Ferruccio Colombini
email: colombini@dm.unipi.it

Daniele Del Santo
email: delsanto@units.it

Francesco Fanelli
email: francesco.fanelli@math.cnrs.fr

Guy Métivier
email: guy.metivier@math.u-bordeaux1.fr

A WELL-POSEDNESS RESULT FOR HYPERBOLIC OPERATORS WITH ZYGMUND COEFFICIENTS

In this paper we prove an energy estimate with no loss of derivatives for a strictly hyperbolic operator with Zygmund continuous second order coecients both in time and in space. In particular, this estimate implies the well-posedness for the related Cauchy problem. On the one hand, this result is quite surprising, because it allows to consider coecients which are not Lipschitz continuous in time. On the other hand, it holds true only in the very special case of initial data in H 1/2 × H -1/2 . Paradierential calculus with parameters is the main ingredient to the proof.

Introduction

This paper is devoted to the study of the Cauchy problem for a second order strictly hyperbolic operator dened in a strip [0, T ] × R N , for some T > 0 and N ≥ 1. Consider a second order operator of the form def:op def:op [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF]

Lu := ∂ 2 t u - N j,k=1
∂ j (a jk (t, x) ∂ k u)

(with a jk = a kj for all j, k) and assume that L is strictly hyperbolic with bounded coecients, i.e. there exist two constants 0 < λ 0 ≤ Λ 0 such that

λ 0 |ξ| 2 ≤ N j,k=1
a jk (t, x) ξ j ξ k ≤ Λ 0 |ξ| 2 for all (t, x) ∈ [0, T ] × R N and all ξ ∈ R N . It is well-known (see e.g. [START_REF] Hörmander | Linear partial dierential operators[END_REF] or [START_REF] Mizohata | The Theory of Partial Dierential Equations[END_REF]) that, if the coecients a jk are Lipschitz continuous with respect to t and only measurable in x, then the Cauchy problem for L is well-posed in H 1 × L 2 . If the a jk 's are Lipschitz continuous with respect to t and C ∞ b (i.e. C ∞ and bounded with all their derivatives) with respect to the space variables, one can recover the well-posedness in H s × H s-1 for all s ∈ R. Moreover, in the latter case, one gets, for all s ∈ R and for a constant C s depending only on it, the following energy estimate:

sup 0≤t≤T u(t, •) H s+1 + ∂ t u(t, •) H s ≤ est:no-loss (2) ≤ C s u(0, •) H s+1 + ∂ t u(0, •) H s + T 0 Lu(t, •) H s dt
for all u ∈ C([0, T ]; H s+1 (R N )) ∩ C 1 ([0, T ]; H s (R N )) such that Lu ∈ L 1 ([0, T ]; H s (R N )). Let us explicitly remark that the previous inequality involves no loss of regularity for the function u: estimate (2) holds for every u ∈ C 2 ([0, T ]; H ∞ (R N )) and the Cauchy problem for L is well-posed in H ∞ with no loss of derivatives.

If the Lipschitz continuity (in time) hypothesis is not fullled, then [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] is no more true, in general. Nevertheless, one can still try to recover H ∞ well-posedness, with a nite loss of derivatives in the energy estimate.

The rst case to consider is the case of the coecients a jk depending only on t:

Lu = ∂ 2 t u - N j,k=1
a jk (t) ∂ j ∂ k u .

In [START_REF] Colombini | Sur les équations hyperboliques avec des coecients qui ne dépendent que du temps[END_REF], Colombini, De Giorgi and Spagnolo assumed the coecients to satisfy an integral log-Lipschitz condition:

hyp:int-LL hyp:int-LL [START_REF] Chemin | Fluides parfaits incompressibles[END_REF] T -ε

0 |a jk (t + ε) -a jk (t)| dt ≤ C ε log 1 + 1 ε ,
for some constant C > 0 and all ε ∈ ]0, T ]. More recently (see paper [START_REF] Tarama | Energy estimate for wave equations with coecients in some Besov type class[END_REF]), Tarama analysed instead the problem when coecients satisfy an integral log-Zygmund condition: there exists a constant C > 0 such that, for all j, k and all ε ∈ ]0, T /2[ , one has hyp:int-LZ hyp:int-LZ [START_REF] Cicognani | Modulus of continuity of the coecients and loss of derivatives in the strictly hyperbolic Cauchy problem[END_REF] T -ε ε |a jk (t + ε) + a jk (t -ε) -2 a jk (t)| dt ≤ C ε log 1 + 1 ε .

On the one hand, this condition is somehow related, for a function a ∈ C 2 ([0, T ]), to the pointwise condition |a(t)| + |t a ′ (t)| + |t 2 a ′′ (t)| ≤ C (considered in [START_REF] Yamazaki | On the L 2 (R n ) well-posedness of some singular or degenerate partial differential equations of hyperbolic type[END_REF] by Yamazaki). On the other hand, it's obvious that if the a jk 's satisfy [START_REF] Chemin | Fluides parfaits incompressibles[END_REF], then they satisfy also [START_REF] Cicognani | Modulus of continuity of the coecients and loss of derivatives in the strictly hyperbolic Cauchy problem[END_REF]: so, a more general class of functions is considered.

Both in [START_REF] Colombini | Sur les équations hyperboliques avec des coecients qui ne dépendent que du temps[END_REF] and [START_REF] Tarama | Energy estimate for wave equations with coecients in some Besov type class[END_REF], the authors proved an energy estimate with a xed loss of derivatives: there exists a constant δ > 0 such that, for all s ∈ R, the inequality

sup 0≤t≤T u(t, •) H s+1-δ + ∂ t u(t, •) H s-δ ≤ est:c-loss (5) ≤ C s u(0, •) H s+1 + ∂ t u(0, •) H s + T 0 Lu(t, •) H s dt
holds true for all u ∈ C 2 ([0, T ]; H ∞ (R N )), for some constant C s depending only on s. Also the case of dependence of the a jk 's both in time and space was deeply studied. In paper [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coecients[END_REF], Colombini and Lerner assumed an isotropic pointwise log-Lipschitz condition, i.e. there exists a constant C > 0 such that, for all ζ = (τ, ξ) ∈ R × R N , ζ = 0, one has Mixing up a Tarama-like hypothesis (concerning the dependence on time) with the previous one of Colombini and Lerner was instead considered in [START_REF] Colombini | A note on hyperbolic operators with log-Zygmund coecients[END_REF] in the case of space dimension 1, and then in [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coecients[END_REF] in the more general situation of N ≥ 1. The authors supposed the coecients to be log-Zygmund continuous in the time variable t, uniformly with respect to x, and log-Lipschitz continuous in the space variables, uniformly with respect to t. This hypothesis reads as follow: there exists a constant C such that, for all τ > 0 and all y ∈ R N \{0}, one has

|a jk (t, x + y) -a jk (t, x)| ≤ C |y| log 1 + 1 |y| .
In all these cases, one can prove an energy estimate with a loss of derivatives increasing in time: for all s ∈ ]0, s 0 [ (the exact value of s 0 changes from statement to statement), there exist positive constants β and C s and a time

T * ∈ ]0, T ] such that sup 0≤t≤T * u(t, •) H -s+1-βt + ∂ t u(t, •) H -s-βt ≤ est:t-loss (6) ≤ C s u(0, •) H -s+1 + ∂ t u(0, •) H -s + T * 0 Lu(t, •) H -s-βt dt for all u ∈ C 2 ([0, T ]; H ∞ (R N )).
In particular, from both inequalities ( 5) and ( 6), if coecients a jk are C ∞ b with respect to x, one can still recover the H ∞ well-posedness for the associated Cauchy problem, but, as already pointed out, with a nite loss of derivatives.

Such a loss, in a certain sense, cannot be avoided. As a matter of fact, Cicognani and Colombini proved in [START_REF] Cicognani | Modulus of continuity of the coecients and loss of derivatives in the strictly hyperbolic Cauchy problem[END_REF] that, if the regularity of the coecients is measured by a modulus of continuity, then any intermediate modulus of continuity between the Lipschitz and the log-Lipschitz ones necessarily entails a loss of regularity, which however can be made arbitrarly small. Moreover, they showed also that, in the log-Lipschitz instance, a loss of derivatives proportional to time, as found in [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coecients[END_REF], actually has to occur.

Nevertheless, in the case of dependence of coecients only on time, a special fact happens. In the above mentioned paper [START_REF] Tarama | Energy estimate for wave equations with coecients in some Besov type class[END_REF], Tarama considered also a jk 's satisfying an integral Zygmund condition: there exists a constant C > 0 such that, for all j, k and all ε ∈ ]0, T /2[ , one has hyp:int-Z hyp:int-Z [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coecients[END_REF] T -ε

ε |a jk (t + ε) + a jk (t -ε) -2 a jk (t)| dt ≤ C ε .
Under this assumption, he was able to prove an energy estimate which involves no loss of derivatives, and so well-posedness in H 1 × L 2 and, more in general, in H s × H s-1 for all s ∈ R. To get this result, he resorted to the main ideas of paper [START_REF] Colombini | Sur les équations hyperboliques avec des coecients qui ne dépendent que du temps[END_REF]: he smoothed out the coecients by use of a convolution kernel, and he linked the approximation parameter (say) ε with the dual variable, in order to perform dierent regularizations in dierent zones of the phase space. However, the key to the proof was dening a new energy, which involves (by dierentiation in time) also second derivatives of the approximated coecients a ε (t). In particular, his idea was to delete, in dierentiating energy in time, the terms presenting both the rst derivative a ′ ε (t), which has bad behaviour, and ∂ t u, for which one cannot gain regularity.

Now, what does it happen if we consider coecients depending also on the space variable? In this case, the condition becomes the following: there exists a positive constant C such that, xed any 1 ≤ i, j ≤ N , for all τ ≥ 0 and all y ∈ R N one has hyp:point-Z hyp:point-Z ( 8)

sup (t,x) a jk (t + τ, x + y) + a jk (t -τ, x -y) -2 a jk (t, x) ≤ C τ + |y| .
On the one hand, keeping in mind the strict embeddings emb emb [START_REF] Colombini | The Cauchy problem for wave equations with non-Lipschitz coecients; application to continuation of solutions of some nonlinear wave equations[END_REF] Lip ֒→ Zyg ֒→ log-Lip , the result of [START_REF] Cicognani | Modulus of continuity of the coecients and loss of derivatives in the strictly hyperbolic Cauchy problem[END_REF] implies that (a priori) a loss, even if arbitrarly small, always occur. On the other hand, Zygmund regularity is a condition on second variation, hence it is not related to the modulus of continuity and it runs o the issue of Cicognani and Colombini. Moreover, Lipschitz (in time) assumption is only a sucient condition to get estimate (2), and Tarama's result seems to suggest us that well-posedness in H s × H s-1 can be recovered also in this case, at least for some special s.

In the present paper we give a partial answer to the previous question. We assume hypothesis (8) on the a jk 's, i.e. a pointwise Zygmund condition with respect to all the variables, and we get an energy estimate without any loss of derivatives, but only in the space H 1/2 × H -1/2 . In fact, we are able to prove our result considering a complete second order operator: we take rst order coecients which are θ-Hölder continuous (for some θ > 1/2) with respect to the space variable, and the coecients of the 0-th order term only bounded. Let us point out that from this issue it immediately follows the H ∞ well-posedness with no loss of derivatives for an operator whose coecients are C ∞ b with respect to x. The rst fundamental step to obtain the result is passing from Zygmund continuous functions to more general symbols having such a regularity, and then analysing the properties of the related paradierential operators. In doing this, we make a heavy use of the paradierential calculus with parameters, as introduced and developed in [START_REF] Métivier | Interactions de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace[END_REF] and [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF]. In particular, it allows us to recover positivity of the paradierential operator associated to a positive symbol: this is a crucial point in our analysis. The second key ingredient to our proof is dening a new energy. It is only a slight modication of the original one of Tarama: we change the weight-functions involved in it and we replace product by them with action of the related paradierential operators. The last basic step relies in approximating the operator L, dened in (1), with a paradierential operator of order 2. The price to pay is a remainder term, which is however easy to control by use of the energy. All these operations have the eect to produce, in energy estimates, very special cancellations at the level of principal and subprincipal parts of the operators involved in the computations. These deletions allow us to get the result, but they seem to occur only in the H 1/2 × H -1/2 framework.

Therefore, considerations made before, under hypothesis [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coecients[END_REF], have not found an answer, yet, and it is not clear at all if well-posedness in H s × H s-1 , for s which varies in some interval containing 1/2, holds true or not.

Before entering into the details of the problem, let us give an overview of the paper. In the rst section, we will present our work setting, giving the main denitions and stating our results: a basic energy estimate for operator (1) under hypothesis [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coecients[END_REF], and a well-posedness issue which immediately follows from it.

The next section is devoted to the tools we need to handle our problem. They are mostly based on Littlewood-Paley Theory and classical Paradierential Calculus, introduced rst by J.-M. Bony in [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF]. Here we will follow the presentation given in [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF]. Moreover, we need also to introduce new classes of Sobolev spaces, of logarithmic type, already studied in [START_REF] Colombini | The Cauchy problem for wave equations with non-Lipschitz coecients; application to continuation of solutions of some nonlinear wave equations[END_REF]. Then we will quote some basic properties of Zygmund continuous functions and we will study their convolution with a smoothing kernel. A presentation of a new version of Paradierential Calculus, depending on some parameter γ ≥ 1 (see papers [START_REF] Métivier | Interactions de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace[END_REF] and [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF]) will follow. This having been done, we will make immediately use of the Paradierential Calculus with parameters to pass from such functions to more general symbols, having Zygmund regularity with respect to time and space and smooth in the ξ variable. Moreover, we will associate to them new paradierential operators, for which we will develop also a symbolic calculus.

In the end, we will be able to takle the proof of our energy estimate. The main eorts are dening a new energy and replacing the elliptic part of L with a suitable paradierential operator. Then, the rest of the proof is classical: we will dierentiate the energy with respect to time and we will estimate this derivative in terms of the energy itself. Gronwall's Lemma will enable us to get the thesis.

Finally, section 5 will be devoted to the well-posedness in the space H ∞ of the Cauchy problem related to L, when its coecients are assumed smooth enough. This result is a straightforward consequence of the previous one, and can be recovered following the same steps of the proof. Therefore, we will restrict ourselves to point out only the main dierencies, without repeating the complete argument.

Basic denitions and main result

This section is devoted to the presentation of our main result, i.e. an energy estimate for a complete hyperbolic operator with Zygmund continuous second order coecients. First of all, let us introduce a denition.

d:zyg Denition 2.1. A function f ∈ L ∞ (R N ) belongs to the Zygmund space Z(R N ) if the quantity |f | Z := sup ζ∈R N ,|ζ|<1 sup z∈R N |f (z + ζ) + f (z -ζ) -2 f (z)| • |ζ| -1 < +∞ .

Moreover we dene f

Z := f L ∞ + |f | Z .
Let us consider now the operator over [0, T ] × R N (for some T > 0 and N ≥ 1) dened by eq:op eq:op (10)

Lu = ∂ 2 t u - N i,j=1 ∂ i (a ij (t, x) ∂ j u) + b 0 (t, x) ∂ t u + N j=1 b j (t, x) ∂ j u + c(t, x) u ,
and let us suppose L to be strictly hyperbolic with bounded coecients, i.e. there exist two positive constants 0 < λ 0 ≤ Λ 0 such that, for all (t, x) ∈ [0, T ] × R N and all ξ ∈ R N , one has h:hyp h:hyp (11)

λ 0 |ξ| 2 ≤ N i,j=1 a ij (t, x) ξ i ξ j ≤ Λ 0 |ξ| 2 .
Moreover, we assume the coecients of the principal part of L to be isotropically Zygmund continuous, uniformly over [0, T ] × R N . In particular, there exists a constant K 0 such that, xed any 1 ≤ i, j ≤ N , for all τ ≥ 0 and all y ∈ R N , one has h:Z_tx h:Z_tx ( 12)

sup (t,x) a ij (t + τ, x + y) + a ij (t -τ, x -y) -2 a ij (t, x) ≤ K 0 τ + |y| .
Finally, let us suppose also that, for some θ > 1/2, we have :lower-order :lower-order [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF]

b j ∈ L ∞ ([0, T ]; C θ (R N )) ∀ 0 ≤ j ≤ N and c ∈ L ∞ ([0, T ] × R N ) .
Under these hypothesis, one can prove the following result. th:en_est Theorem 2.2. Let L be the operator dened by [START_REF] Hörmander | Linear partial dierential operators[END_REF], and assume it is strictly hyperbolic with bounded coecients, i.e. relation [START_REF] Métivier | Interactions de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace[END_REF] holds true. Moreover, let us suppose the coecients a ij to fulll condition [START_REF] Métivier | Para-dierential calculus and applications to the Cauchy problem for nonlinear systems[END_REF], and the b j 's and c to verify hypothesis [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF], for some θ > 1/2.

Then there exist positive constants C, λ such that the inequality

sup 0≤t≤T u(t, •) H 1/2 + ∂ t u(t, •) H -1/2 ≤ est:thesis (14) ≤ C e λT u(0, •) H 1/2 + ∂ t u(0, •) H -1/2 + T 0 e -λt Lu(t, •) H -1/2 dt holds true for all u ∈ C 2 ([0, T ]; H ∞ (R N )).
From the previous estimate, which involves no loss of derivatives, one can recover, in a standard way, the well-posedness issue in the space H 1/2 × H -1/2 . c:wp Corollary 2.3. Let us consider the Cauchy problem

(CP ) Lu = f u |t=0 = u 0 , ∂ t u |t=0 = u 1 ,
where L is dened by conditions ( 10), ( 11), ( 12) and ( 13), and

f ∈ L 1 ([0, T ]; H -1/2 ).
Then (CP ) is well-posed in the space H 1/2 × H -1/2 , globally on the time interval [0, T ].

Tools

In this section we want to introduce the main tools, from Fourier Analysis, we will need to prove Theorem 2.2. Most of them are the same we resorted to in the recent paper [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coecients[END_REF], where we considered the case of coecients log-Zymung continuous with respect to time, and log-Lipschitz continuous in space variables. Nevertheless, for a seek of completeness, we will give here the most of the details.

The rst part is devoted to the classical Littlewood-Paley Theory and to the presentation of new Sobolev spaces, of logarithmic type, introduced rst in [START_REF] Colombini | The Cauchy problem for wave equations with non-Lipschitz coecients; application to continuation of solutions of some nonlinear wave equations[END_REF].

Then we will analyse some properties of the Zygmund continuous functions. We will consider also convolution in time with a smoothing kernel.

In the next subsection we will present the Littlewood-Paley Theory depending on a parameter γ ≥ 1: this modication permits a more rened study of our problem. In particular, we will introduce the new class of low regularity symbols we will deal with, and we will show how one can associate to them a paradierential operator. As pointed out in the introduction, passing from multiplication by functions to action by operators is just the fundamental step which allows us to improve the result of Tarama. A wide analysis of symbolic calculus in this new class will end the present section.

Littlewood-Paley decomposition ss:L-P

Let us rst dene the so called Littlewood-Paley decomposition, based on a non-homogeneous dyadic partition of unity with respect to the Fourier variable. We refer to [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF], [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] and [START_REF] Métivier | Para-dierential calculus and applications to the Cauchy problem for nonlinear systems[END_REF] for the details.

So, x a smooth radial function χ supported in the ball B(0, 2), equal to 1 in a neighborhood of B(0, 1) and such that r → χ(r e) is nonincreasing over R + for all unitary vectors e ∈ R N . Set also ϕ (ξ) = χ (ξ) -χ (2ξ) .

The dyadic blocks (∆ j ) j∈Z are dened by 1

∆ j := 0 if j ≤ -1, ∆ 0 := χ(D) and ∆ j := ϕ(2 -j D) if j ≥ 1.
1 Throughout we agree that f (D) stands for the pseudo-dierential operator u → F -1 (f Fu).

We also introduce the following low frequency cut-o:

S j u := χ(2 -j D) = k≤j ∆ k for j ≥ 0.
The following classical properties will be used freely throughout the paper:

• for any u ∈ S ′ , the equality u = j ∆ j u holds true in S ′ ;

• for all u and v in S ′ , the sequence (S j-3 u ∆ j v) j∈N is spectrally supported in dyadic annuli.

Let us also mention a fundamental result, which explains, by the so-called Bernstein's inequalities, the way derivatives act on spectrally localized functions. l:bern Lemma 3.1. Let 0 < r < R. A constant C exists so that, for any nonnegative integer k, any couple (p, q) in [1, +∞] 2 with p ≤ q and any function u ∈ L p , we have, for all λ > 0,

supp u ⊂ B(0, λR) =⇒ ∇ k u L q ≤ C k+1 λ k+N 1 p -1 q u L p ; supp u ⊂ {ξ ∈ R N | rλ ≤ |ξ| ≤ Rλ} =⇒ C -k-1 λ k u L p ≤ ∇ k u L p ≤ C k+1 λ k u L p .
Let us recall the characterization of (classical) Sobolev spaces via dyadic decomposition: for all s ∈ R there exists a constant C s > 0 such that est:dyad-Sob est:dyad-Sob (15)

1 C s +∞ ν=0 2 2νs u ν 2 L 2 ≤ u 2 H s ≤ C s +∞ ν=0 2 2νs u ν 2 L 2 ,
where we have set u ν := ∆ ν u.

So, the H s norm of a tempered distribution is the same as the ℓ 2 norm of the sequence (2 sν ∆ ν u L 2 ) ν∈N . Now, one may ask what we get if, in the sequence, we put weights dierent to the exponential term 2 sν . Before answering this question, we introduce some denitions. For the details of the presentiation, we refer also to [START_REF] Colombini | The Cauchy problem for wave equations with non-Lipschitz coecients; application to continuation of solutions of some nonlinear wave equations[END_REF].

Let us set Π(D) := log(2 + |D|), i.e. its symbol is π(ξ) := log(2 + |ξ|). d:log-H^s Denition 3.2. For all α ∈ R, we dene the space H s+α log as the space Π -α H s , i.e.

f ∈ H s+α log ⇐⇒ Π α f ∈ H s ⇐⇒ π α (ξ) 1 + |ξ| 2 s/2 f (ξ) ∈ L 2 .
From the denition, it's obvious that the following inclusions hold for s 1 > s 2 , α 1 ≥ α 2 ≥ 0:

H s 1 +α 1 log ֒→ H s 1 +α 2 log ֒→ H s 1 ֒→ H s 1 -α 2 log ֒→ H s 1 -α 1 log ֒→ H s 2 .
We have the following dyadic characterization of these spaces (see [START_REF] Métivier | Para-dierential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]Prop. 4.1.11]). p:log-H Proposition 3.3. Let s, α ∈ R. A tempered distribution u belongs to the space H s+α log if and only if:

(i) for all k ∈ N, ∆ k u ∈ L 2 (R N ); (ii) set δ k := 2 ks (1 + k) α ∆ k u L 2 for all k ∈ N, the sequence (δ k ) k belongs to ℓ 2 (N). Moreover, u H s+α log ∼ (δ k ) k ℓ 2 .
Hence, this proposition generalizes property [START_REF] Tarama | Energy estimate for wave equations with coecients in some Besov type class[END_REF].

Even if energy estimate [START_REF] Mizohata | The Theory of Partial Dierential Equations[END_REF] involves no loss of derivatives, in our analysis we will need this new spaces, which are intermediate between the classical ones. As a matter of fact, action of operators associated to Zygmund symbols often entails a logarithmic loss of derivatives. We will formally justify in a while what we have just said; rst of all, let us recall some properties of Zygmund continuous functions.

3.2

Zygmund continuous functions ss:Zyg

We have already introduced the space Z(R N ) in denition 2.1. Let us now analyse some of its properties.

Let us recall that this class of functions coincides (see e.g. [START_REF] Chemin | Fluides parfaits incompressibles[END_REF] for the proof) with the Besov space C 1 * ≡ B 1 ∞,∞ , which is characterized by the condition est:Z est:Z ( 16)

sup ν≥0 ( 2 ν ∆ ν f L ∞ ) < +∞ .
Moreover, we have (see e.g. [1, Ch. 2] for the proof) the continuous embedding Z ֒→ LL, where we denote with LL the space of log-Lipschitz functions. As a matter of fact, for all f ∈ Z there exists a constant C > 0 such that, for any 0 < |y| < 1, est:Z->LL est:Z->LL (17)

sup x∈R N |f (x + y) -f (x)| ≤ C |y| log 1 + γ + 1 |y| ,
where γ ≥ 1 is a xed real parameter. In what follows, we will make a broad use of paradierential calculus with parameters (see subsection 3.3), which will come into play in a crucial way in our computations. So, we prefer performing immediately such a change. Now, given a f ∈ Z, we can regularize it by convolution. As, in the sequel, we are interested in smoothing out coecients of our hyperbolic operator only with respect to the time variable, let us immediately focus on the 1-dimensional case.

So, x a f ∈ Z(R). Take an even function [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF] and such that ρ(t) dt = 1, and dene the mollier kernel

ρ ∈ C ∞ 0 (R), 0 ≤ ρ ≤ 1, whose support is contained in the interval [-1,
ρ ε (t) := 1 ε ρ t ε ∀ ε ∈ ]0, 1] .
Then, for all ε ∈ ]0, 1] we set eq:f_e eq:f_e (18)

f ε (t) := (ρ ε * f ) (t) = Rs ρ ε (t -s) f (s) ds .
Let us state some properties about the family of functions we obtain in this way. The most important one is that we can't expect to control the rst derivative uniformly on ε: our starting function is not Lipschitz. Nevertheless, second derivative behaves well again. p:Z-approx Proposition 3.5. Let f be a Zygmund continuous function such that 0 < λ 0 ≤ f ≤ Λ 0 , for some positive real numbers λ 0 and Λ 0 .

Then there exists a constant C > 0, depending only on the Zygmund seminorm of f , i.e. |f | Z , such that the following facts hold true for all ε ∈ ]0, 1]:

0 < λ 0 ≤ f ε ≤ Λ 0 est:ell (19) |f ε (t) -f (t)| ≤ C ε est:f_e-f (20) |∂ t f ε (t)| ≤ C log 1 + γ + 1 ε est:d_t-f (21) ∂ 2 t f ε (t) ≤ C 1 ε . est:d_tt-f (22)
Proof. ( 19) is obvious. Using the fact that ρ is even and has unitary integral, we can write

f ε (t) -f (t) = 1 2 ε ρ s ε (f (t + s) + f (t -s) -2f (t)) ds ,
and inequality (20) immediately follows. For (22) we can argue in the same way, recalling that ρ ′′ is even and that ρ ′′ = 0.

We have to pay attention to the estimate of the rst derivative. As

ρ ′ ≡ 0, one has ∂ t f ε (t) = 1 ε |s|≤ε ρ ′ s ε (f (t -s) -f (t)) ds .
Keeping in mind (17) and noticing that the function σ → σ log(1 + γ + 1/σ) is increasing, we get inequality (21). The proposition is now completely proved.

Paradierential calculus with parameters ss:pd_param

Let us present here the paradierential calculus depending on some parameter γ. One can nd a complete and detailed treatement in [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF] (see also [START_REF] Métivier | Interactions de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace[END_REF]).

Fix γ ≥ 1 and take a cut-o function ψ ∈ C ∞ (R N ×R N ) which veries the following properties:

• there exist 0 < ε 1 < ε 2 < 1 such that ψ(η, ξ) = 1 for |η| ≤ ε 1 (γ + |ξ|) 0 for |η| ≥ ε 2 (γ + |ξ|) ; • for all (β, α) ∈ N N × N N , there exists a constant C β,α such that ∂ β η ∂ α ξ ψ(η, ξ) ≤ C β,α (γ + |ξ|) -|α|-|β| .
We will call such a function an admissible cut-o .

For instance, if γ = 1, one can take

ψ(η, ξ) ≡ ψ -3 (η, ξ) := +∞ k=0 χ k-3 (η) ϕ k (ξ) ,
where χ and ϕ are the localization (in phase space) functions associated to a Littlewood-Paley decomposition, see [START_REF] Métivier | Para-dierential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]Ex. 5.1.5]. Similarly, if γ > 1 it is possible to nd a suitable integer µ ≥ 0 such that d_eq:pp_symb d_eq:pp_symb (23)

ψ µ (η, ξ) := χ µ (η) χ µ+2 (ξ) + +∞ k=µ+3 χ k-3 (η) ϕ k (ξ)
is an admissible cut-o function.

r:gamma-dyad Remark 3.6. Let us immediately point out that we can also dene a dyadic decomposition depending on the parameter γ. First of all, we set def:Lambda def:Lambda (24) Λ(ξ, γ)

:= γ 2 + |ξ| 2 1/2 .
Then, taken the usual smooth function χ associated to a Littlewood-Paley decomposition, we dene

χ ν (ξ, γ) := χ 2 -ν Λ(ξ, γ) , S γ ν := χ ν (D x , γ) , ∆ γ ν := S γ ν+1 -S γ ν .
The usual properties of the support of the localization functions still hold, and for all xed γ ≥ 1 and all tempered distributions u, we have

u = +∞ ν=0 ∆ γ ν u in S ′ .
Moreover, we can introduce logarithmic Besov spaces using the new localization operators S γ ν , ∆ γ ν . For the details see section 2.1 of [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF]. What is important to retain is that, once we x γ ≥ 1, the previous construction is equivalent to the classical one, and one can still recover previous results. For instance, if we dene the space H s+α log γ as the set of tempered distributions for which eq:gH-def eq:gH-def (25)

u 2 H s+α log γ := R N ξ Λ 2s (ξ, γ) log 2α (1 + γ + |ξ|) | u(ξ)| 2 dξ < +∞ ,
for every xed γ ≥ 1 it coincides with H s+α log , the respective norms are equivalent and the characterization given by proposition 3.3 still holds true.

Let us come back to the admissible cut-o function ψ introduced above. Thanks to it, we can dene more general paradierential operators, associated to low regularity functions: let us explain how.

Dene the function G ψ as the inverse Fourier transform of ψ with respect to the variable η:

G ψ (x, ξ) := F -1 η ψ (x, ξ) .
The following properties hold true.

l:G Lemma 3.7. For all (β, α)

∈ N N × N N , ∂ β x ∂ α ξ G ψ (•, ξ) L 1 (R N x ) ≤ C β,α (γ + |ξ|) -|α|+|β| , pd_est:G_1 (26) | • | log 2 + 1 | • | ∂ β x ∂ α ξ G ψ (•, ξ) L 1 (R N x ) ≤ C β,α (γ + |ξ|) -|α|+|β|-1 log(1 + γ + |ξ|).
pd_est:G_2 (27)

Proof. See [START_REF] Métivier | Para-dierential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]Lemma 5.1.7].

Thanks to G, we can smooth out a symbol a in the x variable and then dene the paradierential operator associated to a as the classical pseudodierential operator associated to this smooth function.

First of all, let us dene the new class of symbols we are dealing with.

d:symbols Denition 3.8. Let m and δ be two given real numbers.

(i) We denote with Z (m,δ) the space of functions a(t, x, ξ, γ) which are locally bounded over [0, T 0 ]×R N ×R N ×[1, +∞[ and of class C ∞ with respect to ξ, and which satisfy the following properties:

for all α ∈ N N , there exists a C α > 0 such that, for all (t, x, ξ, γ), est:g-symb est:g-symb (28)

∂ α ξ a(t, x, ξ, γ) ≤ C α (γ + |ξ|) m-|α| log δ (1 + γ + |ξ|) ;
there exists a constant K > 0 such that, for any τ ≥ 0 and y ∈ R N , one has, for all

ξ ∈ R N and γ ∈ [1, +∞[ , sup (t,x)
a(t + τ, x + y, ξ, γ) + a(t -τ, x -y, ξ, γ) -2a(t, x, ξ, γ) ≤ est:Z_symb (29)

≤ K τ + |y| (γ + |ξ|) m log δ (1 + γ + |ξ|) .
(ii) Σ (m,δ) is the space of symbols σ of Z (m,δ) for which there exists a 0 < ǫ < 1 such that, for all (t, ξ, γ) ∈ [0, T ]×R N ×[1, +∞[ , the spectrum (i.e. the support of the Fourier transform with respect to x) of the function x → σ(t, x, ξ, γ) is contained in the ball {|η| ≤ ǫ (γ + |ξ|)}.

In a quite natural way, we can equip Z (m,δ) with the seminorms

|a| (m,δ,k) := sup |α|≤k sup R N ξ ×[1,+∞[ (γ + |ξ|) -m+|α| log -δ (1 + γ + |ξ|) ∂ α ξ a(•, •, ξ, γ) L ∞ (t,x)
, eq:seminorms (30)

a Z := inf K > 0 relation (29) holds true . eq:Z_sem (31) Moreover, by spectral localization and Paley-Wiener Theorem, a symbol σ ∈ Σ (m,δ) is smooth also in the x variable. So, we can dene the subspaces Σ (m,δ) (µ,̺) (for µ and ̺ ∈ R) of symbols σ which verify (28) and also, for all β > 0, est:g-s_x est:g-s_x (32)

∂ β x ∂ α ξ σ(•, •, ξ, γ) L ∞ (t,x) ≤ C β,α (γ + |ξ|) m-|α|+|β|+µ log δ+̺ (1 + γ + |ξ|) .
Now, given a symbol a ∈ Z (m,δ) , we can dene eq:symb-def eq:symb-def (33)

σ ψ a (t, x, ξ, γ) := ( ψ(D x , ξ) a ) (t, x, ξ, γ) = G ψ (•, ξ) * x a(t, •, ξ, γ) (x) .
p:par-op Proposition 3.9. (i) For all m, δ ∈ R, the smoothing operator

R : a(t, x, ξ, γ) → σ ψ a (t, x, ξ, γ) is bounded from Z (m,δ) to Σ (m,δ) .
(ii) The dierence a -σ ψ a ∈ Z (m-1,δ+1) . (iii) In particular, if ψ 1 and ψ 2 are two admissible cut-o functions, then the dierence of the two smoothed symbols, σ ψ 1 a -σ ψ 2 a , belongs to Σ (m-1,δ+1) . r:psi-ind Remark 3.10. As we will see in a while, part (ii) of previous proposition says that the dierence between the original symbol and the classical one associated to it is more regular. Part (iii), instead, infers that the whole construction is independent of the cut-o function xed at the beginning.

General paradierential operators

As already mentioned, we can now dene the paradierential operator associated to a using the classical symbol corresponding to it: eq:T-def eq:T-def (34)

T ψ a u(t, x) := σ ψ a (t, • , D x , γ) u (x) = 1 (2π) N R N ξ e ix•ξ σ ψ a (t, x, ξ, γ) u(ξ) dξ .
Note that T ψ a u still depends on the parameter γ ≥ 1. For instance, if a = a(x) ∈ L ∞ and if we take the cut-o function ψ -3 , then T ψ a is actually the usual paraproduct operator. If we take ψ µ as dened in (23), instead, we get a paraproduct operator which starts from high enough frequencies, which will be indicated with T µ a (see section 3.3 of [START_REF] Colombini | The Cauchy problem for wave equations with non-Lipschitz coecients; application to continuation of solutions of some nonlinear wave equations[END_REF]).

Let us now study the action of general paradierential operators on the class of logarithmic Sobolev spaces. First of all, a denition is in order. d:op_order Denition 3.11. We say that an operator P is of order m + δ log if, for every (s, α) ∈ R 2 and every γ ≥ 1, P maps H s+α log γ into H (s-m)+(α-δ) log γ continuously. With slight modications to the proof of Proposition 2.9 of [START_REF] Métivier | Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems[END_REF], stated for the calssical Sobolev class, we get the next fundamental result. l:action Lemma 3.12. For all σ ∈ Σ (m,δ) , the corresponding operator σ( • , D x ) is of order m + δ log. Lemma 3.12 immediately implies the following theorem, which describes the action of the new class of paradierential operators. t:action Theorem 3.13. Given a symbol a ∈ Z (m,δ) , for any admissible cut-o function ψ, the operator

T ψ a is of order m + δ log.
As already remarked, the construction does not depends on the cut-o function ψ used at the beginning. Next result says that main features of a paradierential operator depend only on its symbol.

p:act-psi Proposition 3.14. If ψ 1 and ψ 2 are two admissible cut-o functions and a ∈ Z (m,δ) , then the dierence

T ψ 1 a -T ψ 2 a
is of order (m -1) + (δ + 1) log. Therefore, changing the cut-o function ψ doesn't change the paradierential operator associated to a, up to lower order terms. So, in what follows we will miss out the dependence of σ a and T a on ψ.

Symbolic calculus in the Zygmund class Z (m,δ)

For convenience, in what follows we will temporarily consider δ = 0: the geneeral case δ = 0 easily follows with slight modications.

So, let us now take a Zygmund symbol a ∈ Z (m,0) (for some m ∈ R). Assume moreover that it satises a strictly ellipticity condition: there exists a constant λ 0 > 0 such that, for all (t, x, ξ, γ), a(t, x, ξ, γ) ≥ λ 0 (γ + |ξ|) m .

Finally, let us smooth a out with respect to the rst variable, as we have seen in paragraph 3.2, and let us denote by a ε the result of the convolution. Obviously, also the a ε 's satisfy the ellipticity condition with the same λ 0 (by relation ( 19)), so in particular independent of ε. In addition, next estimates hold true. l:symb Lemma 3.15. The classical symbol associated to a ε , which we will denote by σ a (we drop the dependence on ε to simplify notations), satisfy the following inequalities:

∂ α ξ σ a ≤ C (γ + |ξ|) m-|α| ∂ β x ∂ α ξ σ a ≤ C (γ + |ξ|) m-|α|+|β|-1 log (1 + γ + |ξ|) if |β| = 1 ∂ β x ∂ α ξ σ a ≤ C (γ + |ξ|) m-|α|+|β|-1 if |β| ≥ 2 .
Moreover, the classical symbol associated to ∂ t a ε coincides with ∂ t σ a and veries, instead,

∂ α ξ σ ∂ta ≤ C (γ + ξ) m-|α| log 1 + γ + 1 ε ∂ β x ∂ α ξ σ ∂ta ≤ C (γ + |ξ|) m-|α|+|β| + C ε (γ + |ξ|) m-|α|+|β|-1 . Finally, σ ∂ 2 t a ≡ ∂ 2
t σ a and one has

∂ α ξ σ ∂ 2 t a ≤ C ε (γ + |ξ|) m-|α| ∂ β x ∂ α ξ σ ∂ 2 t a ≤ C ε (γ + |ξ|) m-|α|+|β| .
Proof. The rst inequality is obvious by the chain rule and the properties of a ε , G ψ . For second and third ones, we have to observe that

∂ i G(x -y, ξ)dx = ∂ i G(z, ξ)dz = F -1 η (η i ψ(η, ξ)) dz = (η i ψ(η, ξ)) |η=0 = 0 ,
and the same still holds if we keep dierenciating with respect to x. Hence if we dierentiate only once with respect to the space variable, what we get is the following:

∂ i σ a = ∂ i G(x -y) a ε (y) dy = ∂ i G(y) ρ ε (t -s) (a(s, x -y, ξ) -a(s, x, ξ)) ds dy ,
and the embedding Z ֒→ LL implies second inequality. For second derivatives we can use also the parity of G and write

∂ i ∂ j σ a = 1 2 ∂ i ∂ j G(y) ρ ε (t -s) (a(s, x + y, ξ) + a(s, x -y, ξ) -2a(s, x, ξ)) ds dy ,
and the thesis immediately follows. Recalling the spectral localization, the estimate for higher order derivatives follows from the just proved one, combined with Bernstein's inequalities. Now, let us consider the rst time derivative. Former inequality concerning ∂ t a is obvious: as ρ ′ = 0, we have

σ ∂ta = G(x -y) 1 ε 2 ρ ′ (s/ε) (a(t -s, y, ξ) -a(t, y, ξ)) ds dy .
If we dierentiate the classical symbol also in space, instead, the behaviour is better: both ∂ i G and ρ ′ are odd, hence

∂ i σ ∂ta = 1 4 ∂ i G(y) 1 ε 2 ρ ′ (s/ε) (a(t + s, x + y, ξ) -a(t + s, x -y, ξ)- -a(t -s, x + y, ξ) + a(t -s, x -y, ξ)) ds dy .
Now, adding and subtracting the quantity 2 a(t, x, ξ) and taking advantage of the Zygmund regularity condition, we have

|∂ i σ ∂ta | ≤ C |∂ i G| 1 ε 2 ρ ′ s ε (|s| + |y|) ds dy ,
and so we get the expected control. Let us remark that the two terms in the right-hand side of the inequality are the same once we set ε = (γ + |ξ|) -1 . Finally, arguing as before, the last two inequalities can be easily deduced from the fact that ρ ′′ is even and has null integral. r:symb Remark 3.16. It goes without saying that, with obvious changes, an analogous statement holds true also for symbols of class Z (m,δ) , for any δ ∈ R.

From Lemma 3.15, properties of paradierential operators associated to a ε and its time derivatives immediately follow, keeping in mind Theorem 3.13. Now, we want to state an accurate result on composition and adjoint operators associated to symbols in the Zygmund class Z (m,δ) . As a matter of fact, the proof of our energy estimate is based on very special cancellations at the level of principal and subprincipal parts of the involved operators: hence, we need to understand the action of the terms up to the next order.

With a little abuse of notation, for a symbol a we will write ∂ x a meaning that the space derivative actually acts on the classical symbol associated to a. t:symb_calc Theorem 3.17. (i) Let us take two symbols a ∈ Z (m,δ) and b ∈ Z (n,̺) and denote by T a , T b the respective associated paradierential operators. Then eq:comp_op eq:comp_op (35)

T a • T b = T a b -i T ∂ ξ a ∂xb + R • .
The principal part T a b is of order (m + n) + (δ + ̺) log.

The subprincipal part T ∂ ξ a ∂xb has order (m + n -1) + (δ + ̺ + 1) log.

The remainder operator R • , instead, has order (m + n -1) + (δ + ̺) log.

(ii) Let a ∈ Z (m,δ) . The adjoint operator (over L 2 ) of T a is given by the formula eq:adj_op eq:adj_op (36)

(T a ) * = T a -i T ∂ ξ ∂xa + R * .
The order of T a is still m + δ log.

The order of T ∂ ξ ∂xa is instead (m -1) + (δ + 1) log.

Finally, the remainder term R * has order (m -1) + δ log.

This theorem immediately follows from Lemma 3.15. Let us end this subsection stating a basic positivity estimate.

p:pos Proposition 3.19. Let a(t, x, ξ, γ) be a real-valued symbol in Z (2m,0) , such that

a(t, x, ξ, γ) ≥ λ 0 (γ + |ξ|) 2m .
Then, there exists a constant λ 1 , depending only on the seminorm |a| Z and on λ 0 , such that, for γ large enough, one has

Re (T a u, u) L 2 ≥ λ 1 u 2 H m γ .
Proof. Let us set a = b 2 : we note that b ∈ Z (m,0) . Thanks to symbolic calculus, we can write:

Re (T a u, u) L 2 = Re (T b T b u, u) L 2 + Re R ′ u, u L 2 = Re (T b u, (T b ) * u) L 2 + Re R ′ u, u L 2 = Re (T b u, T b u) L 2 + Re R ′ u, u L 2 + Re R ′′ u, u L 2 ,
where the remainder operators R ′ and R ′′ have principal symbols respectively equal to ∂ ξ b ∂ x b and ∂ ξ ∂ x b. Hence they have order (2m -1) + log and (m -1) + log respectively. Therefore, using also Lemma 3.15, we get (for all γ ≥ 1)

Re (T a u, u) L 2 ≥ u 2 H m γ -R ′ u H -(2m-1)/2-(1/2) log γ u H (2m-1)/2+(1/2) log γ - -R ′′ u H -(m-1)/2-(1/2) log γ u H (m-1)/2+(1/2) log γ ≥ u 2 H m γ -u 2 H (2m-1)/2+(1/2) log γ -u 2 H (m-1)/2+(1/2) log γ .
Now, by denition of H s+α log γ norms, it's easy to see that the second and third terms in the last line can be absorbed by the rst one, for γ large enough. r:pos Remark 3.20. Let us expressly point out the following fact. If the positive symbol a has low regularity in time and we smooth it out by convolution with respect to this variable, we obtain a family (a ε ) ε of positive symbols, with same constant λ 0 . Now, all the paradierential operators associated to these symbols will be positive operators, uniformly in ε: i.e. the constant λ 1 of previous inequality can be choosen independently of ε.

Previous proposition allows us to recover positivity of paradierential operators associated to positive symbols. This fact will be fundamental in energy estimates.

Proposition 3.19, together with Theorem 3.17, implies the following corollary.

c:pos Corollary 3.21. Let a be a positive symbol in the class Z (1,0) .

Then there exists γ ≥ 1, depending only on |a| Z , such that

T a u L 2 ∼ ∇u L 2 for all u ∈ H 1 (R N ).
4 Proof of the energy estimate s:proof

Let us now tackle the proof of Theorem 2.2. It relies on dening a suitable energy associated to u and on splitting operator L into a principal part, given by a paradierential operator, and a remainder term, which is easy to control by the energy.

The rest is classical: we will control the time derivative of the energy by the energy itself, and we will get inequality ( 14) by use of Gronwall's Lemma.

Energy

Let us smooth out the coecients of the operator L with respect to the time variable, as done in (18), and let us dene the second order symbol eq:a eq:a (37)

α ε (t, x, ξ) := j,k a jk,ε (t, x) ξ j ξ k + γ 2 .
By analogy with what done in [START_REF] Colombini | Sur les équations hyperboliques avec des coecients qui ne dépendent que du temps[END_REF] (see also [START_REF] Colombini | Hyperbolic operators with non-Lipschitz coecients[END_REF], [START_REF] Colombini | A note on hyperbolic operators with log-Zygmund coecients[END_REF] and [START_REF] Colombini | Time-dependent loss of derivatives for hyperbolic operators with non-regular coecients[END_REF] for the case of localized energy), we immediately link the approximation parameter ε with the dual variable ξ, setting eq:param eq:param (38

) ε = γ 2 + |ξ| 2 -1/2 .
For notation convenience, in the sequel we will miss out the index ε. Now, by use of Corollary 3.21, let us x a positive γ, which will depend only on λ 0 and on sup j,k |a jk | Zx , such that the operators T α -1/4 and T α 1/4 are positive, i.e. for all w ∈ H ∞ one has

T α -1/4 w L 2 ≥ λ 0 2 w H -1/2 , T α 1/4 w L 2 ≥ λ 0 2 w H 1/2 .
r:pos-eps Remark 4.1. Keeping in mind Remark 3.20, it's easy to see that the xed γ doesn't depend on the approximation parameter ε.

This having been done, let us take a u ∈ H ∞ , solution of [START_REF] Hörmander | Linear partial dierential operators[END_REF]. Let us dene

v(t, x) := T α -1/4 ∂ t u -T ∂t(α -1/4 ) u w(t, x) := T α 1/4 u
and the Tarama's energy associated to u:

eq:E eq:E (39)

E(t) := v(t) 2 L 2 + w(t) 2 L 2 .
Using positivity of involved operators, it's easy to see that w(t)

L 2 ∼ u(t) H 1/2 and that v(t) L 2 ≤ C ( ∂ t u(t) H -1/2 + u(t) H 1/2 ) ∂ t u(t) H -1/2 ≤ C v(t) L 2 + T ∂t(α -1/4 ) u L 2 ≤ C (E(t)) 1/2 .
So, we gather that there exists a constant C for which The aim of this subsection is to show that, roughly speaking, we can approximate our striclty hyperbolic operator L with a paradierential operator, up to a remainder term of order 1. The latter can be immediately bounded by the energy, while the former represents the principal part of L, but it is easier to deal with.

(E(0)) 1/2 ≤ C ( ∂ t u(0) H -1/2 + u(0) H 1/2 ) est:E_0 (40) (E(t)) 1/2 ≥ C -1 ( ∂ t u(t) H -1/2 + u(t) H 1/2
For convenience, let us dene another second order symbol:

eq:a-tilde eq:a-tilde (42)

α(t, x, ξ) := j,k a jk (t, x) ξ j ξ k ,
i.e. α is analogous to α, but functions a jk are not regularized in time.

l:L->T Lemma 4.2. Let us dene the operator R in the following way:

R u := j,k ∂ j (a jk (t, x) ∂ k u) + Re T α u .
Then R maps continuously H s into H s-1 , for all 0 < s < 1.

Proof. Given u ∈ H s , rst of all we want to prove that the dierence

a jk ∂ k u -T a jk ∂ k u = a jk ∂ k u -i T a jk ξ k u = ν≥µ S ν+2 ∂ k u ∆ ν a jk = ν≥µ R ν is still in H s .
As each R ν is spectrally supported in a ball of radius proportional to 2 ν , and as s > 0, we can apply Lemma 2.84 of [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF]. So, it's enough to estimate the L 2 norm of each term R ν .

Using also characterization ( 16), we have

R ν L 2 ≤ S ν+2 ∂ k u L 2 ∆ ν a jk L ∞ ≤ C S ν+2 ∂ k u L 2 2 -ν .
Let us note that the constant C depends on the Zygmund seminorm of a jk . As ∇u ∈ H s-1 , with s < 1, Proposition 2.79 of [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF] applies, and it nally gives us

R ν L 2 ≤ C ∇u H s-1 2 -sν c ν ,
for a sequence (c ν ) ν ∈ ℓ 2 (N) of unitary norm. So, the above mentioned Lemma 2.84 implies that a jk ∂ k u -i T a jk ξ k u ∈ H s , as claimed. Therefore,

j,k ∂ j (a jk ∂ k u) -i ∂ j T a jk ξ k u ∈ H s-1 .
Now, some computations are needed. With a little abuse of notation, we will write ∂ j α meaning that the space derivative actually acts on the classical symbol associated to α.

Noting that j,k a jk ξ k = (∂ ξ j α)/2, we get

i j,k ∂ j T a jk ξ k = i j,k T a jk ξ k ∂ j + j 1 2 T ∂ j ∂ ξ j α = -T α + i 2 j T ∂ j ∂ ξ j α = - 1 2 (T α + (T α ) * ) + R ′ ,
where we have used also Theorem 3.17. The operator R ′ has symbol ∂ 2

x ∂ 2 ξ α, and so, by Lemma 3.15, it has order 1.

In the end, we have discovered that

j,k ∂ j (a jk ∂ k ) + Re T α : H s -→ H s-1
is a continuous operator of order 1, provided that 0 < s < 1. The lemma is now proved.

Let us now focus on the rst order terms, and x an index 0 ≤ j ≤ N . As b j ∈ C θ (R N ), it is in particular bouned. Therefore, the corresponding paraproduct operator has order 0 (see e.g. Theorem 2.82 of [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF]), and then

Re v(t) , T α -1/4 T b j ∂ j u L 2 ≤ C v(t) L 2 ∂ j u(t) H -1/2 ≤ C E(t) .
For the remainder operator, as θ > 1/2 we can apply Lemma 4.3, and we get

Re v(t) , T α -1/4 B j ∂ j u L 2 ≤ C v(t) L 2 B j ∂ j u H -1/2 ≤ C v(t) L 2 B j ∂ j u H θ-1/2 ≤ C v(t) L 2 ∂ j u(t) H -1/2 ≤ C E(t) .
The analysis of the term of order 0 is instead straightforward:

Re (v(t) , T α -1/4 cu) L 2 ≤ C v(t) L 2 cu H -1/2 ≤ C v(t) L 2 u(t) L 2 ≤ C E(t) .
So, it remains us to handle only the last term of relation ( 44): this will be done in a while. For the moment, let us dierentiate the second part of the energy with respect to time: eq:d_t-w eq:d_t-w (45)

d dt w(t) 2 L 2 = 2Re w(t) , T ∂t(α 1/4 ) u L 2 + 2Re (w(t) , T α 1/4 ∂ t u) L 2 .
We couple each term of this relation with the respective one coming from the last item of (44) and, by use of symbolic calculus (recall in particular Theorem 3.17), we will try to control them.

Let us be more precise and make rigorous what we have just said. First of all, we consider

T 1 := 2Re -T ∂t(α -1/4 ) u , -T α -1/4 Re T α u L 2 + 2Re T α 1/4 u , T ∂t(α 1/4 ) u L 2 .
Noticing that ∂ t α 1/4 = -α 1/2 ∂ t α -1/4 , we can write

2Re T α 1/4 u , T ∂t(α 1/4 ) u L 2 = 2Re T α 1/4 u , -T α 1/2 T ∂t(α -1/4 ) u + 2Re (T α 1/4 u , M u) L 2 ,
where M has principal symbol equal to ∂ ξ α 1/2 ∂ x ∂ t α -1/4 . Therefore, we get

T 1 = 2Re T ∂t(α -1/4 ) u , T α -1/4 Re T α -(T α 1/2 ) * T α 1/4 u L 2 + 2Re (T α 1/4 u , M u) L 2 .
By Lemma 3.15, the remainder term can be controlled by the energy:

Re (T α 1/4 u , M u) L 2 ≤ C E(t) .
Let us now consider the operator

P := T α -1/4 Re T α -(T α 1/2 ) * T α 1/4 .
A straightforward computation shows that the principal symbol of P is 0; moreover, as Re Op = (Op + (Op) * )/2, from Theorem 3.17 we gather that its subprincipal symbol is given by

-i ∂ ξ α -1/4 ∂ x α + 1 2 α -1/4 ∂ ξ ∂ x α + ∂ ξ α 1/2 ∂ x α 1/4 + ∂ ξ ∂ x α 1/2 α 1/4 ,
and so it has order 1/2 + log. Therefore, we nally get the control for T 1 :

|T 1 | ≤ C E(t) + T ∂t(α -1/4 ) u H log P u H -log ≤ C E(t) + u 2 H 1/2 ≤ C E(t) .
Now, let us handle the term

T 2 := 2Re (T α 1/4 u , T α 1/4 ∂ t u) L 2 + 2Re (T α -1/4 ∂ t u , -T α -1/4 Re T α u) L 2 eq:T_2 (46) = 2Re (∂ t u , Qu) L 2 ,
where we have dened the operator

Q := Q 1 -Q 2 = (T α 1/4 ) * T α 1/4 -(T α -1/4 ) * T α -1/4 Re T α .
To compute the order of Q, let us proceed with care and analyse the symbol of each of its terms. Once again, by use of Theorem 3.17 one infers

Q 1 = T α 1/2 -i T ∂ ξ (α 1/4 )∂x(α 1/4 ) + T ∂ ξ ∂x(α 1/4 )α 1/4 - eq:Q_1 (47) -T 1 2 ∂ 2 ξ (α 1/4 )∂ 2 x (α 1/4 ) + T ∂ 2 ξ ∂x(α 1/4 )∂x(α 1/4 ) + T 1 2 ∂ 2 ξ ∂ 2 x (α 1/4 )α 1/4 + l.o.t. .
In the same way, one gets also

T α -1/4 Re T α = T α 3/4 -i T ∂ ξ (α -1/4 )∂xα + 1 2 T α -1/4 ∂ ξ ∂xα - - 1 2 T ∂ 2 ξ (α -1/4 )∂ 2 x α + T ∂ ξ (α -1/4 )∂ξ∂ 2 x α + T α -1/4 ∂ 2 ξ ∂ 2 x α + l.o.t. ,
and so nally we arrive to the formula

Q 2 = T α 1/2 -i T ∂ ξ (α -1/4 )∂x(α 3/4 ) + ∂ ξ ∂x(α -1/4 )α 3/4 + α -1/4 ∂ ξ (α -1/4 )∂xα + 1 2 α -1/2 ∂ ξ ∂xα - eq:Q_2 (48) -T 1 2 ∂ 2 ξ (α -1/4 )∂ 2 x (α 3/4 ) + ∂ 2 ξ ∂x(α -1/4 )∂x(α 3/4 ) + ∂ ξ (α -1/4 )∂x(∂ξ(α -1/4 )∂xα) + + T 1 2 ∂ ξ (α -1/4 )∂x(α -1/4 ∂ ξ ∂xα) + 1 2 ∂ 2 ξ ∂ 2 x (α -1/4 )α 3/4 + 1 2 α -1/4 ∂ 2 ξ (α -1/4 )∂ 2 x α + + T 1 2 α -1/4 ∂ ξ (α -1/4 )∂ξ∂ 2 x α + 1 2 α -1/2 ∂ 2 ξ ∂ 2 x α + + T ∂ ξ ∂x(α -1/4 )∂ξ(α -1/4 )∂xα + 1 2 ∂ ξ ∂x(α -1/4 )α -1/4 ∂ ξ ∂xα + l.o.t. .
Now, it's evident that Q has null principal symbol. Its subprincipal symbol, instead, has order log; nevertheless, comparing equalities (47) and (48), we discover that it is identically 0, too. r:new_en Remark 4.4. Let us stress again this point: requiring also the subprincipal part of the operator Q to be null forces us to take the symbol α of order 1/2.

In particular, one can try to introduce a new energy, dening

v(t, x) := T β ∂ t u -T ∂tβ u , w(t, x) := T βα 1/2 u and E(t) := v(t) 2 L 2 + w(t) 2 L 2 ,
where β is a generic symbol of order s (not necessarily s = 1/2). The presence of α 1/2 in w is due to the fact that we want w L 2 ∼ u H s+1 .

Repeating the same computations as before, one can see that the following constraints appear:

• β has to be of the form β = α -1/4 f , for some symbol f (x, ξ) of order s -1/2;

• even in the simplest case, i.e. f = f (ξ), the subprincipal part of the operator Q is not null, if f is not null, and its order is (2s + 1) + log, which is not suitable for us, due to the logarithmic loss.

Let us come back to the operator Q. Its sub-subprincipal part is not void, but, analysing one term by one, we discover that it has order 0. r:order Remark 4.5. This fact can be seen also without all the previous complicated computations. Let us compare the subprincipal symbol of Q, of order log, with the one of the next order part. The latter presents one more derivative both in ξ and in x with respect to the former. The derivative ∂ ξ makes the order decrease of 1; the space derivative, instead, acts as described in Lemma 3.15. Therefore, two dierent kinds of terms occur in the sub-subprincipal symbol:

(i) terms which present a product of the type ∂ x (α µ ) ∂ x (α ν ): in this case, the order of these terms increases of log;

(ii) terms which have a factor of the type ∂ 2 x (α µ ): in this case, instead, the order increases of

1 -log.
Therefore, rst terms have total order equal to log -1 + log = 2 log -1, while the other ones are of order log -1 + 1 -log = 0. Hence, the sub-subprincipal part of Q is at most of order 0.

In any case, we nally obtain the following control on T 2 : The aim of this section is to prove the following result.

|T 2 | ≤ C ∂ t u
th:wp Theorem 5.1. Let L be the operator dened by [START_REF] Hörmander | Linear partial dierential operators[END_REF], and assume it is strictly hyperbolic with bounded coecients, i.e. relation [START_REF] Métivier | Interactions de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace[END_REF] holds true. Suppose that its coecients a ij , b j and c are all of class C ∞ b (R N ), and that, in addition, the a ij 's are Zygmund continuous with respect to the time variable: for all τ ≥ 0 one has sup (t,x) a ij (t + τ, x) + a ij (t -τ, x) -2 a ij (t, x) ≤ K 0 τ .

Then the related Cauchy problem is well-posed in the space H ∞ , globally in time.

Proof. The proof is based on the computations performed in Section 4, so we will limit ourselves to point out only the main dierencies.

First of all, we set v(t, x) := T Λ σ α -1/4 ∂ t u -T ∂t(Λ σ α -1/4 ) u and w(t, x) := T Λ σ α 1/4 u ,

where the symbol Λ(ξ, γ) was dened in (24). As before the energy associated to u is the quantity

E(t) := v(t) 2 L 2 + w(t) 2 L 2 .
At this point, one has to notice that, thanks to the additional regularity of the coecients, the thesis of Lemmas 4.2 and 4.3 are true for any s > 0, and also the multiplication by c maps H s into itself for the same s.

Therefore, if we take a σ > -1/2, all the previous computations hold true, with no changes.

We have to pay attention only to the analysis of the term T 2 , dened by (46). The principal symbol of Q is still 0, but the subprincipal one doesn't cancel anymore (recall also Remark 4.4). Nevertheless, it's easy to see that this time its order is 2σ. As a matter of fact, also Lemma 3.15 still holds true, but in the second estimate (i.e. that one where |β| = 1) the logarithmic loss disappears, due to the additional regularity of the a ij 's.

So, we have

|T 2 | ≤ C ∂ t u H σ-1/2 Qu H -σ+1/2 ≤ C ∂ t u H σ-1/2 u H σ+1/2 ≤ C E(t) .
In the end, we arrive to an inequality of the form

  sup z=(t,x)∈R×R N |a jk (z + ζ) -a jk (z)| ≤ C |ζ| log 1 + 1 |ζ| .

  Z_norm Remark 3.18. Let us stress this fundamental fact: the operator norms of all the subprincipal part terms in the previous theorem (i.e. T ∂ ξ a ∂xb and T ∂ ξ ∂xa ) depend only on the seminorms |a| Z and |b| Z , and not on γ.

r:

  ) .

	4.2	Changing the operator
	est:E_t (41)

  H -1/2 Qu H 1/2 ≤ C E(t) . ≤ C 1 E(t) + C 2 (E(t)) 1/2 Lu(t) H -1/2 ,for some positive constants C 1 , C 2 depending only on λ 0 , Λ 0 and on the Zygmund norms, both in space and time, of the coecients a jk .Applying Gronwall's inequality to previous estimate entails So, remembering inequalities (40) and (41), we manage to bound the norm of the solution in H 1/2 × H -1/2 in terms of initial data and external force only: for all t ∈ [0, T ],∂ t u(t) H -1/2 + u(t) H 1/2 ≤ C e λ t ∂ t u(0) H -1/2 + u(0) H 1/2 + Lu(τ ) H -1/2 dτ ,which is actually the thesis of Theorem 2.2. Moreover, this relation implies, in particular, well-posedness in the space H 1/2 × H -1/2 .

	4.4	Final estimates
	Putting all the proved inequalities together, we get the estimate
	d dt E(t) est:sol-fin (50)
		+	t	e -λ τ
		0

est:E-final est:E-final (49)

(E(t)) 1/2 ≤ C e λ t (E(0)) 1/2 + t 0 e -λ τ Lu(τ ) H -1/2 dτ .

5

On the H ∞ well-posedness s:H^inf

  H σ-1/2 dt , which holds true for any σ > -1/2 and for positive constants λ and C depending only on σ and on the norms of the coecients of L on the respective functional spaces.From this relation we immediately gather the H ∞ well-posedness of the Cauchy problem related to L.

sup 0≤t≤T u(t, •) H σ+1/2 + ∂ t u(t, •) H σ-1/2 ≤ ≤ C e λT u(0, •) H σ+1/2 + ∂ t u(0, •) H σ-1/2 + T 0 e -λt Lu(t, •)

In the same spirit of Lemma 4.2, we have also the next result. l:b Lemma 4.3. Given a Hölder continuous function b ∈ C θ (R N ), for some θ > 0, let us dene the remainder operator

Then, B maps H -s (R N ) into H θ-s (R N ) continuously for all s ∈ ]0, θ[ . Proof. As just done, let us write

Hence, thanks to Lemma 2.84 of [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF], it's enough to estimate le L 2 norm of each B ν .

Using the dyadic characterization of Hölder spaces, we have

while, as s > 0, Proposition 2.79 of [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Dierential Equations[END_REF] gives

where the sequence (d ν ) ν belongs to the unitary sphere in ℓ 2 (N).

Therefore, we nally gather

and this implies Bv ∈ H θ-s .

Thanks to Lemmas 4.2 and 4.3, equation ( 10) can be rewritten in the following way:

with the notations ∂ 0 = ∂ t and B j = b j -T b j .

Energy estimates

Now we are nally ready to compute the time derivative of the energy. Thanks to Tarama's cancellations and identity (43), we have

By use of Lemma 3.15, keeping in mind the choice of the parameter ε in (38), it's quite easy to see that the following estimates hold true: where, in the last inequality, we have used also relation (20).


Aubert Gilles

J A Laboratoire

Dieudonné

Daniele Graziani
email: daniele.graziani@inria.fr

A relaxation result for an inhomogeneous functional preserving point-like and curve-like singularities in image processing

Keywords: AMS subject Classication: 49J45 49M20 49Q20 W 1, p loc -lower semicontinuity, divergence measure elds, Leibniz rule, p-capacity, convexity, relaxation, image processing

In the present paper we address a relaxation theorem for a new integral functional of the calculus of variations dened on the space of functions in W 1,p loc whose gradient is an L p -vector eld with distributional divergence given by a Radon measure.

The result holds for integrand of type f (x, ∆u) without any coerciveness condition, with respect to the second variable, and C 1 -continuity assumptions with respect to the spatial variable x.

Introduction

In a recent paper (see [START_REF] Aubert | Blanc-Féraud Analysis of a new variational model to restore point-like and curve-like singularities in imaging[END_REF]) the authors have introduced a new functional of calculus of variations to preserve point-like and curve-like singularities in biological images corrupted by noise. The energy, they deal with, was the following

F(u, Ω ) : = Ω f (∆u)dx + Ω f ∞ ( dµ a d|µ a | )d|µ a | + Ω f ∞ ( dµ 0 d|µ 0 | )d|µ 0 | + Ω g(∇u)dx + Ω |u -u 0 | 2 dx, (1.1)
1 Corresponding author.

1
where Ω is an arbitrary open set such that Ω ⊂⊂ Ω ⊂ R N . N ∈ {2, 3}, u ∈ ∆M p loc (Ω)

with 2N N +2 ≤ p < N N -1
. ∆M p (Ω) is the space of W 1,p loc -functions whose gradient is an L p -vector eld with distributional divergence given by a Radon measure; the measures µ a , µ 0 are given by the p-capacitary decomposition applied to the singular part of the divergence measure of ∇u, that is (1.2)

Div∇u = ∆udx + (∆u) s = ∆udx + µ a + µ 0 ; dµ a d|µ a | , dµ 0 d|µ 0 |
are the Radon-Nikodym derivatives of the measures µ a and µ 0 with respect to their total variation. The restriction p < N N -1 is needed to allow singularities on curves and points (see [START_REF] Aubert | Variational approximation of detecting point-like target problems[END_REF][START_REF] Graziani | A formal Γ-convergence approach for the detection of points in 2-D images[END_REF] on this issue). The integrands f , g are convex functions and f ∞ is the recession function (see [START_REF] Aubert | Blanc-Féraud Analysis of a new variational model to restore point-like and curve-like singularities in imaging[END_REF] for a precise denition of all these quantities and references on p-capacity). Finally u 0 is a given data and here the restriction p ≥ 2N N +2

is assumed to give sense to the discrepancy term Ω |u -u 0 | 2 dx. In [START_REF] Aubert | Blanc-Féraud Analysis of a new variational model to restore point-like and curve-like singularities in imaging[END_REF], under suitable growth assumptions on the integrands f and g, an existence result was proven. It was also shown that functional (1.1) coincides with the lower semicontinuous envelope, with respect to W 1,p loc (Ω)-weak convergence of the following functional F (u, Ω) := Ω f (∆u)dx + Ω g(∇u)dx + Ω |u -u 0 | 2 dx on C 2 (Ω), +∞ on ∆M p loc (Ω) \ C 2 (Ω).

For the applications it can be crucial to allow a dependence with respect to the spatial variable x in the integrand f . Indeed in the damaged image reconstruction problem one might like to emphasize the singularities contained in a given region of Ω by giving appropriate values to the integrand. Moreover such x-dependence, in most cases, does not satisfy strong regularity property, as strong dierentiability for instance. Typical examples are integrand of type f (x, ξ) = a(|x|)f (ξ). Therefore it makes sense, even from an experimental point of view, to consider integrand of type f (x, ∆u). However in this work we have limited ourselves to a pure theoretical analysis. We refer to [START_REF] Aubert | Blanc-Féraud Analysis of a new variational model to restore point-like and curve-like singularities in imaging[END_REF] for numerical applications in the homogeneous case.

In view of integral representation formula, it is natural to investigate lower semicontinuity property when such an x-dependence is allowed. Indeed the lower semicontinuity of functional F corresponds to obtain the so called lim inf inequality (see subsection 2.6) for the relaxed functional of F . Here we disregard the data term

Ω |u -u 0 | 2 ,
since it is a continuous perturbation that does not aect the integral representation of the relaxed functional. Moreover, since decomposition (1.2) plays no role in the proof of the relaxation result, we split the Radon measure Div∇u by using the canonical Radon-Nikodym decomposition:

Div∇u = ∆udx + (∆u) s .
Therefore in this paper we rst study the lower semicontinuity, with respect to the W 1,p loc (Ω)-weak convergence, of the following inhomogeneous version of functional (1.1):

(1.3)

F(u, Ω ) := Ω f (x, ∆u)dx + Ω f ∞ (x, d(∆u) s d|(∆u) s | )d|(∆u) s | + Ω g(∇u)dx.
with Ω ⊂⊂ Ω. In order to prove W 1,p loc (Ω)-weak lower semicontinuity result we use a successful technique, developed in these last years, to address the L 1 -lower semicontinuity of integral functionals (even if the integrand depends on the variable u) dened on the space BV (Ω). It permits to prove lower semicontinuity theorems by dropping the coerciveness assumptions and under weak dierentiability requirements on the integrand f (x, u, ∇u). The main tools of this approach are a chain rule formula and an approximation result for families of convex functions depending on a parameter due to E.De Giorgi (see Theorem 2.2). There is by now a vast literature on this topic. Without claiming of being exhaustive we refer to [START_REF] Graziani | On the L 1 -lower semicontinuity and relaxation in BV (Ω; R M ) for integral functionals with discontinuous integrand[END_REF][START_REF] Maggi | On Serrin's lower semicontinuty theorem[END_REF] and references therein for an overview on this subject. Very roughly speaking, by De Giorgi's Theorem, one can write functional F as a supremum of ane functionals involving the scalar product between certain coecients (the so-called De Giorgi's coecients) and the derivatives of u. Then if {u n } ⊂ ∆M p loc (Ω) is a sequence which converges to a function u ∈ ∆M p loc (Ω), one can recover the convergence of the derivatives of u n , by switching the derivatives to suitable test functions and so proving the continuity of those ane functionals and therefore the lower semicontinuity of F.

In this paper we adapt this strategy to our dierent variational framework in order to obtain W 1,p loc -weak lower semicontinuity under weak regularity conditions on the in- tegrand with respect to the spatial variable x. In particular a new Leibniz rule for the product between a proper scalar function b and u ∈ ∆M p (Ω) is established. Besides in order to deal with the right duality involving the W 1,p loc (Ω)-weak convergence, we assume that the integrand f belongs to W 1,p (Ω

) with ∇ x f ∈ L p loc (Ω × R), 1 p + 1 p =1 (see assumption (4.1)).
The last part of the paper is devoted to provide the so called lim sup inequality for the relaxed functional of F , which, combined with the lim inf inequality, gives the integral representation F = F (see subsection 2.6 and Theorem 5.1). The proof is based on an approximation result for lower semicontinuous functions contained in [START_REF] Fusco | A remark on Serrin's Theorem[END_REF]. This result, under a suitable uniform lower semicontinuity condition with respect to the spatial variable (see Theorem 2.3), permits to write a convex integrand f as a supremum of functions which are split as a product of a function depending only on the spatial variable times a function only depending on the second variable. Then in order to prove lim sup inequality we can adapt the technique used in [START_REF] Aubert | Blanc-Féraud Analysis of a new variational model to restore point-like and curve-like singularities in imaging[END_REF] for the isotropic functional (1.1). Moreover to attain the upper bound we need, as in the case of integral functional dened on BV -space, to require a linear growth from above on the integrand f .

The paper is organized as follows. Section 2 is devoted to notations, preliminary denitions and results. In Section 3 we address the new Leibniz rule formula. In section 4 we prove the lower semicontinuity result. Finally in section 5 we provide the integral representation formula.

Definition and main properties

2.1. Distributional divergence and classical spaces. In this subsection we recall the denition of the distributional space L p,q (Div; Ω) and DM p (Ω), 1 ≤ p, q ≤ +∞,

(see [START_REF] Anzellotti | Pairings between measures and bounded functions and compensated compactness[END_REF][START_REF] Chen | Frid Divergence-measure elds and conservation laws[END_REF][START_REF] Chen | On the theory of divergence-measure elds and its applications[END_REF]). In all the paper Ω ⊂ R N is an open bounded set with Lipschitz boundary. N ∈ {2, 3} is the space dimension. L n and dx will denote the Lebesgue measure on R N .

The bracket •, • stands for the duality product in some distributional space.

Denition 2.1. We say that U ∈ L p,q (Div; Ω) if U ∈ L p (Ω; R N ) and if its distributional divergence DivU ∈ L q (Ω). If p = q the space L p,q (Div; Ω) will be denoted by L p (Div; Ω).

Denition 2.2. For

U ∈ L p (Ω; R N ), 1 ≤ p ≤ +∞, set |DivU |(Ω) := sup{ U, ∇ϕ : ϕ ∈ C ∞ 0 (Ω), |ϕ| ≤ 1}.
We say that U is an L p -Divergence measure eld, i.e. U ∈ DM p (Ω), if

U DM p (Ω) := U L p (Ω;R N ) + |DivU |(Ω) < +∞.
We say that U ∈ DM p loc (Ω) if U ∈ DM p (Ω ) for every Ω ⊂⊂ Ω.

We recall that U ∈ L p (Ω; R N ) belongs to DM p (Ω) if and only if there exists a Radon measure denoted by DivU such that

U, ∇ϕ = - Ω DivUϕ ∀ϕ ∈ C ∞ 0 (Ω),
and the total variation of the measure DivU is given by |DivU |(Ω)

Let us recall the following classical result (see [START_REF] Chen | On the theory of divergence-measure elds and its applications[END_REF] Proposition 3.1).

Theorem 2.1. Let {U

h } h ⊂ DM p (Ω) be such that (2.1) U h U in L p (Ω; R N ), as h → +∞ for 1 ≤ p < +∞.
Then

U L p (Ω;R N ) ≤ lim inf h→+∞ U h L p (Ω;R N ) , |DivU |(Ω) ≤ lim inf h→+∞ |DivU h |(Ω).
Finally we dene the following space (2.2)

∆M p loc (Ω) := {u ∈ W 1,p loc (Ω), ∇u ∈ DM p loc (Ω)}.
For u ∈ ∆M p loc (Ω) the measure Div∇u is also denoted with ∆u.

l ∈ N, u l : Ω → [0, ∞]. Then Ω sup l u l dµ = sup {A k } k∈N sup l∈N l k=1 A k u k dµ},
where A k ⊂ Ω are open and pairwise disjoint sets with compact closure in Ω.

2.3. Convex Functions. We briey recall the classical approximation theorem due to E.De Giorgi (see [START_REF] Giorgi | Semicontinuity theorems in the calculus of variations[END_REF]). We state this result in the particular form we will use in the sequel. For more general statement we refer to [START_REF] Giorgi | Semicontinuity theorems in the calculus of variations[END_REF].

Theorem 2.2. Let f : Ω × R → [0, +∞), (x, t) → f (x, t), be a Borel function convex with respect to t for all x ∈ Ω. There exists a sequence

{ξ l } ⊂ C ∞ 0 (R) with ξ l ≥ 0 and R ξ l (t)dt = 1, such that f (x, t) = sup l∈N (α l (x) + β l (x)t) + , where α l (x) := R f (x, t) 2ξ l (t) + ξ l (t)t dt (2.3) b l (x) := - R f (x, t)ξ l (t)dt. (2.4)
It is worth noticing that the coecients α l and β l explicitly depend on the function f . The explicit formulas permit to deduce regularity properties of the coecients α l and β l from proper hypotheses satised by f . We conclude this section with another approximation result for convex function contained in [START_REF] Fusco | A remark on Serrin's Theorem[END_REF]. Also in this case we state the result in a simpler case and we refer the reader to Lemma 8 of [START_REF] Fusco | A remark on Serrin's Theorem[END_REF] for a more general statement.

Theorem 2.3. Let f : Ω × R → [0, +∞) be a lower semicontinuous function in (x, t) such that f (x, •) is convex for all x ∈ Ω. Assume in addition that for all > 0 and for all x 0 ∈ Ω there exists δ > 0 such that

(2.5) f (x 0 , t) ≤ f (x, t) + (1 + f (x, t)) for all (x, t) ∈ Ω × R such that |x -x 0 | < δ.
Then there exist two sequences

{a k } ⊂ C ∞ 0 (Ω) and {ψ k } ⊂ C ∞ 0 (Ω) such that, for all k ∈ N, 0 ≤ a k ≤ 1, ψ k is a convex function satisfying 0 ≤ ψ k (t) ≤ Λ k (1 + |t|),
for some Λ k ≥ 0, and (2.6)

f (x, t) = sup k∈N a k (x)ψ k (t) for all (x, t) ∈ Ω × R.
F (u, Ω) := Ω f (x, ∆u)dx + Ω g(∇u)dx on C 2 (Ω), +∞ on ∆M p loc (Ω) \ C 2 (Ω);
(2.7)

F(u, Ω ) := Ω f (x, ∆u)dx + Ω f ∞ (x, d(∆u) s d|(∆u) s | )d|(∆u) s | + Ω g(∇u)dx,
where f ∞ is the recession function given by lim t→+∞ f (x, tξ) t , with ξ ∈ R and the measure (∆u) s are given by classical decomposition applied to the Radon measure ∆u. we dene the lower semicontinuous envelope or relaxed functional with respect to the W 1,p loc -weak convergence of F given by:

(2.8)

F (u, Ω ) := inf u h ⊂C 2 (Ω ) {lim inf h→+∞ F (u h , Ω ) u h u},
where Ω ⊂⊂ Ω is an arbitrary open set.

For the convenience we recall here that a sequence {u h } ⊆ W 1,p loc (Ω) W 1,p loc -weakly converges to u ∈ W 1,p loc (Ω ) if and only if for all ϕ ∈ W 1,-p (Ω ) and Ω ⊂⊂ Ω we have

u h , ϕ W 1,p (Ω )×W 1,-p (Ω ) → u, ϕ W 1,p (Ω )×W 1,-p (Ω ) .
If L : ∆M p loc (Ω) → R is a continuous operator with respect to the W 1,p loc (Ω)-weak convergence we have:

(2.9)

(F + L) = F + L.
For general properties of the relaxation we refer to [START_REF] Braides | Γ-convergence for beginners[END_REF][START_REF] Maso | Introduction to Γ-convergence[END_REF].

Leibniz Rule

For our purpose the Leibniz rule here below plays a crucial role.

Lemma 3.1. Let 1 < p < N N -1 . Let b be a scalar function belonging to W 1,p (Ω) ∩ C 0 (Ω), with 1 p + 1 p = 1. Then for every u ∈ ∆M p loc (Ω) and for every φ ∈ C ∞ 0 (Ω), the following formula holds:

b(x)φ(x)d(Div∇u) = - Ω ∇u(x) • ∇ b(x)φ(x) dx.
Remark 3.1. The continuity assumption is needed in order to identify b with its continuous representative, otherwise the previous formula is false.

The proof is based on approximation argument.

Lower semicontinuity

In this section the lower semicontinuity result for functional (2.7) is addressed. The result is obtained by means of Theorem 2.2 and Lemma 3.1.

Let Ω ⊂ R N be a bounded open set. Let f : Ω × R → [0, +∞) be a Borel function, convex in the second variable, which satises the following condition:

(4.1) f (•, t) ∈ W 1,p (Ω) ∩ C 0 (Ω) with: ∇ x f ∈ L p loc (Ω × R). Theorem 4.1. Let f : Ω × R → [0,
lim inf h→+∞ F(u h , A) ≥ F(u, A). If A ⊂⊂ Ω is an arbitrary open set, we set G(u, A) := A f (x, ∆u)dx + A f ∞ (x, d(∆u) s d|(∆u) s | )d|(∆u) s |.
By Theorem(2.2) there exists a sequence {ξ l } ⊂ C ∞ 0 (R) with ξ l ≥ 0 and R ξ l dx = 1 such that for any (x, t) ∈ R we have

f (x, t) = sup l∈N α l (x) + β l (x)t + and f ∞ (x, t) = sup l∈N β l (x)t + ,
where, recalling (2.3) and (2.4)

α l (x) = R f (x, t) 2ξ l (t) + ξ l (t) dt β l (x) = - R f (x, t)ξ l (t)dt. (4.2)
Then for every u ∈ ∆M p loc (Ω), we have

G(u, A) = A sup l∈N α l (x) + β l (x)∆u(x) + dx + A sup l∈N β l (x) d(∆u) s d|(∆u) s | + d|(∆u) s |.
Since the measures dx and |(∆u) s | are mutually singular we have:

G(u, A) = A sup l∈N α l (x) + β l (x) ∆u(x) |∆u(x)| + + β l (x) d(∆u) s d|(∆u) s | (x) + d|Divu|.
Hence, by applying Lemma 2.1, with µ = |Div∇u|(A), we obtain

G(u, A) = sup {A k } k∈N sup l∈N l k=1 A k α k (x) + β k (x)∆u(x) + dx + A k β k (x) d(∆u) s d|(∆u) s | (x) + d|(∆u) s | , (4.3)
where A k ⊂ A are open and pairwise disjoint.

Let now for all k, η k be a test function in

C ∞ 0 (A k ) with 0 ≤ η k ≤ 1. As dx and |(∆u) s | are mutually singular measures G(u, A) = sup {A k } k∈N sup l∈N l k=1 sup 0≤η k ≤1 A k α k (x)η k (x)dx + A k β k (x)η k (x)∆u(x)dx + A k β k (x) d(∆u) s d|(∆u) s | (x)η k (x)d|(∆u) s | . (4.4)
Let us dene

H(u, A k ) := A k α k (x)η k (x)+ A k β k (x)∆u(x)η k (x)dx+ A k β k (x) d(∆u) s d|(∆u) s | (x)η k (x)d|(∆u) s
We are going to prove the continuity of H with respect to the W 1,p loc (Ω)-weak convergence for every k xed, by applying Lemma 3.1 with β k and u. Therefore, we need to check rst that β k (x) satises the hypotheses of Lemma 3.1. For every test Φ ∈ C ∞ 0 (Ω; R N )

we have

∇β k (x), Φ(x) D (Ω) = - Ω β k (x)DivΦ(x)dx = Ω R f (x, t)ξ k (t)dt DivΦ(x)dx. Since ∇ x f ∈ L p (Ω × R) we can apply Fubini's theorem to get that ∇β k (x), Φ(x) D (Ω) = R ξ k (t)dt Ω f (x, t)DivΦ(x)dx = - R ξ k (t)dt Ω ∇ x f (x, t) • Φ(x)dx.
Hence, we conclude that

∇β k (x), Φ(x) D (Ω) = - Ω R ∇ x f (x, t)ξ k (t)dt • Φ(x)dx,
and therefore we have the identication:

∇β k = - R ∇ x f (x, t)ξ k (t)dt. Then, since ξ k ⊂ C ∞ 0 (R) and ∇ x f ∈ L p loc (Ω × R) we infer β k ∈ W 1,p (Ω). Finally it is not dicult that, since f is continuous, β k is continuous too.
We are now position of proving the continuity of H. Let {u h } ⊂ ∆M p loc (Ω) be a sequence W 1,p loc -weakly converging to u ∈ ∆M p loc (Ω). By applying Lemma 3.1

lim h→+∞ H(u h , A k ) = A k α k (x)η k (x)dx + lim h→+∞ - A k β k (x)∇u h (x) • ∇η k (x)dx - A k ∇β k (x) • ∇u h (x)η k (x)dx = A k α k (x)η k (x)dx + A K β k (x)∇u(x) • ∇η k (x)dx - A K ∇β k (x) • ∇u(x)η k (x)dx = H(u, A k ). (4.5)
where in the last equality we have used again Lemma 3.1.

Therefore (4.5) implies that G, being the supremum of the sum of supremum of lower continuous functionals is lower semicontinuous itself. Then since g is convex it is not dicult to see that F is lower semicontinuous too.

RELAXATION

This section is devoted to the relaxation result. This result will be attained, once we will have proven the lower bound F ≥ F and the upper bound F ≤ F. The rst one is a consequence of the lower semicontinuity result proved in Theorem 4.1. In order to achieve upper bound we will strengthen the assumptions on the integrand f , by requiring uniform lower semicontinuity condition with respect to x.

5.1. Integral representation formula. We will assume that the integrands f : Ω × R → [0, +∞) and g : R N → [0, +∞) satisfy the following assumptions:

(5.1)

f (x, t) ≤ C 1 (1 + |t|) ∀(x, t) ∈ Ω × R, where 0 < C 1 < +∞ is a constant; (5.2) g(ξ) ≤ C 2 (1 + |ξ| p ) ∀ξ ∈ R N ,
where 0 < C 2 < +∞ is a constant. Moreover from assumption (5.1) it follows that

(5.3) f ∞ (x, t) ≤ C 1 |t| ∀(x, t) ∈ Ω × R.
We also assume that for all > 0 there exists δ > 0 such that for all x ∈ Ω (5.4)

f (x, t) ≤ f (y, t) + (1 + f (y, t)) ∀(y, t) ∈ Ω × R such that |x -y| ≤ δ
which implies that for all > 0 there exists δ > 0 such that for all x ∈ Ω (5.5)

f ∞ (x, t) ≤ f ∞ (y, t) + f ∞ (y, t) ∀(y, t) ∈ Ω × R such that |x -y| ≤ δ.
Remark 5.1. Let us note that assumption (5.4) holds whenever the integrand f is coercive with respect to the variable t for every xed x and satises the following stronger condition: for all > 0 there exists δ > 0 such that for all

x ∈ Ω f (x, t) -f (y, t) < |t| ∀(y, t) ∈ Ω × R such that |x -y| ≤ δ.
This is the case, for instance, for f (x, t) = |xt|, if 0 / ∈ Ω.

Theorem 5.1. Let f : Ω × R → [0, +∞) be a Borel lower semicontinuous function convex in the second variable satisfying (4.1). Assume that (5.1),(5.2) and (5.4) hold.

Then we have

F(u, Ω ) = F (u, Ω ) ∀u ∈ ∆M p loc (Ω) ∀Ω ⊂⊂ Ω.
Proof.

Step one: We start by proving the lower bound F ≤ F .

Since F is the greatest lower semicontinuous functional not greater than F , F ≤ F , and by Theorem 4.1, F is W 1,p loc -weak lower semicontinuous, it follows that F(u, Ω ) ≤ F (u, Ω ) for all u ∈ ∆M p loc (Ω) and for all Ω ⊂⊂ Ω.

Step two: We now prove the upper bound F ≥ F .

By taking into account assumption (5.4) we can apply Theorem 2.3 to the function f . So we have for all (x, t)

∈ Ω × R (5.6) f (x, t) = sup k a k (x)ψ k (t), f ∞ (x, t) = sup k a k (x)ψ ∞ k (t),
where ψ k are convex functions. Therefore it follows that there exist {α

l k }, {β l k } ⊂ R such that for all (x, t) ∈ Ω × R (5.7) f (x, t) = sup k a k (x) sup l (α l k + β l k t) f ∞ (x, t) = sup k a k (x) sup l β l k t . Let now u ∈ ∆M p loc (Ω) and Ω h = {x ∈ Ω such that dist(x, ∂Ω) > 1 h }. Let {u h } ⊂ C ∞ (Ω h ) dened as u h = u * ρ h where ρ h a standard mollifying sequence, with suppρ h = {x ∈ Ω; |x| ≤ 1 h }. For x ∈ Ω h we have that the support of ρ h (x -•) is contained in Ω.
Then for every x ∈ Ω h we have

a k (x) α k l + β l k ∆u h (x) = a k (x) Ω ρ h (x -y) α l k + β l k ∆u(y) dy + Ω ρ h (x -y)β l k d(∆u) s |d(∆u) s | (y)d|(∆u) s | .
By taking into account (5.6) and (5.7) we get:

a k (x) α k l + β l k ∆u h (x) ≤ Ω ρ h (x -y)f (x, ∆u(y))dy + Ω ρ h (x -y)f ∞ (x, d(∆u) s |d(∆u) s | (y))d|(∆u) s |.
Let > 0 and δ be consequently given by assumption (5.4). Then for h large enough, for x ∈ Ω h and y ∈ Ω such that |x -y| ≤ 1 where we have used that Ω ρ h (x -y)dx ≤ R N ρ h (x -y)dx = 1.

As in the proof of Theorem 3.1,2 in [START_REF] Gomann | Sublinear functions of measures and variation integrals[END_REF], we also get that (5.12)

This, being Ω arbitrary, concludes the step two and the whole proof.

2. 4 .

 4 Functionals and their properties. We denote by A(Ω), the family of all open bounded subsets Ω ⊂⊂ Ω.Let f : Ω × R → [0, +∞) be a Borel function continuous with respect to the spatial variable and convex in the second variable. We shall consider the following functionals dened for u ∈ ∆M p loc (Ω) and A ∈ A(Ω) :

  Proof. Let {ρ } be a standard sequence of molliers. We set b = b * ρ . Since the support of b is strictly contained in Ω, b is well dened in Ω. Then, by denition of distributional divergence and taking into account that b φ ∈ C ∞ 0 (Ω), we have (3.1) b (x)φ(x)d(Div∇u) = -Ω ∇u(x) • ∇ b (x)φ(x) dx. Now we have b φ → bφ uniformly in Ω and ∇(b φ) → ∇(bφ) in L p (Ω; R N ). Then we achieve the thesis by taking the limit as → 0 in (3.1).

h,

  we have that |x -y| ≤ 1 h ≤ δ. Hence from (5.4),(5.5) it follows that for all > 0a k (x) α k l + β l k ∆u h (x) ≤ Ω ρ h (x -y)f (y, ∆u(y))dy + Ω ρ h (x -y)f ∞ (y, d(∆u) s |d(∆u) s | (y))d|(∆u) s | + Ω ρ h (x -y)(1 + f (y, ∆u(y))dy + Ω ρ h (x -y)f ∞ (y, d(∆u)s |d(∆u) s | (y))d|(∆u) s |.

(5. 8 )

 8 Let now Ω ⊂⊂ Ω. Since Ω h is a sequence of open sets invading Ω, we have for h large enough Ω ⊂ Ω h . Then by taking the supremum rstly over l and over k on the left hand side of (5.8), performing an integration over Ω ,with respect to the variable x, and by taking into account the growth conditions (5.1) and (5.3) we get for h large enoughΩ f (x, ∆u h (x))dx ≤ Ω dx Ω ρ h (x -y)f (y, ∆u(y))dy + Ω dx Ω ρ h (x -y)f ∞ (y, d(∆u) s |d(∆u) s | (y))d|(∆u) s | x -y)f (y, ∆u(y))dy + Ω dx Ω ρ h (x -y)f ∞ (y, d(∆u) s |d(∆u) s | (y))d|(∆u) s | + L n (Ω)C 1 (1 + |Div∇u|(Ω )).

Finally

  by applying Fubini's Theorem and taking into account that is arbitrary we conclude (5.9)Ω f (x, ∆u h )dx ≤ Ω f (x, ∆u)dx + Ω f ∞ (x, d(∆u) s d|(∆u) s | )d|(∆u) s |,

f

  (x, ∆u h )dx + Ω g(∇u h )dx ≤ F(u, Ω ).Therefore since {u h } h ⊂ C 2 (Ω ) and u h goes to u with respect to the W 1,p loc -convergence, from (5.11) it follows thatF (u, Ω ) = inf u h ⊂C 2 (Ω ) {lim inf h→+∞ F (u h , Ω ) u h u} ≤ lim inf h→+∞ F (u h , Ω ) ≤ F(u, Ω ).

  d(∆u) s d|(∆u) s | is the Radon-Nikodym derivatives of the singular measure (∆u) s with respect to its total variation. We recall that since f is convex f ∞ is a well dened Borel function convex in the second variable. .5. Relaxation. Let F be the functional dened in (2.6). For every u ∈ ∆M p

	Remark 2.1. The continuity assumption with respect to x is is needed in order to iden-
	tify

f with its continuous representative, otherwise the term Ω f ∞ (x, d(∆u) s d|(∆u) s | )d|(∆u) s | in functional (2.7) may dier if one considers dierent continuous representative.

2loc (Ω),

  +∞) be a Borel function convex in the second variable satisfying (4.1). Let g : R N → [0, +∞) be a convex function. Then functional (2.7) is lower semicontinuous on ∆M p loc (Ω) with respect to the W 1,p loc -weak convergence. Proof.We have to show that for an arbitrary open set A ⊂⊂ Ω and for every sequence {u h } ⊆ W1,p

loc (A) W 1,p loc -weakly convergent to u ∈ W 1,p loc (A), th following equality holds:

Acknowledgments: We warmly thank anonymous referee for its careful reading of the paper and numerous remarks. Its advices have permitted to signicantly improve our work.