Jacques M Bahi

Christophe Guyeux
email: christophe.guyeux@univ-fcomte.fr

Michel Salomon

Building a Chaotic Proven Neural Network

Chaotic neural networks have received a great deal of attention these last years. In this paper we establish a precise correspondence between the so-called chaotic iterations and a particular class of artificial neural networks: global recurrent multi-layer perceptrons. We show formally that it is possible to make these iterations behave chaotically, as defined by Devaney, and thus we obtain the first neural networks proven chaotic. Several neural networks with different architectures are trained to exhibit a chaotical behavior.

Devaney's chaos; chaotic iterations; mathematical topology

I.

INTRODUCTION

Due to the widespread use of the Internet and new digital technologies in nowadays life, security in computer applications and networks never was such a hot topic. Digital rights managements, e-voting security, anonymity protection, and denial of services are examples of new security concerns appeared this last decade. Tools on which this security is based are, among others: hash functions, pseudo-random number generators, cryptosystems, and digital watermarking schemes. Due to their wide use in security protocols, these tools are targeted everyday by hackers and new threats are frequently revealed. For example, security flaws have been recently identified in the previous standard in hash functions called SHA-1 [START_REF] Rijmen | Update on sha-1[END_REF]. As the new standards (SHA-2 variants) are algorithmically similar to SHA-1, stronger hash functions using new concepts are desired.

New approaches based on chaos are frequently proposed as an alternative to solve concerns which recurrently appear in the computer science security field [START_REF] Bahi | Topological chaos and chaotic iterations. Application to hash functions[END_REF]- [START_REF] Wang | Randomness quality of CI chaotic generators. Application to internet security[END_REF]. The advantage of the use of chaotic dynamics for security problems lies in their unpredictability proven by mathematical theory of chaos. This theory brings many qualitative and quantitative tools, namely ergodicity, entropy, expansivity, and sensitive dependence to initial conditions [START_REF] Guyeux | Le désordre des itérations chaotiques et leur utilité en sécurité informatique[END_REF]. These tools allow the study of the randomness of the disorder generated by the considered system [START_REF] Bahi | Improving random number generators by chaotic iterations. Application in data hiding[END_REF].

Recently, many researchers have built chaotic neural networks in order to use it as a component of new proposed hash functions [START_REF] Li | A novel hash algorithm construction based on chaotic neural network[END_REF], pseudo-number generators, cryptosystems [START_REF] Dalkiran | Artificial neural network based chaotic generator for cryptology[END_REF], [START_REF] Lian | A block cipher based on chaotic neural networks[END_REF], and digital watermarking schemes. Since the first introduction by McCulloch and Pitts in 1959, artificial neural networks have been shown to be efficient nonlinear statistical data modeling tools which can implement complex mapping functions. They may be trained *Authors in alphabetical order to learn a chaotic process and also, by construction, exhibit suitable properties: data confusion and diffusion, one-way function and compression. Security is not the only application domain of such new tools: the existence of chaos in our brain has been recently revealed, and the use of a chaotic artificial neural network as a model can serve, for example, neuroscientists in their attempts to understand how the brain works.

However, using an element of chaos as a component of the scheme in not sufficient, in our opinion, to be able to claim that the whole process behaves chaotically. We believe that this claim is not so evident and must be proven. Let us notice that up to now the proposed chaotical neural networks have failed to convince the mathematics community due to a lack of proof. This is why it is explained in this paper how it is possible to build an artificial neural network that behaves chaotically, as it is defined by Devaney [START_REF] Devaney | An Introduction to Chaotic Dynamical Systems[END_REF]. We will establish a correspondence between particular neural networks and chaotic iterations, which leads to the definition of the first artificial neural networks proven chaotic, according to Devaney.

The remainder of this paper is organized as follows. The next section is devoted to some recalls on chaotic iterations and Devaney's chaos followed by a brief description of artificial neural networks (ANNs). Section III presents a review of some works related to chaotic neural networks. Our approach, which consists in building a global recurrent ANN whose iterations are chaotic, is formalized and discussed in Section IV. Concrete examples of chaotic neural networks also show the relevance of our method. Finally in Section V we conclude and outline future work.

II. BASIC RECALLS

In the sequel S n denotes the n th term of a sequence S and E i denotes the i th component of a vector E. In other words, at the n th iteration, only the S n -th cell is "iterated". Note that in a more general formulation, S n can be a subset of components and )) ( ( where k<n, describing for example, delays transmission [START_REF] Robert | Discrete Iterations: A Metric Study[END_REF]. Finally, let us remark that the term "chaotic", in the name of these iterations, has a priori no link with the mathematical theory of chaos, recalled below.

f f f k    is for the k th composition of a function f.
2) Devaney's Chaotic Dynamical Systems: Consider a topological space ( , ) and a continuous function f on . Definition 2 f is said to be topologically transitive if, for any pair of open sets U, V , there exists k>0 such that

V u f k ) ( . Definition 3 An element (a point) x is a periodic element (point) for f of period n N * , if f n (x) = x.
Definition 4 f is said to be regular on ( , ) if the set of periodic points for f is dense in : for any point x in , any neighborhood of x contains at least one periodic point. Definition 5 f is said to be chaotic on ( , ) if f is regular and topologically transitive.

The chaos property is strongly linked to the notion of "sensitivity", defined on a metric space ( ,d) by: Definition 6 f has sensitive dependence on initial conditions if there exists > 0 such that, for any x and any neighborhood V of x, there exists y and 0 n such that d(f n (x),f n (y)) > . is the constant of sensitivity of f. Indeed, Banks et al. have proven in [START_REF] Banks | On devaney's definition of chaos[END_REF] that when f is chaotic and ( ,d) is a metric space, then f has the property of sensitive dependence on initial conditions (this property was formerly an element of the definition of chaos). To sum up, quoting Devaney in [START_REF] Devaney | An Introduction to Chaotic Dynamical Systems[END_REF], a chaotic dynamical system "is unpredictable because of the sensitive dependence on initial conditions. It cannot be broken down or simplified into two subsystems which do not interact because of topological transitivity. And in the midst of this random behavior, we nevertheless have an element of regularity". Fundamentally different behaviors are consequently possible and occur in an unpredictable way.

3) Chaotic Iterations and Devaney's Chaos:

In this section we give outline proofs of the properties on which our study of chaotical neural networks is based. The complete theoretical framework is detailed in [START_REF] Guyeux | Hash functions using chaotic iterations[END_REF].

Denote by the discrete boolean metric, y x y x 0 ) , (

. Given a function f:

B N B N , define the function F f : ]] ; 1 [[ B N B N such that ]] ; 1 [[ ) ) , ( ) ( ) , ( ( ) , ( j k j f j k E f j k E E k F
, where + and are the boolean addition and product operations, x is the negation of x.

Consider the phase space

= ]] ; 1 [[ N B N and the map )) ), ( ( ), ( ( ) , ( E S i F S E S G f f where : (S n ) n N S  (S n+1
) n N S is the shift function, and the initial function i is the map which associates to a sequence, its first term:

i : (S n ) n N S  (S 0 ) [[1;N]].
Thus chaotic iterations can be described by the following iterations [START_REF] Guyeux | Hash functions using chaotic iterations[END_REF] )

( 1 0 k f k X G X X . Let us define a new distance between two points ) , ( ), , ( E S E S   by ) , ( ) , ( )) , ( ); , (( S S d E E d E S E S d s e     , where 1 ]] ; 0 [[ ) , ( ) , ( k k k e E E E E d   ] 1 ; 0 [ 10 9 ) , ( 1 k k k k s S S S S d   .
This new distance has been introduced in [START_REF] Guyeux | Hash functions using chaotic iterations[END_REF] to satisfy the following requirements. When the number of different cells between two systems is increasing, then their distance should increase too. In addition, if two systems present the same cells and their respective strategies start with the same terms, then the distance between these two points must be small because the evolution of the two systems will be the same for a while. The distance presented above follows these recommendations. Indeed, if the floor value d(X,Y) is equal to n, then the systems E, Ĕ differ in n cells. In addition, d(X,Y) is a measure of the differences between strategies S and Š. More precisely, this floating part is less than 10 -k if and only if the first k terms of the two strategies are equal. Moreover, if the k th digit is nonzero, then the k th terms of the two strategies are different.

It is proven in [START_REF] Guyeux | Hash functions using chaotic iterations[END_REF] by using the sequential continuity that the vectorial negation ) , , ( ) , , (

1 1 0 x x x x f   satisfies the following proposition: Proposition 1 0 f G is a continuous function on ( ,d).
It is then checked in [START_REF] Guyeux | Hash functions using chaotic iterations[END_REF], that in the metric space ( ,d), the vectorial negation fulfills the three conditions for Devaney's chaos: regularity, transitivity, and sensitivity. This has led to the following result. Proposition 2 0 f G is a chaotic map on ( ,d) in the sense of Devaney.

B. Neural Networks

An artificial neural network is a set of simple processing elements called neurons that are interconnected, usually with a layer structure. It takes some input values and produces some output ones. Like a biological neural network, the connections between neurons influence the outputs given by the artificial network. Thanks to a training process, an ANN is able to learn complex relationships between inputs and Figure 1. Description of a neuron outputs. A neuron j computes an output y= (x,w) where () is the activation function, x is the input vector, and w the parameter vector. w can be used to parameterize or the neuron inputs. In this last case it means that the connections are weighted and a vector w component is then referred to as a synaptic weight. Fig. 1 describes a neuron j with weighted connections. Its output y j satisfies: Neural networks have a layered architecture, but they may differ in the way the output of a neuron affect himself. In fact, based on the connection graph, two kinds of networks can be distinguished: those having a least one loop and those without any one. A neural network which exhibits a loop is called a feedback (or recurrent) network, whereas a network belonging to the second class is said feed-forward. Obviously, a feedback network can be seen as a dynamical system. In the following sections, we use a recurrent version of the multi-layer perceptron (MLP), a well-known ANN architecture for which the universal approximation property has been proven in the feed-forward context [START_REF] Cybenko | Dynamic load balancing for distributed memory multiprocessors[END_REF]. Typically, a MLP consists in layer of input neurons, in one or more layers of hidden neurons, and a layer of output neurons. Since an input neuron is simply used as a channel to dispatch an input to each neuron of the first hidden layer, we will not further consider the input layer. Usually, the neurons of a given layer have similar characteristics and each one is fully connected to the next layer. Finally, note that the number of inputs and output neurons is completely specified by the considered problem, while the number of hidden neurons depends directly on the complexity of the relationships to be learned by the ANN.

n i i ij j n i j i ij j
As said previously, a neural network is designed to model relationships between and outputs. In order to find a proper modeling, an ANN must be trained so that it provides the desired set of output vectors. The training (or learning) process consists mainly in feeding the network with some input vectors and updating the neurons parameters (weights and bias value) using a learning rule and some information which reflects the quality of the current modeling. When the expected output vectors (D k ) are known in advance, the quality can be expressed through the Mean Square Error [START_REF] Lecun | Efficient backprop[END_REF]:

N k k k Y D N MSE 1 2 2 1
where N is the number of input-output vector pairs used to train the ANN (the pair set is called the training or learning set) and Y k denotes an output vector produced by the output layer for a given input vector X k . Consequently, in that case the training process, which is said supervised, results in an optimization algorithm targeted to find the weights and biases that minimize the MSE. Various optimization techniques exist, they have given raise to distinct training algorithms performing iterative parameters update. Gradientbased methods are particularly popular to the backpropagation algorithm, but they are sensitive to local minima. Heuristics like simulated annealing or differential evolution permit to find a global minimum, but they have a slow convergence. To control the training process, two methods are the most commonly used: firstly the number of iterations, also called epochs, reaches an upper bound; secondly the MSE goes below a threshold value.

III. RELATED WORK

Since a while neuroscientists discuss the existence of chaos in the brain. In the context of artificial neural networks, this interest has given raise to various works studying the modeling of chaos in neurons. The chaotic neuron model designed by Aihara et al. [START_REF] Aihara | Chaotic neural networks[END_REF] is particularly used to build chaotic neural networks. For example, in [START_REF] Crook | A novel chaotic neural network architecture[END_REF] is proposed a feedback ANN architecture which consists of two layers (apart from the input layer) with one of them composed of chaotic neurons. In their experiments, the authors showed that without any input sequence the activation of each chaotic neuron results in a positive average Lyapunov exponent, which means a true chaotic behavior. When an input sequence is given iteratively to the network the chaotic neurons reach unstable periodic orbits with different periods, and thus potentially provide a recognition state. Similarly, the same authors have recently introduced another model of chaotic neuron: the nonlinear dynamic state (NDS) neuron, and used it to build a neural network which is able to recognize learned unstable periodic orbits identifying patterns [START_REF] Crook | Pattern recall in networks of chaotic neurons[END_REF].

Today, another field of research in which chaotic neural networks have received a lot of attention is data security. In fact, chaotic cryptosystems are an appealing alternative to classical ones due to properties such as sensitivity to initial conditions or topological transitivity. Thus chaotic ANNs have been considered to build ciphering methods, hash functions, digital watermarking schemes, pseudo-random number generators, etc. In [START_REF] Dalkiran | Artificial neural network based chaotic generator for cryptology[END_REF] such a cipher scheme based on the dynamics of Chua's circuit is proposed. More precisely, a feed-forward MLP with two hidden layers is built to learn about 1500 input-output vector pairs, where each pair is obtained from the three nonlinear ordinary differential equations modeling the circuit. Hence, the proposed chaotic neural network is a network which is trained to learn a true chaotic physical system. In the cipher scheme the ANN plays the role of chaos generator with which the plain-text will be merged. Untrained neural networks have also been considered to define block ciphering [START_REF] Lian | A block cipher based on chaotic neural networks[END_REF] or hash functions [START_REF] Li | A novel hash algorithm construction based on chaotic neural network[END_REF]. The background idea is to exploit the inherent properties of the ANNs architecture such as diffusion and confusion.

A. Defining a First Chaotic Recurrent Neural Network

We will now explain how to build a chaotic neural network using chaotic iterations.

Let us reconsider the vectorial negation function denoted by f 0 : B N B N and its associated map

0 f F : ]] ; 1 [[ B N B N .
Firstly, it is possible to define a MLP which recognizes

0 f F . That means, for all (k,x) ]] ; 1 [[ B N
, the response of the output layer to the input (k,x) is ) , ( 0 x k F f . Secondly, the output layer can be connected to the input layer as it is depicted in Fig. 2 leading to a global recurrent neural network working as follows:

At the initialization stage, the ANN receives a boolean vector x 0 B N as input state, and

S 0 [[1;N]] in its input integer channel i(). Thus, ) , ( 0 0 1 0 x S F x f B N is computed by the network.
This state x 1 is published as an output. Additionally, x 1 is sent back to the input layer, to act as Boolean state in the next iteration. At iteration number n, the recurrent neural network receives the state x n B N from its output layer and

i(S n ) [[1;N]] from its input integer channel i(). It can thus calculate ) ), ( ( 0 1 n n f n x S i F B N , which
will be the new output of the network. In this way, if the initial state x 0 B N is sent to the network with a sequence S [[1;N]] N applied in the input channel i(), then the sequence (x n )

n N* of the outputs is exactly the same than the sequence obtained from the following chaotic iterations:

x 0 B N and n N * , . )) ( ( ]], ; 1 [[ 1 0 1 i S i S x f x x i n n S n n i n i n
From a mathematical viewpoint, the MLP defined in this subsection and chaotic iterations recalled above have the same behavior. In particular, given the same input vector ((S n )

n N ,x 0 ), they produce the same output vector (x n ) n N* : they are two equivalent reformulations of the iterations of 0 f G in . As a consequence, the behavior of our MLP faithfully reflects the behavior of 0 f G which is chaotic according to Devaney.

B. Improving the Variety of Chaotic Recurrent Neural Networks

The approach proposed to build chaotic neural networks, explained in the subsection, is not restricted to an adhoc function f 0 : B N B N , it can be generalized as follows. The function 0 f F associated to the vectorial negation f 0 , which has been recognized by the neural network, can be replaced by any functions

(i,x) iff ) , ( x i F f is N(i,x).
We have proven in [START_REF] Guyeux | Le désordre des itérations chaotiques et leur utilité en sécurité informatique[END_REF] that: Theorem 1 Functions f : B n B n such that G f is chaotic according to Devaney, are functions such that the graph (f) is strongly connected.

Since it is easy to check whether a graph is strongly connected, we can use this theorem to discover new functions f : B N B N such that the neural network associated to G f behaves chaotically, as defined by Devaney.

C. The Discovery of New Chaotic Neural Networks

Considering Theorem 1, it is easy to check that ) , , , , ( ) , , ( Due to Theorem 1, the ANNs associated to f 0,1 , f 0,2 and f 1,1 behave chaotically, as defined by Devaney. Whereas it is not the case for the networks based on the boolean function g 0 and g 1 , since (g 0 ) and (g 1 ) are not strongly connected.

1 2 1 1 1 1 x x x x x x f n   is such that 1 f G behaves chaotically,

D. Experimental Results

Among the five neural networks evoked in the previous subsection we decided to study the training process of three of them. Note also that for each neural network we have considered MLP architectures with one and two hidden layers, with in the first case different numbers of hidden neurons (sigmoidal activation). Thus we will have different versions of a neural network modeling the same iteration function. Only the size and number of hidden layer may change, since the numbers of inputs and output neurons (linear activation) are fully specified by the function. The neural networks are trained using the quasi-Newton L-BFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno) algorithm in combination with the Wolfe linear search. The training is performed until the learning error (MSE) is lower than a chosen threshold value (10 -2 ).

Table I gives for each considered neural network the mean number of epochs needed to train it and a success rate which reflects a successful training in less than 1000 epochs. Both values are computed considering 25 trainings with random weights and biases initialization. These results highlight several points. Firstly, various MLP architectures can learn a same iteration function, with obviously a best suited one (a hidden layer composed of ten sigmoidal neurons). In particular the two hidden layer structure seems to be too complex for the functions to be learned. Secondly, training networks so that they behave chaotically seems to be more difficult, since they need in average more epochs to be correctly trained. However, the relevance of this point needs to be further investigated. Similarly, there may be a link between the training difficulty and the disorder (evaluation of their constants of sensitivity, expansivity, etc.) induced by a chaotic iteration function.

V. CONCLUSION AND FUTURE WORK

Many chaotic neural networks have been developed for different fields of application, in particular for data security purpose where they are used to define ciphering methods, hash functions and so on. Unfortunately, the proposed networks are usually claimed to be chaotic without any proof. In this paper we have presented a rigorous mathematical framework which allows us to construct artificial neural networks proven chaotic, according to Devaney. More precisely, a correspondence between chaotic iterations, which are a particular case of topological chaos in sense of Devaney, and MLP neural networks with a global feedback is established. In fact, we have shown that an iteration function is chaotic if its graph of iteration is strongly connected (a property easily checked), and that a global recurrent MLP can learn such a function. Future research will study more carefully the performance of the training process and alternative neural network architectures.

Figure 2 .

 2 Figure 2. Example of a global recurrent neural network modeling function 0 f F such that

  N such that the chaotic iterations G f are chaotic, as defined by Devaney. To be able to define functions that can be used in this situation, we must firstly introduce the graph of iterations of a given function f : B N B N , given a configuration x. In what follows the configuration N(i,x) switching the i-th component of x. Intuitively, x and N(i,x) are neighbors. The chaotic iterations of the function f can be represented by the graph (f) defined below. Definition 7 (Graph of iterations) In the oriented graph of iterations (f), vertices are configurations of B N , and there is an arc labeled i from x to N

  as defined by Devaney. Consequently, we can now obtain two chaotic neural networks by learning either

	F or	F .
	f	0	1 f
		To support our approach, a set of illustrative examples
	composed of five neural networks is given. The three first
	networks are respectively defined by:
			f	0	1 ,	(	1 x	,	2 x	,	3 x	,	4 x	)	(	1 x	,	2 x	,	3 x	,	4 x	)	,
			f	0	,	2	(	1 x	,	2 x	,	3 x	)	(	1 x	,	2 x	,	3 x	)	,
			1 , 1 f	(	1 x	,	2 x	,	3 x	)	(	1 x	,	1 x	,	2 x	)	,
	while the last ones are defined by:
			g	0	(	1 x	,	2 x	,	3 x	)	(	1 x	,	2 x	,	3 x	)	,
			1 g	(	1 x	,	2 x	,		3 x	)	(	1 x	,	2 x	,	3 x	)	.

TABLE I .

 I OUTLINE OF THE RESULTS FROM SEVERAL ITERATION FUNCTIONS LEARNING USING DIFFERENT RECURRENT MLP ARCHITECTURES

			One hidden layer
		8 neurons	10 neurons
	Function	Mean epoch	Success rate	Mean epoch	Success rate
	f 0,2	82.21	100%	73.44	100%
	f 1,1		100%	59.84	100%
	g 1	36.24	100%	37.04	100%
		Two hidden layers: 8 and 4 neurons
		Mean epoch number	Success rate
	f 0,2	203.68	76%
	f 1,1	135.54	96%
	g 1	72.56	100%


Vincent Strubel
email: vincent.strubel@insa-lyon.fr

Nicolas Fillot

Fabrice Ville

Philippe Vergne

Alexandre Mondelin

Yves Maheo

Debris

Debris Entrapment in Elliptical EHD Contacts

Keywords: debris entrapment, hybrid bearings, elliptical contacts, numerical simulations, twin-disc test rig 1. Overview

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Lubricating contacts is essential in a wide range of mechanical applications but warranty a safe and clean lubrication all along the bearing life has proved to be impossible to achieve. Debris entrapment inevitably occurs and thus has to be well understood in order to predict and limit premature failures. Although entrapment phenomena were already widely investigated in classical steel point contacts 1-3) , it is essential to quantify the influence on the entrapment phenomena by using different contacting materials, such as hybrid couple of materials, or different geometries (with different ellipticity radii). In order to find and explain the influence of these parameters on particle entrapment, numerical simulations as well as experimental tests are performed. Respectively simulating particle trajectories upstream the contact area and revealing effective entrapment ratios on a twin-disc machine.

Numerical simulations

Numerical simulations focusing on the lubricant layer upstream the contact area (Fig. 1), allow to simulate both the lubricant flow and the motion of suspended particle debris. Considering that particles are entrained by the lubricant flow and possibly, caught between the contacting surfaces, this numerical work allows to determine particle trajectories which highlight critical entrapment cases.

Twin disc experiments

Experiments are performed on a twin-disc machine 4) . It is composed of two contacting discs a flat and a crowned one, supplied by a contaminated lubricant (Fig. 2). The nature and the geometry of the discs can vary so that several types of contact can be studied (Table 1).

Table 1 Operating conditions

An observation of the contacting surfaces after the tests reveals dents which correspond to the marks generated by the particle entrapment. In this way effective entrapment ratios are obtained and quantitative comparisons can be performed.

Case study

The dual approach allows to validate numerical simulations with experimental tests and then to confirm the observed tendencies by performing a larger number of numerical calculations based on wider conditions.

Operating conditions

Pure rolling Curvature radius of the crowned disc (mm) Hence the nature of the contacting materials was found to have a relative weak influence on the entrapment phenomenon. On the contrary the variation of the ellipticity ratio significantly affects the entrapment ratios.

As well the independence of results by using different couple of materials as the ellipticity effect were first numerically assumed, they were then confirmed by the experimental work. Finally these tendencies were validated by a numerical analysis of the lubricant flow. It allowed to describe specific reverse flows occurring within elliptical contacts.

Conclusions

Bringing face to face experiments with numerical simulations leaded to focus on a key parameter governing the entrapment phenomenon in elliptical contact, which is the intensity of the reverse flow.

[1] Dwyer-Joyce, R. S., and Heymer, J., "The entrainment of solid particles into rolling elastohydrodynamic contacts," Tribology Series, 31, 1996, 135-140. [4] Ville, F., Nelias, D., Tourlonias, G., Flamand, L., and Sainsot, P. , "On the two-disc machine: A polyvalent and powerful tool to study fundamental and industrial problems related to elastohydrodynamic lubrication," Tribology Series, 39, 2001, 393-402.

Elliptical contact

[ 2 ]

 2 Ville, F., and Nelias, D., "An Experimental Study on the Concentration and Shape of Dents Caused by Spherical Metallic Particles in EHL Contacts," Tribology Transactions, 42(1), 1999, 231-240. [3] Nikas, G. K., "Mathematical analysis of the entrapment of solid spherical particles in non-conformal contacts," Journal of Tribology, 123(1), 2001, 83-93.

Fig. 1

 1 Fig.1 Numerical simulation upstream from the contact area


David Vicente

Anisotropic Mumford-Shah Model and its Approximation with Γ-convergence

published or not. The documents may come

Anisotropic Mumford-Shah Model and its approximation with Gamma-convergence

Introduction

This work is motivated by the problem of segmentation of sets strongly elongated in some directions as, for example, tubes or thin plates in an image of dimension n ∈ {2; 3}. Let Ω ⊂ R n be an open bounded domain and g ∈ L ∞ (Ω). We denote by H n-1 the (n -1)-dimensional Hausdorff measure. The model we introduce in this paper consists in minimizing

E(u, K, M) = Ω\K (u -g) 2 dx + Ω\K |∇u| 2 dx + K Mν, ν 1/2 dH n-1 + Ω DM n+α dx (0.1)
where K is compact and H n-1 -rectifiable with unitary normal vector ν, α > 0 and M takes its values in a compact subset G of symmetric definite positive matrices. The associated minimizing problem is (P) : min{E(u, K, M) : Kcompact and H n-1 -rectifiable, u ∈ W 1,2 (Ω \ K), M ∈ W 1,n+α (Ω; G)}.

If we consider M ≡ Id n , we recognize the well-known Mumford-Shah model (see [START_REF] Mumford | Optimal approximations by piecewise smooth functions and associated variationnal problems[END_REF] for the seminal paper). In this sense, our model is the anisotropic version of the Mumford-Shah energy.

At any point x ∈ K, the directions associated to the main eigenvalues of M(x) must represent the directions of elongation of the set K. For example, if we want to detect a tube in a twodimensional image, we may consider G as the subset of symmetric definite positive matrices with fixed eigenvalues {1; µ}, such that 1 µ. For the detection of thin plates in a three dimensional image, we may consider the fixed eigenvalues {1; µ; µ}. By this way, the third term of (0.1) will force ν to be in the same direction as the first eigenvector and then K to be elongated in the orthogonal directions. Moreover, we set α > 0 in the last term of (0.1) in order to force M to be at least continuous (α-Hölder), which corresponds to the assumption that the image g admits a local geometrical coherence.

This model does not consist in the detection of sets with codimension higher than one, as it has been done in [START_REF] Alberti | Variational convergence for functionals of Ginzburg-Landau type[END_REF] and [START_REF] Ghiraldin | Variational approximation of a functional of Mumford-Shah type in codimension higher than one[END_REF] for vector-valued functions. In our case, the sets we want to detect are with (small) positive volume.

In order to prove that (P) is a well posed problem, for u ∈ SBV(Ω), we introduce

E(u, M) = Ω (u -g) 2 dx + Ω |∇u| 2 dx + Ju Mν u , ν u 1/2 dH n-1 + Ω DM n+α dx (0.2)
where ∇u is the derivative of u with respect to the Lebesgue measure, J u is its jump set and ν u is unitary and orthogonal vector to J u . The associated minimizing problem is

(P ) : min{E(u, M) : u ∈ SBV(Ω), M ∈ W 1,n+α (Ω; G)},
We verify that (P ) admits a solution and, with a regularity result of the jump set of a minimizer, which has been established in a joint paper [START_REF] Vicente | Anisotropic Minkowski content and application to minimizers of free discontinuity problems[END_REF], we prove that a minimizer of (P ) naturally provides a solution for (P). In order to get a functional more adapted for a numerical implementation, we will approximate (0.2) by a family (E ε ) ε which only depends on the integration with respect to the Lebesgue measure. More precisely, we set

E ε (u, z, M) = Ω (u -g) 2 dx+ Ω |∇u| 2 (1 -z 2 ) 2 dx+ Ω ε M∇z, ∇z + z 2 4ε dx+ Ω DM n+α dx.
As it has been done in [START_REF] Ambrosio | Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence[END_REF] for the initial Mumford-Shah model, the function z takes its values in [0; 1] and plays the role of a control on the gradient of u. The approximation takes place for ε → 0 + in the sense of the Γ-convergence.

In section 1 we recall some results on spaces of functions with bounded variation and we mention a regularity result that we have proved in a previous article. In section 2, we prove that the problem admits a solution. Section 3 in completely devoted to the approximation process. We introduce the family of functionals (E ε ) ε with their domains and we give the complete proof of Γ-convergence to E when ε → 0 + .

Functional framework and regularity result

We adopt the notations:

• v 1 , v 2 ∈ R for the canonical scalar product of v 1 , v 2 ∈ R n , • |v| for the euclidean norm of v ∈ R n , • M for the induced norm of M ∈ M n (R), • n-1 i=1 v i ∈ R n for the canonical vectorial product of v 1 , . . . , v n-1 ∈ R n , • dist for the euclidean distance in R n , • S + n (R) ⊂ M n (R)
for the subset of symmetric definite positive matrices, • GL n (R) ⊂ M n (R) for the subset of invertible matrices,

• O n (R) ⊂ GL n (R) for the subgroup of orthogonal matrices,

• B(Ω) the class of Borelian subsets of Ω,

• B(Ω) for the space of Borelian functions defined in Ω,

• L n for the Lebesgue measure in R n ,

• H k for the k-dimensional Hausdorff measure,

• M(Ω; R n ) for the space of vectorial Radon measures defined in Ω ⊂ R n ,

• A f (x)dx = 1 L n (A) A f (x)dx, for A ∈ B(Ω) and L n (A) > 0.
Let f be a function defined on open sets, we adopt the following vocabulary:

• f is superadditive if f (A ∪ B) ≥ f (A) + f (B)

for any disjoints sets A, B,

• f is non decreasing if f (A) ≤ f (B) for any sets A, B such that A ⊂ B.

Functional spaces

We assume throughout this paper that the following constraint is satisfieded by Ω which is obviously satisfied in the context of applications in Image Processing because the domain is a parallelepiped.

Definition 1.1. We say that Ω ⊂ R n satisfies the reflexion condition (R) if Ω is an open and bounded domain with Lipschitz regular boundary ∂Ω such that there exists a neighborhood U of ∂Ω and a bi-Lipschitzian homeomorphism ϕ : U ∩ Ω → U \ Ω such that, for any x ∈ ∂Ω, we have

lim y→x ϕ(y) = x.
For the classical definitions and results on BV and SBV we refer to [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF]. In particular, for the definition of weak* convergence in BV, we refer to [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF] (Definition 1.58). However, for the need of notations, we mention that, if u ∈ BV(Ω), then its derivative Du belongs to the space M(Ω; R n ) of vectorial Radon measure. Moreover, if u ∈ SBV(Ω) then the Cantor part of Du is null and we obtain

Du = ∇u • L n + (u + -u -)ν u • H n-1 J u ,
where ∇u is the density of Du with respect to the Lebesgue measure L n , u + (resp. u -) is the approximate upper (resp. lower) limit and H n-1 J u is the restriction of H n-1 to its jump set J u . Now, we focus on some results which will be useful throughout the paper. First, we need the rectifiability of J u .

Theorem 1.1. Let u be a given function in BV(Ω). There exists a countable family

(C i ) i∈N of compact C 1 -hypersurfaces such that J u = N ∪ i∈N C i , where H n-1 (N ) = 0.
Then, we will need the following chain rule ( [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF], Theorem 3.99).

Theorem 1.2. Let u ∈ SBV(Ω) and let

f : R → R be a Lipschitz function. Then, v = f • u belongs to SBV(Ω) and Dv = f (u)∇u • L n + (f (u + ) -f (u -))ν u • H n-1 J u . (1.1)
The following is a straightforward consequence of [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF], Corollary 3.89.

∀x ∈ U \ Ω, u(x) = u(ϕ -1 (x)),
Then, we have

H n-1 (J u ∩ ∂Ω) = 0.
We also need slicing results.

Definition 1.2. Let ν ∈ S n-1 be fixed. We denote by Π ν the hyperplane

{x ∈ R n : x • ν = 0} . If x ∈ Π ν , we set Ω x = {t ∈ R : x + tν ∈ Ω} , Ω ν = {x ∈ Π ν : Ω x = ∅} .
For any function u defined on Ω and any x ∈ Ω ν , we set

(u) x : Ω x → R t → u(x + tν).
The following Theorem is proved in [START_REF] Goffman | Sublinear functions of measures and variational integrals[END_REF].

Theorem 1.3. Let u ∈ L ∞ (Ω) be a function such that, for all ν ∈ S n-1 , i) (u) x ∈ SBV(Ω x ) for H n-1 a.e. x ∈ Ω ν , ii) Ων Ωx |∇(u) x |dt + H 0 (J (u)x ) dH n-1 (x) < +∞; then, u ∈ SBV(Ω) and H n-1 (J u ) < +∞. Conversely, let u ∈ SBV(Ω) ∩ L ∞ (Ω) be such that H n-1 (J u ) < +∞. Then i) and ii) are satisfied. Moreover, we have iii) ∇u(x + tν), ν = ∇(u) x (t), for a.e. t ∈ Ω x and H n-1 -a.e. x ∈ Ω ν , iv) Ju ν u , ν dH n-1 (x) = Ων H 0 (J (u)x )dH n-1 (x).
We need a generalization of the Coarea formula. For that, we introduce the following

Definition 1.3. The function f : Ω × R n → R + is sublinear with respect to the second variable,if i) f (x, v 1 + v 2 ) ≤ f (x, v 1 ) + f (x, v 2 ) for any (x, v 1 , v 2 ) ∈ Ω × R n × R n , ii) f (x, tv) = tf (x, v) for any (x, v, t) ∈ Ω × R n × R + .
Suppose that µ 1 is a Radon measure and µ 2 is a vectorial Radon measure on Ω. According to Besicovitch derivation Theorem,

lim r→0 µ 2 (B(x, r)) µ 1 (B(x, r))
exists and is finite for µ 1 almost every x, we denote by dµ2 dµ1 (x) this limit when it exists. We recall that µ 2 is absolutely continuous with respect to µ 1 if µ 2 (A) = 0 whenever µ 1 (A) = 0. When this holds, we write µ 2 µ 1 . We consider the convex functional defined on the space M(Ω; R n ) by

Φ : µ 2 ∈ M(Ω; R n ) → Ω f x, dµ 2 dµ 1 dµ 1 , (1.2)
where µ 1 is a positive measure such that µ 2 µ 1 . It is shown in [START_REF] Maso | Integral representation on bounded variation spaces of Γ-limits of variational integrals[END_REF] that the integral in (1.2) does not depend on the choice of µ 1 . For that reason, we will write it in the condensed form

Φ(µ 2 ) = Ω f (x, µ 2 ).
We give a variant of the Coarea formula extended to the sublinear functionals which can be found in [START_REF] Bellettini | Some results on surface measures in calculus of variations[END_REF].

Proposition 1.2. Let Φ(x, s, v) a Borel function of Ω × R × R n which is sublinear in v.
Let p be a Lipschitz continuous function on Ω and, for t > 0, we set S t = {x ∈ Ω; p(x) < t}. Then, for almost all t ∈ R, 1 St belongs to BV(Ω) and we have

Ω Φ(x, p, Dp)dx = R dt Ω Φ(x, t, D1 St ).

Minkowski content and regularity result for the jump set

For M : Ω → S + n (R) and (x, v) ∈ R n × R n , we set φ(x, v) = M -1 (x)v, v 1/2 .
This functional is a Riemannian metric, so, for S ⊂ Ω and x, y ∈ Ω, we may define its associated distance as

dist φ (x, y) = inf 1 0 φ γ, dγ dt dt : γ ∈ W 1,1 ([0; 1]; R n ), γ(0) = x, γ(1) = y , dist φ (x, S) = inf {dist φ (x, y) : y ∈ S} .
The associated anisotropic Minkowski (n -1)-dimensional upper and lower contents are defined by the limits

M M (S) = lim sup ρ→0 + L n ({x : dist φ (x, S) < ρ}) 2ρ , M M (S) = lim inf ρ→0 + L n ({x : dist φ (x, S) < ρ}) 2ρ .
(1.3) If M M (S) = M M (S), we call their common value the (n -1)-dimensional anisotropic Minkowski content M M (S). In [START_REF] Vicente | Anisotropic Minkowski content and application to minimizers of free discontinuity problems[END_REF], we have proved the following

Theorem 1.4. Let M : Ω → S + n (R) be continuous, h ∈ L ∞ (Ω), γ > 0 and u ∈ SBV(Ω) a minimizer of E γ,h,M (v) = γ Ω (v -h) 2 dx + Ω |∇v| 2 dx + Jv Mν v , ν v 1/2 dH n-1 : v ∈ SBV(Ω) .
Then, we have

H n-1 (J u \ J u ) = 0, M M (J u ) = Jv Mν v , ν v 1/2 dH n-1 .

Existence result for (P)

In order to prove that (P) admits a solution, we prove that (P ) is well posed and that it provides a minimizer for (P).

Existence result for the relaxed problem (P )

The main result of the section is the following Theorem 2.1. The problem (P ) admits a solution.

The proof is a straightforward consequence of Lemmas 2.1 and 2.2.

Lemma 2.1. Let (u k ) k ⊂ SBV(Ω) ∩ L ∞ (Ω) and (M k ) k ⊂ W 1,n+α (Ω; G) be such that sup k E(u k , M k ) < ∞.
Then, there exist subsequences, still denoted by

(u k ) k and (M k ) k , and (u, M) ∈ SBV(Ω) × W 1,n+α (Ω; G), such that (u k ) k is weakly* convergent to u in SBV(Ω) and (M k ) k is weakly conver- gent to M in W 1,n+α (Ω).
Proof. As G ⊂ S + n (R) is compact, there exists 0 < λ < Λ such that, for any (M, v) ∈ G × R n , the following ellipticity condition is satisfied

λ|v| 2 ≤ M v, v ≤ Λ|v| 2 .
(2.1)

It yields Ω |∇u k | 2 dx + H n-1 (J u k ) ≤ max 1; λ -1/2 E(u k , M k ). (2.2)
As E(u k , M k ) is bounded, Theorem 4.8. in [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF] implies that there exists a subsequence (u k ) k weakly* convergent in BV(Ω) to u ∈ SBV(Ω). According to (2.1), we have

M n+α W 1,n+α (Ω) ≤ Λ (n+α)/2 + E(u k , M k ), so (M k ) k is a bounded sequence of W
E(u, M) ≤ lim inf k→∞ E(u k , M k ).
Proof. We may assume that lim inf E(u k , M k ) < +∞, otherwise the result is ensured. Weak* convergence in SBV(Ω) and weak convergence in

W 1,n+α (Ω) give lim inf k→∞ Ω (u k -g) 2 dx + Ω DM k n+α dx ≤ Ω (u -g) 2 dx + Ω DM n+α dx. (2.3) According to inequality (2.2), Ω |∇u k | 2 dx + H n-1 (J u k ) is bounded with respect to k. With [7],
Theorem 4.7., we deduce that

Ω |∇u| 2 dx ≤ lim inf k→∞ Ω |∇u k | 2 dx, H n-1 (J u ) ≤ lim inf k→∞ H n-1 (J u k ). (2.4)
According to (2.3) and (2.4), it remains to prove that

Ju Mν u , ν u 1/2 dH n-1 ≤ lim inf k→∞ Ju k M k ν u k , ν u k 1/2 dH n-1 .
This result is proved in [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF], Theorem 5.2., for an homogeneous and fixed media (if M k does not depend on x ∈ Ω nor in k). In order to generalize this result, we introduce a piecewise constant approximation of M. For any vector e i ∈ S n-1 of the canonical basis, we set

Π i t = {x ∈ Ω : x, e i = t} , N i t = t ∈ R : H n-1 (J u ∩ Π i t ) > 0 ∪ t ∈ R : ∃k ∈ N, H n-1 (J u k ∩ Π i t ) > 0 .
As H n-1 (J u ) < ∞ (2.4) and H n-1 (J u k ) < ∞ for any k ∈ N, then N i t is at most countable. We fix η > 0 arbitrary small. Intersecting Ω with disjoint cubes with edges orthogonal to the axes of the canonical basis, there exists A a finite partition of Ω which, for any A ∈ A and for any k ∈ N, satisfies

diam(A) < η, H n-1 (J u ∩ ∂A) = 0, H n-1 (J u k ∩ ∂A) = 0. (2.

5)

For any A ∈ A, we fix one point x A ∈ A and we set M A (x) = M(x A ) for any x ∈ A. We have the decomposition

Ju k M k ν u k , ν u k 1/2 dH n-1 = Ju k M k ν u k , ν u k 1/2 -Mν u k , ν u k 1/2 dH n-1 + Ju k Mν u k , ν u k 1/2 -M A ν u k , ν u k 1/2 dH n-1 + Ju k M A ν u k , ν u k 1/2 dH n-1 .
(2.6)

In the three following Claim, we will estimate the limit of those three terms.

Claim 1:

lim k→∞ Ju k M k ν u k , ν u k 1/2 -Mν u k , ν u k 1/2 dH n-1 = 0.
According to Ellipticity inequality (2.1), we have

| M k ν u k , ν u k 1/2 -Mν u k , ν u k 1/2 | ≤ 1 2 √ λ | (M k -M)ν u k , ν u k |, ≤ 1 2 √ λ M k -M L ∞
and then

Ju k | M k ν u k , ν u k 1/2 -Mν u k , ν u k 1/2 |dH n-1 ≤ 1 2 √ λ M k -M L ∞ H n-1 (J u k ). As the inclusion W 1,n+α (Ω) ⊂ L ∞ (Ω) is compact and (M k ) k weakly converges to M, then (M k ) k uniformly converges to M. Moreover, (H n-1 (J u k )) k is bounded, it concludes the proof of the claim.
Claim 2: The sequence

lim η→0 + Ju k Mν u k , ν u k 1/2 -M A ν u k , ν u k 1/2 dH n-1 = 0
and the convergence takes place uniformly with respect to k ∈ N.

For A ∈ A and x ∈ A, ellipticity inequality (2.1) gives

| M(x)ν u k , ν u k 1/2 -M A (x)ν u k , ν u k 1/2 | ≤ 1 2 √ λ M(x) -M(x A ) .
As M ∈ W 1,n+α (Ω), there exist a constant C > 0 such that, for any x ∈ A, we have

M(x) -M(x A ) ≤ Cη α .

It yields

Ju k Mν u k , ν u k 1/2 -M A ν u k , ν u k 1/2 dH n-1 ≤ Cη α H n-1 (J u k ) 2 √ λ .
As (H n-1 (J u k )) k is a bounded sequence, it concludes the proof of the claim.

Claim 3:

Ju

M A ν u , ν u 1/2 dH n-1 ≤ lim inf k→∞ Ju k M A ν u k , ν u k 1/2 dH n-1 .

We denote by

• A the interior of the set A. According to [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF], Theorem 5.2., we have

Ju∩ • A M(x A )ν u , ν u 1/2 dH n-1 ≤ lim inf k→∞ Ju k ∩ • A M(x A )ν u k , ν u k 1/2 dH n-1 .
According to (2.5), the contribution of the boundaries is null, it gives

Ju∩A M(x A )ν u , ν u 1/2 dH n-1 ≤ lim inf k→∞ Ju k ∩A M(x A )ν u k , ν u k 1/2 dH n-1
and then

A∈A Ju∩A M(x A )ν u , ν u 1/2 dH n-1 ≤ A∈A lim inf k→∞ Ju k ∩A M(x A )ν u k , ν u k 1/2 dH n-1 , ≤ lim inf k→∞ A∈A Ju k ∩A M(x A )ν u k , ν u k 1/2 dH n-1 .
As A is a partition of Ω, it concludes the proof of Claim 3 :

Ju M A ν u , ν u 1/2 dH n-1 ≤ lim inf k→∞ Ju k M A ν u k , ν u k 1/2 dH n-1 .
Let δ > 0 be an arbitrary small number. With the same arguments as for Claim 2, we get

Ju Mν u , ν u 1/2 -M A ν u , ν u 1/2 dH n-1 ≤ Cη α H n-1 (J u ) 2 √ λ .
So, according to Claim 2, there exists a partition A which satisfies

         lim sup k→∞ Ju k Mν u k , ν u k 1/2 -M A ν u k , ν u k 1/2 dH n-1 ≤ δ, Ju Mν u , ν u 1/2 -M A ν u , ν u 1/2 dH n-1 ≤ δ.
According to (2.6), Claim 1 and Claim 3, we have

Ju Mν u , ν u 1/2 dH n-1 ≤ 2δ + lim inf k→∞ Ju k M k ν u k , ν u k 1/2 dH n-1 .
As δ > 0 is arbitrary, it concludes the proof of Lemma 2.2.

We now prove Theorem 2.1.

Proof. We denote by (u k , M k ) k ⊂ SBV(Ω) a minimizing sequence for E. As g ∈ L ∞ (Ω), we set

ϕ(t) =    -g L ∞ (Ω) if t ≤ -g L ∞ (Ω) , t if |t| ≤ g L ∞ (Ω) , g L ∞ (Ω) if t ≥ g L ∞ (Ω) .
(2.7)

We denote v k = ϕ • u k . As the function ϕ is 1-Lipshitz, we may apply Theorem 1.2 and then

v k ∈ SBV(Ω) ∩ L ∞ (Ω) with the decomposition Dv k = ϕ (u k )∇u k • L n + (ϕ(u + k ) -ϕ(u - k ))ν u k • H n-1 J u k . For any k, it is easy to check that E(v k , M k ) ≤ E(u k , M k ), so (v k , M k ) k is also a minimizing sequence for E.
According to Theorem 2.1, there exists v ∈ SBV(Ω) and a subsequence, still denoted (v k ) k weakly* convergent to v. With Theorem 2.2, we have

E(v, M) ≤ lim inf E(v k , M k ) and then (v, M) is a minimizer of E.

Existence result for (P)

In this section we prove that the problems (P) and (P ) have common solutions and then (P) is also well posed. First, we consider (u, K, M) in the domain of E. As in (2.7), we may define the truncated function v = ϕ(u) and then we have |v -g| ≤ |u -g| and |∇v| ≤ |∇u|. In particular, we get E(v, K, M) ≤ E(u, K, M). Thus, for any (u , K , M ) in the domain of E, we may assume that u ∈ L ∞ (Ω \ K). On the other hand, in [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF] (Proposition 4.4), is given the following

Proposition 2.1. Let Ω ⊂ R n be open and bounded, K ⊂ R n be closed, H n-1 (K ∩ Ω) < ∞ and u ∈ L ∞ (Ω \ K) ∩ W 1,1 (Ω \ K). Then, we have u ∈ SBV(Ω) and H n-1 (S u \ K) = 0.
As Ω is bounded and

u ∈ W 1,2 (Ω\K ), then we get u ∈ W 1,1 (Ω\K ). As K M ν, ν 1/2 dH n-1 < +∞, then Ellipticity condition (2.1) gives H n-1 (K ∩ Ω) < ∞. According to Proposition 2.1, we deduce that u ∈ SBV(Ω) and H n-1 (S u \ K ) = 0. It yields E(u , M ) ≤ E(u , K , M ),
and then min E ≤ min E. Conversely, according to Theorem 2.1, there exists a minimizer (u * , M * ) of E. In particular, with the notations of Theorem 1.4, u is a minimizer of E 1,g,M and then

H n-1 (J u \ J u ) = 0.
So, we set K = J u and then K is compact and

H n-1 -rectifiable, Ω \ K is open, u ∈ W 1,2 (Ω \ K ) and E(u , M ) = E(u , K , M ).
We may conclude that min E = min E and their minimizers coincide. Moreover, we have Proposition 2.2. Let u ∈ SBV(Ω) be a minimizer of (P),

then u ∈ C 1 (Ω \ J u ). Proof. Let B r (x) ⊂ Ω \ J u ; then u ∈ W 1,2 (B r (x)
) and it is a minimizer of the functional

I(v) = Br(x) (v -g) 2 dx + Br(x) |∇v| 2 dx among the functions v in u + W 1,2 0 (B r (x)
) and then classical regularity results give u ∈ C 1 (B r (x)).

Γ-convergence result

This section is entirely devoted to the approximation process. In 3.1 we define the domain for E ε and give the main Theorem of this paper. In 3.2 we prove, for ε > 0 fixed, that the minimization of E ε admits a solution. In 3.3, the most technical part of the paper, we give the complete proof of Γ-convergence. Finally, in 3.4, we conclude the proof of the main Theorem.

The functionals, their domain and the main Theorem

Formally, we define the functional E ε (u, z, M) as

E ε (u, z, M) = Ω (u -g) 2 dx+ Ω |∇u| 2 (1 -z 2 ) 2 dx+ Ω ε M∇z, ∇z + z 2 4ε dx+ Ω DM n+α dx.
As in [START_REF] Ambrosio | Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence[END_REF], the function z : Ω → [0; 1] plays the role of control of the gradient of u. We need to introduce a domain for E ε that ensures the existence of a minimizer. If u, z ∈ W 1,2 (Ω), then this functional is well defined. However, the coefficient (1 -z 2 ) 2 removes the coercivity with respect to u and the existence result can not be achieved according to the Sobolev norm. If, by addition, u is bounded, we have

|∇(u(1 -z 2 ))| 2 = |∇u(1 -z 2 ) -2uz∇z| 2 , ≤ 2|∇u| 2 (1 -z 2 ) 2 + 4 u L ∞ (Ω) |∇z| 2 .
According to ellipticity inequality (2.1), it gives

Ω |∇(u(1 -z 2 ))| 2 dx ≤ 2 + 4 u L ∞ (Ω) λε E ε (u, z, M).
So, it is natural to set

D n (Ω) = (u, z) : u ∈ B(Ω), z ∈ W 1,2 (Ω; [0; 1]), ∀N ∈ N u N (1 -z 2 ) ∈ W 1,2 (Ω) ,
where u N is the truncated function defined, for any x ∈ Ω, by

u N (x) =    -N if u(x) ≤ -N, u(x) if |u(x)| ≤ N, N if u(x) ≥ N. (3.1) Assuming (u, z) ∈ D n (Ω) does not ensure that u ∈ W 1,2
(Ω) and ∇u can not be defined as the gradient of u in the Sobolev sense. However, we can define ∇u in the following sense.

Definition 3.1. Let u ∈ L 1 (Ω) and x ∈ Ω a Lebesgue point of u; we say that u is approximately differentiable at x if there exists L ∈ R n such that lim r→0 + B(x,r) |u(y) -u(x) -L, y -x | r dy = 0. (3.2)
If u is approximately differentiable at x then L, uniquely determined by (3.2), is called the approximate differential of u at x.

The following ensures that E ε (u, z, M) is well defined for (u, z) ∈ D n (Ω).

Proposition 3.1. If (u, z) ∈ D n (Ω), then u is approximately differentiable in {x ∈ Ω : z(x) = 1}
and z is approximately differentiable in Ω.

Proof.

As Ω is open and bounded then W

∈ L ∞ (Ω), then v 1 v 2 is approximately differentiable almost everywhere; • if v 2 is approximately differentiable almost everywhere, then v 2 -1
is also approximately differentiable almost everywhere in {x : v 2 (x) = 0} (Proposition 3.71 in [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF]).

We deduce that u N is approximately differentiable almost everywhere in {x : z(x) = 1}. This is true for any N ∈ N, so this is also true for u.

The main result of the paper is the following Theorem 3.1. Let Ω ⊂ R n be an open and bounded domain which satisfies reflexion condition (1.1), G be a compact subset of S + n (R) and α > 0 be fixed. We consider H : B(Ω)×W 1,n+α (Ω; G) → [0; +∞] defined as

H(u, M) = Ω (u -g) 2 dx + Ω DM n+α dx, E : B(Ω) × W 1,n+α (Ω; G) → [0; +∞] defined as E(u, M) =        H(u, M) + Ω |∇u| 2 dx + Ju Mν u , ν u 1/2 dH n-1 if u ∈ SBV(Ω), +∞ otherwise and E ε : B(Ω) × B(Ω) × W 1,n+α (Ω; G) → [0; +∞] defined as E ε (u, z, M) =        H(u, M) + Ω |∇u| 2 (1 -z 2 ) 2 dx + Ω ε M∇z, ∇z + z 2 4ε dx if (u, z) ∈ D n (Ω), +∞ otherwise.
Then, the following assertions are satisfied.

i) For any ε > 0, E ε admits a minimizer, denoted by (u ε , z ε , M ε ). Moreover, we can assume that u ε (x) = g(x) on {x ∈ Ω :

z ε (x) = 1}.
ii) For any (ε k ) k converging to 0 + , there exists a subsequence, still denoted by

(ε k ) k , and u ∈ SBV(Ω) such that (u ε k , z ε k ) k converges to (u, 0) almost everywhere, (M k ) k converges weakly to M and (u, M) is a minimizer of E.
We set

(P ε ) : Min{E ε (u, z, M) : (u, z, M) ∈ B(Ω) × B(Ω) × W 1,n+α (Ω; G)},
Theorem 3.1 i) implies that, for ε > 0 fixed, (P ε ) is a well posed problem. Theorem 3.1 ii) implies that, up to the extraction of a subsequence, the sequence of solutions of (P ε k ) converge to a solution of (P ).

Existence result for (P ε )

Theorem 3.1 i) is a straightforward consequence of Propositions 3.2 and 3.3.

Proposition 3.2. Let ε > 0 be fixed. There exists (u k , z k , M k ) k a minimizing sequence of E ε such that (u k ) k is a bounded sequence of L ∞ (Ω), (u k , z k ) k converges almost everywhere to (u, z) ∈ D n (Ω), u(x) = g(x) on {x ∈ Ω : z(x) = 1} and (M k ) k weakly converges to M ∈ W 1,n+α (Ω; G).
To prove it, we need the following Lemma which proof is given in Appendix 4.1.

Lemma 3.1. For (u, z) ∈ D n (Ω) and ν ∈ S n-1 fixed, we have (u x , z x ) ∈ D 1 (Ω x ) for H n-1 -almost every x ∈ Ω ν (see the notations of Definition 1.2), and

∇u x (t) = ∇u(x + tν), ν , ∇z x (t) = ∇z(x + tν), ν ,
for almost every t ∈ Ω x \ {s : z(x + sν) = 1}. Now, we prove Proposition 3.2.

Proof. Let (u k , z k , M k ) k be a minimizing sequence of E ε . We fix N ≥ g L ∞ (Ω) and we consider the truncated functions (u N k ) k defined in (3.1). As (u k , z k ) ∈ D n (Ω), we have u N k (1-z 2 k ) ∈ W 1,2 (Ω). As Ω is bounded, then W 1,2 (Ω) ⊂ SBV(Ω). According to Calderón-Zygmund Theorem ([7], Theorem 3.83), u N k (1 -z 2 k
) is approximately differentiable almost everywhere. For the same reasons, 1 -z 2 k is also approximately differentiable almost everywhere. According to Proposition 3.71 in [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF], we deduce that u N k is approximately differentiable almost everywhere in {x :

z k (x) = 1}. Moreover, ∇u N k (x) = 0 almost everywhere in {x : |u N k (x)| = N } and ∇u N k (x) = ∇u k (x) almost everywhere in {x : |u N k (x)| < N } (Proposition 3.73 in [7]), it gives Ω |∇u N k | 2 (1 -z 2 k ) 2 dx ≤ Ω |∇u k | 2 (1 -z 2 k ) 2 dx. (3.3) so E ε (u N k , z k , M k ) ≤ E ε (u k , z k , M k ) and then (u N k , z k , M k )
k is also a minimizing sequence. According to ellipticity inequality (2.1), we have

Ω |∇z k | 2 dx + Ω z 2 k dx ≤ 1 λε + 4ε E ε (u k , z k , M k ),
and then (z k ) k is a bounded sequence of W 1,2 (Ω). So, there exists a subsequence, still denoted by (z k ) k , which converges almost everywhere to z ∈ W 1,2 (Ω). As (z k ) k takes its values almost everywhere in [0; 1], then z takes also its values in [0; 1]. For

w k = u N k (1 -z 2 k ), we have Ω |∇w k | 2 dx + Ω w 2 k dx ≤ 2 Ω |∇u N k | 2 (1 -z 2 k ) 2 dx + 2N 2 Ω |∇z k | 2 dx + N 2 Ω (1 -z 2 k ) 2 dx
and then (w k ) k is a bounded sequence of W 1,2 (Ω). So, there exists a subsequence, still denoted by (w k ) k , which converges almost everywhere to w ∈ W 1,2 (Ω). In particular, (u N k (x)) k converges for almost every x ∈ {y : z(y) = 1} to u(x). We set u(x) = g(x) for x ∈ {y : z(y) = 1}. This construction ensures that (u, z) 1,n+α (Ω) and there exists a subsequence weakly convergent to M ∈ W 1,n+α (Ω). As weakly convergence in W 1,n+α (Ω) implies uniform convergence and G is closed, then M ∈ W 1,n+α (Ω; G).

∈ D n (Ω). As G ⊂ S + n (R) is compact, Ω is bounded and Ω DM k dx ≤ E ε (u k , z k , M k ), then (M k ) k is a bounded sequence of W
Proposition 3.3. If (u k , z k ) k ⊂ D n (Ω) converges almost everywhere to (u, z) ∈ D n (Ω), (u k ) k is a bounded sequence of L ∞ (Ω) and (M k ) k ⊂ W 1,n+α (Ω; G) weakly converges to M ∈ W 1,n+α (Ω; G), then lim inf k→∞ E ε (u k , z k , M k ) ≥ E ε (u, z, M).
Proof. Fatou Lemma and lower semi-continuity of the Sobolev semi-norm give

lim inf k→∞ Ω (u k -g) 2 dx + Ω z 2 k 4ε dx + Ω DM k n+α dx ≥ Ω (u -g) 2 dx+ Ω z 2 4ε dx+ Ω DM n+α dx.
So, to show Proposition 3.3, it suffices to prove that lim inf

k→∞ Ω ε M k ∇z k , ∇z k dx ≥ Ω ε M∇z, ∇z dx (3.4) and lim inf k→∞ |∇u k | 2 (1 -z 2 k ) 2 dx ≥ Ω |∇u| 2 (1 -z 2 ) 2 dx. (3.5)
Proof of (3.4) On the other hand, we have

If lim inf k Ω |∇z k | 2 dx = +∞,
Ω ε M k ∇z k , ∇z k dx - Ω ε M∇z k , ∇z k dx ≤ ε M k -M L ∞ Ω |∇z k | 2 dx.
As the weakly convergence of (M k ) k in W 1,n+α (Ω) implies the uniform convergence, we may conclude that lim inf

k Ω ε M k ∇z k , ∇z k dx - Ω ε M∇z k , ∇z k dx = 0.
To prove inequality (3.4), it remains to verify that lim inf

k→∞ Ω ε M∇z k , ∇z k dx ≥ Ω ε M∇z, ∇z dx. As the application W 1,2 (Ω) → L 2 (Ω; R n ), z → ∇z
is continuous for the strong topology, it remains to prove that the application

L 2 (Ω; R n ) → R, Z → Ω MZ, Z dx is lower semi-continuous for the weak topology of L 2 (Ω; R n ). Let (Z k ) k ⊂ L 2 (Ω; R n ) be weakly convergent to Z ∈ L 2 (Ω; R n ). We set L : L 2 (Ω; R n ) → R, U → Ω MZ, U dx
According to ellipticity inequality (2.1), L ∈ (L 2 (Ω; R n )) and then (L(Z k )) k converges to L(Z). Moreover, for k fixed, the following polynomial function is positive

t → Ω M(Z + tZ k ), Z + tZ k dx.
Thus, its discriminant is negative and we deduce the following anisotropic Cauchy-Schwarz inequality

Ω MZ, Z k dx ≤ Ω MZ, Z dx 1 2 Ω MZ k , Z k dx 1 2
.

As (L(Z k )) k converges to L(Z), passing through the lim inf in the previous inequality yields

Ω MZ, Z dx ≤ Ω MZ, Z dx 1 2 lim inf k→∞ Ω MZ k , Z k dx 1 2
and then we may conclude the Proof of (3.4) by taking Z k = ∇z k , Z = ∇z in the previous inequality

Ω M∇z, ∇z dx ≤ lim inf k→∞ Ω M∇z k , ∇z k dx.
Proof of (3.5) We first consider the one-dimensional case n = 1 and then by a slicing argument we get the lower semi-continuity for the general case n ≥ 1. Let A ⊂ {x ∈ Ω : z(x) < 1} be an open and relatively compact subset of Ω ⊂ R. As (z k ) k weakly converges to z in W 1,2 (Ω), then (z k ) k uniformly converges to z. In particular, there exists δ > 0 and

k 0 ∈ N k ≥ k 0 ⇒ A ⊂ {x ∈ Ω : z k (x) ≤ 1 -δ}.
Thus, for any k ≥ k 0 , we have

A |∇u k | 2 dx ≤ 1 1 -(1 -δ) 2 E ε (u k , z k )
and then we deduce that (u k ) k is a bounded sequence of W 1,2 (A). As u k (1 -z 2 k ) converges almost everywhere to u(1 -z 2 ) in Ω, there exists a subsequence, still denoted by (u k ) k , which weakly converges to u in W 1,2 (A). In particular, (∇u k ) k weakly converges to ∇u in L 2 (A). For ξ ∈ L 2 (Ω), we decompose

A ξ ∇u k (1 -z 2 k ) -∇u(1 -z 2 ) dx = A ξ∇u k (z 2 -z 2 k )dx+ A ξ∇u k (1 -z 2 )dx+ A ξ∇u(z 2 -z 2 k )dx. (3.6) As (1 -z 2 )ξ ∈ L 2 (A) and (∇u k ) k weakly converges to ∇u in L 2 (A), then we have A ξ∇u k (1 -z 2 )dx → A ξ∇u(1 -z 2 )dx. (3.7)
Moreover, we have

A ξ∇u k (z -z k )dx ≤ ξ L 2 (A) ∇u k L 2 (A) z 2 -z 2 k L ∞ (A)
and

A ξ∇u(z -z k )dx ≤ ξ L 2 (A) ∇u L 2 (A) z 2 -z 2 k L ∞ (A) .
As a weakly convergent sequence is bounded, then (∇u k ) k is bounded in L 2 (A) and we deduce that

A ξ∇u k (z 2 -z 2 k )dx → 0, A ξ∇u(z 2 -z 2 k )dx → 0. (3.8)
According to (3.6), (3.7) and (3.8), we get

A ξ∇u k (1 -z 2 k )dx → A ξ∇u(1 -z 2 )dx and then (∇u k (1 -z 2 k )) k weakly converges to ∇u(1 -z 2 ) in L 2 (A).
As the norm is lower semicontinuous, we deduce

A |∇u| 2 (1 -z 2 ) 2 dx ≤ lim inf k→∞ A |∇u k | 2 (1 -z 2 k ) 2 dx, ≤ lim inf k→∞ Ω |∇u k | 2 (1 -z 2 k ) 2 dx.
Passing to the limit A ↑ {x ∈ Ω : z(x) < 1} gives

Ω |∇u| 2 (1 -z 2 ) 2 dx ≤ lim inf k→∞ Ω |∇u k | 2 (1 -z 2 k ) 2 dx.
We generalize this result to the dimension n ≥ 1. With the notation u x introduced in (1.2), using the previous result obtained in dimension 1, Lemma 3.1 and Fatou Lemma, give

A | ∇u, ν | 2 (1 -z 2 ) 2 dx = Aν Ax |∇u x (t)| 2 (1 -z x (t) 2 ) 2 dtdx, ≤ Aν lim inf k→∞ Ax |∇(u k ) x (t)| 2 (1 -(z k ) x (t) 2 ) 2 dtdx, ≤ lim inf k→∞ Aν Ax |∇(u k ) x (t)| 2 (1 -(z k ) x (t) 2 ) 2 dtdx, ≤ lim inf k→∞ Aν Ax | ∇u k (x + tν), ν | 2 (1 -z k (x + tν) 2 ) 2 dtdx, ≤ lim inf k→∞ A | ∇u k , ν | 2 (1 -z 2 k ) 2 dx, ≤ lim inf k→∞ A |∇u k | 2 (1 -z 2 k ) 2 dx,
for any open set A ⊂ Ω and every ν ∈ S n-1 . The function x → ∇u(x) |∇u(x)| is measurable in U = {x ∈ Ω : z(x) = 1, ∇u(x) = 0}. According to Lusin Theorem (1.45 of [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF]), there exists an increasing sequence of compacts (K l ) l ⊂ U such that

L n (U \ K l ) ≤ 1 l , x → ∇u(x) |∇u(x)| is continuous in K l .
Thus, for any x ∈ K l , there exists r > 0 such that

y ∈ B(x, r) ⇒ ∇u(x) |∇u(x)| - ∇u(y) |∇u(y)| ≤ 1 l , ( 3.9)
As a consequence of Besicovitch Covering Theorem (2.18 of [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF]), there exists a countable, pairwise disjoint collection of balls (B i ) i∈I satisfying (3.9) such that

B i ⊂ Ω, L n K l \ i∈I B i = 0, for any ∀i ∈ I.
For any i ∈ I, we fix x i ∈ B i and we set

ν i = ∇u(xi) |∇u(xi)| ; then Bi | ∇u, ν i | 2 (1 -z 2 ) 2 dx ≤ lim inf k→∞ Bi |∇u k | 2 (1 -z 2 k ) 2 dx. As (B i ) i is pairwise dijoint, we deduce ∪iBi | ∇u, ν i | 2 (1 -z 2 ) 2 dx = i∈I Bi | ∇u, ν i | 2 (1 -z 2 ) 2 dx, ≤ i∈I lim inf k→∞ Bi |∇u k | 2 (1 -z 2 k ) 2 dx, ≤ lim inf k→∞ i∈I Bi |∇u k | 2 (1 -z 2 k ) 2 dx, ≤ lim inf k→∞ ∪iBi |∇u k | 2 (1 -z 2 k ) 2 dx, ≤ lim inf k→∞ Ω |∇u k | 2 (1 -z 2 k ) 2 dx.
Moreover, we have

∀x ∈ B i ∩ K l , |∇u| 2 -| ∇u, ν i | 2 ≤ ∇u, ∇u |∇u| 2 -∇u, ν i 2 , ≤ ∇u, ∇u |∇u| -ν i ∇u, ∇u |∇u| + ν i , ≤ 2 l |∇u| 2 .

It gives

Bi∩K l |∇u| 2 (1 -z 2 ) 2 dx ≤ l l -2 Bi∩K l | ∇u, ν i | 2 (1 -z 2 ) 2 dx. As L n (K l \ ∪ i B i ) = 0 and (B i ) i is pairwise disjoint, we get K l |∇u| 2 (1 -z 2 ) 2 dx = i Bi∩K l |∇u| 2 (1 -z 2 ) 2 dx, ≤ l l -2 i Bi∩K l | ∇u, ν i | 2 (1 -z 2 ) 2 dx, ≤ l l -2 ∪iBi∩K l | ∇u, ν i | 2 (1 -z 2 ) 2 dx, ≤ l l -2 ∪iBi | ∇u, ν i | 2 (1 -z 2 ) 2 dx, ≤ l l -2 lim inf k→∞ Ω |∇u k | 2 (1 -z 2 k ) 2 dx.
As (K l ) l is an increasing sequence such that L n (U \ K l ) → 0, passing to the limit l → ∞ gives

U |∇u| 2 (1 -z 2 ) 2 dx ≤ lim inf k→∞ Ω |∇u k | 2 (1 -z 2 k ) 2 dx
and we may conclude

Ω |∇u| 2 (1 -z 2 ) 2 dx ≤ lim inf k→∞ Ω |∇u k | 2 (1 -z 2 k ) 2 dx.

Γ-convergence result for ε → 0 +

This section is dedicated to the proof of Theorem 3.1 ii). For that, we will prove the following Γ-convergence result.

Theorem 3.2. Assume that the conditions of Theorem 3.1 are satisfied. Let (ε k ) k be a sequence which converges to 0 + , we have

i) if u ∈ B(Ω) ∩ L ∞ (Ω), (u k , z k ) k ⊂ D n (Ω), (u k ) k is bounded in L ∞ (Ω), (u k , z k ) k converges to (u, 0
) almost everywhere and (M k ) k weakly converges to M, then

lim inf k→∞ E ε k (u k , z k , M k ) ≥ E(u, M); (3.10) ii) for any u ∈ B(Ω) ∩ L ∞ (Ω) and M ∈ W 1,n+α (Ω; G), there exists a sequence (u k , z k ) k ⊂ D n (Ω) such that (u k ) k is bounded in L ∞ (Ω), (u k , z k ) k converges to (u, 0
) almost everywhere and

lim sup k→∞ E ε k (u k , z k , M) ≤ E(u, M).
(3.11)

The inequality for the lower Γ-limit

We now prove the first inequality of Γ-convergence (3.10

). Let u ∈ B(Ω) ∩ L ∞ (Ω) and (u k , z k ) k ⊂ D n (Ω) such that (u k ) k is bounded in L ∞ (Ω), (u k , z k ) k converges to (u, 0
) almost everywhere and (M k ) k weakly converges to M. In the sequel, we emphasize on the domain of the function: for U an open subset of Ω, we adopt the following notation

F (u, M; U ) = U |∇u| 2 dx + Ju∩U Mν u , ν u 1/2 dH n-1 , F ε k (u k , z k , M k ; U ) = U |∇u k | 2 (1 -z 2 k ) 2 dx + U ε k M k ∇z k , ∇z k + z 2 k 4ε k dx,
Fatou Lemma and lower continuity for the Sobolev semi-norm yields

lim inf k→∞ Ω (u k -g) 2 dx + Ω DM k n+α dx ≥ Ω (u -g) 2 dx + Ω DM n+α dx
and then it suffices to prove that lim inf

F ε k (u k , z k , M k ; Ω) ≥ F (u, M; Ω). Moreover, if lim inf k Ω ε k |∇z k | 2 dx =
Ω ε k M k ∇z k , ∇z k dx - Ω ε k M∇z k , ∇z k dx ≤ M k -M L ∞ ε k Ω |∇z k | 2 dx.
As the weakly convergence of (M k ) k to M in W 1,n+α (Ω) implies the uniform convergence, we may conclude that lim inf

k Ω ε k M k ∇z k , ∇z k dx - Ω ε k M∇z k , ∇z k dx = 0.
To prove inequality (3.10), it remains to verify that

lim inf k→∞ F ε k (u k , z k , M; Ω) ≥ F ε k (u k , z k , M; U ).
We perform the proof in two steps: the first step deals with dimension 1. The second generalizes it for dimension n ≥ 2.

The one-dimensional case

In this section, we assume that Ω = I is an open interval and the metric M is simply a constant m > 0. We assume in this section that m is fixed. To avoid confusion, we denote the approximating functional by

G ε (u, z; I) = I |∇u(t)| 2 (1 -z(t) 2 ) 2 dt + I mε|∇z(t)| 2 + z(t) 2 4ε dt, (3.12)
where the domain is

D 1 (I) = {(u, z) : u ∈ B(I), z ∈ W 1,2 (I; [0; 1]), ∀N ∈ N u N (1 -z 2 ) ∈ W 1,2 (I)}.
We denote the lower Γ-limit, by

G -(u; I) = inf lim inf k→∞ G ε k (u k , z k ; I) ,
where the inf is taken over all sequence (u k , z k ) k ⊂ D 1 (I) such that (u k , z k ) converges almost everywhere to (u, 0) in I. We need the following Lemma which proof is given in appendix 4.2.

Lemma 3.2. Let I ⊂ R be an open interval, J ⊂ I be a set with finite cardinal. We have

u ∈ W 1,2 (I \ J) ⇒ u ∈ SBV(I), J u ⊂ J.
The main result of this subsection is given by the following.

I |∇u(t)| 2 dt + m 1 2 H 0 (J u ∩ I) ≤ G -(u; I).
The proof of this Proposition consists in showing the two following Lemmas.

Lemma 3.3. If u ∈ W 1,2 (B η (x)), then we have G -(u; B η (x)) ≥ Bη(x) |∇u(t)| 2 dt. Lemma 3.4. If u ∈ W 1,2 (B ρ (x)) for any ρ ∈]0; η[, then for any ρ ∈]0; η[ we have G -(u; B ρ (x)) ≥ m 1/2 .
Suppose that Lemma 3.3 and 3.4 are proved, we deduce Proposition 3.4.

Proof. We set

J = x ∈ I : ∀ρ > 0, u ∈ W 1,2 (B ρ (x)) .
Let {x 1 , . . . , x N } ⊂ J and ρ > 0 be such that {B ρ (x i ) : i = 1, . . . , N } is pairwise disjoint. According to Lemma 3.4 we have

∀i ∈ {1, . . . , N }, G -(u; B ρ (x i )) ≥ m 1/2 and then N i=1 G -(u; B ρ (x i )) ≥ N m 1/2 . As G -(u; •) is superadditive, we have G -(u; ∪ N i=1 B ρ (x i )) ≥ N m 1/2 . and G -(u; •) is non decreasing, it gives G -(u; I) ≥ N m 1/2 .
As G -(u; •) < +∞, the set J is finite. So, there exists ρ > 0 such that {B ρ (x) : x ∈ J} is pairwise disjoint. As G -(u; • ) is superadditive and non decreasing, we have

x∈J G -(u; B ρ (x)) + G -(u; I \ ∪ x∈J B ρ (x)) ≤ G -(u; I).
According to Lemma 3.3 and 3.4, it gives

H 0 (J)m 1 2 + I\∪ x∈J Bρ(x) |∇u(t)| 2 dt ≤ G -(u; I).
Taking the limit ρ → 0 + yields

H 0 (J)m 1 2 + I\J |∇u(t)| 2 dt ≤ G -(u; I).
In particular u ∈ W 1,2 (I \ J) and, according to Lemma 3.2, we get u ∈ SBV(I), J u ⊂ J and then

H 0 (J)m 1 2 + I |∇u(t)| 2 dt ≤ G -(u; I).
Now, we prove lemma 3.3.

Proof. We can assume that G -(u; B η (x)) < +∞, otherwise the result is ensured. By a diagonal extraction, there exists a sequence (u k , z k ) k ⊂ D 1 (B ρ (x)) converging almost everywhere to (u, 0) and

G ε k (u k , z k ; B ρ (x)) → G -(u; B ρ (x)).
As G -(u; B η (x)) is finite, there exists C > 0 such that ∀k ∈ N,

Bη(x) ε k |∇z k | 2 + z 2 k 4ε k dt ≤ C. (3.13) Applying the inequality 2ab ≤ a 2 + b 2 with a 2 = ε k |∇z k | 2 and b 2 = z 2 k 4ε k gives ∀k ∈ N, Bη(x) |∇z k |z k dt ≤ C. (3.14) We set c k = 1 -z 2 k . As z k ∈ W 1,2 (B η (x)), then c k ∈ BV (B η (x)) and (3.14) is ∀k ∈ N, Bη(x) |∇c k |dt ≤ 2C.
Coarea formula (see [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF]) yields ∀k ∈ N, Let σ < 1 in an arbitrary neighborhood of 1 and δ ∈]0; σ[ be fixed numbers. According to (3.15) and mean value Theorem, there exits

δ k ∈]δ; σ[ such that ∀k ∈ N, H 0 ({y ∈ B η (x) : c k (y) = δ k }) ≤ 2C σ -δ . (3.16) We set A k = {y ∈ B η (x) : c k (y) ≥ δ k }.
As (ε k ) k converges to 0, inequality (3.13) implies that (z k ) k converges to 0 and c k to 1 almost everywhere. As δ k < σ and σ < 1 then (L

1 (A k )) k converges to L 1 (B η (x)). Sobolev embedding Theorem ensures that W 1,2 (B η (x)) ⊂ C(B η (x))
, so c k is continuous and A k is a countable union of closed intervals of B η (x). According to (3.16), this union is finite and its cardinality is uniformly bounded by N . For any k, there exits a disjoint family of closed intervals

(I i k ) i=1...N such that A k = N i=1 I i k , ∀i ∈ {1, . . . , N -1}, max(I i k ) < min(I i+1 k )
. There exists a subsequence, still denoted by (I i k ) i=1...N , such that (min(I i k )) k and (max(I i k )) k converge for any i ∈ {1, . . . , N }. We set a i ∞ and b i ∞ the previous limits,

I i ∞ =]a i ∞ ; b i ∞ [ and A = N i=1 I i ∞ . As (L 1 (A k )) k converges to L 1 (B η (x)),
|∇u k | 2 (1 -z 2 k ) 2 dt ≥ lim inf k→∞ O |∇u k | 2 (1 -z 2 k ) 2 dt.
As z k takes its values in [0; 1], we get lim inf

k→∞ Bη(x) |∇u k | 2 (1 -z 2 k ) 2 dt ≥ lim inf k→∞ O |∇u k | 2 δ 2 k dt, ≥ δ 2 lim inf k→∞ O |∇u k | 2 dt.
Moreover, there exists k 0 such that:

k ≥ k 0 ⇒ O ⊂ A k , then we have ∀x ∈ O 1 -z 2 k ≥ δ. As u k (1 -z 2 k ) ∈ W 1,2 (Ω) and √ δ > 0, we get u k ∈ W 1,
|∇u k | 2 (1 -z 2 k ) 2 dt ≥ δ 2 Bη(x)
|∇u| 2 dt.

Letting δ to 1 -, it concludes the proof of lemma 3.3.

We prove lemma 3.4.

Proof. We can assume that G -(u; B ρ (x)) < +∞ for any ρ ∈]0; η[, otherwise the result is ensured.

As u ∈ W 1,2 (B ρ (x)), there exists three sequences (y 1 k ) k∈N , (y 2 k ) k∈N and (y 3 k ) k∈N such that:

       y 1 k → x, z k (y 1 k ) → 0, y 2 k → x, z k (y 2 k ) → 1, y 3 k → x, z k (y 3 k ) → 0, ∀k ∈ N, y 1 k < y 2 k < y 3 k . We have G(u k , z k ; B ρ (x)) ≥ x+ρ x-ρ ε k m|∇z k | 2 + z 2 k 4ε k dt.
The inequality a 2 + b 2 ≥ 2ab gives:

G(u k , z k ; B ρ (x)) ≥ x+ρ x-ρ m 1/2 |∇z k |z k dt. As [y 1 k , y 2 k ] ⊂ B ρ (x)
, we obtain:

G(u k , z k ; B ρ (x)) ≥ y 3 k y 1 k m 1/2 |∇z k |z k dt.
We have

G(u k , z k ; B ρ (x)) ≥ m 1/2 y 2 k y 1 k |∇z k (t)|z k (t) dt + m 1/2 y 3 k y 2 k |∇z k (t)|z k (t) dt. Since z k ∈ W 1,2 (B η (x))
, we may use the change of variable s = z k (t). This yields:

( ) 2 k ≥ m 1/2 z k (y 2 k ) z k (y 1 k ) sds + m 1/2 z k (y 3 k ) z k (y 2 k ) sds, ≥ m 1/2 z 2 k (y 2 k ) -z 2 k (y 1 k ) 2 + z 2 k (y 2 k ) -z 2 k (y 3 k ) 2
By assumption, we have z k (y 1 k ) → 0, z k (y 2 k ) → 1 and z k (y 3 k ) → 0, so that we deduce:

z 2 k (y 2 k ) -z 2 k (y 1 k ) 2 + z 2 k (y 2 k ) -z 2 k (y 3 k ) 2 → 1.
We can conclude :

lim inf k→∞ G(u k , z k ; B ρ (x)) ≥ m 1/2 .

Generalization to dimension n ≥ 2

We give the proof of the first inequality of Γ-convergence (3.10) for n ≥ 2.

Proof. Let u ∈ SBV(Ω) ∩ L ∞ (Ω) and (u k , z k ) k ⊂ D n (Ω) converging almost everywhere to (u, 0) such that (u k ) k is bounded in L ∞ (Ω). We have to prove lim inf k→∞ F ε k (u k , z k , M; Ω) ≥ F (u, M; Ω). (3.17)
We assume that lim inf F ε k (u k , z k , M; Ω) is finite, otherwise the result is ensured. As M ∈ W 1,n+α (Ω), then M is α-Hölder regular. So, there exists θ ≥ 0 such that, for any x, y ∈ Ω, we have

M(x) -M(y) ≤ θ|x -y| α . ( 3

.18)

Claim: There exists

C(δ) such that i) lim δ→0 + C(δ) = 1, ii) for A ⊂ Ω open, a ∈ A, diam(A) ≤ δ and ν ∈ S n-1 , we have lim inf k→∞ F ε k (u k , z k , M; A) ≥ A ∇u, ω 2 dx + C(δ) Ju∩A |M(a)ν| M(a)ν, ν 1/2 ω, ν u dH n-1 ,
where ω = M(a)ν |M(a)ν| .

We denote by A an arbitrary open subset of Ω such that diam(A) ≤ δ and we fix a ∈ A. Let ν ∈ S n-1 be fixed. According to Hölder regularity (3.18) and ellipticity inequality (2.1), we have

∀(x, v) ∈ A × R n , | M(x)v, v -M(a)v, v | ≤ θδ α |v| 2 , ≤ θδ α λ -1 M(a)v, v . Then, we get ∀(x, v) ∈ A × R n , M(x)v, v ≥ (1 -θδ α λ -1 ) M(a)v, v
We set C(δ) = 1 -θδ α λ -1 . Then, we may write

∀x ∈ A, M(x)∇z k (x), ∇z k (x) 2 ≥ C(δ) M(a)∇z k (x), ∇z k (x) 2 .
As M(a) is a symmetric definite positive matrix, Cauchy-Schwartz inequality gives

∀v ∈ R n , M(a)ν, ν M(a)v, v ≥ M(a)ν, v 2 , which is equivalent to ∀v ∈ R n , M(a)v, v ≥ |M(a)ν| 2 M(a)ν, ν M(a)ν |M(a)ν| , v 2 . (3.19) We set ω = M(a)ν |M(a)ν| . If we apply inequality (3.19) to F ε k (u k , z k , M; A), we have F ε k (u k , z k , M; A) ≥ A |∇u k | 2 (1 -z 2 k ) 2 + C(δ) |M(a)ν| 2 M(a)ν, ν ε k ω, ∇z k 2 + z 2 k 4ε k dx.
With the notation introduced in (1.2), (v) y is the function defined on A y ω as (v) y (t) = v(y + tω). According to Lemma 3.1, we have ∇(u k ) y (t) = ∇u(y + tω), ω and ∇(z k ) y (t) = ∇z(y + tω), ω , so Fubini Theorem gives

F ε k (u k , z k , M; A) ≥ Aω A y ω |∇(u k ) y | 2 (1 -((z k ) y ) 2 ) 2 + C(δ) |M(a)ν| 2 M(a)ν, ν ε k |∇(z k ) y | 2 + ((z k ) y ) 2 4ε k dt dH n-1 (y).
With the one-dimensional notations (3.12), it gives

F ε k (u k , z k , M; A) ≥ Aω G ε k ((u k ) y , (z k ) y ; A y ω ) dH n-1 (y), where m = C(δ) |M(a)ν| 2 M(a)ν,ν for any x ∈ A. Fatou lemma yields lim inf k→∞ F ε k (u k , z k , M; A) ≥ Aω lim inf k→∞ G ε k ((u k ) y , (z k ) y ; A y ω ) dH n-1 (y)
and then lim inf

k→∞ F ε k (u k , z k , M; A) ≥ Aω G -((u) y ; A y ω ) dH n-1 (y). As lim inf F ε k (u k , z k , M; A) is finite, we deduce that G -((u) y ; A y ω )
is finite for H n-1 almost every y ∈ A ω . We may apply Proposition 3.4 with I = A y ω and u = (u) y , it gives that (u) y ω ∈ SBV(A y ω ) for H n-1 almost every y ∈ A ω and we have

lim inf k→∞ F ε k (u k , z k , M; A) ≥ Aω A y ω |∇(u) y | 2 dt + H 0 (J (u)y ∩ A y ω )m 1 2
dH n-1 (y).

As lim inf

F ε k (u k , z k , M; A) is finite, Theorem 1.3 implies Aω A y ω |∇(u) y | 2 dt + H 0 (J (u)y ∩ A y ω )m 1/2 dH n-1 (y) = Ω | ∇u, ω | 2 dx+ Ju∩A m 1/2 ω, ν u dH n-1 .
We deduce

lim inf k→∞ F ε k (u k , z k , M; A) ≥ A | ∇u, ω | 2 dx + Ju∩A m 1/2 ω, ν u dH n-1 .
If we replace m and ω by their values, it gives

lim inf k→∞ F ε k (u k , z k , M; A) ≥ A ∇u, ω 2 dx + C(δ) Ju∩A M(a)ν, ν u M(a)ν, ν 1/2 dH n-1
and the Claim is proved.

The function x → ∇u(x) |∇u(x)| is measurable in U = {x ∈ Ω : ∇u(x) = 0}.
According to Lusin Theorem (1.45 of [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF]), there exists an increasing sequence of compacts (K l ) l ⊂ U such that

L n (U \ K l ) ≤ 1 l , x → ∇u(x) |∇u(x)| is continuous in K l .
Thus, for any x ∈ K l , there exists r > 0 such that

y ∈ B(x, r) ⇒ ∇u(x) |∇u(x)| - ∇u(y) |∇u(y)| ≤ 1 l , ( 3.20)
As a consequence of Besicovitch Covering Theorem (2.18 of [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF]), there exists a countable, pairwise disjoint collection of balls (B i ) i∈I satisfying (3.20) such that

∀i ∈ I, B i ⊂ Ω, diam(B i ) ≤ δ, L n K l \ i∈I B i = 0.
For any i ∈ I, we fix x i ∈ B i and we set

ν i = (M(a)) -1 ∇u(xi) |(M(a)) -1 ∇u(xi)| . According to First Step, with A = B i , a = x i and ν = ν i , we get lim inf k→∞ F ε k (u k , z k , M; B i ) ≥ Bi ∇u, ∇u(x i ) |∇u(x i )| 2 dx.
Moreover, we have

∀x ∈ B i ∩ K l , |∇u| 2 -∇u, ∇u(x i ) |∇u(x i )| 2 ≤ ∇u, ∇u |∇u| 2 -∇u, ∇u(x i ) |∇u(x i )| 2 , ≤ ∇u, ∇u |∇u| - ∇u(x i ) |∇u(x i )| ∇u, ∇u |∇u| + ∇u(x i ) |∇u(x i )| , ≤ 2 l |∇u| 2 .

It gives

Bi∩K l ∇u, ∇u(x i ) |∇u(x i )| 2 dx ≥ l l + 2 Bi∩K l |∇u| 2 dx. As lim inf F ε k (u k , z k , M; • ) is supperadditive and non decreasing, we have lim inf k→∞ F ε k (u k , z k , M; Ω) ≥ i∈I lim inf k→∞ F ε k (u k , z k , M; B i ), ≥ i∈I Bi ∇u, ∇u(x i ) |∇u(x i )| 2 dx, ≥ i∈I Bi∩K l ∇u, ∇u(x i ) |∇u(x i )| 2 dx, ≥ l l + 2 i∈I Bi∩K l |∇u| 2 dx, ≥ l l + 2 ∪iBi∩K l |∇u| 2 dx, As L n (K l \ ∪ i B i ) = 0, we deduce lim inf k→∞ F ε k (u k , z k , M; Ω) ≥ l l + 2 K l |∇u| 2 dx and taking the limit l → ∞ gives lim inf k→∞ F ε k (u k , z k , M; Ω) ≥ U |∇u| 2 dx = Ω |∇u| 2 dx.
In particular, Ω |∇u| 2 dx is finite. As u belongs to SBV(Ω), according to Theorem 1.1, there exists a pairwise disjoint family (C i ) i∈N of C 1 compact manifolds and M ∈ Ω such that:

J u = N ∪ i∈N C i , H n-1 (N ) = 0.
As lim inf F ε k (u k , z k , M; Ω) is finite, First Step and Theorem 1.3 imply that H n-1 (J u ) is also finite. According to ellipticity inequality (2.1), we deduce that Ju Mν u , ν u 1/2 dH n-1 is finite. Then, for a fixed δ > 0, there exists N ∈ N such that

Ju\ N i=1 Ci Mν u , ν u 1/2 dH n-1 ≤ δ. ( 3
y ∈ B(x, r) ∩ K ⇒ M(x)ν u (x) M(x)ν u (x), ν u (x) 1/2 - M(y)ν u (y) M(y)ν u (y), ν u (y) 1/2 ≤ δ. (3.24)
As a consequence of Besicovitch Covering Theorem (2.18 of [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF]), there exists a countable, pairwise disjoint, collection of balls ( B j ) j∈ Ĩ satisfying (3.24) such that, for any j ∈ Ĩ, it satisfies

B j ⊂ K τ , diam( B j ) ≤ δ, H n-1   K \ j∈ Ĩ B j   = 0.
For any j ∈ Ĩ, we fix xj ∈ B j . According to First Step, with A = B j , a = xj and ν = ν u (x j ), we get

lim inf k→∞ F ε k (u k , z k , M; B j ) ≥ C(δ) Ju∩ Bj M(x j )ν u (x j ), ν u M(x j )ν u (x j ), ν u (x j ) 1/2 dH n-1 .
For any x ∈ B j ∩ K, we have

M(x j )ν u (x j ), ν u (x) M(x j )ν u (x j ), ν u (x j ) 1/2 -M(x)ν u (x), ν u (x) 1/2 ≤ M(x j )ν u (x j ) M(x j )ν u (x j ), ν u (x j ) 1/2 - M(x)ν u (x) M(x)ν u (x), ν u (x) 1/2 , ≤ δ.

It gives

Bj ∩K M(x j )ν u (x j ), ν u M(x j )ν u (x j ), ν u (x j ) 1/2 dH n-1 ≥ Bj ∩K Mν u , ν u 1/2 dH n-1 -δH n-1 ( B j ∩ K).
As lim inf F ε k (u k , z k , M; • ) is supperaddditive and non decreasing, we have

lim inf k→∞ F ε k (u k , z k , M; K τ ) ≥ j∈ Ĩ lim inf k→∞ F ε k (u k , z k , M; B j ), ≥ j∈ Ĩ C(δ) Bj ∩K Mν u , ν u 1/2 dH n-1 -δH n-1 ( B j ∩ K) , ≥ C(δ) ∪j Bj ∩K Mν u , ν u 1/2 dH n-1 -δH n-1 (∪ j B j ∩ K) . As L n (K \ B j ) = 0, we get lim inf k→∞ F ε k (u k , z k , M; K τ ) ≥ C(δ) K Mν u , ν u 1/2 dH n-1 -δH n-1 (K) . ( 3
F ε k (u k , z k , M; Ω) ≥ lim inf k→∞ F ε k (u k , z k , M; K τ ) + lim inf k→∞ F ε k (u k , z k , M; Ω \ K τ ), ≥ C(δ) K Mν u , ν u 1/2 dH n-1 -δH n-1 (K) + Ω\Kτ |∇u| 2 dx.
According to (3.22) and (3.21), we have

lim inf k→∞ F ε k (u k , z k , M; Ω) ≥ C(δ) Ju Mν u , ν u 1/2 dH n-1 -δ -δH n-1 (K) + Ω |∇u| 2 dx -δ. Letting δ → 0 + concludes the proof lim inf k→∞ F ε k (u k , z k , M; Ω) ≥ Ju Mν u , ν u 1/2 dH n-1 + Ω |∇u| 2 dx.

The inequality for the higher Γ-limit

In this section we prove the upper inequality of Γ-convergence, that is ii) of Theorem 3.2. Let u ∈ B(Ω) ∩ L ∞ (Ω) and M ∈ W 1,n+α (Ω; G) be fixed. It is sufficient to prove that there exists a sequence (

u k , z k ) k ⊂ D n (Ω) such that (u k , z k ) k converges to (u, 0) almost everywhere, (u k ) k is bounded in L ∞ (Ω) and lim sup k→∞ Ω |∇u k | 2 (1 -z 2 k ) 2 dx + Ω ε k M k ∇z k , ∇z k + z 2 k 4ε k dx ≤ Ω |∇u| 2 dx + Ju Mν u , ν u 1/2 dH n-1 .
We first prove a weaker result, where Ju Mν u , ν u 1 2 dH n-1 is replaced by its approximation with a Minkowski content. Then, with Theorem 1.4, we extend this result to the general setting.

Approximation with anisotropic Minkowski content

We set

F ε k (u, z, M) = Ω |∇u| 2 (1 -z 2 ) 2 dx + Ω ε k M∇z, ∇z + z 2 4ε k dx
and we prove the following Proposition 3.5. Assume that the conditions of Theorem 3.1 are satisfied. For u ∈ SBV(Ω) ∩ L ∞ (Ω) and M ∈ W 1,n+α (Ω; G), there exists a sequence

(u k , z k ) k ⊂ D n (Ω) such that (u k , z k ) k converges to (u, 0) almost everywhere (u k ) k is bounded in L ∞ (Ω) and lim sup k→∞ F ε k (u k , z k , M) ≤ Ω |∇u| 2 dx + M M (J u ),
where M M is defined in (1.3).

Proof. Without loss of generality, we may assume that |∇u| ∈ L 2 (Ω) and M M (J u ) < +∞, otherwise the result is obvious. If u ∈ W 1,2 (Ω), then J u = ∅ and the stationary sequence

u k = u, z k = 0 is a solution. If u ∈ W 1,2 (Ω), then J u = ∅ and (1 -z 2 k ) 2
has to be infinitesimal near of J u . For ρ > 0, we set (J u ) ρ = {x : d Ju φ (x) < ρ}. We separate Ω in three parts:

(J u ) b k , (J u ) a k +b k \ (J u ) b k , Ω \ (J u ) a k +b k with a k = -4ε k ln(ε k ), b k = ε 2 k . (3.26) Let Ψ k ∈ C ∞ 0 (Ω) such that Ψ k = 1 in (J u ) b k 2 and Ψ k = 0 in Ω \ (J u ) b k . We set u k = (1 -Ψ k )u and then u k = u in Ω \ (J u ) b k . As (b k ) k converges to 0 then u k converges to u almost everywhere. We set z k = 1 in (J u ) b k and z k = ε 2 k in Ω \ (J u ) a k +b k . In (J u ) a k +b k \ (J u
) b k we adopt the following construction: we introduce

θ k (t) = ε 2 k exp t 2ε k and we set zk (t) =    1 ∀t ∈ [0; b k ], θ k (a k + b k -t) ∀t ∈]b k ; a k + b k ], ε 2 k ∀t ∈]a k + b k ; +∞[. (3.27)
This is a continuous and decreasing function defined on [0; +∞[, moreover, for any

t ∈]b k ; a k + b k [, it satisfies ε k (z k (t)) 2 = (z k (t)) 2 4ε k . (3.28) We set z k = zk • d Ju φ . As z k is constant in (J u ) b k ∪ (Ω \ (J u ) a k +b k ), we have F ε k (u k , z k , M) = Ω\(Ju) a k +b k |∇u| 2 (1 -ε 4 k ) 2 dx + (Ju) a k +b k \(Ju) b k |∇u| 2 (1 -z 2 k ) 2 dx + (Ju) a k +b k \(Ju) b k ε k M∇z k , ∇z k + z 2 k 4ε k dx + ε 3 k 4 L n (Ω \ (J u ) a k +b k ) + 1 4ε k L n ((J u ) b k ) (3.29)
As |∇u| ∈ L 2 (Ω) and (a k + b k ) k converges to 0, the first term of (3.29) converges to Ω |∇u| 2 dx. As z k L ∞ ≤ 1, the second term converges to 0. As Ω is a bounded domain, the fourth term converges to 0. As M M (J u ) < +∞, there exists (ω k ) k a sequence which converges to 0 + such that

L n ((J u ) b k ) ≤ 2b k (M M (J u ) + ω k ) (3.30)
and then the fifth term is lower than 1 2 ε k (M M (J u ) + ω k ). So, the fifth term converges to 0. To compute the limit of (F ε k (u k , z k , M)) k , it remains to study the convergence of

A k (z k ) = (Ju) a k +b k \(Ju) b k ε k M∇z k , ∇z k + z 2 k 4ε k dx.
Ellipticity inequality (2.1) yields

|d Ju φ (x) -d Ju φ (y)| ≤ d φ (x, y), ≤ λ -1 2 |x -y|.
So, d Ju φ is Lipschitzian and Rademacher Theorem ensures that d Ju φ exists for almost every x ∈ Ω, in the sense of the approximate differentiability 3.2. Thus, for almost every

x ∈ (J u ) a k +b k \ (J u ) b k , we have ∇z k = z k • d Ju φ ∇d Ju φ . It gives A k (z k ) = (Ju) a k +b k \(Ju) b k ε k (z k • d Ju φ ) 2 M∇d Ju φ , ∇d Ju φ + (z k • d Ju φ ) 2 4ε k dx.
In [12], Theorem 3.2, it is proved that M(x)∇d Ju φ (x), ∇d Ju φ (x) = 1 for almost every x. So, we may write

A k (z k ) = (Ju) a k +b k \(Ju) b k ε k (z k • d Ju φ ) 2 + (z k • d Ju φ ) 2 4ε k M∇d Ju φ , ∇d Ju φ 1/2 dx.
We may apply Proposition 1.2 with Φ = φ and p = d Ju φ , it gives

A k (z k ) = a k +b k b k ε k z k (t) 2 + zk (t) 2 4ε k Ω MD1 (Ju)t , D1 (Ju)t 1/2 dt. (3.31) We set          H M (t) = Ω MD1 (Ju)t , D1 (Ju)t 1/2 , A M (s) = s 0 H M (t)dt.
Applying another time Proposition 1.2 gives

A M (s 2 ) -A M (s 1 ) = s2 s1 Ω MD1 (Ju)t , D1 (Ju)t 1/2 dt, = (Ju)s 2 \(Ju)s 1 M∇d Ju φ , ∇d Ju φ 1/2 dx, = L n ((J u ) s2 \ (J u ) s1 ). So, A M ∈ W
A k (z k ) = a k +b k b k ε k z k (t) 2 + zk (t) 2 4ε k H M (t)dt, = a k +b k b k zk (t) 2 2ε k H M (t)dt, = (a k + b k ) 2 2ε k A M (a k + b k ) - b k 2ε k A M (b k ) - 1 ε k a k +b k b k z k (t)z k (t)A M (t).
The first term obviously converges to 0. As for (3.30), we have

A M (b k ) ≤ 2b k (M M (J u ) + ω k )
and then the second term converges to 0 too. As s → A M (s) is non decreasing, then

A M (t) ≤ 2t(M M (J u ) + ω k ) for any t ∈ [b k ; a k + b k ].
For the last term, we apply another time this inequality, it gives

- 1 ε k a k +b k b k z k (t)z k (t)A M (t)dt ≤ - (M M (J u ) + ω k ) ε k a k +b k b k 2tz k (t)z k (t)dt. (3.32)
Integrating by parts yields

a k +b k b k 2tz k (t)z k (t)dt = (a k + b k )z k (a k + b k ) 2 -b k zk (b k ) 2 - a k +b k b k zk (t) 2 dt. (3.33)
According to the definitions of (a k , b k , z k ) (3.26) and (3.27), we have

(a k + b k )z k (a k + b k ) 2 -b k zk (b k ) 2 = o(ε k ) (3.34)
and equation (3.28) gives

a k +b k b k zk (t) 2 dt = 2ε k a k +b k b k |z k (t)|z k (t)dt, = ε k (1 -ε 2 k ). ( 3
F ε k (u k , z k , M) ≤ Ω |∇u| 2 + M M (J u ).
To conclude the proof, it suffices to notice that (u k , z k ) k ⊂ D n (Ω).

Approximation in the general setting

The goal of this section is to replace M M (J u ) by Ju Mν u , ν u

(u k , z k ) k ⊂ D n (Ω) such that (u k , z k ) k converges to (u, 0) almost everywhere (u k ) k is bounded in L ∞ (Ω) and lim sup k→∞ F ε k (u k , z k , M) ≤ F (u, M).
To prove this result, we need to introduce the following Definition 3.2. Let F(Ω) be the set of functions u ∈ SBV(Ω) for which, if F (u, M) < +∞, then there exists a sequence

(u k ) k ⊂ SBV(Ω)∩L ∞ (Ω) converging almost everywhere to u, lim k→∞ F (u k , M) = F (u, M) and ∀k ∈ N, M M (J u k ) = Ju k Mν u k , ν u k dH n-1 .
Proof. Assume F(Ω) = SBV(Ω). According to Proposition 3.5, by a diagonal extraction we may exhibit a sequence (u k , z k ) k ⊂ D n (Ω) such that (u k , z k ) k converges to (u, 0) almost everywhere and lim sup k→∞

F ε k (u k , z k , M) ≤ F (u, M).
So, to prove the upper inequality of Γ-convergence, it suffices to show that F(Ω) = SBV(Ω). We divide the proof in three Claims. By a direct diagonal extraction process, we have the following. For u ∈ SBV(Ω) and N > 0, we denote by u N the truncated function defined in (3.1). So, (u N ) N converges to u almost everywhere for N → ∞. Moreover, Theorem 1.2 gives

Du N = 1 -N ≤u≤N ∇uL n + ((u N ) + -(u N ) -)ν u H n-1 J u .
and then we deduce lim N →∞ F (u N ) = F (u, M). According to Claim 1, it suffices to prove that SBV(Ω) ∩ L ∞ (Ω) ⊂ F(Ω). We may assume that F (u, M) < +∞, otherwise the result is ensured. Let us extend M and u in Ω = Ω ∪ U as in Proposition 1.1, so we have

H n-1 (J u ∩ ∂Ω) = 0.
With the same arguments as for Theorem 2.1, we may prove that there exists v k ∈ SBV(Ω ) a minimizer of the following functional:

E u,k (v) = k Ω (v -u) 2 dx + Ω |∇v| 2 dx + Jv Mν v , ν v 1/2 dH n-1 .
In particular, E u,k (v k ) ≤ E u,k (u) for any k ∈ N, gives

M M (J v k ) = Jv k Mν v k , ν v k 1/2 dH n-1 .
We introduce the sequence of positive Radon measures (µ k ) k and µ defined for any B ∈ B(Ω ) by

µ k (B) = B |∇v k | 2 dx + Jv k ∩B Mν v k , ν v k 1/2 dH n-1 , µ(B) = B |∇u| 2 dx + Ju∩B Mν u , ν u 1/2 dH n-1 .
With the same arguments as for Theorem 2. follows by the definition of v k . According to [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF], Proposition 1.80, the measures (µ k ) k weakly converge to µ. Thus, (µ k (B)) k converges to µ(B) if µ(∂B) = 0, and then (µ k (Ω)) k converges to µ(Ω), that is (F (v k )) k converges to F (u, M). According to Claim 1, we deduce that u ∈ F(Ω).

Proof of Theorem 3.1 ii)

Proof. With the same arguments as for Theorem 3.1 i), for any ε > 0, there exists (u ε , z ε ) a minimizer of E ε (•, •, M). According to (3.3), with N ≥ g L ∞ (Ω) , we have

L n ({x ∈ Ω : |u ε (x)| > N }) > 0 ⇒ E ε (u N ε , z ε , M) < E ε (u ε , z ε , M).
We deduce that u ε L ∞ (Ω) ≤ N for any ε > 0. For ω ε = u ε (1 -z 2 ε ), we get

∇ω ε = ∇u ε (1 -z 2 ε ) -2u ε z ε ∇z ε .
It yields 2 (E ε (u ε , z ε , M))

Ω |∇ω ε |dx ≤ L n (Ω) 1 2 Ω |∇u ε | 2 (1 -z 2 ε ) 2 dx
1 2 + 1 + 1 λ E ε (u ε , z ε , M).
According to Proposition 3.6, we deduce that (E ε k (u ε k , z ε k , M)) k is a bounded sequence. So, (ω ε k ) k is bounded in BV(Ω) and there exists a subsequence, still denoted by (ω ε k ) k which converges almost everywhere to ω ∈ BV(Ω). As Ω z 2 k dx ≤ ε k E ε k (u ε k , z ε k , M), then (z k ) k converges to 0 in L 2 (Ω) and there exists a subsequence, still denoted by (z k ) k , which converges almost everywhere to 0. As ω ε k = u ε k (1 -z 2 ε k ), then (u ε k ) k converges almost everywhere to u ∈ B(Ω) ∩ L ∞ (Ω). With the same arguments as for Theorem 2.1, E(•, M) admits a minimizer v ∈ SBV(Ω) and v ∈ L ∞ (Ω). According to Theorem 3.2 ii), there exists (v ε k , zε k ) k ⊂ D n (Ω) such that (v ε k , zε k ) k converges to (v, 0) almost everywhere and lim sup k→∞

E ε k (v ε k , zε k , M) ≤ E(v, M).
According to Theorem 3.2, i), we get

lim inf k→∞ E ε k (u ε k , z ε k , M) ≥ E(u, M). As (u ε k , z ε k ) is a minimizer of E ε k , we have ∀k ∈ N, E ε k (v ε k , zε k , M) ≥ E ε k (u ε k , z ε k , M).
We conclude that E(v, M) ≥ E(u, M) and then (u, M) is also a minimizer of E.

Appendix

Proof of Lemma 3.1

Proof. As Ω is bounded, then W 1,2 (Ω) ⊂ SBV(Ω) and then, according to Calderón-Zygmund Theorem (3.83 in [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF]), the derivative in the Sobolev sense is equal to the approximate differential for almost every point in Ω. Moreover, according to Theorem 3.107 of [START_REF] Ambrosio | A compactness theorem for a new class of functions of bounded variation[END_REF], for v ∈ SBV(Ω) and ν ∈ S n- exist for any x ∈ I. As J is finite, then W 1,2 (I \ J) ⊂ W 1,1 (I \ J). According to the jump formula, it gives that Du ∈ M(I) and we have the decomposition Du = ∇uL 1 + (u(x + ) -u(x -))H 0 J.

So, we can conclude that u ∈ SBV(I) and J u ⊂ J.

Proposition 1 . 1 .

 11 Let Ω be an open and bounded domain satisfying (R) and Ω = U ∪ Ω. For u ∈ BV(Ω), we consider an extension in Ω by the following way

  according to Ellipticity condition (2.1), we have lim inf k Ω M k ∇z k , ∇z k dx = +∞ and the result of Proposition 3.3 is ensured. So, we may assume that lim inf Ω |∇z k | 2 dx < +∞.

Proposition 3 . 4 .

 34 Let I ⊂ R be an open interval and u ∈ B(I). If G -(u; I) < ∞, then u ∈ SBV(I) and

1 0H 0

 10 ({y ∈ B η (x) : c k (y) = t})dt ≤ 2C. (3.15)

  .21) We set K = N i=1 C i and K τ = {x ∈ Ω : dist(x, K) < τ }. As Ω |∇u| 2 dx is finite, there exists τ > 0 such that Ω\Kτ |∇u| 2 dx ≤ δ. (3.22)With the same arguments as before, we getlim inf k→∞ F ε k (u k , z k , M; Ω \ K τ ) νu(x),νu(x) 1/2 is continuous in K, for any x ∈ K there exists r > 0 such that

1 2 5 . 3 . 6 .

 1536 dH n-1 in Proposition 3.Proposition Assume that the conditions of Theorem 3.1 are satisfied. For u ∈ SBV(Ω) ∩ L ∞ (Ω) and M ∈ W1,n+α (Ω; G), there exists a sequence

Claim 1 :

 1 If u ∈ SBV(Ω) and (u k ) k ⊂ SBV(Ω) satisfy i) (u k ) k ⊂ F(Ω), ii) lim k→∞ F (u k , M) = F (u, M) and F (u, M) < ∞,iii) (u k ) k converges to u almost everywhere, then u ∈ F(Ω).

Claim 2 :

 2 It suffices to prove that SBV(Ω) ∩ L ∞ (Ω) ⊂ F(Ω).

Claim 3 :

 3 Let u ∈ SBV(Ω) ∩ L ∞ (Ω), we have u ∈ F(Ω).

  -u) 2 dx ≤ F (u, M) and then (v k ) k converges to u almost everywhere. As u ∈ L ∞ (Ω), Theorem 1.4 gives

1 ,

 1 F is lower semi-continuous in SBV, for any openA ⊂ Ω , it gives lim inf k→∞ µ k (A) ≥ µ(A).The inequality lim sup k→∞ µ k (Ω ) ≤ µ(Ω )

Ω 2 ε 2ε 1 2 and b = ε 1 2

 221 |∇z ε |z ε dx.(3.36)Applying the inequality 2ab ≤ a 2 + b 2 with a = z |∇z ε | gives Ω |∇z ε |z ε dx ≤ Ω ε|∇z ε | 2 dx + to ellipticity inequality (2.1), we get Ω ε|∇z ε | 2 dx ≤ 1 λ E ε (u ε , z ε , M). (3.38) By (3.36), (3.37) and (3.38), we deduce Ω |∇ω ε |dx ≤ L n (Ω)

1

 1

v 2 x 4 . 2

 242 + |∇v x (t)| 2 dtdx < ∞, ⇒ v x ∈ W 1,2 (Ω x )for a.e. x ∈ Ω. Applying this property with v = z and v = u(1 -z 2 ) gives the result of Lemma 3.1. Proof of Lemma 3.2 Proof. For ]a; b[⊂ I \J, according to Theorem 2.8. of [7], there exists a unique function ũ ∈ C([a; b]) such that u(x) ∈ ũ(x) for L 1 -a.e. x ∈ I and then u(x -) := lim y→x - u(y) and u(x + ) := lim y→x + u(y)