Fakhrielddine Bader

Mostafa Bendahmane
email: mostafa.bendahmane@u-bordeaux.fr

AND Mazen Saad
email: mazen.saad@ec-nantes.fr

Raafat Talhouk
email: rtalhouk@ul.edu.lb

MICROSCOPIC TRIDOMAIN MODEL OF ELECTRICAL ACTIVITY IN THE HEART WITH DYNAMICAL GAP JUNCTIONS. PART 1-MODELING AND WELL-POSEDNESS

Keywords: 1991 Mathematics Subject Classification. 65N55, 35A01, 35A02, 65M, 92C30 Tridomain model, Global existence, Uniqueness, Weak solution, Gap junctions, Cardiac electro-physiology

We present a novel microscopic tridomain model describing the electrical activity in cardiac tissue with dynamical gap junctions. The microscopic tridomain system consists of three PDEs modeling the tissue electrical conduction in the intra-and extra-cellular domains, supplemented by a nonlinear ODE system for the dynamics of the ion channels and the gap junctions. We establish the global existence and uniqueness of the weak solutions to our microscopic tridomain model. The global existence of solution, which constitutes the main result of this paper, is proved by means of an approximate non-degenerate system, the Faedo-Galerkin method, and an appropriate compactness argument.

The heart study started since more than two millennia back. This organ, about the size of its owner's clenched fist, contracts rhythmically to circulate blood throughout the body, while other organs like the brain and lungs, were thought to exist to cool the blood. Until this day the heart keeps the position of one of the most important and the most studied organs in the human body. Especially, cardiovascular disease (CVD) leading to heart attack, is the top cause of death in the worldwide as announced by the "World Health Organization" in 2019. Given the large number of related pathologies, there is an important need for understanding the chemical and electrical phenomena taking place in the cardiac tissue.

In fact, the heart is a muscular organ can be viewed as double pump consisting of four chambers: upper left and right atria, and lower left and right ventricles. These four chambers are surrounded by a cardiac tissue that is organized into muscle fibers. These fibers form a network of cardiac muscle cells called "cardiomyocytes" connected end-to-end by junctions called intercalated discs. Intercalated discs contain gap junctions and desmosomes. Gap junctions transverse of contiguous cells and connect the cytoplasm of one cell to the cytoplasm of the adjacent cell. Cardiac tissue use gap junctions to spread action potential to nearby cells. This allows the heart to generate a single continuous and forceful contraction that pumps the blood throughout the body [START_REF] James | [END_REF][START_REF] Arnold | Physiology of the Heart[END_REF].

The structure of cardiac tissue (myocarde) studied in this paper is characterized at two different scales (see Figure 1). At microscopic scale, the cardiac tissue consists of two intracellular media which contains the contents of the cardiomyocytes (the cytoplasm) that are connected by gap junctions and the other is called extracellular and consists of the fluid outside the cardiomyocytes cells. Each intracellular medium and the extracellular one are separated by a cellular membrane (the sarcolemma). While at the macroscopic scale, this domain is well considered as a single domain (homogeneous). It should be noted that there is a difference between the chemical composition of the cytoplasm and that of the extracellular medium. This difference plays a very important role in cardiac activity. On the one hand, the sarcolemma allows the penetration of inorganic ions (sodium, potassium, calcium,...) and proteins, some of which play a passive role and others play an active role powered by cellular metabolism. In particular, the concentration of anions (negative ions) in cardiomyocytes is higher than in the external environment. This difference of concentrations creates a transmembrane potential, which is the difference in potential at the sarcolemma between each intracellular medium and the extracellular one. On the other hand, gap junctions allows the movement of not only inorganic ions but also organic ions between two adjacent cells [START_REF] James | [END_REF]. It provide the pathways for intracellular current flow, enabling coordinated action potential propagation. So, the difference of chemical through the gap junction creates a gap potential, which is the difference in potential between these two intracellular media. Our model that describes the electrical activity in the cardiac tissue including the gap junctions, is called by "tridomain model". From the mathematical viewpoint, the microscopic tridomain model consists of three quasi-static equations, two for the electrical potential in the intracellular medium and one for the extracellular medium, coupled by a nonlinear ODE system at each membrane (the sarcolemma) and by a linear one at gap junction for the dynamics of the ion channels. These equations depend on scaling parameter ε whose is the ratio between the microscopic scale and the macroscopic one. The microscopic tridomain model was proposed three years ago [START_REF] Tveito | A cell-based framework for numerical modeling of electrical conduction in cardiac tissue[END_REF][START_REF] Jaeger | Properties of cardiac conduction in a cell-based computational model[END_REF] in the case of just two coupled cells compared to our model which is defined on larger collections of cells.

The goal of the present paper is to investigate existence and uniqueness of solutions of the tridomain equations, coupled with an ionic model, namely the FitzHugh-Nagumo model. We mention some works in the literature on the bidomain model that gives a macroscopic description of the cardiac tissue from two inter-penetrating domains which are the intracellular and extracellular domains at the microscopic scale. The first mathematical formulation of this model was constructed by Tung [START_REF] Tung | A bi-domain model for describing ischemic myocardial dc potentials[END_REF]. This variant leads to two quasi-static whose unknowns are intra-and extracellular electric potentials coupled with non linear ordinary differential equations called ionic models at the membrane. Next, Krassowska and Neu [START_REF] Neu | Homogenization of syncytial tissues[END_REF] have proposed to represent cells by large cylinders connected to each other by narrow channels and then have applied the two-scale asymptotic method to formally obtain the bidomain model from the microscopic problem. In particular, they are considered that these narrow channels precisely model gap-junctions ("low-resistance connections between cells"). There are some references dealing with the well-posedness of this model. First, global existence in time and uniqueness for the solution of the micro-and macroscopic bidomain model coupled with FitzHugh-Nagumo simplification for the ionic currents, is proven in [START_REF] Colli | Degenerate evolution systems modeling the cardiac electric field at micro-and macroscopic level[END_REF][START_REF] Pennacchio | Multiscale modeling for the bioelectric activity of the heart[END_REF]. It is based on a reformulation of the bidomain problem as a Cauchy system for an evolution variational inequality in a properly chosen Sobolev space. Next, the authors in [START_REF] Veneroni | Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field[END_REF] used Schauder's fixed point theorem to establish the well-posedness of the macroscopic bidomain problem with a generalized phase-I Luo-Rudy ionic model [START_REF] Luo | A dynamic model of the cardiac ventricular action potential. i. simulations of ionic currents and concentration changes[END_REF]. The authors in [START_REF] Boulakia | A coupled system of pdes and odes arising in electrocardiograms modeling[END_REF] have studied the well-posedness of the macroscopic bidomain model coupled to a third PDE that describes the electrical potential of the surrounding tissue within the torso. The existence of a global solution of the latter model is proved using the Faedo-Galerkin method for a wide class of ionic models (including Mitchell-Schaeffer model [START_REF] Colleen | A two-current model for the dynamics of cardiac membrane[END_REF], FitzHugh-Nagumo [START_REF] Fitzhugh | Impulses and physiological states in theoretical models of nerve membrane[END_REF][START_REF] Nagumo | An active pulse transmission line simulating nerve axon[END_REF], Aliev-Panfilov [START_REF] Rubin | A simple two-variable model of cardiac excitation[END_REF], and Roger-McCulloch [START_REF] Jack | A collocation-galerkin finite element model of cardiac action potential propagation[END_REF]). Furthermore, in [START_REF] Bourgault | Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology[END_REF], existence of a global solution of the macroscopic bidomain model is proved only for the last three ionic models, using a semi-group approach and the Galerkin technique. While uniqueness, however, is achieved only for the FitzHugh-Nagumo ionic model in the two previous works. Moreover, the authors in [START_REF] Bendahmane | Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue[END_REF] proved the existence and uniqueness of solution of the macroscopic bidomain model by using the Faedo-Galerkin method (see for instance [START_REF] Bendahmane | Unfolding homogenization method applied to physiological and phenomenological bidomain models in electrocardiology[END_REF] where the authors prove the well-posedness of solution for the microscopic bidomain model using the same technique). In the present work, we prove the existence of solution for the novel microscopic tridomain model by a constructive method based on Faedo-Galerkin approach without the restrictive assumption, usually found in the literature, on the conductivity matrices to have the same basis of eigenvectors or to be diagonal matrices (see for instance [START_REF] Bourgault | Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology[END_REF] where the authors prove the existence of a local in time strong solution of the bidomain equations). It is worth to mention that our approach is innovative and cannot be found in the literature in the context of existence of solutions to the microscopic tridomain model.

The main contribution of the present paper. The cardiac tissue structure studied at micro-macro scales. We start by modeling the microscopic tridomain model by taking account the presence of gap junctions as connection between adjacent cardiac cells. Next, we formulate our tridomain model in dimensionless form with the hope to get more insight in the meaning of the microscopic and macroscopic scales. Finally, we end by proving the well-posedness of the microscopic tridomain problem by using Faedo-Galerkin method, a priori estimates and L 2 -compactness argument on the membrane surface.

The outline of the paper is as follows. In Section 2, we describe the geometry of cardiac tissue in the presence of gap junction and some notations and explanations on the boundary conditions are introduced. Furthermore, we introduce in detail our microscopic tridomain model in the cardiac tissue structure. In Section 3, our main result is stated: existence and uniqueness of a weak solutions.

In Section 4, we shall completely define and prove existence, uniqueness of a weak solutions. It is based on Faedo-Galerkin technique, a priori estimates and compactness results. The results are obtained under minimal regularity assumptions on the data.

Tridomain modeling of the cardiac tissue

The aim of this section is to describe the geometry of cardiac tissue and to present the microscopic tridomain model of the heart. Following the standard approach of the homogenization theory, this structure is featured by mic characterizing the microscopic length of a cell. Under the one-level scaling, the characteristic length mic is related to a given macroscopic length L (of the cardiac fibers), such that the scaling parameter ε introduced by:

ε = mic L .
Physiologically, the cardiac cells are connected by many gap junctions. Therefore, geometrically, the domain Ω consists of two intracellular media Ω k i,ε for k = 1, 2, that are connected by gap junctions Γ 1,2 ε = ∂Ω 1 i,ε ∩ ∂Ω 2 i,ε and extracellular medium Ω e,ε (for more details see [START_REF] Tveito | A cell-based framework for numerical modeling of electrical conduction in cardiac tissue[END_REF][START_REF] Jaeger | Properties of cardiac conduction in a cell-based computational model[END_REF]). Each intracellular medium Ω k i,ε and the extracellular one Ω e,ε are separated by the surface membrane Γ k ε (the sarcolemma) which is expressed by:

Γ k ε = ∂Ω k i,ε ∩ ∂Ω e,ε , with k = 1, 2,
while the remaining (exterior) boundary is denoted by ∂ ext Ω. We can observe that the intracellular domains as a perforated domain obtained from Ω by removing the holes which correspond to the extracellular domain Ω e,ε .

We can divide Ω into N ε small elementary cells Y ε = ). Therefore, for each macroscopic variable x that belongs to Ω, we define the corresponding microscopic variable y ≈ x ε that belongs to Y with a translation. Indeed, we have:

x ∈ Ω ⇒ ∃h ∈ Z d such that x ∈ Y h ε ⇒ x = ε(h + y) ⇒ y = x ε -h ∈ Y.
Since, we will study the behavior of the functions u(x, y) which are y-periodic, so by periodicity

we have u x, x ε -h = u x, x ε
. By notation, we say that y = x ε belongs to Y. We are assuming that the cells are periodically organized as a regular network of interconnected cylinders at the microscale. The microscopic unit cell Y is also divided into three disjoint connected parts: two intracellular parts Y k i for k = 1, 2, that are connected by an intercalated disc (gap junction) Γ 1,2 and extracellular part Y e . Each intracellular parts Y k i and the extracellular one are separated by a common boundary Γ k for k = 1, 2. So, we have:

Y := Y 1 i ∪ Y 2 i ∪ Y e , Γ k := ∂Y k i ∩ ∂Y e , Γ 1,2 := ∂Y 1 i ∩ ∂Y 2 i , with k = 1, 2.
In a similar way, we can write the corresponding common periodic boundary as follows:

(2) Γ ε,h = T h ε + εΓ = {εξ : ξ ∈ h + Γ}, with T h ε denote the same previous translation, Γ ε,h := Γ k ε,h , Γ 1,2 ε,h and Γ := Γ k , Γ 1,2 for k = 1, 2. In summary, the intracellular and extracellular media can be described as follows:

Ω k i,ε = Ω ∩ h∈Z d Y k i,ε,h , Ω e,ε = Ω ∩ h∈Z d Y e,ε,h , Γ k ε = Ω ∩ h∈Z d Γ k ε,h and Γ 1,2 ε = Ω ∩ h∈Z d Γ 1,2 ε,h , where Y k i,ε,h , Y e,ε,h and Γ k ε , Γ 1,2 ε are respectively defined as (1)-(2) for k = 1, 2. Both sets Ω k i,ε , k = 1, 2
and Ω e,ε are assumed to be connected Lipschitz domains so that a Poincaré-Wirtinger inequality is satisfied in both domains. The boundaries Γ k , k = 1, 2 and Γ 1,2 are smooth manifolds such that Γ k ε , k = 1, 2 and Γ 1,2 ε are smooth and connected.

Microscopic tridomain model.

A vast literature exists on the bidomain modeling of the heart, we refer to [START_REF] Pierre | Modélisation et simulation de l'activité électrique du coeur dans le thorax[END_REF][START_REF] Colli | Degenerate evolution systems modeling the cardiac electric field at micro-and macroscopic level[END_REF][START_REF] Pennacchio | Multiscale modeling for the bioelectric activity of the heart[END_REF][START_REF] Colli-Franzone | Mathematical and numerical methods for reactiondiffusion models in electrocardiology[END_REF] for more details. Here, we define a novel microscopic tridomain model described in detail in [START_REF] Tveito | A cell-based framework for numerical modeling of electrical conduction in cardiac tissue[END_REF][START_REF] Jaeger | Properties of cardiac conduction in a cell-based computational model[END_REF] and used in our investigations, as well the models chosen for the membrane and gap junctions dynamics. In the sequel, the space-time set (0, T ) × O is denoted by O T in order to simplify the notation. There are a few references dealing with the tridomain model for other cells types, e.g. cardiomyocytes and fibroblasts [START_REF] Frank B Sachse | A model of electrical conduction in cardiac tissue including fibroblasts[END_REF] and for simulating bioelectric gastric pacing [START_REF] Sathar | A multiscale tridomain model for simulating bioelectric gastric pacing[END_REF][START_REF] Du | Progress in mathematical modeling of gastrointestinal slow wave abnormalities[END_REF].

Basic equations. The basic tridomain equations modeling the propagation of cardiac action potentials at cellular level in the presence of gap junctions which can be formulated as follows. First, we know that the structure of the cardiac tissue can be viewed as composed by two intracellular spaces Ω k i for k = 1, 2, that are connected by gap junction Γ 1,2 and the extracellular space Ω e . The membrane Γ k is defined by the intersection between each intracellular domain Ω k i and the extracellular one with k = 1, 2.

Thus, the membrane Γ k is pierced by proteins whose role is to ensure ionic transport between the two media (intracellular and extracellular) through this membrane. So, this transport creates an electric current. Using Ohm's law, the intracellular electrical potentials u k i and extracellular one u e are respectively related to the current volume densities J k i and J e for k = 1, 2 :

J k i = M i ∇u k i , in Ω k i,T := (0, T ) × Ω k i , J e = M e ∇u e , in Ω e,T := (0, T ) × Ω e ,
where M j represents the corresponding conductivity of the tissue for j = i, e (which are assumed to be isotropic at the microscale and are given in mS/cm).

In addition, the transmembrane potential v k is known as the potential at the membrane Γ k which is defined as follows:

v k = (u k i -u e )| Γ k : (0, T ) × Γ k → R for k = 1, 2
. Moreover, we assume the intracellular and extracellular spaces are source-free and thus the intracellular and extracellular potentials are solutions to the elliptic equations:

(3) -divJ k i = 0, in Ω k i,T , -divJ e = 0, in Ω e,T , with k = 1, 2.
According to the current conservation law, the surface current density I k m is now introduced: (4)

I k m = -J k i • n k i = J e
• n e , on Γ k T := (0, T ) × Γ k , with n k i denotes is the (outward) normal pointing out from Ω k i,ε for k = 1, 2 and n e is the normal pointing out from Ω e,ε .

The membrane has both a capacitive property schematized by a capacitor and a resistive property schematized by a resistor. On the one hand, the capacitive property depends on the formation of the membrane which can be represented by a capacitor of capacitance C m (the capacity per unit area of the membrane is given in µF/cm 2 ). We recall that the quantity of the charge of a capacitor is q k = C m v k . Then, the capacitive current I k c for k = 1, 2 is the amount of charge that flows per unit of time:

I k c = ∂ t q k = C m ∂ t v k .
On the other hand, the resistive property depends on the ionic transport between the intracellular and extracellular media. Then, the resistive current I r is defined by the ionic current I k ion measured from the intracellular to the extracellular medium which depends on the transmembrane potential v k and the gating variable w k : Γ k → R with k = 1, 2. Moreover, the total transmembrane current I k m (see [START_REF] Colli-Franzone | Mathematical and numerical methods for reactiondiffusion models in electrocardiology[END_REF]) is given by:

I k m = I k c + I k r -I k app on Γ k T , with I k
app is the applied current of the membrane surface for k = 1, 2 (given in µA/cm 2 ). Consequently, due to the dynamics of the ionic fluxes through the cell membrane, its electrical potential v k satisfies the following dynamic condition on Γ k involving the gating variable w k :

(5)

I k m = C m ∂ t v k + I ion v k , w k -I k app on Γ k T , ∂ t w k -H v k , w k = 0 on Γ k T .
Furthermore, the functions H and I ion correspond to the ionic model of membrane dynamics. All surface current densities I k m for k = 1, 2 and I ion are given in µA/cm 2 . Moreover, time is given in ms and length is given in cm.

In addition, we represent the gap junction between intra-neighboring cells by a passive model. This model includes several state variables in addition to the gap junction potential s which is defined as follows:

s = (u 1 i -u 2 i )| Γ 1,2 : (0, T ) × Γ 1,2 → R.
The ionic current I 1,2 through the gap junction Γ 1,2 defined by: ( 6)

I 1,2 = -J 1 i • n 1 i = J 2 i • n 2 i , on Γ 1,2 T := (0, T ) × Γ 1,2 .
Similarly, the ionic current I 1,2 at a gap junction Γ 1,2 represents the sum of the capacitive and resistive currents. Consequently, regarding the dynamic structure of the gap junction, its electrical potential s satisfies the following dynamic condition on Γ 1,2 :

(7) I 1,2 = C 1,2 ∂ t s + I gap (s) on Γ 1,2
T , where C 1,2 represents the capacity per unit area of the intercalated disc and I gap represents the corresponding resistive current. In general, the value of C 1,2 is set to C m /2 because the intercalated disc is assumed to be a membrane of thickness twice as large as the cell membrane, and the specific capacitance of a capacitor C m formed by two parallel plates separated by an insulator may be assumed to be inversely proportional to the thickness of the insulator [START_REF] Jaeger | Properties of cardiac conduction in a cell-based computational model[END_REF].

Non-dimensional analysis. We use the microscopic model given in the previous part without the parameter scaling ε. In the non-dimensionalization procedure, ε will appear also in each boundary conditions due to the scaling of the involved quantities (see [START_REF] Craig | The bidomain model of cardiac tissue: from microscale to macroscale[END_REF][START_REF] Colli-Franzone | Mathematical and numerical methods for reactiondiffusion models in electrocardiology[END_REF][START_REF] Bader | Derivation of a new macroscopic bidomain model including three scales for the electrical activity of cardiac tissue[END_REF] for the bidomain case).

As a natural assumption for homogenization, we want to formulate the tridomain equations ( 3)-( 7) in dimensionless form with the hope to get more insight in the meaning of the parameter ε. We define the dimensionless parameter ε as the ratio between the microscopic length mic and the macroscopic length L, i.e.

ε = mic L .
Using all fundamental material constants, several additional time and length constants can be formulated. For convenience, the macroscopic length is defined as L = √ R m λ mic , the membrane time constant τ m is given by:

τ m = R m C m ,
where R m is the resistance of the passive membrane and λ is a normalization of the conductivity matrix M j for j = i, e.

After that, we can convert the microscopic tridomain problem into a non-dimensional form by scaling space and time with the constants, such as,

x = L x and t = τ m t.
We take x to be the variable at the macroscale (slow variable), y := x ε to be the microscopic space variable (fast variable) in the unit cell Y . We also scale the electric potentials for k = 1, 2:

u k i = δv u k i
, u e = δv u e , and w k = δw w k where δv, δw are respectively the convenient units to measure the electric potentials and the gating variable. Furthermore, we normalize the conductivities matrices as follows

M j = 1 λ M j , for j = i, e,
and we nondimensionalize the ionic functions I ion , H, the applied current I k app , k = 1, 2, and the gap current I gap by using the following scales:

I ion v k , w k = R m δv I ion v k , w k , H v k , w k = τ m δw H v k , w k , I k app = R m δv I k app , and
I gap ( s) = R m C m δvC 1,2 I gap (s)
,

where v k = u k i -u e for k = 1, 2 and s = u 1 i -u 2 i .
Remark 1. Recalling that the dimensionless parameter ε, given by ε := mic R m λ , is the ratio between the microscopic cell length mic and the macroscopic length L, i.e. ε = mic /L and solving for ε, we obtain

ε = L R m λ .
As pointed in [START_REF] Craig | The bidomain model of cardiac tissue: from microscale to macroscale[END_REF][START_REF] Colli-Franzone | Mathematical and numerical methods for reactiondiffusion models in electrocardiology[END_REF], we can consider for a typical cardiac cell that mic = 100µm, λ = 5mS/cm and R m = 10 000 Ωcm 2 , leading to ε = 7.1 × 10 -3 .

Remark 2. Using all scaling parameters, we obtain the dimensionless of gap boundary condition (7) as follows

ε C 1,2 C m ∂ t s + I gap ( s) = I 1,2 on Γ 1,2 ε,T .
As previously stated, we can consider C 1,2 = C m /2 so we rewrite the above equation as follows

ε 2 ∂ t s + I gap ( s) = I 1,2 on Γ 1,2 ε,T .
Cardiac tissue exhibits a number of significant inhomogeneities in particular those related to cell-to-cell communications. Rescaling the equations ( 3)- [START_REF] Bourgault | Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology[END_REF] in the intracellular and extracellular media and omitting the superscript • of the dimensionless variables, we obtain the following nondimensional form:

-∇ • M ε i ∇u k i,ε = 0 in Ω k i,ε,T := (0, T ) × Ω k i,ε , (8a) -∇ • (M ε e ∇u e,ε ) = 0 in Ω e,ε,T := (0, T ) × Ω e,ε , (8b)
u k i,ε -u e,ε = v k ε on Γ k ε,T := (0, T ) × Γ k ε , (8c) -M ε i ∇u k i,ε • n k i = M ε e ∇u e,ε • n e = I k m on Γ k ε,T , (8d) ε ∂ t v k ε + I ion v k ε , w k ε -I k app,ε = I k m on Γ k ε,T , (8e)
∂ t w k ε -H v k ε , w k ε = 0 on Γ k ε,T , (8f) u 1 i,ε -u 2 i,ε = s ε on Γ 1,2 ε,T := (0, T ) × Γ 1,2 ε , (8g) -M ε i ∇u 1 i,ε • n 1 i = M ε i ∇u 2 i,ε • n 2 i = I 1,2 on Γ 1,2 ε,T , (8h) ε 2 (∂ t s ε + I gap (s ε )) = I 1,2 on Γ 1,2 ε,T , (8i)
with k = 1, 2 and each equation corresponds to the following sense: (8a) Intra quasi-stationary conduction, (8b) Extra quasi-stationary conduction, (8c) Transmembrane potential, (8d) Continuity equation at cell membrane, (8e) Reaction condition at the corresponding cell membrane, (8f) Dynamic coupling, (8g) Gap junction potential, (8h) Continuity equation at gap junction, (8e) Reaction condition at gap junction.

Observe that the tridomain equations (8a)-(8b) are invariant with respect to the above scaling. We define now the rescaled electrical potential as follows:

u k i,ε (t, x) := u k i t, x, x ε , u e,ε (t, x) := u e t, x, x ε , for k = 1, 2.
Analogously, we obtain the rescaled transmembrane potential v k ε , the rescaled gap junction potential s ε and the corresponding gating variable w k ε for k = 1, 2. Furthermore, the conductivity tensors are considered dependent both on the slow and fast variables, i.e. for j = i, e, we have

(9) M ε j (x) := M j x, x ε ,
satisfying the elliptic and periodicity conditions: there exist constants α, β ∈ R, such that 0 < α < β and for all λ ∈ R d :

M j λ • λ ≥ α |λ| 2 , (10a) |M j λ| ≤ β |λ| , (10b) M j y-periodic, for j = i, e. (10c)
Remark 3. Finally, we assume that each M j is symmetric: M T j = M j .

We complete system [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations andRelated Models[END_REF] with no-flux boundary conditions on ∂ ext Ω:

M ε i ∇u k i,ε • n = (M ε e ∇u e,ε ) • n = 0 on (0, T ) × ∂ ext Ω,
where k = 1, 2 and n is the outward unit normal to the exterior boundary of Ω. We impose initial conditions on transmembrane potential v k ε , gap junction potential s ε and gating variable w k ε as follows:

(11)

v k ε (0, x) = v k 0,ε (x), w k ε (0, x) = w k 0,ε (x) a.e. on Γ k ε,T , and s ε (0, x) = s 0,ε (x)
a.e. on Γ 1,2 ε,T , with k = 1, 2. We mention some assumptions on the ionic functions, the source term and the initial data. Assumptions on the ionic functions. The ionic current I ion (v k , w k ) at each cell membrane Γ k can be decomposed into I a,ion v k and I k b,ion w k , where

I ion v k , w k = I a,ion v k + I b,ion w k with k = 1, 2.
Furthermore, the nonlinear function I a,ion : R → R is considered as a C 1 function and the functions I b,ion : R → R and H : R 2 → R are considered as linear functions. Also, we assume that there exists r ∈ (2, +∞) and constants α 1 , α 2 , α 3 , α 4 , α 5 , C > 0 and β 1 > 0, β 2 ≥ 0 such that:

1 α 1 |v| r-1 ≤ |I a,ion (v)| ≤ α 1 |v| r-1 + 1 , |I b,ion (w)| ≤ α 2 (|w| + 1), (12a) |H(v, w)| ≤ α 3 (|v| + |w| + 1), and I b,ion (w) v -α 4 H(v, w)w ≥ α 5 |w| 2 , (12b) Ĩa,ion : v → I a,ion (v) + β 1 v + β 2 is strictly increasing with lim v→0 Ĩa,ion (v)/v = 0, (12c) ∀v, v ∈ R, Ĩa,ion (v) -Ĩa,ion (v ) (v -v ) ≥ 1 C (1 + |v| + |v |) r-2 |v -v | 2 , (12d) with (v, w) := v k , w k for k = 1, 2.
Remark 4. In the mathematical analysis of bidomain equations, several paths have been followed in the literature according to the definition of the ionic currents. We summarize below the phenomenological models:

Other non-physiological models have been introduced as approximations of ion current models. They can be used in large problems because they are typically small and fast to solve, although they are less flexible in their response to variations in cellular properties such as concentrations or cell size. We take in this paper the FitzHugh-Nagumo model [START_REF] Fitzhugh | Impulses and physiological states in theoretical models of nerve membrane[END_REF][START_REF] Nagumo | An active pulse transmission line simulating nerve axon[END_REF] that satisfies assumptions [START_REF] Colli | Degenerate evolution systems modeling the cardiac electric field at micro-and macroscopic level[END_REF] which reads as

H (v, w) = a 1 v -b 1 w, (13a)
I ion (v, w) = [ρv(1 -v) (v -θ)] -ρw := I a,ion (v) + I b,ion (w) (13b)
where a 1 , b 1 , ρ, θ are given parameters with a 1 , b 1 > 0, ρ < 0 and θ ∈ (0, 1). According to this model, the functions I ion and H are continuous and the non-linearity I a,ion is of cubic growth at infinity then the most appropriate value is r = 4. Using Young's inequality, we have

(14) |v| 2 ≤ 2 |v| 3 3 + 1 3 , |v| ≤ |v| 3 3 + 2 3 , |v| ≤ |v| 2 2 + 1 2
and then assumption (12a) holds for r = 4 :

|I a,ion (v)| = |ρv(1 -v) (v -θ)| ≤ 2 3 θ + 1 3 (1 + θ) |ρ| + 1 3 θ + 2 3 (1 + θ) + 1 |ρ| |v| 3 , |I b,ion (w)| = |ρ| |w| , |H (v, w)| = |a 1 v -b 1 w| ≤ a 1 |v| + b 1 |w| .
Now, we compute the function E(u, v) := I b,ion (w) v -α 4 H(v, w)w defined in R 2 . So, the second assumption (12b) holds with α 4 = -ρ a 1 :

(15) E(u, v) = ρ a 1 w 2 .
Moreover, the conditions (12c)-(12d) are automatically satisfied by any cubic polynomial I ion with positive leading coefficient. We end this remark by mentioning other reduced ionic models: the Roger-McCulloch model [START_REF] Jack | A collocation-galerkin finite element model of cardiac action potential propagation[END_REF] and the Aliev-Panfilov model [START_REF] Rubin | A simple two-variable model of cardiac excitation[END_REF], may consider more general that the previous model but still rise some mathematical difficulties. Furthermore, the Mitchell-Schaeffer model [START_REF] Colleen | A two-current model for the dynamics of cardiac membrane[END_REF] has been studied in [START_REF] Boulakia | A coupled system of pdes and odes arising in electrocardiograms modeling[END_REF][START_REF] Kunisch | Well-posedness for the mitchell-schaeffer model of the cardiac membrane[END_REF] and its regularized version have a very specific structure. In particular, no proof of uniqueness of solutions for these models exists in the literature. Now, we represent the gap junction Γ 1,2 ε between intra-neighboring cells by a passive membrane:

(16)

I gap (s) = G gap s,
where G gap = 1 Rgap is the conductance of the gap junctions. A discussion of the modeling of the gap junctions is given in [START_REF] Hogues | A model study of electric field interactions between cardiac myocytes[END_REF]. Assumptions on the source term. There exists a constant C independent of ε such that the source term I k app,ε satisfies the following estimation for k = 1, 2:

(17) ε 1/2 I k app,ε L 2 (Γ k ε,T ) ≤ C.
Assumptions on the initial data. The initial condition v k 0,ε , s 0,ε and w k 0,ε satisfy the following estimation: [START_REF] Arnold | Physiology of the Heart[END_REF] k=1,2

ε 1/r v k 0,ε L r (Γ k ε ) + ε 1/2 s 0,ε L 2 (Γ 1,2 ε ) + k=1,2 ε 1/2 w k 0,ε L 2 (Γ k ε ) ≤ C,
for some constant C independent of ε. Moreover, v k 0,ε , s 0,ε and w k 0,ε are assumed to be traces of uniformly bounded sequences in C 1 (Ω) with k = 1, 2.

Finally, one can observe that Equations in [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations andRelated Models[END_REF] are invariant under the change of u k i,ε , k = 1, 2 and u e,ε into u k i,ε + c, u e,ε + c, for any c ∈ R. Therefore, we may impose the following normalization condition: [START_REF] James | [END_REF] Ωe,ε u e,ε dx = 0, for a.e. t ∈ (0, T ).

Main results

In this part, we highlight our main results obtained in our paper. First, we define the weak solutions of the microscopic tridomain model. Next, we find a priori estimates and we supply our existence and uniqueness results by using Faedo-Galerkin method, compactness argument and monotonicity.

We start by stating the weak formulation of the microscopic tridomain model as given in the following definition.

Definition 5 (Weak formulation of microscopic system). A weak solution to problem (8)-( 11) is a collection (u 1 i,ε , u 2 i,ε , u e,ε , w 1 ε , w 2 ε ) of functions satisfying the following conditions: (A) (Algebraic relation).

v k ε = (u k i,ε -u e,ε )| Γ k ε,T a.e. on Γ k ε,T , for k = 1, 2, s ε = (u 1 i,ε -u 2 i,ε )| Γ 1,2 ε,T
a.e. on Γ 1,2 ε,T .

(B) (Regularity).

u k i,ε ∈ L 2 0, T ; H 1 Ω k i,ε , u ε e ∈ L 2 0, T ; H 1 (Ω e,ε ) , Ωe,ε u e,ε (t, x) dx = 0, for a.e. t ∈ (0, T ), v k ε ∈ L 2 0, T ; H 1/2 Γ k ε ∩ L r Γ k ε,T , r ∈ (2, +∞) s ε ∈ L 2 Γ 1,2 ε,T , w k ε ∈ L 2 (Γ k ε,T ), ∂ t v k ε ∈ L 2 0, T ; H -1/2 Γ k ε + L r/(r-1) Γ k ε,T , ∂ t s ε ∈ L 2 (Γ 1,2 ε,T ), ∂ t w k ε ∈ L 2 (Γ k ε,T ) for k = 1, 2.
(C) (Initial conditions).

v k ε (0, x) = v k 0,ε (x), w k ε (0, x) = w k 0,ε (x)
a.e. on Γ k ε , and s ε (0, x) = s 0,ε (x)

a.e. on Γ 1,2 ε .

(D) (Variational equations).

(

) k=1,2 Γ k ε,T ε∂ t v k ε ψ k dσ x dt + 1 2 Γ 1,2 ε,T ε∂ t s ε Ψ dσ x dt + k=1,2 Ω k i,ε,T M ε i ∇u k i,ε • ∇ϕ k i dxdt + Ω e,ε,T M ε e ∇u e,ε • ∇ϕ e dxdt + k=1,2 Γ k ε,T εI ion v k ε , w k ε ψ k dσ x dt + 1 2 Γ 1,2 ε,T εI gap (s ε )Ψ dσ x dt = k=1,2 Γ k ε,T εI k app,ε ψ k dσ x dt 20
Γ k ε,T ∂ t w k ε e k dσ x dt = Γ k ε,T H v k ε , w k ε e k dσ x dt for all ϕ k i ∈ L 2 0, T ; H 1 Ω k i,ε (21)
, ϕ e ∈ L 2 0, T ; H 1 (Ω e,ε ) with

• ψ k = ψ k i -ψ k e := ϕ k i -ϕ e | Γ k ε,T ∈ L 2 0, T ; H 1/2 Γ k ε ∩ L r Γ k ε,T for k = 1, 2, • Ψ = Ψ 1 i -Ψ 2 i := ϕ 1 i -ϕ 2 i | Γ 1,2 ε,T ∈ L 2 (Γ 1,2 ε,T ), • e k ∈ L 2 (Γ k ε,T ) for k = 1, 2.
Remark 6. Due to Lions-Magenes theorem (see [START_REF] Boyer | Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations andRelated Models[END_REF] p. 101), the following injection

V := u ∈ L 2 0, T ; H 1/2 Γ k ε ∩ L r Γ k ε,T and ∂ t u ∈ L 2 0, T ; H -1/2 Γ k ε + L r/(r-1) Γ k ε,T ⊂ C 0 [0, T ]; L 2 (Γ ε ) , for k = 1, 2 is continuous with r ∈ (2, +∞). Then, v k ε ∈ C 0 [0, T ]; L 2 (Γ k ε ) for k = 1, 2.
Therefore, the initial data of v k ε for k = 1, 2 in Definition 5 is well defined. In the same manner, the initial condition on s ε and on w k ε for k = 1, 2 makes sense. Theorem 7 (Microscopic Tridomain Model). Assume that the conditions (10)-( 18) hold. Then, System (8)-( 11) possesses a unique weak solution in the sense of Definition 5 for every fixed ε > 0.

Furthermore, this solution verifies the following energy estimates: there exists constants C 1 , C 2 , C 3 , C 4 , independent of ε such that:

(22) k=1,2 √ εv k ε 2 L ∞ (0,T ;L 2 (Γ k ε )) + k=1,2 √ εw k ε 2 L ∞ (0,T ;L 2 (Γ k ε )) + √ εs ε 2 L ∞ (0,T ;L 2 (Γ 1,2 ε )) ≤ C 1 (23) k=1,2 u k i,ε L 2 (0,T ;H 1 (Ω k i,ε )) + u ε e L 2 (0,T ;H 1 (Ωe,ε)) ≤ C 2 , (24)
k=1,2

ε 1/r v k ε L r (Γ k ε,T ) ≤ C 3 and k=1,2 ε (r-1)/r I a,ion (v k ε ) L r/(r-1) (Γ k ε,T ) ≤ C 4 . Moreover, if v k ε,0 ∈ H 1/2 (Γ k ε ) ∩ L r (Γ k ε ), k = 1, 2
, then there exists a constant C 5 independent of ε such that:

(25) k=1,2 √ ε∂ t v k e 2 L 2 (Γ k ε,T ) + k=1,2 √ ε∂ t w k ε 2 L 2 (Γ k ε,T ) + √ ε∂ t s ε 2 L 2 (Γ 1,2 ε,T ) ≤ C 5 .
The proof of Theorem 7 is treated in Section 4.

Remark 8. The authors in [START_REF] Bendahmane | Unfolding homogenization method applied to physiological and phenomenological bidomain models in electrocardiology[END_REF][START_REF] Bader | Derivation of a new macroscopic bidomain model including three scales for the electrical activity of cardiac tissue[END_REF][START_REF] Bader | Three scale unfolding homogenization method applied to cardiac bidomain model[END_REF] treated the microscopic bidomain problem where the gap junction is ignored. They considered that there are only intra-and extracellular media separated by the membrane (sarcolemma). Comparing to [START_REF] Bendahmane | Unfolding homogenization method applied to physiological and phenomenological bidomain models in electrocardiology[END_REF], the microscopic tridomain model in our work consists of three elliptic equations coupled through three boundary conditions, two on each cell membrane and one on the gap junction which separates between two intracellular media.

Existence and Uniqueness of solutions for the microscopic tridomain model

This section is devoted to proving existence and uniqueness of solutions to the heterogeneuous microscopic tridomain model presented in Section 2 for fixed ε > 0. The proof of Theorem 7 is based on the Faedo-Galerkin method and carried out in several steps:

• Construction of the basis on the intra-and extracellular domains.

• Construction and local existence of approximate solutions.

• Find some a priori estimates of the approximate solutions.

• Existence and uniqueness of solution to the microscopic tridomain model.

We refer the reader to the well-posedness results for weak solutions of the microscopic bidomain model, established in [START_REF] Bourgault | Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology[END_REF][START_REF] Bendahmane | Unfolding homogenization method applied to physiological and phenomenological bidomain models in electrocardiology[END_REF] by using a Faedo-Galerkin technique. See also [START_REF] Bendahmane | Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue[END_REF][START_REF] Boulakia | A coupled system of pdes and odes arising in electrocardiograms modeling[END_REF] for a similar approach, based on a parabolic regularization technique.

In this proof, we will remove the ε-dependence in the solution

u 1 i,ε , u 2 i,ε , u e,ε , v 1 ε , v 2 ε , s ε , w 1 ε , w 2 ε
for simplification of notation. The demonstration is described as follows:

Step 1: Construction of the basis. We first consider functions φ, φ ∈ C 0 (Ω k i,ε ) and we let V k 0,i denote the completion of C 0 (Ω k i,ε ) under the norm induced by the inner product

•, • V k 0,i which defined by Θ, Θ V k 0,i := Ω k i,ε φ φ dx + Γ k ε φ| Γ k ε φ| Γ k ε dσ + Γ 1,2 ε φ| Γ 1,2 ε φ| Γ 1,2 ε dσ, for k = 1, 2,
where

Θ = t φ φ| Γ k ε φ| Γ 1,2 ε , Θ = t φ φ| Γ k ε φ| Γ 1,2 ε . Similarly, for functions φ, φ ∈ C 1 (Ω k i,ε ) and we let V k 1,i denote the completion of C 1 (Ω k i,ε
) under the norm induced by the inner product

•, • V k 1,i which defined by Θ, Θ V k 1,i := Ω k i,ε M ε i ∇φ • ∇ φ dx + Γ k ε φ| Γ k ε φ| Γ k ε dσ + Γ k ε ∇ Γ k ε φ • ∇ Γ k ε φ dσ + Γ 1,2 ε φ| Γ 1,2 ε φ| Γ 1,2 ε dσ + Γ 1,2 ε ∇ Γ 1,2 ε φ • ∇ Γ 1,2 ε φ dσ, for k = 1, 2
where ∇ Γ denotes the tangential gradient operator on Γ (Γ := Γ k ε , Γ 1,2 ε ). We note that the following injections hold:

V k 0,i ⊂ L 2 (Ω k i,ε ), and V k 1,i ⊂ H 1 (Ω k i,ε
). Moreover, the injection from V k 1,i to V k 0,i is continous and compact for k = 1, 2. We refer the reader to [START_REF] Gal | Well-posedness and long time behavior of the non-isothermal viscous cahn-hilliard equation with dynamic boundary conditions[END_REF][START_REF] Racke | The cahn-hilliard equation with dynamic boundary conditions[END_REF] for similar approaches. It follows from a well-known result (see e.g. [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF] p. 54) that the bilinear form a(Θ, Θ) := Θ, Θ

V k 1,i
defines a strictly positive self adjoint unbounded operator

B k i B k i : D(B k i ) = {Θ ∈ V k 1,i : B k i Θ ∈ V k 1,i } → V k 0,i such that, for any Θ ∈ V k 1,i , we have B k i Θ, Θ V k 0,i = a(Θ, Θ)
. Thus, for n ∈ N, we take a complete system of eigenfunctions

{Θ k i,n = t φ k i,n ψ k i,n Ψ k i,n } n of the problem B k i Θ k i,n = λ n Θ k i,n , in V k 0,i , for k = 1, 2, with • {λ n } n be a sequence such that 0 < λ 1 ≤ λ 2 , . . . , λ n → ∞ as n → ∞. • Θ k i,n ∈ D(B k i ), ψ k i,n := φ k i,n | Γ k ε and Ψ k i,n := φ k i,n | Γ 1,2 ε
where φ k i,n , ψ k i,n and Ψ k i,n are regular enough for k = 1, 2.

Moreover, the eigenvectors {Θ k i,n } n turn out to form an orthogonal basis in V k 1,i and V k 0,i , and they may be assumed to be normalized in the norm of

V k 0,i for k = 1, 2. Since C 1 (Ω k i,ε ) ⊂ V k 1,i ⊂ H 1 (Ω k i,ε ) and C 1 (Ω k i,ε ) is dense in H 1 (Ω k i,ε ) then V k 1,i is dense in H 1 (Ω k i,ε ) for the H 1 -norm. Therefore, {Θ k i,n } n is a basis in H 1 (Ω k i,ε
) for the H 1 -norm. On the other hand, we consider a basis {ζ k n } n , n ∈ N that is orthonormal in L 2 (Γ k ε ) and orthogonal in H 1 (Γ k ε ) and we set the spaces

P k i, = span{Θ k i,1 , . . . , Θ k i, }, P k i,∞ = ∞ =1 P k i, , K k i, = span{ζ k 1 , . . . , ζ k }, K k i,∞ = ∞ =1 K k i, ,
where P k i,∞ and K k i,∞ are respectively dense subspaces of V k 1,i and H 1 (Γ k ε ) for k = 1, 2.

Remark 9. Analogously, we construct a basis on the extracellular domain. We let V p,e denote the completion of C p (Ω e,ε ) under the norm induced by the inner product •, • Vp,e for φ, φ ∈ C p (Ω e,ε ), p = 0, 1 which respectively defined by

Θ , Θ V0,e := Ωe,ε φ φ dx + k=1,2 Γ k ε φ| Γ k ε φ| Γ k ε dσ, and
Θ , Θ V1,e := Ωe,ε M ε e ∇φ • ∇ φ dx + k=1,2 Γ k ε φ| Γ k ε φ| Γ k ε dσ + Γ k ε ∇ Γ k ε φ • ∇ Γ k ε φ dσ , where Θ = t φ φ| Γ 1 ε φ| Γ 2 ε , Θ = t φ φ| Γ 1 ε φ| Γ 2 ε
. Similarly, we take a complete basis which is orthogonal in V 1,e and orthonormal in V 0,e and we set the spaces P e, = span{Θ e,1 , . . . , Θ e, }, P e,∞ =

∞ =1 P e, ,
where P e,∞ is a dense subspace of V 1,e .

Step 2: Construction and local existence of approximate solutions. Supplied with the basis introduced in the first step, we look for the approximate solutions as sequences {u k i,n } n>1 , {u e,n } n>1 and {w k n } n>1 , k = 1, 2 defined for t > 0 and x ∈ Ω by: (26)

U k i =       u k i,n u k i,n u k i,n       := n =1 d k i, (t)       φ k i, ψ k i, Ψ k i,       , U e =       u e,n u 1 e,n u 2 e,n       := n =1 d e, (t)       φ e, ψ 1 e, ψ 2 e,       and w k n := n =1 c k (t)ζ k (x), with φ k i, | Γ k ε = ψ k i, , φ k i, | Γ 1,2 ε = Ψ k i, and φ e, | Γ k ε = ψ k e, for k = 1, 2.
To apply the Faedo-Galerkin scheme, we first regularize the microscopic tridomain system ( 8)-( 11) using specific approximation as follows (recall that our system is degenerate)

(27) (ε + δ n ) Γ 1 ε ∂ t u 1 i,n ψ 1 i dσ x -ε Γ 1 ε ∂ t u 1 e,n ψ 1 i dσ x + δ n Ω 1 i,ε ∂ t u 1 i,n φ 1 i dx + ( ε 2 + δ n ) Γ 1,2 ε ∂ t u 1 i,n Ψ 1 i dσ x - ε 2 Γ 1,2 ε ∂ t u 2 i,n Ψ 1 i dσ x = Γ 1 ε ε -I ion (v 1 n , w 1 n ) + I 1 app,ε ψ 1 i dσ x - 1 2 Γ 1,2 ε εI gap (s n )Ψ 1 i dσ x - Ω 1 i,ε M ε i ∇u 1 i,n • ∇φ 1 i dx (28)
(ε + δ n ) Γ 2 ε ∂ t u 2 i,n ψ 2 i dσ x -ε Γ 2 ε ∂ t u 2 e,n ψ 2 i dσ x + δ n Ω 2 i,ε ∂ t u 2 i,n φ 2 i dx - ε 2 Γ 1,2 ε ∂ t u 1 i,n Ψ 2 i dσ x + ( ε 2 + δ n ) Γ 1,2 ε ∂ t u 2 i,n Ψ 2 i dσ x = Γ 2 ε ε -I ion (v 2 n , w 2 n ) + I 2 app,ε ψ 2 i dσ x + 1 2 Γ 1,2 ε εI gap (s n )Ψ 2 i dσ x - Ω 2 i,ε M ε i ∇u 2 i,n • ∇φ 2 i dx (29)
-ε k=1,2 Γ k ε ∂ t u k i,n ψ k e dσ x + (ε + δ n ) k=1,2 Γ k ε ∂ t u k e,n ψ k e dσ x + δ n Ωe,ε ∂ t u e,n φ e dx = k=1,2 Γ k ε ε I ion (v k n , w k n ) -I k app,ε ψ k e dσ x - Ωe,ε M ε e ∇u e,n • ∇φ e dx (30)
Γ k ε ∂ t w k n ζ k dσ x = Γ k ε H v k n , w k n ζ k dσ x ,
where the regularization parameter

δ n = 1 n , Θ k i = t φ k i ψ k i Ψ k i ∈ P k i,n , ζ k ∈ K k n for k = 1, 2,
and Θ e = t φ e ψ 1 e ψ 2 e ∈ P e,n . The regularization terms multiplied by δ n have been added to overcome degeneracy in [START_REF] Kunisch | Well-posedness for the mitchell-schaeffer model of the cardiac membrane[END_REF]. Moreover, the resulting regularized problem is supplemented with initial conditions: (31)

u k i,n (0, x) = u k 0,i,n (x) := n =1 d k i, (0)φ k i, (x), u k i,n (0, x) = u k 0,i,n (x) := n =1 d k i, (0)ψ k i, (x), u k i,n (0, x) = u k 0,i,n (x) := n =1 d k i, (0)Ψ k i, (x), d k i, (0) := U k 0,i , Θ k i, V k 0,i , u e,n (
w k n (0, x) = w k 0,n := n =1 c k (0)ζ k (x), c k (0) := w k 0 , ζ k L 2 (Γ k ε ) ,
where U k 0,i := U k i (0, x), for k = 1, 2 and U 0,e := U e (0, x). Next, we prove in the following lemma the local existence of solutions for the previous regularized problem:

Lemma 10 (Local existence of solutions for the regularized problems). Assume that the conditions (10)-( 18) hold. Then, there exits a positive time 0 < t 0 ≤ T such that System 26) into ( 27)- [START_REF] Frank B Sachse | A model of electrical conduction in cardiac tissue including fibroblasts[END_REF]. Then, the problem ( 27)-( 30) is equivalent to the system of ordinary differential equations (ODE) in the following compact form: [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF] c 2 ) with the ( , m) entry of matrix:

(ε + δ n )A 1 ii (d 1 i ) -εA 1 ie d e + δ n A 1 ii (d 1 i ) + ( ε 2 + δ n )A 1 ii (d 1 i ) - ε 2 A 1,2 ii (d 2 i ) = F 1 i (t, d 1 i , d 2 i , d e , c 1 , c 2 ) (ε + δ n )A 2 ii (d 2 i ) -εA 2 ie d e + δ n A 2 ii (d 2 i ) - ε 2 A 1,2 ii (d 1 i ) + ( ε 2 + δ n )A 2 ii (d 2 i ) = F 2 i (t, d 1 i , d 2 i , d e , c 1 , c 2 ) k=1,2 -εA k ie (d k i ) + (ε + δ n )A k ee d e + δ n A ee d e = F e (t, d 1 i , d 2 i , d e , c 1 , c 2 ) G k (c k ) = H k (t, d 1 i , d 2 i , d e , c 1 ,
• A k ii is φ k i, , φ k i,m L 2 (Ω k i,ε ) resp. of A ee is φ e, , φ e,m L 2 (Ωe,ε) , • A k ii is ψ k i, , ψ k i,m L 2 (Γ k ε ) resp. of A k ee is ψ k e, , ψ k e,m L 2 (Γ k ε ) , • A k ie is ψ k i, , ψ k e,m L 2 (Γ k ε ) , • A k ii is Ψ k i, , Ψ k i,m L 2 (Γ k ε ) resp. of A 1,2 ii is Ψ 1 i, , Ψ 2 i,m L 2 (Γ 1,2 ε ) , • G k is ζ k , ζ k m L 2 (Γ k ε )
, for 1 ≤ , m ≤ n and k = 1, 2. Herein, the vectors F k i , F e and H k for k = 1, 2 correspond to the right hand sides of the equations given in ( 27)- [START_REF] Frank B Sachse | A model of electrical conduction in cardiac tissue including fibroblasts[END_REF]. Furthermore, the first three equations in ODE system (32) can be written as follows:

(33) M       (d 1 i ) (d 2 i ) d e (c 1 ) (c 2 )       =       F 1 i F 2 i F e H 1 H 2      
, with M := M 1 + εM 2 and each matrix defined by: ( 34)

M 1 =            δ n A 1 ii + A 1 ii + A 1 ii 0 0 0 0 0 δ n A 2 ii + A 2 ii + A 2 ii 0 0 0 0 0 δ n A 1 ee + A 2 ee + A ee 0 0 0 0 0 G 1 0 0 0 0 0 G 2            and (35) M 2 =         A 1 ii + 1 2 A 1 ii -1 2 A 1,2 ii -A 1 ie 0 0 -1 2 t A 1,2 ii A 2 ii + 1 2 A 2 ii -A 2 ie 0 0 -t A 1 ie -t A 2 ie A 1 ee + A 2 ee 0 0 0 0 0 0 0 0 0 0 0 0        
.

In order to write

      (d 1 i ) (d 2 i ) d e (c 1 ) (c 2 )       = M -1       F 1 i F 2 i F e H 1 H 2      
, one needs to prove that the matrix M is invertible. According to Lemma 11, given below, the matrix M is symmetric positive definite, hence invertible. Consequently, we can write the ODE system [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF] in the form z (t) = F (t, z(t)). Finally, we prove the existence of a local solution [0, t 0 ) to this ODE system with t 0 ∈ (0, T ) (independent of the initial data). To this end, we show that th entries of F k i , F e and H k for k = 1, 2 are Caratheodory functions bounded by L 1 functions using the assumptions ( 10)-( 18) by following the same strategy in [START_REF] Bendahmane | Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue[END_REF].

Lemma 11. For all n ∈ N * , the matrix M is positive definite.

Proof. Since we have M = M 1 + εM 2 with M 1 and M 2 defined respectively by ( 34)- [START_REF] Veneroni | Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field[END_REF]. Note that by the orthonormality of the basis, the matrices

A k ii , A k ii , A k ii , A k ee ,
A ee and G k are equal to the identity matrix I n×n for k = 1, 2. So, the matrix

M 1 =       3δ n I n×n 0 0 0 0 0 3δ n I n×n 0 0 0 0 3δ n I n×n 0 0 0 0 0 I n×n 0 0 0 0 0 I n×n       It suffices to show that the matrix M 2 is positive semi-definite. Let d = t d 1 i d 2 i d e c 1 c 2 where d k i = t d k i,1 , . . . , d k i,n ∈ R n , d e = t (d e,1 , . . . , d e,n ) ∈ R n and c k = t c k 1 , . . . , c k n ∈ R n for k = 1, 2,
we prove that t dM 2 d ≥ 0. Indeed, we have:

t dM 2 d = t d 1 i A 1 ii + 1 2 A 1 ii d 1 i + t d 2 i A 2 ii + 1 2 A 2 ii d 2 i + t d e A 1 ee + A 2 ee d e -t d 1 i A 1,2 ii d 2 i -2 t d 1 i A 1 ie d e -2 t d 2 i A 2 ie d e = t d 1 i A 1 ii d 1 i -2 t d 1 i A 1 ie d e + t d e A 1 ee d e + t d 2 i A 2 ii d 2 i -2 t d 2 i A 2 ie d e + t d e A 2 ee d e + 1 2 t d 1 i A 1 ii d 1 i -t d 1 i A 1,2 ii d 2 i + 1 2 t d 2 i A 2 ii d 2 i := E 1 + E 2 + E 3
We complete by showing that E 1 ≥ 0 and the proof of the other terms E 2 , E 3 is similar. Due the form of matrices and the orthonormality of basis, we obtain:

E 1 = t d 1 i A 1 ii d 1 i -2 t d 1 i A 1 ie d e + t d e A 1 ee d e = n , m=1
d 1 i, d 1 i,m Γ 1 ε ψ 1 i, ψ 1 i,m -2d 1 i, d e,m Γ 1 ε ψ 1 i, ψ 1 e,m + d e, d e,m Γ 1 ε ψ 1 e, ψ 1 e,m dσ x = Γ 1 ε n d 1 i, ψ 1 i, -d e, ψ 1 e, 2 dσ x ≥ 0.
Remark 12. The above proof of the matrix M points out the role of the regularization term M 1 . It allows to obtain a matrix M in (33) which is nonsingular, so that the resulting system of ODE is non-degenerate.

To prove global existence of the Faedo-Galerkin solutions on [0, T ), we derive a priori estimates, independent of the regularization parameter n, bounding u k i,n , u e,n , v k n , w k n for k = 1, 2 and s n in the next step.

Step 3: Energy estimates. The Faedo-Galerkin solutions satisfy the following weak formulations:

(36) (ε + δ n ) Γ 1 ε ∂ t u 1 i,n ψ 1 i,n dσ x -ε Γ 1 ε ∂ t u e,n ψ 1 i,n dσ x + δ n Ω 1 i,ε ∂ t u 1 i,n ϕ 1 i,n dx + ( ε 2 + δ n ) Γ 1,2 ε ∂ t u 1 i,n Ψ 1 i,n dσ x - ε 2 Γ 1,2 ε ∂ t u 2 i,n Ψ 1 i,n dσ x = Γ 1 ε ε -I ion (v 1 n , w 1 n ) + I 1 app,ε ψ 1 i,n dσ x - 1 2 Γ 1,2 ε εI gap (s n )Ψ 1 i,n dσ x - Ω 1 i,ε M ε i ∇u 1 i,n • ∇ϕ 1 i,n dx (37) (ε + δ n ) Γ 2 ε ∂ t u 2 i,n ψ 2 i,n dσ x -ε Γ 2 ε ∂ t u e,n ψ 2 i,n dσ x + δ n Ω 2 i,ε ∂ t u 2 i,n ϕ 2 i,n dx - ε 2 Γ 1,2 ε ∂ t u 1 i,n Ψ 2 i,n dσ x + ( ε 2 + δ n ) Γ 1,2 ε ∂ t u 2 i,n Ψ 2 i,n dσ x = Γ 2 ε ε -I ion (v 2 n , w 2 n ) + I 2 app,ε ψ 2 i,n dσ x + 1 2 Γ 1,2 ε εI gap (s n )Ψ 2 i,n dσ x - Ω 2 i,ε M ε i ∇u 2 i,n • ∇ϕ 2 i,n dx (38)
-ε k=1,2 Γ k ε ∂ t u k i,n ψ k e,n dσ x + (ε + δ n ) k=1,2 Γ k ε ∂ t u k e,n ψ k e,n dσ x + δ n Ωe,ε ∂ t u e,n ϕ e,n dx = k=1,2 Γ k ε ε I ion (v k n , w k n ) -I k app,ε ψ k e,n dσ x - Ωe,ε M ε e ∇u e,n • ∇ϕ e,n dx (39)
Γ k ε ∂ t w k n e k n dσ x = Γ k ε H v k n , w k n e k n dσ x ,
where

ϕ k i,n (t, x) := n =1 a k i, (t)φ k i, (x), ϕ e,n (t, x) := n =1 a e, (t)φ e, (x), e k n (t, x) :=
n =1 b (t)ξ k (x),
for some given (absolutely continuous) coefficients a k i, (t), a e, (t), b k (t) with = 1, . . . , n and k = 1, 2. Moreover, we recall that ψ k i,n (resp. ψ k e,n ) is the trace of ϕ k i,n (resp. of ϕ e,n ) on Γ k ε and Ψ k i,n is the trace of ϕ k i,n on Γ 1,2 ε for k = 1, 2. We find now the a priori estimates of the solution of approximate problem (36)-(39). First, we sum the three equations (36)-(38) to obtain the following weak formulation:

(40) k=1,2 Γ k ε ε∂ t v k n ψ k n dσ x + k=1,2 Γ k ε δ n ∂ t u k i,n ψ k i,n dσ x + k=1,2 Γ k ε δ n ∂ t u k e,n ψ k e,n dσ x + 1 2 Γ 1,2 ε ε∂ t s n Ψ n dσ x + k=1,2 Γ 1,2 ε δ n ∂ t u k i,n Ψ k i,n dσ x + k=1,2 Ω k i,ε δ n ∂ t u k i,n ϕ k i,n dx + Ωe,ε δ n ∂ t u e,n ϕ e,n dx + k=1,2 Ω k i,ε M ε i ∇u k i,n • ∇ϕ k i,n dx + Ωe,ε M ε e ∇u e,n • ∇ϕ e,n dx + k=1,2 Γ k ε εI ion v k n , w k n ψ k n dσ x + 1 2 Γ 1,2 ε εI gap (s n ) Ψ n dσ x = k=1,2 Γ k ε εI k app,ε ψ k n dσ x , (41)
Γ k ε ∂ t w k n e k n dσ x = Γ k ε H v k n , w k n e k n dσ x ,
where

ψ k n = ψ k i,n -ψ k e,n for k = 1, 2 and Ψ n = Ψ 1 i,n -Ψ 2 i,n .
Next, we substitute ϕ k i,n = u k i,n , ϕ e,n = u e,n and e k n = εα 4 w k n , respectively, in (40)-(41) to get the following equality:

(42) 1 2 d dt k=1,2 Γ k ε √ εv k n 2 dσ x + k=1,2 Γ k ε δ n u k i,n 2 dσ x + k=1,2 Γ k ε δ n u k e,n 2
dσ x + 1 2 Γ 1,2 ε √ εs n 2 dσ x + k=1,2 Γ 1,2 ε δ n u k i,n 2 dσ x + k=1,2 Ω k i,ε δ n u k i,n 2 dx + Ωe,ε δ n u e,n 2 dx + k=1,2 Ω k i,ε M ε i ∇u k i,n • ∇u k i,n dx + Ωe,ε M ε e ∇u e,n • ∇u e,n dx + k=1,2 Γ k ε εI ion v k n , w k n v k n dσ x + 1 2 Γ 1,2 ε εI gap (s n ) s n dσ x = k=1,2 Γ k ε εI k app,ε v k n dσ x , α 4 2 d dt Γ k ε √ εw k n 2 dσ x = Γ k ε εα 4 H v k n , w k n w k n dσ x , for k = 1, 2. (43)
Integrating ( 42)-( 43) over (0, t) for t ∈ (0, t 0 ] in each equation and then summing the resulting equations, we procure the following equality using the assumption [START_REF] Colli | Degenerate evolution systems modeling the cardiac electric field at micro-and macroscopic level[END_REF] on

I ion : (44) 1 2 k=1,2 √ εv k n 2 L 2 (Γ k ε ) + α 4 k=1,2 √ εw k n 2 L 2 (Γ k ε ) + 1 2 √ εs n 2 L 2 (Γ 1,2 ε ) + k=1,2 δ n u k i,n 2
L 2 (Γ k ε ) + k=1,2 δ n u k e,n 2
L 2 (Γ k ε ) + k=1,2 δ n u k i,n 2
L 2 (Γ 1,2 ε ) + k=1,2 δ n u k i,n 2
L 2 (Ω k i,ε ) + δ n u e,n 2
L 2 (Ωe,ε) + k=1,2 t 0 Ω k i,ε M ε i ∇u k i,n • ∇u k i,n dxdτ + t 0 Ωe,ε M ε e ∇u e,n • ∇u e,n dxdτ + k=1,2 t 0 Γ k ε ε Ĩa,ion v k n v k n dσ x dτ = 1 2 k=1,2 √ εv k 0,n 2 L 2 (Γ k ε ) + α 4 k=1,2 √ εw k 0,n 2 L 2 (Γ k ε ) + 1 2 √ εs 0,n 2 L 2 (Γ 1,2 ε ) + k=1,2 δ n u k 0,i,n 2 L 2 (Γ k ε ) + k=1,2 δ n u k 0,e,n 2
L 2 (Γ k ε ) + k=1,2 δ n u k 0,i,n 2 L 2 (Γ 1,2 ε ) + k=1,2 δ n u k 0,i,n 2 L 2 (Ω k i,ε ) + δ n u 0,e,n 2
L 2 (Ωe,ε) - 1 2 t 0 Γ 1,2 ε εI gap (s n ) s n dσ x dτ + k=1,2 t 0 Γ k ε ε -I b,ion w k n v k n + α 4 H v k n , w k n w k n dσ x dτ + k=1,2 t 0 Γ k ε ε β 1 v k n + β 2 v k n dσ x dτ + k=1,2 t 0 Γ k ε εI k app,ε v k n dσ x dτ.
We denote by E with = 1, . . . , 9 the terms of the previous equation which is rewritten as follows (to respect the order):

E 1 + E 2 + E 3 + E 4 = E 5 + E 6 + E 7 + E 8 + E 9 .
Now, we estimate E for = 2, . . . , 9 as follows:

• Due the uniform ellipticity (10) of M ε j for j = i, e, we have

E 2 + E 3 ≥ α   k=1,2 t 0 ∇u k i,n 2
L 2 (Ω k i,ε ) dτ + t 0 ∇u e,n 2
L 2 (Ωe,ε) dτ   ≥ 0.
• Using the assumption (12d) on Ĩa,ion , we deduce that E 4 ≥ 0.

• By the assumptions [START_REF] Arnold | Physiology of the Heart[END_REF] on the initial data, we have E 5 ≤ C for some constant independent of n and ε. • By the structure form of I gap defined in [START_REF] Hogues | A model study of electric field interactions between cardiac myocytes[END_REF], we obtain

E 6 ≤ G gap t 0 √ εs n 2 L 2 (Γ 1,2 ε ) dτ.
• Using the assumption on I b,ion and H defined as (12b), then we obtain

E 7 ≤ α 5 k=1,2 t 0 √ εw k n 2 L 2 (Γ k ε ) dτ.
• It easy to estimate E 8 as follows

E 8 ≤ C k=1,2 t 0 √ εv k n 2 L 2 (Γ k ε ) dτ,
with C is constant independent of n and ε. • By Young's inequality with the uniform L 2 boundedness (17) of I k app,ε , there exist constants C 1 , C 2 > 0 independent of n and ε such that

E 9 ≤ C 1 + C 2 k=1,2 t 0 √ εv k n 2 L 2 (Γ k ε ) dτ.
Collecting all the estimates stated above, one obtains from (44) the following inequality for all t ≤ t 0 , (45)

k=1,2 √ εv k n 2 L 2 (Γ k ε ) + k=1,2 √ εw k n 2 L 2 (Γ k ε ) + √ εs n 2 L 2 (Γ 1,2 ε ) ≤ C   1 + k=1,2 t0 0 √ εv k n 2 L 2 (Γ k ε ) dτ + k=1,2 t0 0 √ εw k n 2 L 2 (Γ k ε ) dτ + t0 0 √ εs n 2 L 2 (Γ 1,2 ε ) dτ   .
By an application of Gronwall's lemma in the last inequality, one gets

k=1,2 √ εv k n 2 L 2 (Γ k ε ) + k=1,2 √ εw k n 2 L 2 (Γ k ε ) + √ εs n 2 L 2 (Γ 1,2 ε ) ≤ C.
Hence, we conclude that

k=1,2 √ εv k n 2 L ∞ (0,T ;L 2 (Γ k ε )) + k=1,2 √ εw k n 2 L ∞ (0,T ;L 2 (Γ k ε )) + √ εs n 2 L ∞ (0,T ;L 2 (Γ 1,2 ε )) ≤ C.
Then, we can deduce from this inequality that our approximate weak solution of the microscopic tridomain problem is global on (0, T ). Moreover, one can obtain by exploiting this last inequality along with (44) the following a priori estimates for some constant C > 0 not depending on n and ε:

(46) k=1,2 √ εv k n 2 L ∞ (0,T ;L 2 (Γ k ε )) + k=1,2 √ εw k n 2 L ∞ (0,T ;L 2 (Γ k ε )) + √ εs n 2 L ∞ (0,T ;L 2 (Γ 1,2 ε )) + k=1,2 δ n u k i,n 2
L ∞ (0,T ;L 2 (Γ 1,2 ε )) + k=1,2 δ n u k e,n 2
L ∞ (0,T ;L 2 (Γ k ε )) + k=1,2 δ n u k i,n 2
L ∞ (0,T ;L 2 (Γ k ε )) ≤ C, (47)
k=1,2 ∇u k i,n 2
L 2 (Ω k i,ε,T ) + ∇u e,n 2
L 2 (Ω e,ε,T ) ≤ C, ( 48
) k=1,2 ε Ĩa,ion v k n v k n L 1 (Γ k ε,T ) ≤ C. ( 49
) k=1,2 √ εv k n 2 L 2 (Γ k ε,T ) + k=1,2 √ εw k n 2 L 2 (Γ k ε,T ) + √ εs n 2 L 2 (Γ 1,2 ε,T ) ≤ C,
for some constant C > 0 not depending on n and ε. Furthermore, we deduce from (48) together with assumption (12d) on Ĩa,ion the following estimation:

(50)

k=1,2 ε 1/r v k n r L r (Γ k ε,T ) ≤ C,
for some constant C > 0 not depending on n and ε. The second estimate [START_REF] Nagumo | An active pulse transmission line simulating nerve axon[END_REF] in Theorem 7 is a direct consequence of (50) and assumption (12a) on I a,ion .

It remains to estimate on the L 2 norms of the intracellular and extracellular potentials which are need to complete the proof of Estimate ( 23) on H 1 . To do this end, we will use the next lemma, which is a consequence of the uniform Poincaré-Wirtinger's inequality and the trace theorem for ε-periodic surfaces.

Lemma 13. Let u k i ∈ H 1 Ω k i,ε for k = 1, 2 and u e ∈ H 1 (Ω e,ε ) . Set v k := u k i -u e | Γ k ε for k = 1, 2.
Assume that the condition (19) holds, then there exists a positive constants C, independent of ε, such that

(51) u k i 2 L 2 (Ω k i,ε ) ≤ C √ εv k 2 L 2 (Γ k ε ) + ∇u k i 2 L 2 (Ω k i,ε ) + ∇u e 2 L 2 (Ωe,ε) , with k = 1, 2.
Proof. We follow the same idea to the proof of Lemma 3.7 in [START_REF] Grandelius | The cardiac bidomain model and homogenization[END_REF]. Due the normalization condition [START_REF] James | [END_REF], Poincaré-Wirtinger's inequality implies that

(52) u e,n 2
L 2 (Ωe,ε) ≤ C ∇u e,n 2
L 2 (Ωe,ε) , for some constant C independent on n and ε. Note that in the sequel C is a generic constant whose value can change from one line to another. To estimate on the L 2 norms of u k i,n for k = 1, 2, we write

u k i,n = u k i,n + u k i,n ,
where

u k i,n := 1 Ω k i,ε Ω k i,ε u k i,n dx is constant in Ω k i,ε and u k i,n := u k i,n -u k i,n has zero mean in Ω k i,ε .
Clearly, we see that for k = 1, 2

u k i,n 2
L 2 (Ω k i,ε ) = u k i,n 2
L 2 (Ω k i,ε ) + u k i,n 2
L 2 (Ω k i,ε )
.

In view of Poincaré-Wirtinger's inequality, one has

(53) u k i,n 2
L 2 (Ω k i,ε ) ≤ C ∇ u k i,n 2
L 2 (Ω k i,ε ) = C ∇u k i,n 2
L 2 (Ω k i,ε ) for k = 1, 2. Let us bound now u k i,n 2
L 2 (Ω k i,ε ) = Ω k i,ε |Γ k ε | u k i,n 2 L 2 (Γ k ε ) for k = 1, 2. Since Γ k ε = ε -1 Γ k and Ω k i,ε ≤ |Ω| , we deduce that u k i,n 2
L 2 (Ω k i,ε ) ≤ Cε u k i,n 2
L 2 (Γ k ε ) , for k = 1, 2. It easy to check that u k i,n 2 ≤ C u k i,n -u e,n 2 + u k i,n 2 + |u e,n | 2 .
Finally, we obtain for k = 1, 2

u k i,n 2
L 2 (Ω k i,ε ) ≤ C ε v k n 2 L 2 (Γ k ε ) + ε u k i,n 2
L 2 (Γ k ε ) + ε u e,n 2 L 2 (Γ k ε ) ≤ Cε v k n 2 L 2 (Γ k ε ) + C u k i,n 2
L 2 (Ω k i,ε ) + ε 2 ∇ u k i,n 2
L 2 (Ω k i,ε ) + C u e,n 2
L 2 (Ωe,ε) + ε 2 ∇u e,n 2 L 2 (Ωe,ε) ≤ C √ εv k n 2 L 2 (Γ k ε ) + ∇u k i,n 2
L 2 (Ω k i,ε ) + ∇u e,n 2
L 2 (Ωe,ε) ,
where the second inequality is a direct consequence of the trace theorem and the final one is a result of ( 52) and ( 53). This completes the proof of this lemma. Now, Estimate (47) and (51) imply that (54) u e,n L 2 (0,T ;H 1 (Ωe,ε)) ≤ C, for some constant C independent on n and ε. Furthermore, we have

√ εv k n 2 L 2 (Γ k ε,T ) ≤ C for k = 1, 2.
Then Estimates (51), (47) and (54) ensure that for k = 1, 2, (55)

u k i,n L 2 (0,T ;H 1 (Ω k i,ε )) ≤ C.
This completes the proof of ( 22)-( 24) in Theorem 7. Now we turn to find some uniform estimates on the time derivatives by following [START_REF] Bendahmane | Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue[END_REF] which will be useful for the passage to the limit. We notice first for k = 1, 2 that,

Ω k i,ε,T M ε i ∇u k i,n • ∇ ∂ t u k i,n dx = 1 2 T 0 ∂ t Ω k i,ε M ε i ∇u k i,n • ∇u k i,n dx dt = 1 2 Ω k i,ε M ε i ∇u k i,n (T, •) • ∇u k i,n (T, •) dx - Ω k i,ε M ε i ∇u k i,n (0, •) • ∇u k i,n (0, •) dx , and
Γ k ε,T I a,ion v k n ∂ t v k n dσ x dt = T 0 ∂ t Γ k ε v k n 0 I a,ion ṽk n dṽ k n dσ x dt = Γ k ε v k n (T,•) 0 I a,ion v k n dv k n dσ x - Γ k ε v k n (0,•) 0 I a,ion v k n dv k n dσ x .
Next, we substitute

ϕ k i,n = ∂ t u k i,
√ ε∂ t v k n 2 L 2 (Γ k ε,T ) + α 4 k=1,2 √ ε∂ t w k n 2 L 2 (Γ k ε,T ) + 1 2 √ ε∂ t s n 2 L 2 (Γ 1,2 ε,T ) + k=1,2 δ n ∂ t u k i,n 2
L 2 (Γ k ε,T ) + k=1,2 δ n ∂ t u k e,n 2
L 2 (Γ k ε,T ) + k=1,2 δ n ∂ t u k i,n 2
L 2 (Γ 1,2 ε,T ) + k=1,2 δ n ∂ t u k i,n 2
L 2 (Ω k i,ε,T ) + δ n ∂ t u e,n 2
L 2 (Ω e,ε,T ) + 1 2 k=1,2 Ω k i,ε M ε i ∇u k i,n • ∇u k i,n (T, •) dx + Ωe,ε M ε e ∇u e,n (T, •) • ∇u e,n (T, •) dx + k=1,2 Γ k ε v k n (T,•) 0 ε Ĩa,ion v k n dv k n dσ x = 1 2 k=1,2 Ω k i,ε M ε i ∇u k i,n (0, •) • ∇u k i,n (0, •) dx + Ωe,ε M ε e ∇u e,n (0, •) • ∇u e,n (0, •) dx + k=1,2 Γ k ε v k n (0,•) 0 εI a,ion v k n dv k n dσ x + k=1,2 Γ k ε v k n (T,•) 0 ε β 1 v k n + β 2 dv k n dσ x - 1 2 Γ 1,2 ε,T εI gap (s n ) ∂ t s n dσ x dτ + k=1,2 Γ k ε,T ε -I b,ion w k n ∂ t v k n + α 4 H v k n , w k n ∂ t w k n dσ x dτ + k=1,2 Γ k ε,T εI k app,ε ∂ t v k n dσ x dτ.
We denote by E with = 1, . . . , 6 the terms of the previous equation which is rewritten as follows (to respect the order):

E 1 + E 2 = E 3 + E 4 + E 5 + E 6 ,
where

E 1 := k=1,2 √ ε∂ t v k n 2 L 2 (Γ k ε,T ) + α 4 k=1,2 √ ε∂ t w k n 2 L 2 (Γ k ε,T ) + 1 2 √ ε∂ t s n 2 L 2 (Γ 1,2 ε,T ) + k=1,2 δ n ∂ t u k i,n 2
L 2 (Γ k ε,T ) + k=1,2 δ n ∂ t u k e,n 2
L 2 (Γ k ε,T ) + k=1,2 δ n ∂ t u k i,n 2
L 2 (Γ 1,2 ε,T ) + k=1,2 δ n ∂ t u k i,n 2
L 2 (Ω k i,ε,T ) + δ n ∂ t u e,n 2
L 2 (Ω e,ε,T )
. Now, we estimate E for = 2, . . . , 6 as follows:

• Due the uniform ellipticity (10) of M ε j for j = i, e, with the monotonicity (12d) on Ĩa,ion , then we have

E 2 ≥ α   k=1,2 ∇u k i,n (T, •) 2 L 2 (Ω k i,ε ) + ∇u e,n (T, •) 2 L 2 (Ωe,ε)   + k=1,2 Γ k ε v k n (T,•) 0 ε Ĩa,ion v k n (T, •) dσ x dv k n ≥ 0.
• Furthermore, using the a priori estimate (46) with the assumption on I a,ion and on the initial data, one gets

E 3 ≤ β   k=1,2 ∇u k i,n (0, •) 2 L 2 (Ω k i,ε ) + ∇u e,n (0, •) 2 L 2 (Ωe,ε)   + α 1 k=1,2 Γ k ε ε v k n (0, •) r + v k n (0, •) dσ x + β 1 2 k=1,2 Γ k ε ε v k n (T, •) 2 dσ x + β 2 k=1,2 Γ k ε ε v k n (T, •) dσ x ≤ C 3
for some constant C 3 independent of n and ε. • By the structure form of I gap defined in ( 16), we obtain using Young's inequality with estimate (49)

E 4 ≤ G gap 2 √ εs n 2 L 2 (Γ 1,2 ε,T ) + 1 4 √ ε∂ t s n 2 L 2 (Γ 1,2 ε,T ) . ≤ C 4 + 1 4 √ ε∂ t s n 2 L 2 (Γ 1,2 ε,T )
with C 4 independent of n and ε. • Similarly, using the assumption on I b,ion and H defined as (12a)-(12b), then we obtain using Young's inequality with the estimate (49)

E 5 ≤ C 5 + 1 2 k=1,2 √ ε∂ t v k n 2 L 2 (Γ k ε,T ) + √ ε∂ t w k n 2 L 2 (Γ k ε )
with C 5 independent of n and ε. • By Young's inequality with the uniform L 2 boundedness (17) of I k app,ε , there exist constants C 1 , C 2 > 0 independent of n and ε such that

E 6 ≤ C 6 + 1 2 k=1,2 √ ε∂ t v k n 2 L 2 (Γ k ε,T ) ,
with C 6 independent of n and ε. Exploiting all this estimates along with (56), one obtains

(57) k=1,2 √ ε∂ t v k n 2 L 2 (Γ k ε,T ) + α 4 k=1,2 √ ε∂ t w k n 2 L 2 (Γ k ε,T ) + √ ε∂ t s n 2 L 2 (Γ 1,2 ε,T ) + k=1,2 δ n ∂ t u k i,n 2
L 2 (Γ k ε,T ) + k=1,2 δ n ∂ t u k e,n 2
L 2 (Γ k ε,T ) + k=1,2 δ n ∂ t u k i,n 2
L 2 (Γ 1,2 ε,T ) + k=1,2 δ n ∂ t u k i,n 2
L 2 (Ω k i,ε,T ) + δ n ∂ t u e,n 2
L 2 (Ω e,ε,T ) ≤ C
for some constant C > 0 not depending on n and ε.

The next steps is devoted to completing the proof of Theorem 7 and to passing to the limit when n goes to infinity. Further, it treat the uniqueness of the weak solutions to System (8)- [START_REF] Fitzhugh | Impulses and physiological states in theoretical models of nerve membrane[END_REF] Step 4: Passage to the limit and global existence of solutions. In view of (54)-(55), we can see that v k n , u k j,n are bounded in L 2 0, T ; H 1/2 (Γ k ε ) for j = i, e and k = 1, 2 using the standard trace lemma. Similarly, it easy to check that s n and u k i,n are bounded in L 2 0, T ; H 1/2 (Γ 1,2 ε ) for k = 1, 2. Furthermore, we deduce from (57) the uniform bound on

∂ t v k n in L 2 (Γ k ε,T ) for k = 1, 2 and the uniform bound on ∂ t s n in L 2 (Γ 1,2
ε,T ). Recall that by the Aubin-Lions compactness criterion, the following injection

W := u ∈ L 2 0, T ; H 1/2 (Γ ε ) and ∂ t u ∈ L 2 0, T ; H -1/2 (Γ ε ) ⊂ L 2 (Γ ε,T ) is compact with Γ ε := Γ k ε , Γ 1,2 ε for k = 1, 2.
Hence, we can assume there exist limit functions

u 1 i,ε , u 2 i,ε , u e,ε , v 1 ε , v 2 ε , s ε , w ε with v k ε = u k i,ε -u k e,ε on Γ k ε,T for k = 1, 2 and s ε = u 1 i,ε -u 2 i,ε on Γ 1,2 ε,T
, such that as n → ∞ ( for fixed ε and up to an unlabeled subsequence) (58)

                             v k n → v k ε a.e. in Γ k ε , strongly in L 2 (Γ k ε,T ), and weakly L 2 0, T ; H 1/2 (Γ k ε ) for k = 1, 2, s n → s ε a.e. in Γ 1,2 ε , strongly in L 2 (Γ 1,2 ε,T ), and weakly L 2 0, T ; H 1/2 (Γ 1,2 ε ) , w k n w k ε weakly in L 2 (Γ k ε,T ), u k i,n u k i,ε weakly in L 2 0, T ; H 1 (Ω k i,ε ) for k = 1, 2, u e,n
u e,ε weakly in L 2 0, T ; H 1 (Ω e,ε ) , I a,ion v k n → I a,ion v k ε a.e. in Γ k ε , weakly in L r/(r-1) (Γ k ε,T ), and (59)

     ∂ t v k n ∂ t v k ε weakly in L 2 (Γ k ε,T ), ∂ t w k n ∂ t w k ε weakly in L 2 (Γ k ε,T ) for k = 1, 2, ∂ t s n ∂ t s ε weakly in L 2 (Γ 1,2 ε,T
). Moreover, using again estimate (57), we get for j = i, e and k = 1, 2, (60)

               √ δ n ∂ t u k j,ε 0 in D 0, T ; L 2 (Γ k ε ) for j = i, e, √ δ n ∂ t u k i,ε 0 in D 0, T ; L 2 (Γ 1,2 ε ) , √ δ n ∂ t u k i,ε 0 in D 0, T ; L 2 (Ω k i,ε ) , and √ δ n ∂ t u e,ε 0 in D 0, T ; L 2 (Ω e,ε ) .
The last difficulty is to prove that the nonlinear term I a,ion v k n converges weakly to the term I a,ion v k ε for k = 1, 2. Since v k n converges strongly to v k ε in L 2 (Γ k ε,T ), we can extract a subsequence, such that v k n converges almost everywhere to v k ε in Γ k ε for k = 1, 2. Moreover, since I a,ion is continuous, we have (61) I a,ion v k n → I a,ion v k ε a.e. in Γ k ε for k = 1, 2. However, using a classical result (see Lemma 1.3 in [START_REF] Louis | Quelques méthodes de résolution des problemes aux limites non linéaires[END_REF]):

(62) I a,ion v k n I a,ion v k ε , weakly in L r/(r-1) (Γ k ε,T ) for k = 1, 2.

Remark 14. By our choice of basis, it is clear that u k j,n (0, x) → u k 0,j,ε in L 2 (Γ k ε ) for k = 1, 2 and j = i, e. Furthermore, we have u

εI ion v k ε , w k ε ψ k dσ x + 1 2 Γ 1,2 ε εI gap (s ε ) Ψ dσ x = k=1,2 Γ k ε εI k app,ε ψ k dσ x , (64)
Γ k ε ∂ t w k ε e k dσ x = Γ k ε H v k ε , w k ε e k dσ x ,
for all ϕ k i ∈ H 1 (Ω k i,ε ), ϕ e ∈ H 1 (Ω e,ε ) with

ψ k = ψ k i -ψ k e ∈ H 1/2 (Γ k ε ) ∩ L r (Γ k ε ) for k = 1, 2, Ψ = Ψ 1 i -Ψ 2 i ∈ L 2 (Γ 1,2 ε
) and e k ∈ L 2 (Γ k ε ) for k = 1, 2. Finally, it only remains to be proved that v k ε , w k ε for k = 1, 2 and s ε satisfy the initial conditions stated in Definition 5. Using the weak formulation (36)-(38), we see that v k ε (0, x) = v k 0,ε (x) a.e. on Γ k ε,T , since, by construction, u k j,n (0, x) → u k 0,j,ε in L 2 (Γ k ε ) for k = 1, 2 and j = i, e. The same argument holds for w k ε for k = 1, 2 and s ε .

Step 5: Uniqueness of solutions. This step prove that there there exists at most one weak solution of (63)-(64). We assume that u = u 1, i,ε , u 2, i,ε , u e,ε , w 1, ε , w 2, ε , ∈ { , } are two weak solutions in the sense of Definition 5 with same initial data. Thus, this weak formulations hold respectively for u k, i,ε -u k, i,ε and w k, ε -w k, ε for k = 1, 2.

Firstly, we substitute ϕ k i = u k, i,ε -u k, i,ε , ϕ e = u e,ε -u e,ε , and

e k = ε w k, ε -w k, ε , k = 1, 2,
respectively in (63)-(64). Then, we add the resulting equations and integrate over (0, t) for 0 < t ≤ T to get

1 2 k=1,2 Γ k ε ε v k, ε -v k, ε (t, •) 2 + ε w k, ε -w k, ε (t, •) 2 dσ x + 1 2 Γ 1,2 ε ε s ε -s ε (t, •) 2 dσ x + k=1,2 Ω k i,ε,t M ε i ∇ u k, i,ε -u k, i,ε • ∇ u k, i,ε -u k, i,ε dxdτ + Ωe,ε,t
M ε e ∇ u e,ε -u e,ε • ∇ u e,ε -u e,ε dxdτ Due the uniform ellipticity (10) of M ε j for j = i, e, we have

+ k=1,2 Γ k ε,t ε Ĩa,ion v k, ε -Ĩa,ion v k, ε v k, ε -v k, ε dσ x dτ = 1 2 k=1,2 Γ k ε ε v k, 0,ε -v k, 0,ε 2 + ε w k, 0,ε -w k, 0,ε 2 dσ x + 1 2 Γ 1,2 ε ε s 0,ε -s 0,ε 2 dσ x + β 1 k=1,2 Γ k ε,t ε v k, ε -v k,
k=1,2 Ω k i,ε,t M ε i ∇ u k, i,ε -u k, i,ε • ∇ u k, i,ε -u k, i,ε dxdτ + Ωe,ε,t
M ε e ∇ u e,ε -u e,ε • ∇ u e,ε -u e,ε dxdτ

≥ α   k=1,2 ∇ u k, i,ε -u k, i,ε 2
L 2 (Ω k i,ε,t ) + ∇ u e,ε -u e,ε 2
L 2 (Ωe,ε,t)   ≥ 0.
Furthermore, thanks to the monotonicity assumption (12d) on Ĩa,ion , we deduce that

k=1,2 Γ k ε,t ε Ĩa,ion v k, ε -Ĩa,ion v k, ε v k, ε -v k, ε dσ x dτ ≥ 0.
Moreover, by the linearity of I b,ion , H and I gap , we can deduce using Young's inequality the following estimation (65)

1 2 k=1,2 Γ k ε ε v k, ε -v k, ε (t, •) 2 + ε w k, ε -w k, ε (t, •) 2 dσ x + + 1 2 Γ 1,2 ε ε s ε -s ε (t, •) 2 dσ x + α   k=1,2 ∇ u k, i,ε -u k, i,ε 2
L 2 (Ω k i,ε,t ) + ∇ u e,ε -u e,ε 2
L 2 (Ωe,ε,t)   ≤ 1 2 k=1,2 Γ k ε ε v k, 0,ε -v k, 0,ε 2 + ε w k, 0,ε -w k, 0,ε 2 dσ x + 1 2 Γ 1,2 ε ε s 0,ε -s 0,ε 2 dσ x + C k=1,2 t 0 Γ k ε ε v k, ε -v k, ε 2 + ε w k, ε -w k, ε 2 dσ x + 1 2 Γ 1,2 ε ε s ε -s ε 2 dσ x dτ
where C > 0 is a constant independent of ε. Thus, we obtain by applying Gronwall's inequality which means that u e,ε = u e,ε + c and u k, i,ε = u k, i,ε + c for k = 1, 2. On the one hand, due to the normalization condition [START_REF] James | [END_REF], c = 0 and u e,ε = u e,ε . On the other hand, the estimation (51) holds for u k, i,ε -u k, i,ε which gives

1 2 k=1,2 Γ k ε ε v k, ε -v k, ε (t, •) 2 + ε w k, ε -w k, ε (t, •) 2 dσ x + 1 2 Γ 1,2 ε ε s ε -s ε (t, •) 2 dσ x ≤ C k=1,2 Γ k ε ε v k, 0,ε -v k, 0,ε 2 + ε w k, 0,ε -w k,
u k, i,ε -u k, i,ε 2
L 2 (Ω k i,ε ) ≤ C √ ε v k, ε -v k, ε 2 L 2 (Γ k ε ) + ∇ u k, i,ε -u k, i,ε 2
L 2 (Ω k i,ε ) + ∇ u e,ε -u e,ε 2
L 2 (Ωe,ε)
, with k = 1, 2.

In addition, we have v k, ε = v k, ε so we obtain finally u k, i,ε = u k, i,ε

for k = 1, 2. This gives the uniqueness proof of weak solutions.

Funding. This research was supported by IEA-CNRS in the context of HIPHOP project.

Figure 1 .

 1 Figure 1. Representation of the cardiomyocyte structure http://www.cardio-research.com/cardiomyocytes

2. 1 .

 1 Geometry of heart tissue. The cardiac tissue Ω ⊂ R d (d ≥ 3) is considered as a heterogeneous periodic domain with a Lipschitz boundary ∂Ω. The structure of the tissue is periodic at microscopic scale related to small parameter ε, see Figure2.

Figure 2 .

 2 Figure 2. (Left) Periodic heterogeneous domain Ω. (Right) Unit cell Y at εstructural level.

1 mic 1 ,

 11 small cells are all equal, thanks to a translation and scaling by ε, to the same unit cell of periodicity called the reference cell Y = d n=1 ]0, mic n [. So, the ε-dilation of the reference cell Y is defined as the following shifted set Y ε,h : (1) Y ε,h := T h ε + εY = {εξ : ξ ∈ h + Y }, where T h ε represents the translation of εh with h = (h 1 , . . . , h d ) ∈ Z d and h := (h . . . , h d mic d

  0, x) = u 0,e,n (x) := n =1 d e, (0)φ e, , u k e,n (0, x) = u 0,e,n (x) := n =1 d e, (0)ψ k e, (x), d e, (0) := U 0,e , Θ e, V0,e ,

  (27)-(31) admit a unique solution over the time interval [0, t 0 ]. Proof. The goal is to determine the coefficients d k i = {d k i, } n =1 , d e = {d e, } n =1 and c k = {c k } n =1 for k = 1, 2. For this purpose, if n fixed, we choose Θ k i = Θ k i,m , Θ e = Θ e,m and ζ k = ζ k m for 1 ≤ m ≤ n and substitute the approximate solutions (

  k i,n (0, x) → u k 0,i,ε in L 2 (Γ k ε ) for k = 1, 2.Keeping in mind (58)-(62), we obtain by letting n → ∞ in the weak formulation (40)-(41) (63)

  εI gap s ε -s ε s ε -s ε dσ x dτ +

  s ε = s ε . Moreover, using Estimation (65), we conclude that ∇ u e,ε -u e,ε = 0 a.e. on Ω e,ε,t ,∇ u k, i,ε -u k, i,ε = 0 a.e. on Ω k i,ε,t ,

  n , ϕ e,n = ∂ t u e,n and e k n = εα 4 ∂ t w k

	n , respectively, in (40)-(41)
	then integrate in time to deduce using the previous equalities:
	(56)
	k=1,2


Julien Roth
email: julien.roth@univ-eiffel.fr

Abhitosh Upadhyay
email: abhitosh@iitgoa.ac.in

ON COMPACT EMBBEDED WEINGARTEN HYPERSURFACES IN WARPED PRODUCTS

Keywords: 2010 Mathematics Subject Classification. 53C42, 53A07, 49Q10 Weingerten hypersurfaces, space forms, warped products, pinching

We show that compact embedded starshaped r-convex hypersurfaces of certain warped products satisfying Hr = aH + b with a 0, b > 0, where H and Hr are respectively the mean curvature and r-th mean curvature is a slice. In the case of space forms, we show that without the assumption of starshapedness, such Weingarten hypersurfaces are geodesic spheres. Finally, we prove that, in the case of space forms, if Hr -aH -b is close to 0 then the hypersurface is close to geodesic sphere for the Hausdorff distance. We also prove an anisotropic version of this stability result in the Euclidean space.

Introduction

The well-known Alexandrov theorem [START_REF] Alexandrov | A characteristic property of spheres[END_REF] ensures that a closed embedded hypersurface of the Euclidean space R n+1 with constant mean curvature must be a round sphere. The hypothesis, that the hypersurface is to be embedded, is crucial as proved by the counter examples of Wente [START_REF] Wente | Counterexample to a conjecture of H. Hopf[END_REF], Kapouleas [START_REF] Kapouleas | Constant mean curvature surfaces constructed by fusing Wente tori[END_REF] or Hsiang-Teng-Yu [START_REF] Hsiang | New examples of constant mean curvature immersions of (2k-1)-spheres into Euclidean 2k-space[END_REF] for instance. Further, this result has been extended to scalar curvature and then higher order mean curvatures by Ros [START_REF] Ros | Compact hypersurfaces with constant scalar curvature and a congruence theorem[END_REF][START_REF] Ros | Compact hypersurfaces with constant higher order mean curvatures[END_REF] as well as for any concave function of the principal curvatures by Korevaar [START_REF] Korevaar | Sphere theorems via Alexandrov constant Weingarten curvature hypersurfaces: appendix to a note of A. Ros[END_REF]. For higher order mean curvatures, the necessity of the embedding is still an open question.

Note that Montiel and Ros [START_REF] Montiel | Compact hypersurfaces : the Alexandrov theorem for higher order mean curvature[END_REF] proved that the Alexandrov theorem for the mean curvature as well as for higher order mean curvatures is also true for hypersurfaces of hyperbolic spaces and half-spheres.

On the other hand, very recently, de Lima [START_REF] Lima | A note on compact Weingarten hypersurfaces embedded in R n+1[END_REF] proved a comparable result for the so called linear Weingarten hypersurfaces satisfying H r = aH + b for two real constants a 0 and b > 0, where H and H r are respectively the mean curvature and the r-th mean curvature of the hypersurfaces. The hypersurfaces are supposed to be embedded in this result too and H r is a positive function.

The aim of the present note is to show that Lima's result also holds for a large class of warped products which contains in particular the hyperbolic spaces and the half-spheres.

Let n 2 be an integer and (M n , g M ) be a compact Riemannian manifold of dimension n satisfying Ric M (n -1)kg, for some constant k. Moreover, let t 0 > 0 and h : [0, t 0 ) -→ R be a positive function satisfying the following four conditions (H1) h (0) = 0 and h (0) > 0, (H2) h (t) > 0 for all t ∈ (0, t 0 ), (H3) the function r -→ 2 h (t)

h(t) -(n -1) k -h (t) 2 h(t) 2
is non-decreasing on (0, t 0 ), (H4)

h (t) h(t) + k -h (t) 2 h(t) 2 > 0 for all t ∈ (0, t 0 ).
We consider the warped product P defined by P = [0, t 0 ) × M endowed with the metric g P = dt 2 ⊕ h(t)g M .

Theorem 1.1. Let n 2 and r ∈ {2, • • • , n} are two integers and Σ be a closed, oriented hypersurface embedded into the warped product (P n+1 , g P ). We assume that the four conditions (H1)-(H4) are satisfied. If Σ is star-shaped and H r is a positive function satisfying H r = aH + b for some real constants a 0 and b > 0, then Σ is a slice {t 1 } × M .

Note that in [START_REF] Wu | On rigidity of hypersurfaces with constant curvature functions in warped product manifolds[END_REF], Wu and Xia obtained a slightly different result for another type of relation between higher order mean curvatures.

We also obtain a comparable result for the space forms M n+1 (δ) for which the star-shapeness is not required. Here, M n+1 (δ) denotes the Euclidean space R n+1 if δ = 0, the half-sphere S n+1 + (δ) if δ > 0 and the hyperbolic space H n+1 (δ) if δ < 0. Namely, we have the following. Theorem 1.2. Let n 2 and r ∈ {2, • • • , n} are two integers and let Σ be a closed, connected and embedded hypersurface of M n+1 (δ). Assume that the r-th mean curvature H r is a positive function satisfying H r = aH + b for some real constants a 0 and b > 0. Then, Σ is a geodesic sphere of M n+1 (δ). This result extend the result of Lima [START_REF] Lima | A note on compact Weingarten hypersurfaces embedded in R n+1[END_REF] for real space forms of non-zero sectional curvature. Note also that when a = 0, we recover the Alexandrov theorem of Montiel and Ros [START_REF] Montiel | Compact hypersurfaces : the Alexandrov theorem for higher order mean curvature[END_REF] for hypersurfaces with constant r-th mean curvature.

We also consider the stability of this new characterization of geodesic spheres in space forms, precisely, the following natural question: if a closed embedded and oriented hypersurface Σ of M n+1 (δ) is almost Weingarten in the following sense, H r = aH + b + ε, where ε is a smooth function on Σ which is sufficiently small, is Σ close to a geodesic sphere ?

We answer this question by the following result.

Theorem 1.3. Let n 2 and r ∈ {2, • • • , n} are two integers. Let Σ be a closed embedded and oriented hypersurface of M n+1 (δ) bounding a domain Ω. There exist three constants γ, C and ε 1 , with γ depending only on n; C and ε 1 depending on n, r, δ, min

Σ H r , min Σ (H r;n,1 ), B ∞ , V (Σ) and R so that if Σ is almost linear
Weingarten in the following sense

H r = aH + b + ε,
where a 0 and b > 0 are real constants and ε is a smooth function satisfying

ε 1 ε 1 , then d H (Σ, S ρ0 ) C ε γ 1
, where S ρ0 is a geodesic sphere of a certain radius ρ 0 and d H is the Hausdorff distance between compact sets into M n+1 (δ).

Here, B is the second fundamental form, V (Σ) the volume of Σ, R the extrinsic radius of Σ and H r;n,1 is an extrinsic quantity defined from the second fundamental form (see (27) for the precise definition).

Preliminaries

2.1. Basics about warped products and Brendle's inequality. The classical Heinze-Karcher inequality for compact embedded hypersurfaces of the Euclidean space says that

Σ 1 H dv g (n + 1)V (Ω),
where Σ is the embedded hypersurface which bounds the compact domain Ω, H is the mean curvature of Σ, supposed to be positive, and V (Ω) is the volume of the domain Ω.

In [START_REF] Brendle | Constant mean curvature surfaces in warped product manifolds[END_REF], Brendle proved an analogue of the Heintze-Karcher inequality for a large class of warped products manifolds, namely, the warped products (P, g p ) of the form given in the introduction and satisfying conditions (H1), (H2) and (H3).

Let Σ be a compact embedded and oriented hypersurface of (P n+1 , g P ), we consider the function f = h and the vector field X = h ∂ ∂t . In [START_REF] Brendle | Constant mean curvature surfaces in warped product manifolds[END_REF], Brendle used the fact that X is a conformal vector field, L X g = 2f g, to obtain the following Hsiung-Minkowski formula for hypersurfaces of these warped products

(1) Σ H X, ν dv g = Σ f dv g ,
where g is the induced metric on Σ and ν is the outward normal unit vector field. Using this, he was able to prove the following extension of the Heinze-Karcher inequality:

(2)

Σ f H dv g (n + 1) Ω f dvg.
Moreover, if equality holds, then Σ is umbilical. If in addition, condition (H4) is satisfied, then Σ is a slice {t 1 } × N .

2.2. Higher order mean curvatures and Hsiung-Minkowski formulas. The higher order mean curvatures are extrinsic quantities defined from the second fundamental form and generalising the notion of mean curvature. Up to a normalisation constant the mean curvature H is the trace of the second fundamental form B:

(3) H = 1 n tr (B).
In other words, the mean curvature is

(4) H = 1 n S 1 (κ 1 , . . . , κ n ),
where S 1 is the first elementary symmetric polynomial and κ 1 , . . . , κ n are the principal curvatures. Higher order mean curvatures are defined in a similar way for r ∈ {1, . . . , n} by ( 5)

H r = 1 n r S r (κ 1 , • • • , κ n ),
where S r is the r-th elementary symmetric polynomial, that is for any n-tuple

(x 1 , • • • , x n ) (6) S r (x 1 , . . . , x n ) = 1 i1<•••<ir n x i1 • • • x ir .
By convention we set H 0 = 1 and H n+1 = 0. Finally, for convenience we also set

H -1 = -X, ν .
We also recall some classical inequalities between the H r which are well-known. First, for any r ∈ {0, • • • , n -2}, we have

(7) H r H r+2 H 2 r+1 ,
with equality at umbilical points, cf. [7, p. 104]. Moreover, cf. [START_REF] Barbosa | Stability of hypersurfaces with constant r-mean curvature[END_REF], if

H r+1 > 0, then H s > 0 for any s ∈ {0, • • • , r} and (8)
H 1 r+1 r+1 H 1 r r • • • H 1 2 2 H.
In [START_REF] Kwong | Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces[END_REF], the authors prove a general weighted Hsiung-Minkowski type formula in warped product. Namely, they prove

Σ φ(f H k-1 -H k X, ν )dv g + 1 k n-1 k Σ φ(div Σ T k-1 )(ξ)dv g (9) = - 1 k n-1 k Σ T k-1 (ξ), ∇φ dv g
where φ is a smooth function on Σ and ξ = X T is the tangential part of the conformal vector field X. Moreover, for r 2 we have

(div Σ T k-1 )(ξ) = - n -3 k -2 n-1 j=1 H k-1;j ξ j Ric(e j , ν), with H k-1;j = σ k-1 (λ 1 , • • • , λ j-1 , λ j+1 , • • • , λ n )
and where σ k is the k-th elemntary symmetric polynomial of order k. In addition, if and if conditions (H1)-(H4) and Σ is starshaped, then

Σ f H r-1 dv g Σ H r X, ν dv g . (10)
In the case of space forms M n+1 (δ), we have the classical Hsiung-Mnkowski formulas

(11) M H r Z, ν + c δ (ρ)H r dv g = 0,
where ρ(x) = d(p, x) is the distance function to a base point p (in the sequel, p will be the center of mass of M ), Z is the position vector defined by Z = s δ (ρ)∇ρ, where ∇ is the connection of M n+1 (δ) and the functions c δ and s δ are defined by

c δ (t) =    cos( √ δt) if δ > 0 1 if δ = 0 cosh( |δ|t) if δ < 0 and s δ (t) =      1 √ δ sin( √ δt) if δ > 0 t if δ = 0 1 √ |δ| sinh( |δ|t) if δ < 0.
2.3. Anisotropic mean curvatures. Let F : S n -→ R * + be a smooth function satisfying the following convexity assumption ( 12)

A F = (∇dF + F Id |TxS n ) x > 0,
for all x ∈ S n , where ∇dF is the Hessian of F and > 0 means positive definite in the sense of quadratic forms. Now, we consider the following map

φ : S n -→ R n+1 x -→ F (x)x + (grad |S n F ) x .
The image W F = φ(S n ) is called the Wulff shape of F and is a smooth convex hypersurface of R n+1 due to condition [START_REF] Hsiung | Some integral formulas for closed hypersurfaces[END_REF]. It is to note that if F = 1, then the Wulff shape is the sphere S n . Let X : (M n , g) -→ R n+1 be an isometric immersion of n-dimensional closed, connected and oriented Riemannian manifold M into R n+1 . We denote by ν a normal unit vector field globally defined on M , that is, we have ν : M -→ S n . We set S F = -A F • dν, where A F is defined in [START_REF] Hsiung | Some integral formulas for closed hypersurfaces[END_REF]. The operator S F is called the F -Weingarten operator or anisotropic shape operator. In this anisotropic setting, we can define all the corresponding extrinsic quantities. The anisotropic higher order mean curvatures H F r are defined by

H F r = 1 n r σ r (S F ),
where σ r (S F ) is the r-th elementary symmetric polyniomial with n variables computed for anisotropic principal cruvatures κ F 1 , • • • , κ F n . We denote simply by H F the anisotropic mean curvature H F 1 . Moreover, for convenience, we set H F 0 = 1 and H F n+1 = 0 by convention. For the Wulff shape

W F , κ F 1 = κ F 2 = • • • = κ F n are nonzero constants. Moreover, if κ F 1 = κ F 2 = • • • = κ F n
, then the hypersurface has to be the Wulff shape (up to homotheties and translations). Like in the anisotropic case, we have the following inequalities between higher order mean curvatures. Namely, if H F r+1 > 0 then H F j > 0 for all j ∈ {1, • • • , r} and (13)

H F r 1 r H F r-1 1 r-1 • • • H F 2 1 2 H F .
Moreover, in any of these inequalities, equality occurs at a point p if and only if all the anisotropic principal curvatures at p are equal. Hence, equality occurs everywhere if and only if M is the Wulff shape W F , up to translations and homotheties. Finally, we recall the anisotropic Hsiung-Minkowski formulas. For r ∈ {0, • • • n-1}, we have

(14) M F (ν)H F r + H F r+1 X, ν dv g = 0.
We also have an anisotropic analogue of the Heintze-Karcher inequality (see [START_REF] He | Integral formula of Minkowski type and new characterization of the Wulff shape[END_REF]).

If M is embedded (so bounds a domain Ω) and H F is everywehere positive, then the following inequality holds

(15) M F (ν) H F dv g (n + 1)V (Ω)
with equality if and only if M is the Wulff shape W F (up to translations and homotheties).

All the above mentioned results by Alexandrov and Ros have analogues for anisotropic mean curvatures with the Wulff shape replacing the sphere (see [START_REF] He | Compact embedded hypersurfaces with constant higher order anisotropic mean curvature[END_REF]).

2.4. Michael-Simon extrinsic Sobolev inequality. We conclude this section of preliminaries by recalling the extrinsic Sobolev inequality of Michael and Simon [START_REF] Michael | Sobolev and mean-value inequalities on generalized submanifolds of R n[END_REF]. If (Σ, g) is a closed connected and oriented hypersurface of the Euclidean space, for any C 1 function f on M , the following inequality holds ( 16)

M f n n-1 dv g n-1 n K(n) M (|∇f | + |Hf |) dv g ,
where K(n) is a constant that depends only on n. Applying this inequality for the function f ≡ 1, we get

(17) V (Σ) n-1 n K n Σ |H|dv g .
Now, we consider D ⊂ R n+1 be an open domain and let N n+1 = (D, h) be a conformally flat Riemannian manifold, i.e., h = e 2ϕ h where h is the Euclidean metric and ϕ ∈ C ∞ (D). Let (Σ n , g) → (N n+1 , h) be a closed, connected, oriented and isometrically immersed hypersurface. We deduce from ( 17) that ( 18)

V (Σ) n-1 n c n,ϕ Σ | H|dv g ,
where H is the mean curvature of the isometric immersion (Σ n , g) → (N n+1 , h) with g = e -2ϕ g and c n,ϕ is a constant depending on n and ϕ. Note that here, V (Σ) is the volume of Σ with the metric g which explain the dependence of the constant c n,ϕ on the conformal factor ϕ. Thus, we deduce immediately that

(19) V (Σ) -1 n c n,ϕ H 1 and so (20) V (Σ) -n+1 n c n,ϕ H n+1 n+1 .

Proof of the results

3.1. Proof of Theorems 1.1 and 1.2. We will prove first Theorem 1.1. The proof of Theorem 1.2 differs very slightly so that we will just mention after the minor differences. First, by the generalized Hsiung-Minkowski formula [START_REF] Hopf | Differential Geometry in the large[END_REF], we have

Σ f H r-1 -H r X, ν dv g 0.
Using the assumption that M is a Weingarten hypersurface, that is, H r = aH + b, we get

(21) Σ f H r-1 -aH X, ν -b X, ν dv g 0.
From the Hsiung-Minkowski (1) formula and the divergence theorem, we have

Σ H X, ν dv g = Σ f dv g and Σ X, ν dv g = (n + 1) Ω f dv g ,
respectively. Hence, ( 21) becomes ( 22)

a Σ f dv g + (n + 1)b Ω f dv g Σ f H r-1 dv g .
Now, since H r is supposed to be a positive function, as a consequence of ( 8), we have H r-1 H r-1 r r which after reporting into [START_REF] Scheuer | Stability from rigidity via umbilicity[END_REF] gives

a Σ f dv g + (n + 1)b Ω f dv g Σ f H r-1 r r dv g Σ f H r H -1 r r dv g Σ f H r H dv g (23) a Σ f dv g + b Σ f H dv g , ( 24
)
where we have used in the last two lines the facts that H which means that all the previous inequality are in fact equalities. In particular, equality holds in the Brendle's inequality, which implies that (since condition (H4) is assumed) Theorem 1.1 is proved.

3.2.

Proof of Theorem 1.3. The strategy of the proof consists in showing that the L n+1 -norm of τ is small (compared to ε) and applying the following result of [START_REF] Roth | Explicit rigidity of almost-umbilical hypersurfaces[END_REF] with p = n + 1 where N n+1 is either the Euclidean space, the half-sphere or the hyperbolic space.

Theorem 3.1. (Roth-Scheuer [START_REF] Roth | Explicit rigidity of almost-umbilical hypersurfaces[END_REF]) Let D ⊂ R n+1 be open and let N n+1 = (D, h) be a conformally flat Riemannian manifold, i.e., h = e 2ϕ h where h is the Euclidean metric and ϕ ∈ C ∞ (D). Let Σ n → N n+1 be a closed, connected, oriented and isometrically immersed hypersurface. Let p > n ≥ 2. Then there exist constants c and ε 0 , depending on n, p, V (Σ), B p and ϕ ∞ , as well as a constant α = α(n, p), such that whenever there holds

τ p H p ε 0 ,
there also holds

d H (Σ, S ρ ) cρ H α p τ α p ,
where S ρ is the image of a Euclidean sphere considered as a hypersurface in N n+1 and the Hausdorff distance is also measured with respect to the metric h.

First, we have

τ 2(n+1) n+1 = 1 V (Σ) M τ 2(n+1) dv g 2 1 V (Σ) 2 M τ 2n dv g M τ 2 dv g
by the Cauchy-Schwarz inequality. From this, we deduce immediately that (25) τ

2(n+1) n+1 1 V (Σ) B 2n ∞ M τ 2 dv g .

Now, we estimate

M τ 2 dv g . First, we have this lemma.

Lemma 3.2. There exists a constant positive constant

K 1 = K 1 (n, r, min(H r;n,1 ), B ∞ ) so that τ 2 K 1 H r-1 -H r-1 r r .
Proof: First, as mentioned in the preliminaries section, for any

k ∈ {1, • • • , n -1}, we have H 2 k -H k+1 H k-1 0.
We have a more precise estimate of the positivity of this term. Namely, (26)

H 2 k -H k+1 H k-1 c n τ 2 H 2 k+1;n,1
where c n is a constant depending only on n and (27)

H l;i,j = ∂H l ∂κ i ∂κ j = 1 n l 1 i 1 < • • • < i l n i 1 , • • • , i l = i, j κ i1 • • • • • κ i l .
On can find the proof in [START_REF] Scheuer | Stability from rigidity via umbilicity[END_REF] for instance. Hence, for k = r -1, we have

(28) H 2 r-1 -H r H r-2 c n τ 2 H 2 r;n,1 , which gives, with the fact that H r-2 H r-2 r r , (29)
H 2 r-1 -H 2(r-1) r r c n τ 2 H 2 r;n,1 . Finally, we get (30) H r-1 -H r-1 r r c n τ 2 H 2 r;n,1 H r-1 + H r-1 r r c n τ 2 H 2 r;n,1 2H r-1 .
Thus, bounding H 2 r;n,1 from below by its minimum and H r-1 form above with B ∞ , we get

(31) H r-1 -H r-1 r r c n τ 2 min(H r;n,1 ) 2 2 B r-1 ∞ and finally (32) τ 2 K 1 (H r-1 -H r-1 r r ) by setting K 1 = 2 B r-1 ∞ c n min(H r;n,1 ) 2 .
It is to note that this lemma holds independently of the fact that M is almost Weingarten. Now, using this condition of being almost Weingarten, we will bound from above H r-1 -H r-1 r r

. For this, we begin from the r-th Hsiung-Minkowski formula

Σ (H r X, ν + c δ (ρ)H r-1 ) dv g = 0, which becomes (33) a Σ H X, ν dv g + b Σ X, ν dv g + Σ ε X, ν dv g + Σ c δ (ρ)H r-1 dv g = 0,
after using the assumption that Σ is almost Weingarten. Now, using the first Hsiung-Minkowski formula Σ (H X, ν + c δ (ρ)) dv g = 0, and the divergence theorem, (33) becomes

(34) -a Σ c δ (ρ)dv g +(n+1)b Ω s δ (ρ)dv g + Σ ε X, ν dv g + Σ c δ (ρ)H r-1 dv g = 0.
On the other hand, using H H

) Σ c δ (ρ) τ 2 dv g K 1 Σ c δ (ρ)(H r-1 -H r-1 r r )dv g .
Reporting (34) and ( 35) into (36), we get (37)

Σ c δ (ρ) τ 2 dv g K 1 - Σ ε X, ν dv g - Σ c δ (ρ)εH -1 r r dv g , so that we deduce (38) inf Σ (c δ (ρ)) Σ τ 2 dv g K 1    sup Σ (c δ (ρ)) inf Σ (H 1 r r ) + sup Σ (s δ (ρ))    Σ εdv g .

Now, we set

K 1 =                                          K 1 c δ (R)    1 min Σ (H 1 r r ) + 1 √ δ    if δ > 0 K 1    1 min Σ (H 1 r r ) + R    if δ = 0, K 1    c δ (R) min Σ (H 1 r r ) + s δ (R)    if δ < 0,
where R is the extrinsic radius of Σ. Thus, we have

(39) Σ τ 2 dv g K 2 Σ εdv g ,
where K 1 is a positive constant depending on n, r, δ, min

Σ H r , min Σ (H r;n,1 ), B ∞
and R. Now, combining (39) and (25), we get

(40) τ 2(n+1) n+1 K 2 B 2n+1 ∞ V (Σ) Σ εdv g = K 3 ε 1 ,
where

K 3 = K 2 B 2n+1 ∞ is a constant depending on n, r, δ, min Σ H r , min Σ (H r;n,1 ), B ∞ , V (
Σ) and R. In order to apply Theorem 3.1, we need to compare the L n+1 -norms of τ and the mean curvature H of Σ viewed as a hypersurface of the Euclidean space after the conformal change of metric h = e 2ϕ h. Now, we use [START_REF] Roth | Explicit rigidity of almost-umbilical hypersurfaces[END_REF]

to get (41) 1 c 2 n,ϕ V (Σ) 2(n+1) n H 2(n+1) n+1
.

Hence, (40) gives

τ 2(n+1) n+1 K 3 c 2 n,ϕ V (Σ) 2n+2 n H 2(n+1) n+1 ε 1 = K 4 H 2(n+1) n+1 ε 1 , (42)
where K 4 is a constant depending on n, r, δ, min Σ H r , min Σ (H r;n,1 ), B ∞ , V (Σ) and R. Note that K 4 depends also on ϕ ∞,Ω due to [START_REF] Roth | Explicit rigidity of almost-umbilical hypersurfaces[END_REF], but since ϕ is the conformal change of metric between R n+1 and H n+1 or S n+1 + , this dependence can be replaced by a dependence on δ and R.

Now, if ε 1 is supposed to be smaller than ε 1 = ε 2(n+1) 0 K 4
, where ε 0 is the constant of Theorem 3.

A 2 τ F 2 .
Always proceeding as in Theorem 1.3, we get from the assumption that M is almost anisotropic Weingarten, and using the anisotrpoic version of both Hsiung-Minkowski formula [START_REF] Korevaar | Sphere theorems via Alexandrov constant Weingarten curvature hypersurfaces: appendix to a note of A. Ros[END_REF] and Heinzte-Karcher inequality [START_REF] Kwong | Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces[END_REF] (46)

A 2 Σ F (ν) τ F 2 dv g Σ ε X, ν dv g - Σ F (ν)εH -1 r r dv g .
We deduce immediately from this that

τ F 2 2
A 3 ε , where A 3 is a constant dependin on n, r, F , R, S F ∞ , inf Σ (H F r ) and inf Σ (H F r;n,1 ). Note that the extrinsic radius R appear here since we need to bound from above the term X, ν . Finally, we conclude by applying the following result of De Rosa and Gioffrè. Theorem 4.4 (De Rosa-Gioffrè [START_REF] Rosa | Quantitative stability for anisotropic nearly umbilical hypersurfaces[END_REF][START_REF] Rosa | Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces[END_REF]). Let n > 2, p ∈ (1, +∞) and F : S n -→ R * + satisfying the convexity assumption [START_REF] Hsiung | Some integral formulas for closed hypersurfaces[END_REF]. There exist a constant δ 0 = δ 0 (n, p, F ) > 0 such that if M is closed hypersurface into R n+1 satisfying V ol(M ) = V (W F ) and M τ F p dv g δ 0 then there exist a smooth parametrisation ψ : W F -→ M , a vector c 0 ∈ R n+1 and a constant C depending on n, p and F so that ψ -Id -c 0 W 2,p (W F ) C τ F p .

Moreover, if p ∈ (1, n], then the condition M τ F p dv g δ 0 can be dropped.

Here, it is important to mention that the volume of M is supposed to be equal to V (W F ). If we do not assume this, the same holds replacing W F by the homothetic of W F of volume equal to V (M ), that is for W ρF for ρ = V (M ) V (W F ) 1 n . Since in the statement of Theorem 4.3, we do not assume that the volume is equal to V (W F ), this introduce a dependence of the constant C also on V (M ). Note that this is also the case for Theorem 1.3.

H

  and H r = aH + b respectively. Now, we finish by applying the Brendle inequality

1 rr

 1 ρ)dv g + (n + 1)b Ω c δ (ρ)dv g + Σ c δ (ρ)εH dv g . (35)By multiplying (32) by c δ (ρ) and integrating over Σ, we get (36

  By computations analogous to those of Theorem 1.3, we get first that there exists a constant A 2 depending on n, r, F , S F ∞ and inf

			Σ	(H F r;n,1 ) so that
	(45)	H F r-1 -(H F r )	r-1 r
		1, then we have
		τ n+1	H n+1 ε 0 ,

so that we can apply Theorem 3.1. Note that ε 1 is a positive constant depending on n, r, δ, min Σ H r , min Σ (H r;n,1 ), B ∞ , V (Σ) and R. Thus, there exists ρ 0 > 0 so Proof:

Acknowledgements

Second author gratefully acknowledges the financial support from the Indian Institute of Technology Goa through Start-up Grant (2021/SG/AU/043).

Using (40) once again, we get

where

and R and γ is a positive constant depending only on n. This concludes the proof of Theorem 1.3.

An anisotropic result

We finish this paper by an anisotropic version of Theorem 1.3. In [START_REF] Roth | On compact anisotropic Weingarten hypersurfaces in Euclidean space[END_REF], we prove a new characterization of the Wulff shape which is an anisotropic version of the result of de Lima for linear Weingarten hypersurfaces. Namely, we proved the following. 2 be an integer, F : S n -→ R * + a smooth function satisfying the convexity assumption [START_REF] Hsiung | Some integral formulas for closed hypersurfaces[END_REF] and let M be a closed, connected and embedded hypersurface of R n+1 . Assume that the higher order anisotropic mean curvature H F r , r ∈ {2, • • • n} never vanishes and satisfies H F r = aH F + b for some real constants a 0 and b > 0. Then, up to translations and homotheties, M is the Wulff shape W F .

Also in [START_REF] Roth | On compact anisotropic Weingarten hypersurfaces in Euclidean space[END_REF], we proved the following stability result for r = 2. Theorem 4.2. (Roth-Upadhyay [START_REF] Roth | On compact anisotropic Weingarten hypersurfaces in Euclidean space[END_REF]) Let n 2 be an integer, F : S n -→ R * + a smooth function satisfying the convexity assumption [START_REF] Hsiung | Some integral formulas for closed hypersurfaces[END_REF] and let M be a closed, connected and embedded hypersurface of R n+1 . Assume that the r-th order anisotropic mean curvature H F r never vanishes and satisfies H F 2 = aH F + b + ε for some real constants a 0, b > 0 and ε a smooth function. Set ρ = V (M )

By comparable arguments as those used in the proof of Theorem 1.3, we can extend this result for any r ∈ {2, • • • , r}. Namely, we have the following.