interface representations, a second-order immersed boundary method keeps its convergence order. Concerning the phase advection, a transformed velocity eld has been coupled with the Cartesian frame to perform the advection for curvilinear cases with Cartesian algorithms. Future works will be devoted to nd a way to retrieve the volume conservation property for the VOF-TVD and VOF-PLIC methods. Higher-orders curvilinear to Cartesian transformations for the mesh will be studied too. Concerning the FT method (for all kinds of meshes), the initialisation of [START_REF] Shin | Modelling three-dimensional multiphase ow using a level contour reconstruction method for front tracking without connectivity[END_REF] will be replaced by the TRC and a more accurate reconstruction of the VOF function (using the Level-set function or a geometrical calculation of the volume ratio).

Figure 1 :

 1 Figure 1: Notations in 2D for original and transformed cells

Figure 2 :

 2 Figure 2: The original and transformed meshes T h and Th and the related immersed interfaces Σ and Σ

  operator ∇ = ∂ ∂x , ∂ ∂y , ∂ ∂z T . For each cell, ∂x ∂ξ , ∂y ∂η and ∂z ∂ζ are replaced by the ratios of the local space steps (with ∆x = ∆y = ∆z = 1).

Figure 4 :

 4 Figure 4: A and B meshes

Figure 5 :

 5 Figure 5: Relative L 2 and L ∞ errors for meshes A and B

Figure 6 :

 6 Figure 6: Advection of a cross-shaped phase with a Front-tracking method

Figure 7 :

 7 Figure 7: Relative error on the mass conservation

Table 1 :

 1 2.4 GHz. This time is generally Duration in second and performance ratio for three dierent meshes

	Mesh	Method	Transf. TRC	LS
		Standard	2.06	0.536 270
	Sphere	Octree	1.37	0.332 5.79
		Ratio	1.50	1.61	46.6
		Standard	1.19	0.172 123
	Bunny	Octree	0.88	0.132 1.63
		Ratio	1.35	1.30	75.5
		Standard	45.4	35.9 1970
	Lascaux	Octree	26.1	3.32	56.1
		Ratio	1.73	10.8	35.1

Acknowledgment

The authors gratefully thanks Antoine Lemoine for the implementation of the Octree and kDtree.