Christine Bernardi
email: bernardi@ann.jussieu.fr

Jad Dakroub
email: jad.dakroub@usj.edu.lb

Gihane Mansour
email: gihane.mansour@usj.edu.lb

JAD DAKROUB † ‡ , GIHANE MANSOUR ‡ Farah Rafei
email: farah.rafei@net.usj.edu.lb

Toni Sayah
email: toni.sayah@usj.edu.lb.

Toni Sayah Convergence

CONVERGENCE ANALYSIS OF TWO NUMERICAL SCHEMES APPLIED TO A NONLINEAR ELLIPTIC PROBLEM

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Let Ω be an open polygon in IR 2 , we consider the problem

-∆u + λ|u| 2p u = f in Ω, (1.1)
u = 0 on ∂Ω, (1.2)
where λ and p are two positive real numbers. The right-hand side f belongs to the dual space H -1 (Ω) of the Sobolev space H 1 0 (Ω). The a posteriori error analysis of finite element approximations of the present model problem has been studied by Bernardi, Dakroub, Mansour and Sayah, see [START_REF] Bernardi | A posteriori analysis of iterative algorithms for a nonlinear problem[END_REF]. In fact, let V h ⊂ H 1 0 (Ω) be the P 1 finite element space associated with a regular family of triangulations of Ω, denoted by T h . Using P 1 Lagrange finite elements, the discrete variational problem obtained by the Galerkin method amounts to (from now on, we denote by (•, •) the scalar product of L 2 (Ω))

Find u h ∈ V h such that ∀v h ∈ V h , ∇u h , ∇v h + λ |u h | 2p u h , v h = f, v h . (1.3)
In order to solve the discrete nonlinear problem (1.3), we introduced in [START_REF] Bernardi | A posteriori analysis of iterative algorithms for a nonlinear problem[END_REF] the following linear numerical scheme, called fixed-point algorithm:

Find u i+1 h ∈ V h such that ∀v h ∈ V h , ∇u i+1 h , ∇v h + λ |u i h | 2p u i+1 h , v h = f, v h .
(1.4)

First numerical scheme.

Find u i+1 h ∈ V h such that ∀v h ∈ V h , α u i+1 h -u i h , v h + ∇u i+1 h , ∇v h + λ |u i h | 2p u i+1 h , v h = f, v h , (1.5)
Second numerical scheme.

Find u i+1 h ∈ V h such that ∀v h ∈ V h , α ∇u i+1 h -∇u i h , ∇v h + ∇u i+1 h , ∇v h + λ |u i h | 2p u i+1 h , v h = f, v h , (1.6)
For a parameter α bigger than a specific constant that depends on λ, p and the data f , problem (1.5) and (1.6) always converge. Moreover, our objective is to derive an a posteriori error estimate distinguishing linearization and discretization errors.

In practice, the present problem (1.1) -(1.2) is solved using an iterative method involving a linearization process and approximated by the finite element method. Thus, two sources of error appear, namely linearization and discretization. The main result in [START_REF] Bernardi | A posteriori analysis of iterative algorithms for a nonlinear problem[END_REF] is a two-sided bound of the error distinguishing linearization and discretization errors in the context of an adaptive procedure. This type of analysis was introduced by A.-L. Chaillou and M. Suri [START_REF] Chaillou | Computable error estimators for the approximation of nonlinear problems by linearized models[END_REF][START_REF] Chaillou | A posteriori estimation of the linearization error for strongly monotone nonlinear operators[END_REF] for a general class of problems characterized by strongly monotone operators and developed by L. El Alaoui, A. Ern and M. Vohralík [START_REF] Alaoui | Guaranteed and robust a posteriori error estimate and balancing discretization and linearization errore for monotone nonlinear problems[END_REF] for a class of secondorder monotone quasi-linear diffusion-type problems approximated by piecewise affine, continuous finite elements. We wish to extend these results to the problem that we consider and prove optimal estimates.

The paper is organized as follows:

• Section 2 describes the model problem.

• Section 3 is devoted to the study of the convergence of the schemes.

• Section 4 provides the a posteriori estimates for both problems.

• Section 5 is devoted to the numerical results.

Preliminaries

In this section, we describe the variational formulation associated with the nonlinear problem (1.1)-(1.2) and introduce and recall some corresponding properties which will be used later. We denote by L p (Ω) the space of measurable functions summable with power p, and for all v ∈ L p (Ω), the corresponding norm is defined by

v L p (Ω) = Ω |v(x)| p dx 1/p .
In the case p = 2, we also denote this norm by • 0,Ω . Throughout this paper, we constantly use the classical Sobolev space

H 1 (Ω) = v ∈ L 2 (Ω); ∂v ∂x 1 , ∂v ∂x 2 ∈ L 2 (Ω) ,
which is equipped respectively with the semi-norm and norm

|v| 1,Ω = Ω (| ∂v ∂x 1 | 2 + | ∂v ∂x 2 | 2 )dx 1/2 and v 1,Ω = ||v|| 2 0,Ω + |v| 2 1,Ω ) 1/2 .
In particular, we consider the following space

H 1 0 (Ω) = {v ∈ H 1 (Ω), v | ∂Ω = 0}
, and its dual space H -1 (Ω). We recall the Sobolev imbeddings (see Adams [START_REF] Adams | Sobolev Spaces[END_REF], Chapter 3). Lemma 2.1. For any domain Ω in IR 2 , for all j, 1 ≤ j < ∞, there exists a positive constant

S j such that ∀v ∈ H 1 0 (Ω), v L j (Ω) ≤ S j |v| 1,Ω . (2.1)
Remark 2.2. For domains Ω in IR 3 , inequality (2.1) with standard definition of H 1 0 (Ω) remains valid only for j ≤ 6, whence the interest of working in dimension d = 2.

Setting X = H 1 0 (Ω), the model problem (1.1)-(1.
2) admits the equivalent variational formulation:

Find u ∈ X such that ∀v ∈ X, Ω ∇u∇vdx + Ω λ|u| 2p uvdx = f, v , (2.2)
Theorem 2.3. [2] Problem (2.
2) admits a unique solution u ∈ X.

We now introduce the following technical lemmas:

Lemma 2.4. Let a, b and p be three real numbers. We have the following relation

|a| p -|b| p ≤ p|a -b| |a| p-1 + |b| p-1 .
Proof. The result follows from applying the mean value theorem to f (x) = x p with x > 0.

Lemma 2.5. For all x, y ∈ IR and p ∈ IR, we have

(|x| 2p x -|y| 2p y)(x -y) ≥ 0.
Remark 2.6. In the sequel, we denote by C, C ,... generic constants that can vary from line to line but are always independent of all discretization parameters.

Finite element discretization and convergence

In this section, we begin to collect some useful notation concerning the discrete setting and the a priori estimate. Then, we show the convergence of the schemes (1.5) and (1.6).

Let (T h ) h be a regular family of triangulations of Ω, in the sense that, for each h:

• The union of all elements of T h is equal to Ω.

• The intersection of two different elements of T h , if not empty, is a vertex or a whole edge of both triangles.

• The ratio of the diameter h K of any element K of T h to the diameter of its inscribed circle is smaller than a constant independent of h. As usual, h stands for the maximum of the diameters h K , K ∈ T h . Let V h ⊂ H 1 0 (Ω) be the Lagrange P finite element space associated with T h , more precisely

V h = v h ∈ H 1 0 (Ω); ∀K ∈ T h , v h |K ∈ P (K) ,
where P (K) stands for the space of restrictions to K of polynomial functions of degree ≤ on IR 2 .

Remark 3.1. (Inverse inequality) There exists a constant S I > 0 such that for all v h ∈ V h and K ∈ T h , we have

|v h | 1,K ≤ S I h -1 K ||v h || 0,K . ( 3
u h -u 1,Ω ≤ Ch u 2,Ω .
In the following, we investigate the convergence of the schemes (1.5) and (1.6).

Theorem 3.3. Problem (1.5) admits a unique solution. Furthermore, if the initial value u 0 h satisfies the condition

||u 0 h || 0,Ω ≤ S 2 ||f || -1,Ω , (3.2)
then the solution of the problem (1.5) satisfies the estimates

||u i+1 h || 0,Ω ≤ S 2 ||f || -1,Ω and |u i+1 h | 1,Ω ≤ 1 + αS 2 2 ||f || -1,Ω . (3.3)
Proof. It is readily checked that problem (1.5) has a unique solution as a consequence of the coercivity of the bilinear form. We consider the equation (1.5) with v h = u i+1 h and we obtain :

α 2 ||u i+1 h || 2 0,Ω - α 2 ||u i h || 2 0,Ω + α 2 ||u i+1 h -u i h || 2 0,Ω + |u i+1 h | 2 1,Ω + λ(|u i h | 2p u i+1 h , u i+1 h ) = (f, u i+1 h )
. By using the inequality and we deduce the inequality

(f, u i+1 h ) ≤ 1 2 ||f || 2 -1,Ω + 1 2 |u i+1 h | 2 1,Ω , we deduce the relation α 2 ||u i+1 h || 2 0,Ω - α 2 ||u i h || 2 0,Ω + α 2 ||u i+1 h -u i h || 2 0,Ω + 1 2 |u i+1 h | 2 1,Ω + λ(|u i h | 2p u i+1 h , u i+1 h ) ≤ 1 2 ||f ||
||u i+1 h || 2 0,Ω ≤ S 2 2 |u i+1 h | 2 1,Ω ≤ S 2 2 ||f || 2 -1
,Ω . This gives the first part of (3.3). We now check the second part. We have from (3.4)

|u i+1 h | 2 1,Ω ≤ ||f || 2 -1,Ω + α||u i h || 2 0,Ω ≤ (1 + αS 2 2 )||f || 2 -1
,Ω , whence the desired result.

Theorem 3.4. Problem (1.6) admits a unique solution. Furthermore, if the initial value u 0 h verify the condition

|u 0 h | 1,Ω ≤ ||f || -1,Ω , (3.5)
then the solution of Problem (1.6) satisfies the estimate

|u i+1 h | 1,Ω ≤ ||f || -1,Ω . (3.6)
Proof. We follow the same proof as for Theorem 3.3. It is readily checked that problem (1.6) has a unique solution as a consequence of the coercivity of the bilinear form. We consider the equation (1.6) with v h = u i+1 h and we obtain :

α 2 |u i+1 h | 2 1,Ω - α 2 |u i h | 2 1,Ω + α 2 |u i+1 h -u i h | 2 1,Ω + |u i+1 h | 2 1,Ω + λ(|u i h | 2p u i+1 h , u i+1 h ) = (f, u i+1 h ). We deduce the relation α 2 |u i+1 h | 2 1,Ω - α 2 |u i h | 2 1,Ω + α 2 |u i+1 h -u i h | 2 1,Ω + 1 2 |u i+1 h | 2 1,Ω + λ(|u i h | 2p u i+1 h , u i+1 h ) ≤ 1 2 ||f || 2 -1,Ω . (3.7)
We prove the relation (3.6) recursively. Starting with (3.5), we suppose that we have

|u i h | 1,Ω ≤ ||f || -1,Ω .
We are in one of the following two situations :

• We have |u i+1 h | 1,Ω ≤ |u i h | 1,Ω . We deduce the bound |u i+1 h | 1,Ω ≤ ||f || -1,Ω . • We have |u i+1 h | 1,Ω ≥ |u i h | 1,Ω . It follows from (3.7) that |u i+1 h | 2 1,Ω ≤ ||f || 2 -1,Ω .
We conclude the proof of the theorem.

Unfortunately the proof of the next result is much more technical. Theorem 3.5. Assume that there exists β > 0 such that, for every element K ∈ T h , we have

h K ≥ βh,
(which means that the family of triangulations is uniformly regular). Under the assumptions of Theorem

and for

α > C 2 p 2 λ 2 h -4p (3.8) where C = 4S 4 S 8 S 2p-1 8(2p-1) S 2p I β 2p S 2p 2 ||f || 2p -1,Ω , the sequence of solutions (u i h ) of Problem (1.5) converges in H 1 0 (Ω) to the solution u h of Problem (1.
3). Proof. We take the difference between the equations (1.5) and ( 1

.3) with v h = u i+1 h -u h and we obtain the equation α 2 ||u i+1 h -u h || 2 0,Ω - α 2 ||u i h -u h || 2 0,Ω + α 2 ||u i+1 h -u i h || 2 0,Ω + |u i+1 h -u h | 2 1,Ω +λ(|u i h | 2p u i+1 h -|u h | 2p u h , u i+1 h -u h ) = 0.
The last term in the previous equation, denoted by T , can be decomposed as

T = λ((|u i h | 2p -|u i+1 h | 2p )u i+1 h , u i+1 h -u h ) + λ(|u i+1 h | 2p u i+1 h -|u h | 2p u h , u i+1 h -u h ).
We denote by T 1 and T 2 , respectively, the first and the second terms in the right-hand side of the last equation. Using Lemma 2.5, we have T 2 ≥ 0. Then we derive by using Lemma 2.4 (with p replaced by 2p)

α 2 ||u i+1 h -u h || 2 0,Ω - α 2 ||u i h -u h || 2 0,Ω + α 2 ||u i+1 h -u i h || 2 0,Ω + |u i+1 h -u h | 2 1,Ω + T 2 = -T 1 ≤ 2pλ Ω (|u i+1 h | 2p-1 + |u i h | 2p-1 )|u i+1 h -u i h | |u i+1 h | |u i+1 h -u h |dx ≤ 2pλ||u i+1 h -u i h || 0,Ω |||u i+1 h | 2p-1 + |u i h | 2p-1 || L 8 (Ω) ||u i+1 h || L 8 (Ω) ||u i+1 h -u h || L 4 (Ω) ≤ 2pλS 4 S 8 S 2p-1 8(2p-1) (|u i+1 h | 2p-1 1,Ω + |u i h | 2p-1 1,Ω )|u i+1 h | 1,Ω ||u i+1 h -u i h || 0,Ω |u i+1 h -u h | 1,Ω ≤ 2pλS 4 S 8 S 2p-1 8(2p-1) S 2p I β 2p h -2p ||u i+1 h || 2p-1 0,Ω + ||u i h || 2p-1 0,Ω ||u i+1 h || 0,Ω ||u i+1 h -u i h || 0,Ω |u i+1 h -u h | 1,Ω ≤ 4pλS 4 S 8 S 2p-1 8(2p-1) S 2p I β 2p S 2p 2 h -2p ||f || 2p -1,Ω ||u i+1 h -u i h || 0,Ω |u i+1 h -u h | 1,Ω We denote by C = 4S 4 S 8 S 2p-1 8(2p-1) S 2p I β 2p S 2p 2 ||f || 2p -1,Ω , use the decomposition ab ≤ 1 2ε a 2 + ε 2 b 2 , take ε = 1 Cpλh -2p and obtain the following bound α 2 ||u i+1 h -u h || 2 0,Ω - α 2 ||u i h -u h || 2 0,Ω + α 2 ||u i+1 h -u i h || 2 0,Ω + 1 2 |u i+1 h -u h | 2 1,Ω +T 2 ≤ C 2 p 2 λ 2 2 h -4p ||u i+1 h -u i h || 2 0,Ω . We choice α > C 2 p 2 λ 2 h -4p , denote by C 1 = α -C 2 p 2 λ 2 h -4p 2 and obtain α 2 ||u i+1 h -u h || 2 0,Ω - α 2 ||u i h -u h || 2 0,Ω + C 1 ||u i+1 h -u i h || 2 0,Ω + 1 2 |u i+1 h -u h | 2 1,Ω + T 2 ≤ 0. (3.9)
We deduce that, for all i ≥ 1, we have (if

||u i h -u h || 0,Ω = 0) ||u i+1 h -u h || 0,Ω < ||u i h -u h || 0,Ω ,
and we deduce the convergence of the sequence (u i+1 h -u h ) in L 2 (Ω). By taking the limit of (3.9) we deduce that |u i+1 h -u h | 1,Ω converges to 0 and u i+1 h converges to u h in H 1 0 (Ω). Theorem 3.6. Under the assumptions of Theorem 3.4 and for

α > 4S 2 S 4 S 8 S 2p-1 8(2p-1) ||f || 2p -1,Ω 2 p 2 λ 2 , (3.10)
the sequence of solutions (u i h ) of Problem (1.6) converges in H 1 0 (Ω) to the solution u h of Problem (1.3). Proof. We take the difference between the equations (1.6) and (1.3) with v h = u i+1 h -u h and we obtain the equation

α 2 |u i+1 h -u h | 2 1,Ω - α 2 |u i h -u h | 2 1,Ω + α 2 |u i+1 h -u i h | 2 1,Ω + |u i+1 h -u h | 2 1,Ω +λ(|u i h | 2p u i+1 h -|u h | 2p u h , u i+1 h -u h ) = 0.
The last term in the previous equation, denoted by T , can be decomposed as

T = λ((|u i h | 2p -|u i+1 h | 2p )u i+1 h , u i+1 h -u h ) + λ(|u i+1 h | 2p u i+1 h -|u h | 2p u h , u i+1 h -u h ).
We denote by T 1 and T 2 respectively the first and the second terms in the right-hand side of the last equation. Using Lemma 2.5, we have T 2 ≥ 0. Then we have by using Lemma 2.4

α 2 |u i+1 h -u h | 2 1,Ω - α 2 |u i h -u h | 2 1,Ω + α 2 |u i+1 h -u i h | 2 1,Ω + |u i+1 h -u h | 2 1,Ω + T 2 = -T 1 ≤ 2pλ (|u i+1 h | 2p-1 + |u i h | 2p-1 ), |u i+1 h -u i h | |u i+1 h | |u i+1 h -u h | ≤ 2pλ||u i+1 h -u i h || 0,Ω ||(|u i+1 h | 2p-1 + |u i h | 2p-1 )|| L 8 (Ω) ||u i+1 h || L 8 (Ω) ||u i+1 h -u h || L 4 (Ω) ≤ 4pλS 2 S 4 S 8 S 2p-1 8(2p-1) ||f || 2p -1,Ω |u i+1 h -u i h | 1,Ω |u i+1 h -u h | 1,Ω .
We denote by

C = 4S 2 S 4 S 8 S 2p-1 8(2p-1) ||f || 2p -1,Ω , use the decomposition ab ≤ 1 2ε a 2 + ε 2 b 2 , take ε = 1 Cpλ and
obtain the following bound

α 2 |u i+1 h -u h | 2 1,Ω - α 2 |u i h -u h | 2 1,Ω + α 2 |u i+1 h -u i h | 2 1,Ω + 1 2 |u i+1 h -u h | 2 1,Ω + T 2 ≤ C 2 p 2 λ 2 2 |u i+1 h -u i h | 2 1,Ω .
We choose α > C 2 p 2 λ 2 , denote by

C 1 = α -C 2 p 2 λ 2 2
and obtain

α 2 |u i+1 h -u h | 2 1,Ω - α 2 |u i h -u h | 2 1,Ω + C 1 |u i+1 h -u i h | 2 1,Ω + 1 2 |u i+1 h -u h | 2 1,Ω + T 2 ≤ 0. (3.11)
We derive that, for all i ≥ 1, we have

|u i+1 h -u h | 1,Ω < |u i h -u h | 1,Ω , we obtain the convergence of the sequence (u i+1 h -u h ) in H 1 (Ω)
and, by taking the limit of (3.11) we deduce that u i+1 h converges to u h in H 1 0 (Ω). Remark 3.7. The conditions (3.2) and (3.5) suppose that the initial values of the algorithms are small related to the data f . We can always take u 0 h = 0. Remark 3.8. The previous two theorems bring to light a first difference between the two schemes (1.5) and (1.6): in opposite to (1.5), the convergence of (1.6) is proved when α is larger than a constant independent of h (and does not require the uniform regularity of the family of triangulations).

A posteriori error analysis

We start this section by introducing some additional notation which is needed for constructing and analyzing the error indicators in the sequel.

For any triangle K ∈ T h we denote by E(K) and N (K) the set of its edges and vertices, respectively, and we set

E h = K∈T h E(K) and N h = K∈T h N (K).
With any edge e ∈ E h we associate a unit vector n such that n is orthogonal to e. We split E h and N h in the form

E h = E h,Ω ∪ E h,∂Ω and N h = N h,Ω ∪ N h,∂Ω
where E h,∂Ω is the set of edges in E h that lie on ∂Ω and E h,Ω = E h \ E h,∂Ω . The same goes for N h,∂Ω . Furthermore, for K ∈ T h and e ∈ E h , let h K and h e be their diameter and length, respectively. An important tool in the construction of an upper bound for the total error is Clément's interpolation operator R h with values in V h . The operator R h satisfies, for all v ∈ H 1 0 (Ω), the following local approximation properties (see R. Verfürth, [START_REF] Verfürth | A Posteriori Error Estimation Techniques For Finite Element Methods[END_REF], Chapter 1):

v -R h v L 2 (K) ≤ Ch K |v| 1,∆ K , v -R h v L 2 (e) ≤ Ch 1/2
e |v| 1,∆e , where ∆ K and ∆ e are the following sets:

∆ K = K ∈ T h ; K ∩ K = ∅ and ∆ e = K ∈ T h ; K ∩ e = ∅ .
We now recall the following properties (see R. Verfürth, [START_REF] Verfürth | A Posteriori Error Estimation Techniques For Finite Element Methods[END_REF], Chapter 1):

Proposition 4.1. Let r be a positive integer. For all v ∈ P r (K), the following properties hold

C v L 2 (K) ≤ vψ 1/2 K L 2 (K) ≤ v L 2 (K) , (4.1
)

|v| 1,K ≤ Ch -1 K v L 2 (K) . (4.
2) where ψ K is the triangle-bubble function (equal to the product of the barycentric coordinates associated with the vertices of K).

We also introduce a lifting operator: For each K ∈ T h and any edge e of K, L e,K maps polynomials of fixed degree on e vanishing on ∂e into polynomials on K vanishing on ∂K \ e and is constructed by affine transformation from a fixed lifting operator on the reference triangle. Proposition 4.2. Let r be a positive integer. For all v ∈ P r (e), we have the following property

C v L 2 (e) ≤ vψ 1/2 e L 2 (e) ≤ v L 2 (e) , (4.3)
where ψ e is the bubble function on the edge e, and for all v ∈ P r (e) vanishing on ∂e, we have

||L e,κ v|| L 2 (κ) + h e |L e,κ v| 1,κ ≤ Ch 1/2 e v L 2 (e) , (4.4)
where κ is a triangle of edge e.

Finally, we denote by [v h ] the jump of v h across the common edge e of two adjacent elements K, K ∈ T h .

We have now provided all prerequisites to establish an upper bound and lower bound for the total error. Let u i+1 h and u be the solution of the iterative problem (1.5) or (1.6) and the continuous problem, respectively. They satisfy the identity

Ω ∇(u i+1 h -u)∇vdx = Ω ∇u i+1 h ∇vdx + λ Ω |u| 2p uvdx - Ω f vdx. (4.5)
We now start the a posteriori analysis of our algorithms.

Algorithm (1.5).

In order to prove an upper bound of the error, we introduce an approximation f h of the data f which is constant on each element K of T h . We first write the residual equation

Ω ∇u∇vdx + λ Ω |u| 2p uvdx - Ω ∇u i+1 h ∇vdx -λ Ω |u i h | 2p u i+1 h vdx = Ω (f -f h )(v -v h )dx + K∈T h K (f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h ))(v -v h )dx - 1 2 e∈E(K) e [ ∂u i+1 h ∂n ](v -v h )dτ + α K∈T h K (u i+1 h -u i h )vdx. (4.6)
By adding and subtracting λ K,i at each iteration i by:

Ω |u i+1 h | 2p u i+1 h vdx, we obtain Ω ∇u∇vdx + λ Ω |u| 2p uvdx - Ω ∇u i+1 h ∇vdx -λ Ω |u i+1 h | 2p u i+1 h vdx = Ω (f -f h )(v -v h )dx + K∈T h K (f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h ))(v -v h )dx - 1 2 e∈E(K) e [ ∂u i+1 h ∂n ](v -v h )dτ + λ Ω |u i h | 2p -|u i+1 h | 2p u i+1 h vdx + α K∈T h K (u i+1 h -u i h )vdx.
η (L) K,i = ||u i+1 h -u i h || 1,K , η (D) K,i 2 = h 2 K f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h ) 2 L 2 (K) + e∈E(K) h e [ ∂u i+1 h ∂n ] 2 L 2 (e) .
We are in a position to state the first result of this section:

Theorem 4.3. Upper bound. Let u i+1 h
and u be the solution of the iterative problem (1.5) and the exact problem (2.2) respectively. We have the following a posteriori error estimate

|u i+1 h -u| 1,Ω ≤ C K∈T h ( η (D) K,i 2 + h 2 K f -f h 2 L 2 (K) ) 1/2 + K∈T h η (L) K,i 2 1/2 .
Proof. We consider equation (4.7) with v = u -u i+1 h and we obtain

Ω ∇(u -u i+1 h ) 2 dx + λ Ω (|u| 2p u -|u i+1 h | 2p u i+1 h )(u -u i+1 h )dx = K∈T h K (f -f h )(v -v h )dx + K∈T h K (f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h ))(v -v h )dx - 1 2 e∈E(K) e [ ∂u i+1 h ∂n ](v -v h )dτ + λ Ω |u i h | 2p -|u i+1 h | 2p u i+1 h vdx + α K∈T h K (u i+1 h -u i h )vdx.
(4.8) Then we have by using Lemmas 2.4 and 2.5

|u -u i+1 h | 2 1,Ω ≤ K∈T h ||f -f h || L 2 (K) ||v -v h || L 2 (K) + K∈T h (||f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h )|| L 2 (K) ||v -v h || L 2 (K) + 1 2 e∈E(K) ||[ ∂u i+1 h ∂n ]|| L 2 (e) ||v -v h || L 2 (e ) + λ Ω 2p u i h -u i+1 h | |u i h | 2p-1 + |u i+1 h | 2p-1 )|u i+1 h ||v|dx +α K∈T h ||u i+1 h -u i h || L 2 (K) ||v|| L 2 (K)
We choose v h = R h v, the image of v by the Clément operator and we obtain

|u -u i+1 h | 2 1,Ω ≤ C K∈T h h K ||f -f h || L 2 (K) |v| 1,∆ K + K∈T h (Ch K ||f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h )|| L 2 (K) |v| 1,∆ K + C 2 e∈E(K) h 1 2 e ||[ ∂u i+1 h ∂n ]|| L 2 (e) |v| 1,∆e ) +λ Ω 2p u i h -u i+1 h | |u i h | 2p-1 + |u i+1 h | 2p-1 )|u i+1 h ||v|dx + α K∈T h ||u i+1 h -u i h || L 2 (K) ||v|| L 2 (K)
We begin by bounding the second term of the right-hand side of the last inequality and we obtain by using Theorem 3.3

λ Ω 2p u i h -u i+1 h | |u i h | 2p-1 + |u i+1 h | 2p-1 )|u i+1 h ||v|dx ≤ 2λp|| |u i h | 2p-1 + |u i+1 h | 2p-1 || L 8 (Ω) |u i h -u i+1 h || L 8 (Ω) ||u i+1 h || L 4 (Ω) ||v|| L 2 (Ω) ≤ 2λpS 2 S 4 S 8 || |u i h | 2p-1 || L 8 (Ω) + || |u i+1 h | 2p-1 || L 8 (Ω) |u i h -u i+1 h | 1,Ω |u i+1 h | 1,Ω |v| 1,Ω ≤ 2λpS 2 S 4 S 8 S 2p-1 8(2p-1) (|u i h | 2p-1 1,Ω + |u i+1 h | 2p-1 1,Ω ) u i h -u i+1 h | 1,Ω |u i+1 h | 1,Ω |v| 1,Ω ≤ 4λp(1 + αS 2 2 ) p S 2 S 4 S 8 S 2p-1 8(2p-1) ||f || 2p -1,Ω u i h -u i+1 h | 1,Ω |v| 1,Ω . Let S = 4λp(1 + αS 2 2 ) p S 2 S 4 S 8 S 2p-1 8(2p-1) ||f || 2p -1,Ω , then we have |u -u i+1 h | 2 1,Ω ≤ C K∈T h h K ||f -f h || L 2 (K) |v| 1,∆ K + K∈T h (Ch K ||f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h )|| L 2 (K) |v| 1,∆ K + C 2 e∈E (K) h 1 2
e ||[

∂u i+1 h ∂n ]|| L 2 (e) |v| 1,∆e ) +S|u i h -u i+1 h | 1,Ω |v| 1,Ω + α K∈T h ||u i+1 h -u i h || L 2 (K) ||v|| L 2 (K) .
By using the formula ab ≤ 1 2ε

a 2 + ε 2 b 2 , we obtain |u -u i+1 h | 2 1,Ω ≤ Cε 1 2 K∈T h h 2 K ||f -f h || 2 L 2 (K) + C 1 2ε 1 K∈T h |v| 2 1,K + Cε 2 2 K∈T h h 2 K ||f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h )|| 2 L 2 (K) + C 2 2ε 2 K∈T h |v| 2 1,K + Cε 3 4 K∈T h E∈E (K) h E ||[ ∂u i+1 h ∂n ]|| 2 L 2 (E) + C 3 4ε 3 K∈T h |v| 2 1,K + Sε 4 2 K∈T h ||u i h -u i+1 h || 2 1,K + S 2ε 4 K∈T h |v| 2 1,K + αε 5 2 K∈T h ||u i+1 h -u i h || 2 1,K + α 2ε 5 ||v|| 2 L 2 (Ω)
We choose

ε 1 = 8C 1 , ε 2 = 8C 2 , ε 3 = 4C 3 , ε 4 = 8S et ε 5 = 8αS 2 2 to obtain |u -u i+1 h | 2 1,Ω ≤ C K∈T h h 2 K ||f -f h || 2 L 2 (K) + K∈T h e∈E (K) h e ||[ ∂u i+1 h ∂n ]|| 2 L 2 (e) + K∈T h h 2 K ||f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h )|| 2 L 2 (K) + K∈T h ||u i h -u i+1 h || 2 1,K + K∈T h ||u i+1 h -u i h || 2 1,K + 5 16 |v| 2 1,Ω ,
and then

|u i+1 h -u| 1,Ω ≤ C(( K∈T h ( η (D) K,i 2 + h 2 K ||f -f h || 2 L 2 (K) )) 1 2 + ( K∈T h η (L) K,i 2 ) 1 2 ).
We conclude the proof of the theorem.

We address now the efficiency of the previous indicators.

Theorem 4.4. Lower bound. For each K ∈ T h , there holds

η (L) K,i ≤ u i h -u 1,K + u i+1 h -u 1,K , η (D) K,i ≤ C κ⊂ω K ||u -u i+1 h || 1,κ + η (L) κ,i + h κ f -f h L 2 (κ) ,
where ω K is the union of the triangles sharing at least one edge with K.

Proof. The estimation of the linearization indicator follows easily from the triangle inequality by introducing u in η (L) K,i . We now estimate the discretization indicator η

(D)
K,i . We proceed in two steps:

(i) We start by adding and subtracting λ

Ω |u i+1 h | 2p u i+1 h vdx in (4.

6).

Taking v h = 0, we derive

K∈T h K (f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h ))vdx = Ω ∇(u -u i+1 h )∇vdx - K∈T h K (f -f h )vdx + 1 2 K∈T h e∈E(K) e [ ∂u i+1 h ∂n ]vdτ + λ Ω (|u| 2p u -|u i+1 h | 2p u i+1 h )vdx +λ Ω u i+1 h (|u i+1 h | 2p -|u i h | 2p )vdx -α K∈T h K (u i+1 h -u i h )vdx. (4.9)
We choose v = v K such that

v K = (f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h ))ψ K in K 0 in Ω \ K
where ψ K is the triangle-bubble function.

Using Cauchy-Schwarz inequality, (2.1), (4.1) and (4.2) we obtain

f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h ) 2 L 2 (K) ≤ (1 + λC f 2p -1,Ω ) u -u i+1 h 1,K |v K | 1,K + f -f h L 2 (K) v K L 2 (K) +λC u i h -u i+1 h 1,K |v K | 1,K + α||u i+1 h -u i h || L 2 (K) ||v K || L 2 (K) .
Therefore, we derive the following estimate of the first term of the local discretization estimator η

(D) K,i h K f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h ) L 2 (K) ≤ C( u -u i+1 h 1,K +h K f -f h L 2 (K) ) + C η (L) K,i .
(4.10) (ii) Now we estimate the second term of η (D) K,i . Similarly, using (4.9) we infer

1 2 K∈T h e∈E(K) e [ ∂u i+1 h ∂n ]v dτ = Ω ∇(u i+1 h -u)∇vdx + K∈T h K (f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h ))vdx + Ω (f -f h )vdx -λ Ω (|u i+1 h | 2p u i+1 h -|u i h | 2p u i+1 h )vdx -λ Ω (|u| 2p u -|u i+1 h | 2p u i+1 h )vdx + α K∈T h K (u i+1 h -u i h )vdx. (4.11) We choose v = v e such that v e =      L e,κ ∂u i+1 h ∂n ψ e κ ∈ {K, K } 0 in Ω \ (K ∪ K )
where ψ e is the edge-bubble function, K denotes the other element of T h that share e with K (the operator L e,K was introduced above Proposition 4.2).

Using Cauchy-Schwarz inequality, (2.1), (4.3) and (4.4) we derive

h 1/2 e ∂u i+1 h ∂n 2 L 2 (e) ≤ (1 + λC f 2p -1,Ω ) u -u i+1 h 1,K∪K v e L 2 (e) +h e f -f h L 2 (K∪K ) v e L 2 (e) +h e f h + ∆u i+1 h -λ|u i h | 2p u i+1 h -α(u i+1 h -u i h ) L 2 (K∪K ) v e L 2 (e) +C (eta (L) K,ß + η (L) K ,i )||v e || L 2 (e) .
(4.12) Collecting the two bounds above leads to the following estimation

η (D) K,i ≤ C κ⊂ω K ||u -u i+1 h || 1,κ + η (L) κ,i + h κ f -f h L 2 (κ)
These estimates of the local linearization and discretization indicators are fully optimal.

4.2. Algorithm (1.6). The same calculation is followed as before but in (4.6) and (4.7) we have

α K∈T h K ∇(u i+1 h -u i h )∇v instead of α K∈T h K (u i+1 h -u i h )v.
We are led to define the modified discretization error indicator η(D) K,i by

η(D) K,i ) 2 = h 2 K ||f h + ∆u i+1 h -λ|u i h | 2p u i+1 h + α∆(u i+1 h -u i h )|| 2 L 2 (K) + e∈E(K) h e ||[ ∂u i+1 h ∂n -α ∂(u i+1 h -u i h ) ∂n ]|| 2 L 2 (e) .
The rest of the calculation is similar. We skip the proofs since they are exactly the same as for Theorems 4.3 and 4.4.

Theorem 4.5. Upper bound. Let u i+1 h and u be the solution of the iterative problem (1.6) and the exact problem (2.2) respectively. We have the following a posteriori error estimate

|u i+1 h -u| 1,Ω ≤ C K∈T h ( η(D) K,i 2 + h 2 K f -f h 2 L 2 (K) ) 1/2 + K∈T h η (L) K,i 2 1/2 .
Theorem 4.6. Lower bound. For each K ∈ T h , there holds

η (L) K,i ≤ u i h -u 1,K + u i+1 h -u 1,K , η(D) K,i ≤ C κ⊂ω K ||u -u i+1 h || 1,κ + η (L) κ,i + h κ f -f h L 2 (κ) ,
where ω K is the union of the triangles sharing at least one edge with K.

Numerical results

In this section, we present numerical experiments for our nonlinear problem. These simulations have been performed using the code FreeFem++ due to F. Hecht and O. Pironneau, see [START_REF] Hecht | New development in FreeFem++[END_REF]. For all the numerical investigations and for simplicity, we use the finite element of degree = 1.

5.1.

A priori estimation. We consider the domain Ω =] -1, 1[ 2 , each edge is divided into N equal segments so that Ω is divided into N 2 equal squares and finally into 2N 2 equal traingles . We consider the theoretical solution u = e -5(x 2 +y 2 ) .

For the convergence, we use the classical stopping criterion err L ≤ 10 -5 , where err L is defined by

err L = |u i+1 h -u i h | 1,Ω |u i+1 h | 1,Ω .
We consider λ = 10, p = 50 and N = 50.

A posteriori analysis.

In this section, we test our a posteriori error estimates on our model problem. We consider the same domain Ω with the theoretical solution now given by u = e -100(x 2 +y 2 ) . and we choose λ = 10 and p = 50.

In [START_REF] Bernardi | A posteriori analysis of iterative algorithms for a nonlinear problem[END_REF] and for the adaptive strategy, we define the global indicators (introduced in [START_REF] Alaoui | Guaranteed and robust a posteriori error estimate and balancing discretization and linearization errore for monotone nonlinear problems[END_REF]):

η (D) i = K∈T h η (D) K,i 2 1/2 and η (L) i = K∈T h η (L) K,i 2 1/2
, and we introduce two kinds of stopping criteria :

η (L) i ≤ 10 -5 Classical stopping criterion , (5.1) and η
(L) i ≤ γη (D) i New stopping criterion , (5.2)
where γ is a parameter which balances the discretization and linearization errors. We studied in [START_REF] Bernardi | A posteriori analysis of iterative algorithms for a nonlinear problem[END_REF] the comparison between these two types of stopping criterion and we showed the efficiency of the new one which is considered in this paper with γ = 0.001.

For our numerical investigations, we follow the algorithm described in [START_REF] Bernardi | A posteriori analysis of iterative algorithms for a nonlinear problem[END_REF]. The evolution of the meshes with the new stopping criterion looks like the figures 3 and 4 in [START_REF] Bernardi | A posteriori analysis of iterative algorithms for a nonlinear problem[END_REF]. We note that for λ = 10 and p = 50, the algorithm (1.4) diverges.

Figure 2 gives a comparison in logarithmic scale of the error between the uniform and adaptive methods using the algorithms (1.5) and (1.6) with respect of the number of vertices. We can easily see that the algorithms (1.5) and (1.6) give comparable results but the adaptive method is more powerful than the uniform one.

Table 3 shows comparisons, for approximatively the same precision, of the CPU time between the algorithm (1.5) and (1.6) with respect of α. We remark that algorithm (1.5) is faster than (1.6).

In order to have an idea of the constant on the upper bound in theorem 4.3, Table 4 shows the repartition of the error Err and the sum of the indicators err I = ((η

(D) i ) 2 + (η (L) i ) 2 ) 1/2 |u| 1,Ω η (D) i |u| 1,Ω
, during the refinement level and after the convergence on each one. Even if the errors regularly decrease (for instance from 1 to 0.14 for err I ) with respect to the number of adaptive refinement levels which is consistent with adapted mesh method, the constant remains stable and can be approximated by 2.85.

(4. 7 )

 7 We now define the local linearization indicator η (L) K,i and the local discretization indicator η (D)

2 .

 2 and a fixed α = 22 in our algorithms. In fact, for big values of λ and p, the algorithm (1.4) diverges. We mention that for λ and p where (1.5) and (1.6) diverge, we must take a bigger values of α to obtain the convergence. Figure1shows in logarithmic scale the error Err with respect to h (algorithm 1.5 in the left Algo (1.4) 0.0581392 0.0580725 div div div Algo (1.5) 0.0580467 0.058072 0.0581392 0.0581284 0.0582106 Algo (1.6) 0.0580458 0.0580717 0.0581399 0.0581292 0.0582111 Comparison of the convergence of algorithms (1.5), (1.6) (for α = 22) with (1.4). and algorithm 1.6 in the right). The slope of the error corresponding to (1.5) and (1.6) are respectively 0.92 and 0.96, which validates Theorem 3.2.

Figure 1 .

 1 Figure 1. A priori error with respect of h = 1/N : left (algo 1.5) and right (algo 1.6).

Figure 2 .

 2 Figure 2. Error versus number of vertices in logarithmic scale for adaptive and uniform methods with algorithms (1.5) and (1.6).

Table 1

 1

		shows the error
	Err =	|u i h -u| 1,Ω |u| 1,Ω	,

Table 1 .

 1 Convergence of algorithms (1.5) and (1.6) with respect of α.

Table 3 .

 3 ). Comparison of the precision and the CPU time between the algorithm (1.5) and (1.6) with respect of α.

	α		22		30	40		50
	Algo (1.5) time=5.186 s	time=4.952 s	time=5.505 s	time=5.51 s
			error=0.0487858 error=0.0484306 error=0.048256 error=0.0478781
	Algo (1.6) time=56.625 s	time=59.499 s	time=66.231 s	time=61.268 s
			error=0.0475732 error=0.0486143 error=0.0469877 error=0.0494427
	Itn Err	err I	C =	err I Err	Itn Err	err I	C =	err I Err
	1	0.426417 1.31249 3.07796	1	0.443195 1.29673 2.92587
	2	0.169927 0.524469 3.08644	2	0.175596 0.536429 3.05491
	3	0.138091 0.372655 2.69862	3	0.143715 0.38263 2.66242
	4	0.093948 0.278664 2.96615	4	0.0959932 0.290036 3.02142
	5	0.0806374 0.222064 2.75386	5	0.0848493 0.229053 2.69953
	6	0.063787 0.186245 2.91979	6	0.0643159 0.191061 2.97066
	7	0.0583991 0.160167 2.74262	7	0.0577646 0.157273 2.72265
	8	0.049479 0.14235 2.87698	8	0.0474228 0.137726 2.90422

Table 4 .

 4 Repartition of errors and indicators during the refinement levels (Itn) : Left (algorithm (1.5)) and right (algorithm (1.6)).


Julien Roth
email: julien.roth@u-pem.fr

Julian Scheuer
email: julian.scheuer@math.uni-freiburg.de

EXPLICIT RIGIDITY OF ALMOST-UMBILICAL HYPERSURFACES

Keywords: Mathematics Subject Classification. 53C20, 53C21, 53C24, 58C40. Key words and phrases. Pinching, Almost-umbilical hypersurfaces, Hyperbolic inverse mean curvature flow

We give an explicit estimate of the distance of a closed, connected, orientable and immersed hypersurface of a space form to a geodesic sphere and show that the spherical closeness can be controlled by a power of an integral norm of the traceless second fundamental form, whenever the latter is sufficiently small. Furthermore we use the inverse mean curvature flow in the hyperbolic space to deduce the best possible order of decay in the class of C ∞ -bounded hypersurfaces of the Euclidean space.

Introduction

In this paper we prove two stability theorems of almost-umbilicity type, which give an answer to a question raised in [?], generalise the result in [?] to spaceforms and thereby partially improve [?, Thm. 1.3,Thm. 1.4]. Furthermore we use a recent counterexample for the inverse mean curvature flow in the hyperbolic space, cf. [?], to provide a new counterexample for spherical closeness estimates.

Let us shortly introduce the relevant notation. For an oriented hypersurface of a Riemannian manifold, M n → N n+1 , |M | denotes its surface area, g its induced metric, A its second fundamental form, Å the traceless part of A, (1.1) Å = A -Hg,

x M the center of mass of M and d H the Hausdorff distance of sets.

For a tensor (T j1...j l i1...i k ) on M, we define its L p -norm to be (1.2) T p = ˆM |T j1...j l i1...i k T i1...i k j1...j l | p 2 1 p

, where indices are raised or lowered with the help of g. Let us formulate our first main result.

1.1. Theorem. Let M → R n+1 be a closed, connected, oriented and isometrically immersed C 2 -hypersurface with |M | = 1. Let p > n ≥ 2. Then there exists a constant 0 > 0 depending on n, p and A p , as well as a constant α = α(n, p) ≤ 1, such that whenever there holds

(1.3) Å p ≤ H p 0 , then (1.4) d H (M, S R (x M )) ≤ c α H α p Å α p ≡ α
and M is α -quasi-isometric to a sphere S R with a certain radius R.

By α -quasi-isometric, we mean that the diffeomorphism from M into S R (p) satisfies

(1.5) |dF x (u)| 2 -1 α for any x ∈ M , u ∈ T x M and |u| = 1.
The assumption |M | = 1 is only for simplification. By scaling it is easy to obtain a scale-invariant version of (1.4) for arbitrary M.

In section 3 we generalize this theorem to conformally flat ambient spaces.

The history of the problem to control the closeness to a sphere by curvature quantities is quite long, starting from the well known Nabelpunktsatz. We refer to the bibliography in [?] for a quite detailed overview. Let us only mention several results which have appeared recently. For surfaces, n = 2, a quite straightforward calculation due to Andrews yields an explicit C 0 -estimate for convex hypersurfaces, cf.

[?, Prop. 4, Lemma 5],

(1.6)

x -q, ν -

1 8π ˆM H ≤ C|M ||κ 1 -κ 2 |,
where x is the embedding vector and q is the Steiner point. In section 4 we use the inverse mean curvature flow (IMCF) in the hyperbolic space to prove that the power on the right-hand side of (1.6) can not be improved to α > 1, which is in turn then not possible either in Theorem 1.1. The latter proof relies on a recent example due to Hung and Wang, [?, Thm. 1, Prop. 5], that the convergence after rescaling in the IMCF can not be too fast in the hyperbolic space.

For strictly convex hypersurfaces of R n+1 there is the following estimate of circumradius R minus inradius r due to Leichtweiß,cf. [?,Thm. 1.4,eq. (38)]:

(1.7) R -r ≤ c n max x∈M (R n (x) -R 1 (x)),
where R 1 ≤ • • • ≤ R n are the ordered radii of curvature. However, Theorem 1.1 deals with estimates in dependence of integral pinching. For the case n = 2, an estimate similar to (1.1) with a better constant was obtained by De Lellis and Müller, cf. [?] In [?, Cor. 1.2] Perez derived a qualitative solution and obtained under certain assumptions, for given > 0, a δ > 0, such that

(1.8) Å p < δ implies (1.9) d H (M, S r0 (x)) < .
In [?, p. xvi] the author posed the derivation of an explicit δ as a question of interest.

The second author of the present paper derived a quantitative δ within the class of strictly convex hypersurfaces in [?, Thm. 1.1]. There it is shown that a δ which has the order 2+α , α > 0 provides (1.9). However, the proof in [?] relies on the convexity of the hypersurface. Unfortunately, also in the paper at hand we did not achieve a constant independent on the size of the curvature itself. The constant is only uniform in the class of hypersurfaces with a fixed bound on the curvature of the scaled hypersurface.

The following theorem, due to Grosjean and the first author, [?, Thm. 1.4], already provides this conclusion, however only with the additional assumption of smallness of the oscillation of the mean curvature itself:

1.2. Theorem. [?, Thm. 1.4] Let (M n , g) be a compact, connected and oriented n-dimensional Riemannian manifold without boundary isometrically immersed by φ in R n+1 . Let p be the center of mass of M. Let < 1, r, q > n, s ≥ r and c > 0. Let us assume that |M |

1 n H q ≤ c. Then there exist positive constants C = C(n, q, c), α = α(q, n),

such that if α ≤ 1 C , (1.10) Å r ≤ H r and (1.11) H 2 -H 2 s r 2 ≤ H 2 r , then M is α -Hausdorff close to S 1 H 2 (p). Moreover if |M | 1 n A q ≤ c, then M is diffeomorphic and α -quasi-isometric to S 1 H 2 (p).
Note that in this theorem, L p -norms are defined slightly different, namely such that the L p -norms of scale-invariant functions are scale-invariant. Our notation corresponds to the one in [?]. This ambiguity does not cause any problems, since we prove Theorem 1.1 for |M | = 1. Also note the typo in [?, Thm. 1.4], where the α is missing in the conclusion.

In [?,Thm. 3.1], which also covers other ambient spaces, (1.11) was replaced by an assumption on the gradient of H. However, with the help of the following theorem due to Perez it is possible to get rid of (1.11) completely. 1.3. Theorem. [?,Thm. 1.1] Let p > n ≥ 2 and c 0 > 0 be given. Then there is a constant C > 0, depending only on n, p and c 0 , such that: If Σ ⊂ R n+1 is a smooth, closed and connected n-dimensional hypersurface with

(1.12) |Σ| = 1 and (1.13) A p ≤ c 0 , then (1.14) min λ∈R A -λg p ≤ C Å p .
The proof of Theorem 1.1 is a combination of Theorem 1.2 and Theorem 1.3.

Proof of Theorem 1.1

Without loss of generality we may suppose that M is of class C ∞ , since both sides of the inequality are continuous with respect to the C 2 -norm and hence the general result can then be achieved by approximation.

Using [?, Thm. 1.1], we obtain a λ 0 ∈ R, such that

(2.1) A -λ 0 g p ≤ C Å p , where C = C (n, p, A p ). Let us calculate (2.2) H 2 -H 2 p p 2 ≤ H 2 -λ 2 0 p 2 + λ 2 0 -H 2 p p 2 = ˆM |H -λ 0 | p 2 |H + λ 0 | p 2 2 p + |λ 2 0 -H 2 p | ≤ 2( H p + |λ 0 |) H -λ 0 p ≤ 2 √ n ( H p + |λ 0 |) A -λ 0 g p ≤ c H p Å p ,
where c = c (n, p, A p ). The last inequality is due to the fact that

(2.3) |λ 0 -H p | ≤ c Å p . Defining (2.4) c = max(1, c ), (2.5) = c Å p H p ,

and

(2.6)

0 := min 1, C -1 α 2c then by (1.3), (2.7) ≤ c 0 = 1 2 min 1, C -1 α ,
where α and C are the constants from [?, Thm. 1.4]. Furthermore we have

(2.8) Å p ≤ H p

and

(2.9)

H 2 -H 2 p p 2 ≤ H 2 p .
Thus we may apply [?,Thm. 1.4] to conclude that M is α -close to a sphere.

Generalization to conformally flat manifolds

Using that the property of a submanifold to be totally umbilic is conformally invariant, we easily obtain the following generalization to conformally flat manifolds, which in particular include the space forms and improves the α -proximity statement in [?, Thm. 1.3] in the sense that it removes assumption (2).

3.1. Theorem. Let Ω ⊂ R n+1 be open and let N n+1 = (Ω, ḡ) be a conformally flat Riemannian manifold, i.e.

(3.1) ḡ = e 2ψ g,
where g is the Euclidean metric and ψ ∈ C ∞ (Ω). Let M n → N n+1 be a closed, connected, oriented and isometrically immersed C 2 -hypersurface. Let p > n ≥ 2.

Then there exist constants c and 0 , depending on n, p, |M |, A p and ψ ∞,M , as well as a constant α = α(n, p), such that whenever there holds

(3.2) Å p ≤ 0 ,
there also holds

(3.3) d H (M, SR ) ≤ c Å α p ,
where SR 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 ḡ.

3.2. Remark. Since in conformally flat spaces the scaling behaviour of the second fundamental form with respect to homotheties heavily depends on the nature of the ambient space, in this case there seems to be no way to give a general scale invariant estimate. This is the reason why this closeness estimate is only uniformly valid in the class of C 2 -bounded hypersurfaces. Furthermore note that for example in all space forms the hypersurface SR is actually a geodesic sphere. This follows from the fact that in those spaces totally umbilical hypersurfaces are spheres and total umbilicity is conformally invariant, as will be appearant from the following proof of Theorem 3.1. Thus Theorem 3.1 gives an explicit spherical closeness estimate of almost-umbilical hypersurfaces in the hyperbolic space as well as in the half-sphere of constant positive sectional curvature.

Proof. Under a conformal relation of the metrics as in (3.1) the corresponding induced geometric quantities of the the embedded hypersurface M are related as follows.

(3.4)

g ij = e 2ψ gij and (3.5) h ij e -ψ = hij + ψ β νβ gij .
Those formulae can be found in [?, Prop. 1.1.11]. Hence

(3.6) h ij -Hg ij = e ψ ( hij -H gij )
and hence

(3.7) c Å p ≤ Å p ≤ C Å p ,
where the constants depend on ψ ∞,M . Since the Euclidean and the conformal Hausdorff distances are equivalent whenever |ψ| is bounded, we obtain the result after applying Theorem 1.1.

Due to a well known interpolation theorem for convex hypersurfaces of Riemannian manifolds we obtain the following gradient stability estimate in space forms.

3.3. Corollary. Let N n+1 be a space form and M as in Theorem 3.1 be additionally strictly convex, where we also assume that ḡ is given in geodesic polar coordinates

(3.8) ḡ = dr 2 + ϑ 2 (r)σ ij dx i dx j ≡ dr 2 + ḡij dx i dx j
with suitable ϑ depending on the space form. Let p > n. Then there exist constants c and 0 depending on n, p, |M |, A p and ψ ∞ , as well as a constant α = α(p, n), such that

(3.9) Å p ≤ 0 implies (3.10) v = 1 + ḡij u i u j ≤ e c Å α p ,
where

(3.11) M = {(x 0 , x i ) : x 0 = u(x i ), (x i ) ∈ S 0 }
is a suitable graph representation over a geodesic sphere S 0 → N n+1 and (ḡ ij ) is the inverse of (ḡ ij ).

Proof. It is well known that a strictly convex hypersurface of S n+1 is contained in an open hemisphere, cf.

[?] for the smooth case and also [?, Cor. 1.2] for the C 2 -case. Thus in any space form a strictly convex M is covered by a conformally flat coordinate system as in Theorem 3.1, which is thus applicable. Let S 0 be the corresponding sphere with center p, then we can write M as a graph over S 0 due to the strict convexity. Thus we may apply the well-known interpolation estimate (3.12) v ≤ e κ osc u , cf. [?, Thm. 2.7.10], where κ is a lower bound for the principal curvatures of the coordinate slices {r = const}. The latter, however, only depends on ψ ∞ as well.

3.4. Remark. Note that in the Euclidean case the assumption of the strict convexity is redundant, if (3.9) is satisfied with p = ∞, since then it is possible to deduce strict convexity from (3.9), cf. [?, Lemma 2.2].

An optimality result

We prove the optimality of the the statement in Theorem 1.1 in the sense that there is no hope to derive a uniform estimate of the form

(4.1) d H (M, S R (x 0 )) ≤ c Å α ∞ , α > 1,
in the class of uniformly C ∞ -bounded hypersurfaces M. To be precise, for α > 1 we get the following negation of (1.6) in the class of uniformly convex hypersurfaces and for all n ≥ 2. 4.1. Theorem. Let n ≥ 2 and C = 2 max(|S 2 (0)|, ĀS2 ∞ ). For all α > 1 and for all k ∈ N there exists a uniformly convex smooth hypersurface

M k → R n+1 with (4.2) max( A k ∞ , |M k |) ≤ C, such that (4.3) Åk ∞ < 1 k and for all spheres S ⊂ R n+1 there holds (4.4) d H (M k , S) > k Åk α ∞ .
Here ĀS2 denotes the second fundamental form of the sphere with radius 2.

In a recent paper, Drach gave a counterexample to an improved spherical closeness estimate in the class of C 1,1 hypersurfaces, namely a special spindle shaped hypersurface, cf. [?, Thm. 1]. However since, we consider (1.4) in the space of at least C 2 -hypersurfaces, we need to find a different contradiction to (4.1). This contradiction is deduced along the inverse mean curvature flow in the hyperbolic space.

Before we prove Theorem 4.1, let us for convenience recall the relevant facts about the inverse mean curvature flow in the hyperbolic space H n+1 . There one considers a time parameter family of embeddings of closed, starshaped and mean-convex hypersurfaces (4.5)

x

: [0, T * ) × M → H n+1 , which solves (4.6) ẋ = 1 H ν,
where H = g ij h ij and ν is the outward unit normal to M t = x(t, M ). Note that we have switched the notation of H in this context due to a better comparability with the literature. It is known, cf. [?, Lemma 3.2], that for an initial starshaped and mean-convex hypersurface M 0 the flow exists for all times and all the flow hypersurfaces can be written as a graph over a fixed geodesic spere S 0 , (4.7)

M t = {(x 0 , x i ) : x 0 (t, ξ) = u(t, x i (t, ξ))},
where u describes the radial distance to the center of S 0 . In [?, Thm. 1.2] Gerhardt claimed to have shown convergence of the rescaled hypersurfaces

(4.8) Mt = graph û ≡ graph u - t n
to a geodesic sphere. However, as was pointed out in [?, Thm. 1] with the help of a concrete counterexample, the limit function of û is not constant in general. In particular the authors proved that there is a starshaped and mean-convex initial hypersurface M 0 , such that the limit hypersurface is not of constant curvature, in particular not a geodesic sphere. However, there is a smooth limit function to which the Mt converge smoothly, compare the proof of [?,Thm. 6.11] and also compare [?, Thm. 1.2].

In order to relate the convergence results of the IMCF in the hyperbolic space with the rigidity estimate (1.4) in the Euclidean space, we have to look at the hyperbolic flow in the conformally flat model. In [?] the Poincaré ball model in the ball of radius two was considered. Let r denote the geodesic distance to the center of S 0 in H n+1 , then the by the coordinate change (4.9) ρ = 2 -4 e r + 1 the representation of the hyperbolic metric transforms like The proof of Theorem 4.1 is very similar to the proof of a corresponding positive result in this direction by the second author. In [?] he proved that due to a strong decay of the traceless second fundamental form along the IMCF in R n+1 we indeed obtain spherical roundness in this case without rescaling. The idea how to obtain a negative result in the hyperbolic space is that if we could improve the spherical closeness, then we could mimic the proof in [?] to deduce a roundness result in H n+1 , which is not possible in view of Hung's and Wang's paper.

(4.10) ḡ = dr 2 + sinh 2 (r)σ ij dx i dx j = 1 1 -1 4 ρ 2 2 (dρ 2 + ρ 2 σ ij dx i dx j ) ≡ e 2ψ g,
The idea of the proof of Theorem 4.1 goes as follows: The estimate in (4.12) provides closeness of the flow hypersurfaces to the sphere of radius 2 in the ball model. The order of the closeness is e -t n . The traceless second fundamental form decays correspondingly, as we will point in more detail later in the proof. But if we had this additional exponent α in the spherical closeness estimate, we could even deduce better spherical closeness (to a sphere different from S 2 ) than we have in (4.12) and then we would be able to translate this to a spherical closeness in the hyperbolic space. This would in turn yield a contradiction to Hung's and Wang's result. Now let us prove Theorem 4.1 in detail. First we need some helpful notation and an auxillary result. 4.2. Definition. (i) Let N be either the Euclidean space, the hyperbolic space or an open hemisphere. For a starshaped hypersurface M → N, let M * be the set of points in N, with respect to which M is starshaped. we denote the oscillation of the geodesic distance of the point (u, x i ) to the point p. Here S p denotes a geodesic sphere around p.

By a simple argument we obtain the following alternative for a general expanding sequence of hypersurfaces with controlled oscillation. osc q u(t, •) ≤ ce 1-α n t ∀t ≥ t 0 .

Let t k be a sequence of times with t k → ∞. Due to the compactness of Bδ (p) a subsequence of center points converges, (4.40) q k → q ∈ Bδ (p), where we did not rename the index of the sequence. Since (4.41) | osc q k u(t k , •) -osc q u(t k , •)| ≤ 2 dist(q k , q) ∀k ∈ N, we obtain (4.42) osc q u(t k , •) → 0, k → ∞.

In view of (4.36) and the preservation of starshapedness along IMCF the assumptions of Lemma 4.3 are fulfilled. Applying Lemma 4.3, we obtain that (4.43) osc q (t, •) → 0, in contradiction to the choice of the initial hypersurface.

4.4. Remark. Note that in turn of the proof we even have shown that for given α > 1 and k ∈ N as in Theorem 4.1, such a counterexample M k satisfying (4.3) and (4.4) must actually occur along the inverse mean curvature flow in the conformally flat version of the IMCF in H n+1 . We only used our contrary assumption within this class of flow hypersurfaces.

Concluding remark

We would like to point out that the techniques in section 4 might be useful in other situations. Whenever one would like to estimate the closeness to a sphere in comparison with another geometric quantity, e.g. in comparison with eigenvalue pinching of the Laplacian or also in almost-Schur/almost-CMC type estimates, one could determine how this particular geometric quantity behaves along the IMCF and then determine the best possible roundness estimate using the IMCF in H n+1 . It should often be quite straightforward to derive the best possible decay estimate.

  where σ ij is the standard round metric of the sphere S 0 . is a strictly positive function, due to [?, Lemma 3.1].

(

  ii) For a starshaped hypersurface M → N let p ∈ M * . Then for the graph representation(4.14) M = {(r, x i ) : r = u(x i ), (x i ) ∈ S p },

  4.3. Lemma. Let N be as in Definition 4.2 and M t → N, 0 ≤ t ∈ R, be a family of starshaped hypersurfaces such that and such that for all t 0 ∈ N and p ∈ M * t0 there exists a constant c independent of t 0 , such that (4.17) osc p u t ≤ c osc p u t0 ∀t ≥ t 0 .Then for fixed p, osc p u t does not have zero as a limit value for t → ∞ unless for some suitable sphere S ⊂ R n+1 . According to [?, Thm. 1] for n = 2 and [?, Sec. 4] for n ≥ 3 there exists a starshaped and mean-convex hypersurface M 0 → H n+1 , Now let us switch back to the hyperbolic space. The spheres S Rt are geodesic spheres in H n+1 as well since total umbilicity is preserved under a conformal transformation and in the Euclidean space as well as in the hyperbolic space for closed and embedded hypersurfaces total umbilicity is tantamount to being a geodesic sphere. We denote these spheres in H n+1 by S Rt .Since the inradius of the M t converges to infinity and for large t the M t are strictly convex, for each δ > 0 we find t 0 > 0, such that •) ≤ c osc q u(t 0 , •) ∀t ≥ t 0 . Bδ (p) ∀t ≥ t 0 , because outside Bδ (p) the oscillation is already larger than it is with respect to p.

	(4.16) (4.18) Proof. For given > 0, if zero is a limit point, we may choose t 0 , such that M * t ⊂ M * s ∀s ≥ t osc p u t → 0, t → ∞. (4.19) osc p u t0 ≤ c , then (4.20) osc p u t ≤ c osc p u t0 ≤ ∀t ≥ t 0 . Now we can prove Theorem 4.1. Proof. Assume the contrary, i.e. that there exist α > 1 and k ∈ N, such that for all uniformly convex hypersurfaces M → R n+1 with (4.21) max(| M |, Ã ∞ ) ≤ C we have that (4.22) Å ∞ < 1 k implies such that for no graph representation (4.24) M t = graph u the rescaled IMCF flow hypersurfaces (4.25) Mt = graph u -t n ≡ graph û converge to a geodesic sphere. However, for each graph representation, we obtain smooth convergence of (4.26) û → û∞ . In [?, Thm. 1.2 (2)] it is deduced that (4.27) Å ∞ ≤ ce -2t n , where c = c(n, M 0 ). Now fix a graph representation around p ∈ M * 0 . From (3.4) and (3.5) we obtain that the corresponding Euclidean traceless part decays like (4.28) Å ∞ = e ψ Å ∞ ≤ e ψ max e -2t n , where (4.29) e ψ max = 1 1 -1 4 w 2 max with w as in (4.13) and (4.30) w max = max Due to (4.12) we obtain (4.31) Å ∞ ≤ ce -t n and due to the C ∞ -convergence of w → 2, we are in the situation to apply our assumption (4.23), whenever t is large enough. We obtain a sequence of spheres S Rt ⊂ R n+1 , such that (4.32) dH ( Mt , S Rt ) ≤ ce -α n t , where the Hausdorff distance is measured with respect to the Euclidean metric. Due to (4.12) we even have (4.33) S Rt ⊂ B 2 (0), for large times t. For the corresponding hyperbolic Hausdorff distance we deduce (4.34) Bδ (p) ⊂ M * t0 ⊂ M * t ∀t ≥ t 0 , where the latter inclusion is due to the fact that starshapedness around a given point is preserved. According to [?, Prop. 3.2, Lemma 3.5], there holds for the oscillation of u that for all t 0 and all q ∈ M * t0 we have (4.36) osc q u(t, So in particular, if we choose (4.37) δ = c osc p u(0, •), we find that the oscillation of each M t is minimized within the set Bδ (p) : (4.38) argmin q∈M * Due to (4.34) we obtain d (4.35) (4.39) osc
	(4.23)	dH ( M , S) ≤ k Å α ∞

x∈Sp w(x). H (M t , S Rt ) ≤ e ψ max dH ( Mt , S Rt ) ≤ ce 1-α n t ,

which converges to 0 as t → ∞. t

osc q u(t, •) ∈ qt u(t, •) = min q∈ Bδ (p)


Bernard Kamsu-Foguem
email: bernard.kamsu-foguem@enit.fr

Pierre F Tiako

Laure P Fotso

Clovis Foguem

Modeling for effective collaboration in telemedicine

Keywords:

Telemedicine is a remote medical practice, which utilizes advanced telecommunications and information technologies for the delivery of healthcare and the exchange of heath information across distances. The development of telemedicine has significantly changed the medical collaborative decision making and doctor-patient relationships and has an impact on the responsibilities of physicians to patients and how to treat them. Moreover, in the chain of medical care information exchanged between a requesting physician and medical expert should assist in decision making. In this regard, we propose means for the formalization of exchanges, which is very important because it facilitates a transparent and traceable understanding of the remote process. So, we engage knowledge-based modeling for supporting collaborative activities in telemedicine. This includes the engagement of formal ontologies to ensure structuration of terminology and identification across all entities in a domain so that multiple sources of data can be aggregated through comparable reference terms. The provided benefit is the generated support for logic-based intelligent applications that are able to perform complex reasoning tasks such as checking for errors and inconsistencies and deriving logical inferences.

Introduction

The European Union (EU) and its European Commission (EC) have implemented policies, models with conceptual and technical guidelines for supporting and encouraging research projects related to telemedicine. In addition, the draft directive on patients' rights in cross-border healthcare advocates cooperation in health including the establishment of referral networks, evaluation of medical technologies, data collection, quality, and safety [START_REF] Saliba | Telemedicine across borders: a systematic review of factors that hinder or support implementation[END_REF]. These projects include the creation of a network of Community Technology Assessment health centers to promote collaboration between national health authorities and the provision of timely, objective, reliable, transparent, and transferable solutions. Sharing of health technologies, in the short and long term, enable effective exchange of information within the network and support the decision making by EU Member States related to the effectiveness and sustainability of actions related to improving and controlling quality patient care. This objective has been implemented successfully in recent years with cross-border teleneuromonitoring collaboration between hospitals in the Netherlands, Germany, and Switzerland [START_REF] Doering | A success-story in cross-border telemedicine in Europe: the use of intra-operative teleneuromonitoring during aorta surgery[END_REF].

Especially with a view to the widespread deployment of telemedicine services in Europe, it is necessary to rely as much as possible on standards, infrastructure, and exchange formats. The European Community organizes the implementation of interoperability standards on three levels:

High-level: national or regional projects (e.g., the interoperability framework of ASIP Santé-the shared healthcare information systems agency, European projects such epSOS, adapted to the French context).

Intermediate level: integration Profiles to address key issues (e.g., IHE, Continua Health Alliance). Low level: standards (e.g., HL7, DICOM, IEEE, W3C, Oasis, etc) and interfaces between systems.

The aims of this study include ensuring the structuring of exchanges (e.g., clinical data associated with the request for an opinion, formalizing the response of the expert sought by means of guidelines assuring collaboration, identification of the professionals involved), and guaranteeing security (confidentiality, integrity, traceability, centralized archiving).

The rest of the paper is structured as follows. Section 2 exposes the collaboration in telemedicine with activities and actors involved and types of exchanged data and challenges for data sharing in telemedicine. Section 3 presents requirements and modeling for collaboration in Telemedicine through ontology modeling for semantic specification and knowledge capitalisation from remote collaborative expertise for the effective deployment of telemedicine. Finally, Sections 4 and 5 conclude and discuss lessons learned and future challenges.

Collaboration in telemedicine practices: state and necessities

Activities and actors involved

Telemedicine is a remote medical practice, which utilizes advanced telecommunications and information technologies for the delivery of healthcare and the exchange of heath information across distances, including the following medical activities (see Fig. 1) [START_REF] Simon | Le décret français de télémédecine: une garantie pour les médecins[END_REF]:

Tele-consultation, a procedure whereby medical professionals can consult a patient remotely and interpret the necessary data remotely for medical follow-up;

Tele-expertise, whereby a medical professional can seek remotely an opinion of other medical professionals who have the relevant training or skills;

Tele-monitoring, the ability to monitor and supervise patients remotely; Tele-assistance, a procedure, which enables a medical professional to assist remotely another healthcare professional during the realization of a medical act.

Telemedicine is becoming an integrated tool in the delivery of health care and will soon be a part of mainstream medicine, but the practice of telemedicine is difficult and complex. Telemedicine assists collaborating medical professionals with shared decision-making in medical diagnosis and treatment. Referring (requesting) physicians should involve patients in the decision to seek further medical actions involving the choice of a specialist/consultant, as a starting point for more in-depth investigations into underlying causes. The procedure allows the specialist to investigate more thoroughly the compatibility

Nurse Treating Physician Midwife

Patient

Tele-monitoring Tele-expertise

Requesting Physician

Required Physician

Patient's medical record of the medical scheme with the specific situation and to pursue discussions with the referring (requesting) physicians about the proposed management plan including supposed diagnosis and corresponding treatments. The requesting physician must distinguish groups of patients with similar symptoms (e.g., fever, abdominal pain) and benign disease, from those with more serious health issues who require immediate treatment. It is the duty of the requesting physician to provide the patient with the information necessary for a free and informed consent. For their part, the consulting (required) physician must ensure that he/she has a in hand all the necessary information since the consulting physician is responsible for the diagnosis and treatment formula recommended.

Tele-consultation

Tele-assistance

In many cases, the final diagnosis will be made only secondarily on the basis of a clinical evaluation or the result of further examination, while the prognosis of patients with the most serious disorders may be evident early on. Physicians have the choice either minimizing or maximizing diagnostic procedures (e.g., biology, imaging), providing treatment (e.g., antibiotics, hospitalization, monitoring) or some mixed procedure (e.g., diagnostic or therapeutic laparotomy). A strategy of minimizing procedures reduces the invasiveness of care, risks (e.g., resistant bacteria, nosocomial infections, adverse drug reactions, and the vagaries of interventions) and their costs, but with a hazard of diagnostic and therapeutic delays whose ethical and medico-legal consequences can be cumbersome. In contrast, a strategy for maximizing procedures reduces the risk of missing a severe condition in its early stages (e.g., a disseminated bacterial infection, appendicitis, or appendicular peritonitis); thus, also reducing the individual and collective risks, as well as, costs as described above. Medical decisions are the result of some functions of logistical, ethical, legal, and security constraints and these processes may be conscious or subconscious. To inform the medical decision making process, the clinician may rely on the opinion of other doctors. When it is formalized in a clear set of rules that are conveyed to and understood by medical staff, this opinion may take the form of a clinical protocol of care, practice guidelines, or consensus conferences. These forms of decision aids are based on expert opinion and analysis of the literature in a more or less structured way. Telemedicine is a new kind of decision aid that has experienced sustained development and confirmed its capacity to combine that progress with high quality.

Collaboration is needed among physicians or between physicians and other healthcare professionals involved in telemedicine protocols in accordance with the areas of different and complementary skills so that everyone is responsible for his/her actions and decisions. Several models of tele-consultation are possible depending on the nature of the teleconsultation and location, whether the medical service is public or private, and where services are consumed and where they are realized. When responsibilities are involved we distinguish: tele-consultation among public health care (PHC) institutions, where the patient is a user and medical officers in the public service; tele-consultation performed among health facilities (HF) where the patient has a private contractual relationship with the requesting physician and the physician required; and tele-consultation between PHC institutions, where the patient is hospitalized and a physician who is practicing in a private facility.

In the organization of care, PHC hospitals are responsible for adoption and usage of telemedicine made by doctors. Cooperation among the PHC is a public policy function comprising public service requirements and a social function, involving innovative, cost-effective and diversified organization models. Insofar as the public service mission covers new services, there is, in principle, the same requirement of adaptability to the best technologies and the best care. There is no need to conclude a cooperative agreement among public hospital units and networks to develop telemedicine if it provides better care. It is, therefore, the responsibility of the public practitioner to share their skills with their peers. Telemedicine is a practical tool that allows practitioners to pool their resources, share expertise, exchange ideas, and undertake joint action. However, in every case, for the collaboration provided for by telemedicine to apply, the public service mission needs to be clearly defined and must be explicitly entrusted through an act of public authority (including contracts or cooperative agreements).

Type of data exchanged and challenges for data sharing in telemedicine

Telemedicine applications, which are used to establish a diagnosis and monitor patients for preventive and curative care, require expert advice to develop a therapeutic decision, to prescribe treatment, recommend or perform services or other acts, or monitor the patient's condition. It is in this context that telemedicine has been developed. It allows organized networks to share information and skills, improve the management of patients, both for emergency care and chronic diseases (such as, cardiovascular diseases, diabetes mellitus, respiratory diseases, chronic renal failure (CRF), cancer, nervous system disorders, musculoskeletal disease (MD) or multi-morbid patients) (van den [START_REF] Van Den Berg | Telemedicine and telecare for older patients-a systematic review[END_REF]. The types of telemedical intervention include: measurement of vital signs using telemedical devices, interactive systems, or personal interaction between healthcare provider and patient.

Among other medical acts, telemedicine induces the same obligations for the physician, and patients benefit from the same rights as in a conventional medical practice. Non-compliance may engage professional oversight authorities.

Telemedicine aims are indispensable as the trend of the medical profession grows increasingly towards specialties and subspecialties. Now a patient visiting a general practitioner can consult a specialist via telecommunication facilities, saving both time and money for quality medical care as opposed to the method of transporting patients to specialists for face-toface visits [START_REF] Ekeland | Effectiveness of telemedicine: a systematic review of reviews[END_REF].

Telemedicine highlights mainly two categories of equal importance [START_REF] Adewale | An internet-based telemedicine system in Nigeria[END_REF]:

Increasing diagnostic complexity often requires peer-to-peer consultation based on data (e.g., images), i.e., tele-consultation. Both, a tele-consultation of an expert or of a knowledge based system, are conceivable.

Telemedicine is a tool to ensure equity of healthcare for those people living in rural and remote areas having specialised needs without convenient access to medical services compared to those in urban areas as well as the aspect of home care for specific groups such as the elderly.

Several examples can illustrate the importance of clinical research in telemedicine. In type 1 diabetes, after the first inconclusive results due to insufficient or weak methodologies, the French study TeleDiab 1 brought for the first time a clinical improvement over 6 months in patients followed-up remotely by Diabeo software and tele-consultation [START_REF] Franc | Telemedicine and diabetes: achievements and prospects[END_REF]. This software enables individualized insulin dose adjustments combined with telemedicine support [START_REF] Charpentier | The Diabeo software enabling individualized insulin dose adjustments combined with telemedicine support improves HbA1c in poorly controlled type 1 diabetic patients: a 6-month, randomized, open-label, parallel-group, multicenter trial (TeleDiab 1 Study)[END_REF] and the main medical service to patients was a decreased risk of complications. In the use of telemedicine for chronic heart disease, the results of two meta-analyses showed a significant 30% reduction in morbidity and mortality [START_REF] Clark | Telemonitoring or structured telephone support programmes for patients with chronic heart failure: systematic review and meta-analysis[END_REF][START_REF] Polisena | Home telemonitoring for congestive heart failure: a systematic review and meta-analysis[END_REF] (see Tables 1 and2).

Telemedicine opens up significant opportunities for progress in key medicine areas. The goal is to give patients access to care throughout the nation with real professional interactions among remote medical teams. Telemedicine facilitates the development of real networks where the notion of complementarity and multidisciplinarity become sources of efficiency, quality, and safety. Establishing a network of institutions and health professionals promotes the formation of care channels. Telemedicine removes barriers to collaboration to ensure a better understanding of the interaction of medical programs across country, continent, or world and as it fits into the medical organization's overall service structure. The telemedicine approach is important because health care organizations are moving away from a facility-specific perspective with paperbased records to one that focuses more on considerations relevant to the health care system as a whole with electronic records (see Fig. 2). The Personal Medical Record (PMR) is a tool for telemedicine, which stores all data, reviews, images and consultations in a secured manner.

The development of telemedicine has significantly changed the doctor-patient relationship and has an impact on the responsibility of physicians to patients and how to treat them. The legal consequences of telemedicine are manifold, indeed the responsibility of different stakeholders is difficult to determine and, in general, each physician (whether expert or applicant) must perform certain essential actions. Moreover, in the chain of medical care information exchanged between a requesting physician and medical expert should assist in decision making. In this regard, formalization of exchanges is very important because it facilitates a transparent and traceable understanding of the telemedical process. This requires the definition of a suitable data exchange format adapted to the needs of performance assessment and secure information management. Therefore, confidentiality must be respected, and protection should be guaranteed by law through ethical principles and formal rules concerning the information to be shared, the procedures for sharing it, and the security of confidential information [START_REF] Chopard | Legal, deontological and ethical issues applied to telemedicine. A few insights about telestroke[END_REF]. A balance must be found between the information that the patient wishes to share and that that must be kept private. To organize the legal framework for telemedicine, structures and regulations aimed at identifying the key confidentiality requirements of all system procedures are needed [START_REF] Pattynama | Legal aspects of cross-border teleradiology[END_REF]. Mastering technology issues or providing training remain essential matters for implementation of telemedicine, but legal issues may constrain practical deployment [START_REF] Siegal | Telemedicine: licensing and other legal issues[END_REF]. Health information technology will have to define new management tools that will be required of the information society to allow the health delivery system to adapt to new fields of application, as to maintain the humanist dimension of telemedicine that favors neither excessive centralization nor sectorial specialization.

Requirements and modeling for collaboration in telemedicine

Functional requirements for the effective deployment of telemedicine

Basically, some needs associated with endeavoring to complete such a telemedicine activity involve functional requirements of five types (DGOS, 2012): general organization of the telemedicine activity, remote interaction and exchange (audio/video), temporary sharing of medical data required for an execution of the telemedicine activity, transmission of medical documents useful for an execution of the telemedicine activity, sharing/exchange of useful documents for coordinating cares.

For each proposed establishment of a telemedicine activity at the regional level, the macro-functional requirements should be detailed and adapted to the local environment. The provided information technology system would enable a Synchronous collaboration model with loose coupling: whereby participants use the same application (preliminary remote loading or model in which software and associated data are centrally hosted on the cloud) to access the same information and documents without coupling vision (each actor can move in the document and place the screen on a different page of the other actors involved in the activity).

Asynchronous collaboration model can also be allowed in some situations, such as in reading radiology reports. The request for consultations, sharing of medical information and any action to be taken are asynchronous. A document (patient record, a shuttle service, etc) must trace and draw information exchanges. The exchanges of images are made in an asynchronous form, while audio and video exchanges are synchronous.

An application may propose these models to adapt to the needs of collaboration depending on the nature of the shared information. This action will have to be implemented at both regional and transnational level in order to adapt to the conditions of comprehensive interoperability and collaboration in this field. In the majority of current cases (projects in the experimental phase or still in a development phase), a large number of suppliers ensured a sufficient level of support of their equipment/software, but the operational support (first level) is provided by medical/paramedical staff, involved in collaboration with suppliers of technological and virtualization solutions.

Ontology modeling for semantic specification

A collaborative approach for telemedicine management will allow stakeholders to work together and lead to better decision-making processes in integrating the knowledge of each business domain.

Productive exchanges between health professionals, officials, and operators are a prerequisite for the harmonious development of this new form of medical practice. The transmission of medical data and the possibility for several distant physicians to share their knowledge about a given medical case raise several conceptual and technical questions. Among them there is the problem of the interoperability of the medical information. Finding solutions to interoperability is mandatory to ensure the reliability and sustainability of telemedicine. This study is focused on semantic interoperability that concerns mutual understanding of terminologies and concepts used, but also performing the intended actions comprehensively and safely.

The conceptual semantic interoperability must ensure that the information exchanged is understandable from the point of view of its meaning and its interpretation by the applications that use this information but have been developed for different purposes. The challenge consists in semantic harmonization and the use of concepts and services respecting common rules, constraints, and common conditions. In particular, semantic interoperability will involve understanding the terminologies and concepts, but also the performance of medical procedures considered most completely and efficiently as possible. The research work presented in this paper focuses on the semantic level of interoperability to allow different actors of telemedicine to share and use knowledge models representing best practices. The proposed approach is to articulate the ontologies directories with reasonable axioms and rules for modeling the semantic regulations of telemedicine. The proposed approach therefore concerns a reflection from legal constraints (Right to information and right to civil liability) caused by telemedicine and other technological properties needed to facilitate reasoning and medical decision making. To facilitate the tele-consultation, the use of digital media on the model of the remote practices with medical devices should incorporate intelligent systems designed to help physicians to find out how to support assessments of software service and deal with legal problems. Finding solutions to legal problems and technology (interoperability) is essential to ensure the reliability and sustainability of the Telemedicine. Likewise, it is indispensable to keep a humanistic dimension in order to build confidence to promote mutual respect and a fruitful dialogue among all stakeholders.

Telemedicine is also defined as the use of telecommunication technology in the provision of health care (DOC, 1997). Telemedicine is by nature a teamwork activity. Therefore, we adopted and adapted process models from team-based software engineering work (Tiako, 2011a,b) for defining the proposed telemedicine expert model of Fig. 3.

The telemedicine expert Process model proposed is composed of the following entities: Patient, Activity, Competency, Participant, Tool, and Object.

Patient: a telemedicine process schedules a patient to be treated; Activity: a medical activity is rooted in a telemedicine process. It can have sub-activities and it is constrained by medical directions. An activity has objects as input and output and it needs medical competencies for performing treatments.

Competency: represents the skill set of medical participants involved in an activity; Directions: procedures to be followed when performing an activity, including guidelines that govern the behavior of medical participants involved in an activity; Tools: used by telemedicine experts, which include computers, knowledge-based systems, digital radiography, etc. [START_REF] Stamm | Using telemedicine tools for the administration of a multi-site clinical trial[END_REF] to be used by participants for accessing, forwarding, receiving, and sharing medical objects;

Objects: as defined in telemedicine include information, documents, samples, or organs exchanged among medical participants and/or among a patient and medical participants.

Cardinality (a:b) involved between two relations of our model indicates the number of instances of an entity and the cardinality is attached to in relation to the other entity. The dark square attached to one entity indicates how to read the relation. For instance, cardinality (1, n) attached to entity Activity in relation with entity Competency indicates that each telemedicine activity will need at least one medical expert to be involved. Cardinality shows the patient feeding into the model but not the process, which then provides feedback through the participant to the patient.

Object

Capitalization of knowledge from collaborative expertise

In the Knowledge Acquisition (KA) community, there is the rise of situated cognition. There is growing insistence that much (perhaps all) of human decision making is critically dependent on the context in which it took place, suggesting that human problem solving could never be divorced from the myriad of environmental variables from every angle [START_REF] Compton | Situated cognition and knowledge acquisition research[END_REF].

The research emphasis here highlights the concern about constructing coherent, robust models of problem solving. Intelligent systems will be modelled in terms of different kinds of primitives allowing us to conduct more advanced studies and to propose more comprehensive analysis [START_REF] Musen | The knowledge acquisition workshops: a remarkable convergence of ideas[END_REF]:

(1) An ontology of the domain that captures the classes of entities that compose the application area (e.g., classes that describe medical treatment plans in general); (2) A knowledge base in which the ontology is instantiated to provide the detailed content knowledge required by the system (e.g., the distinctions that define a particular medical treatment plan); (3) A collection of domain-independent problem-solving methods that can automate the reasoning required to perform stereotypical tasks (e.g., a method such as skeletal-plan refinement, which can be used to plan medical treatment); and (4) Mappings that relate the generic inputs and outputs of the problem-solving methods to referents in the domain ontology.

Problem Solving Methods (PSMs) could be re-used, and reuse of models or patterns is a sign of a mature and efficient engineering methodology. However, given the type of problem to be solved and domain knowledge characteristics, a suitable PSM could be selected. For instance, if the task is diagnosis, a PSM that tests and systematically eliminates suspect components following a system structure would be selected ('model based diagnosis). If the domain knowledge consists of classes of malfunctions (e.g., diseases), the major component of the PSM might consist of a (heuristic) classifier that connects symptoms with diseases [START_REF] Breuker | A cognitive science perspective on knowledge acquisition[END_REF]. The first model presented by Musen leaves out an important component regarding the use of the mapping to examine systemic telemedicine problems and to improve care. This requires the analysis of the structure of current (cognitive) work process and we need to go beyond summaries of models in order to provide analysis that contributes to the field of telemedicine. There are also now shared diagnostics, (e.g., by phone or other means), which the new system can potentially build upon. The expectation is that modeling shared diagnostics would run into many barriers, because among other factors, human cognition and diagnostics are very complex.

Tele-expertise involves knowledge models and reasoning mechanisms for distributed collaborative decision-making environments. These reasoning mechanisms are associated with the analysis of diagnosis and/or therapy, carried out in the absence of the patient. Meanwhile, tele-expertise will utilize feedback and capitalize on lessons derived from this work to consolidate and expand expertise in the field of operation.

Each complex situation for constructing advanced reasoning mechanisms, giving new opportunity to ''lessons learned'', can be used for future works that enrich the wealth of experience already established. All knowledge, findings of the work, and the completion of the demonstration are the main support of the valuation by capitalization of acquired expertise.

The management of the acquired knowledge is based on three interrelated components:

Concrete traces stored as elements of knowledge representations are used in the application of expertise. Only traces stored on all media types, but often in information systems, lead to capitalization;

Experts, the actors of expertise, are the main performers of traces during reuse of expertise; these actors also show that the analysis and solutions used as expressions of medical expertise are important because of the meanings they convey.

The context associated with traces and the relevant expertise that gives the necessary understanding of the traces and the expression of their interpretation.

The capitalization of the traces can be facilitated and supported effectively by incorporating, for example, the following:

Specification in a chronological table of all the activities, events, and decisions made during the test;

The steps actually made using the expertise of the intelligence framework;

The structure of expertise with its trees, and the various avenues actually explored; A directed graph over time of the basic knowledge obtained and assumptions (provided their attributes likelihood) chained to each other by ties of causal and / or logical succession;

The balance of expertise or the expert conclusions that can be drawn from either organization specifically related to the case treated, or by validating for other cases of the same family;

Acquired knowledge and know-how to master complex situations and obtain the desired outcomes.

The management of expertise can also be used if the organization is appropriate, the terms of technical meetings for sharing experiences are clarified, and the information systems (building databases with associated internal servers) have suitable user interfaces to ease access of capitalized information (reports, various documents, case files, etc.).

Discussion

This section presents the lessons learned from this work from (a) requirement and processes modeling for collaboration point of views; and from (b) formalization and knowledge-based modeling for collaboration perspectives.

Requirement and process nodeling for collaboration

Telemedicine involves several activities such as (1) the consultation of patients, interpretation of necessary data collected, and follow-up by medical professional; (2) the inquire of opinions from geographically dispersed medical professionals who have the relevant training or skills; (3) the monitoring and remote supervision of patients; and (4) the assistance of a healthcare professional by a remote medical professional to assist remotely during the realization of a medical act.

Collaboration is highly needed among physicians or between physicians and other healthcare professionals involved in telemedicine protocols in accordance with the areas of different and complementary skills so that everyone is responsible for his/her actions and decisions.

Among other medical acts, telemedicine induces the same obligations for the physician, and patients benefit from the same rights as in a conventional medical practice. Non-compliance may engage professional oversight authorities.

Telemedicine applications require expert advice to develop a therapeutic decision, to prescribe treatment, recommend or perform services or other acts, or monitor a patient's condition. It allows organized networks to share information and skills, improve the management of patients, both for emergency care and chronic diseases [START_REF] Kamsu-Foguem | Knowledge-based modelling applied to synucleinopathies[END_REF]Kamsu-Foguem et al., 2014a;Kamsu-Foguem et al., 2014b;Kamsu-Foguem et al., 2014c;Kamsu-Foguem and Foguem, 2014a;[START_REF] Kamsu-Foguem | Conceptual graph-based knowledge representation for supporting reasoning in African traditional medicine[END_REF][START_REF] Tafin-Kampé | Acute osteomyelitis due to Staphylococcus aureus in children: What is the status of treatment today?[END_REF][START_REF] Kamsu-Foguem | User-centered visual analysis using a hybrid reasoning architecture for intensive care units[END_REF].

Telemedical intervention types include: measurement of vital signs using telemedical devices, interactive systems, or personal interaction between healthcare provider and patient. Telemedical final diagnosis on a remote patient is to be made only on the basis of a clinical evaluation or the result of further examination, while the prognosis of such patient with a serious disorder may be evident early on.

Formalization and Knowledge-based Modeling for Collaboration

In telemedicine, formalization of information exchanged between a requesting physician and medical expert is very important because it facilitates a transparent and traceable understanding of the telemedical process. This requires the definition of a suitable data exchange format adapted to the needs of performance assessment and secure information management. Therefore, confidentiality [START_REF] Chopard | Legal, deontological and ethical issues applied to telemedicine. A few insights about telestroke[END_REF] must be respected, and protection should be guaranteed by law through ethical principles and formal rules concerning the information to be shared, the procedures for sharing it, and the security of confidential information.

Depending on the nature of information to be shared in telemedicine, applications deployed may propose a synchronous or an asynchronous approach for implementing schemes of collaboration models. The practice of telemedicine must be exercised in respect of a number of principles to ensure the proper use of telemedicine, its added value in remote healthcare provisions and coordinated care pathways and ensure a reasoned practice and controlled engagement of advanced information and telecommunications technologies. This paper specifies and analyses some of the principles that stakeholders should have in mind throughout the deployment process of the telemedicine activities.

The use of telemedicine requires rigorous organization under the responsibility of doctors and computer scientists, based on clear rules and known to all, formalized by protocols written by the health authorities who will carry them out with the help of legal professionals, involved health structures (public or private health facilities) and biomedical device manufacturers [START_REF] Doumbouya | A framework for decision making on teleexpertise with traceability of the reasoning[END_REF][START_REF] Doumbouya | Telemedicine using mobile telecommunication: towards syntactic interoperability in teleexpertise[END_REF]Kamsu-Foguem 2014a;Kamsu-Foguem 2014b;Kamsu-Foguem and Foguem, 2014b;Kamsu-Foguem and Foguem, 2014c).

Carried out in the absence of patients, telemedical expertises are associated with the analysis of diagnosis and/or therapy and are supported by knowledge models and reasoning mechanisms for distributed collaborative decision-making environments. Such expertises will utilize feedback and capitalize on lessons derived from this work to consolidate and expand expertise in the field of operation.

Conclusion

In this paper, we have shown that the engagement of knowledge-based modeling is interesting for supporting collaborative activities in telemedicine. In fact, the contribution of artificial intelligence techniques is tangible to provide hierarchical organization and perform information structuration and formalization. In telemedicine, the ontologies deliver semantic support to determine the most significant knowledge to assist in the establishment of the most appropriate diagnosis and to eliminate differential diagnoses. In particular, formal ontologies facilitate browsing and searching in an easily accessible fashion (e.g., semantic similarity of ontological entities queried by structural classes and by functional annotations). This promotes feedback on the experiences of users in employing the telemedicine framework and environment and represents a collaborative knowledge platform, which brings together project owners and solution providers with practitioners and professionals.

There are also new research challenges and opportunities in the design and development of collaborative knowledge modeling, based, for example, on experience feedback processes or situations, and on the fundamental knowledgeable properties of such situations in order to manage their understanding, their control and the correct interpretation of the results for continuous improvement.

Fig. 1 .

 1 Fig. 1. Typology of telemedicine activities.

  degree of flexibility that necessary means the assessment of organizational and technical resources for the proper implementation of the collaborative work.The deployment of information sharing can follow three implementation schemes of collaboration models (not exclusive):Synchronous collaboration model with strong coupling: whereby participants have their own identical perspective about one document previously distributed on each local workstation. The distributed application on each workstation provides a shared vision of this document by showing simultaneous identical presentation.

Fig. 2 .

 2 Fig. 2. Technical framework of operating in the telemedicine (Telemd, 2013).

Fig. 3 .

 3 Fig. 3. The proposed telemedicine expert model.

Table 1

 1 Examples of information collected and shared in telemedicine.

	Data from medical devices	Telemedicine activity -specific	Patient record (general	Medical examination results
		clinical data	or specialized medicine)
	Measurements (blood pressure,	Request for an execution of a	Report of medical	Medical imaging (photo or video style)
	blood glucose, pulse and tem-	telemedicine activity	consultation	or	Medical signals
	perature, etc.)	Report from execution of the	hospitalization		Biological outcomes.
	Physiological signals (Stetho-	telemedicine activity	Medication		Gross, microscopic, chemical, immuno-
	scope, ECG, EEG, EMG, etc.)	Recommendations or pre-	prescriptions		logic and molecular examination of
	Static and dynamic imaging (e.g.	scriptions issued in the tele-	Diet and lifestyle	organs and tissues
	the multi-slice CT or the open	medicine activity	recommendations
	MRI system)

Table 2

 2 The functional requirements associated with achieving an effective telemedicine (DGOS, 2012).

	Functionality	Component	Description	Interoperability
	General organization of	Workflow (a depiction of a	Manages the request of the activity, planning,	Produces a guide on the phased
	the telemedicine	sequence of operations) and	assignment final actor, indicators of	implementation of the national health
	activity	planning engine	achievement of the act (to do, done,	identifier (NHI)
			documented, billed)	By connecting to a directory of actors
				integrating target connection to
				repository of social health actors
				(RSHA)
				Can rely on a secure messaging service
		Technical monitoring of the	Allows the actors of caretaking/maintenance	From a syntactic point of view, a
		activity	services to verify the operability of the device	service-oriented architecture is
			and to provide technical traceability in the	encouraging
			realization of the activity	Semantic interoperability depends on
				how the interfaces to a service are
				labeled and how the meaning of the
				information is shared
	Remote interaction and	Videoconferencing/	Non-specific industrial solution to health	Standard videoconference as a means of
	exchange (audio/video)	Audioconferencing		facilitating a comprehensive set of
				collaboration capabilities
		Telepresence	High-end proprietary solution deployed on	none
			dedicated sites
		Webconference	Lightweight videoconferencing solution	Lack of interoperability, client deployed
			between computers equipped with webcams	''on-the-fly''
		Videoconferencing	Functionality included as part of software	Depending on the management
		embedded into a	handling a work area of the telemedicine	software for work area of the
		telemedicine tool		telemedicine
	Temporary sharing of	Telemedicine tool	Solution used in the activity by actors offering	NHI retrieval from workflow engine
	medical data required		package of business functionalities correlated	Authentication of the health
	for an execution of the telemedicine activity		Viewer Digital Imaging and Communica-with an identified patient: Visio	professionals
			tions in Medicine (DICOM)
			Application sharing with streaming or
			with ''whiteboard'' function
			Document storage area replicated on the
			computer of each participant
	Transmission of medical	Communicating medical	Measurement sensors and physiological	Identity retrieval for injection into the
	documents useful for	devices and related software	signals, data routing and forwarding, Alert	whole chain of medical data processing
	an execution of the	useful documents for	Manager, providing data in a business	The responsibility of the technical third
	telemedicine activity	coordinating cares	Graphical User Interface (GUI)	party requires the control of the entire
				chain
		Picture Archiving and	DICOM Storage Service	DICOM
		Communication System
		(PACS)
		Connected biomedical	Device capable of producing an output file	Depends on the transport service
		device	sent by unified messaging or secure
			messaging
	Sharing/exchange of	Personal Medical Record	Consultation of medical documents useful for	NHI
	useful documents for	(PMR)	coordinating cares	Healthcare Professional's Smartcard
	coordinating cares			(HPS)
				Health information system
				interoperability framework and PMR
				compatibility
		Secure messaging structure	Document exchanges between health	The same interoperability requirements
			professionals (with eventually a copy in the	as for PMR
			patient PMR)	Cross-document Messaging (XDM)
				attachments