Joachim Escher

AND Philippe Laurenc ¸ot

Christoph Walker
email: walker@ifam.uni-hannover.de

A PARABOLIC FREE BOUNDARY PROBLEM MODELING ELECTROSTATIC MEMS

Keywords: 2010 Mathematics Subject Classification. 35R35, 35M33, 35Q74, 35B25, 74M05 MEMS, free boundary problem, well-posedness, asymptotic stability, finite time singularity, small aspect ratio limit

The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system (MEMS) is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified.

Introduction

An idealized electrostatically actuated microelectromechanical system (MEMS) consists of a rigid ground plate above which a thin and deformable elastic membrane is suspended that is held fixed along its boundary, see Figure 1. Applying a voltage difference between the two components induces displacements of the membrane and thus transforms electrostatic energy into mechanical energy, a feature that has applications in the design of transistors, switches, or micro-pumps, for instance. There is, however, an upper limit for the applied voltage potential beyond which the electrostatic force cannot be balanced by the elastic response of the membrane which then touches down on the rigid plate. This phenomenon is usually referred to as "pullin" instability. Estimating this threshold value is an important issue in applications as it may be a desirable feature of the device in some situations (e.g. switches, micropumps) or possibly damage the device in others. Mathematical models have been set up for that purpose, and we refer the reader e.g. to [START_REF] Pelesko | Mathematical modeling of electrostatic MEMS with tailored dielectric properties[END_REF][START_REF] Pelesko | Modeling MEMS and NEMS[END_REF][START_REF] Pelesko | Nonlocal problems in MEMS device control[END_REF] and the references therein for a more detailed account of the physical background and the modeling aspects of such devices.

Denoting the displacement of the membrane and the electrostatic potential in the device by u and ψ, respectively, we consider here the idealized situation where the applied voltage and the permittivity of the membrane are constant (normalized to one), and there is no variation in the horizontal direction orthogonal to the x-direction of both ψ and u. Under appropriate scalings, the rigid ground plate is at z = -1, and the undeflected membrane at z = 0 is fixed at the boundary x = -1 and x = 1 of the interval I := (-1, 1), see Figure 1. Letting ε denote the aspect ratio of the device before scaling, i.e. the ratio of the undeformed gap size to the device length, the membrane displacement u = u(t, x) ∈ (-1, ∞) evolves according to

∂ t u -∂ 2 x u = -λ ε 2 |∂ x ψ(t, x, u)| 2 + |∂ z ψ(t, x, u)| 2 , x ∈ I , t > 0 , (1.1)
Figure 1. Idealized electrostatic MEMS device.

with clamped boundary conditions

u(t, ±1) = 0 , t > 0 , (1.2)
and initial condition u(0, x) = u 0 (x) , x ∈ I .

(1.

3) The dimensionless electrostatic potential ψ = ψ(t, x, z) satisfies Laplace's equation

ε 2 ∂ 2
x ψ + ∂ 2 z ψ = 0 , (x, z) ∈ Ω(u(t)) , t > 0 , (1.4) in the region Ω(u(t)) := {(x, z) ∈ I × (-1, ∞) : -1 < z < u(t, x)} between the rigid ground plate at z = -1 and the deflected membrane. The boundary conditions for ψ are then

ψ(t, x, z) = 1 + z 1 + u(t, x)
, (x, z) ∈ ∂Ω(u(t)) , t > 0 .

(1.5)

Equation (1.1) corresponds to the situation in which viscous forces dominate over inertial forces in the system, e.g. see [START_REF] Flores | Dynamics and touchdown in electrostatic MEMS[END_REF][START_REF] Pelesko | Modeling MEMS and NEMS[END_REF]. Also, deformations due to bending are neglected in (1.1). Of particular importance in the model is the parameter λ > 0 which characterizes the relative strengths of electrostatic and mechanical forces and is proportional to the applied voltage. According to the above discussion, the pull-in instability is expected to take place for λ large enough.

The analysis of (1.1)-(1.5) turns out to be rather complex since (1.4) is a free boundary problem: indeed, the domain between the rigid ground plate and the elastic membrane changes with time. Due to this, equations (1.1) and (1.4) are strongly coupled. However, a common assumption made in mathematical analysis hitherto is a vanishing aspect ratio ε that reduces the free boundary problem to a heat equation with a right hand side involving a singularity when the membrane touches down on the ground plate. More precisely, setting ε = 0 allows one to solve (1.4)-(1.5) explicitly for the potential ψ = ψ 0 , that is,

ψ 0 (t, x, z) = 1 + z 1 + u 0 (t, x)
, (t, x, z) ∈ [0, ∞) × I × (-1, 0) , (1.6) where the displacement u = u 0 now satisfies the so-called small aspect ratio model

∂ t u 0 -∂ 2 x u 0 = - λ (1 + u 0 ) 2 ,
x ∈ I , t ∈ (0, ∞) , u 0 (t, ±1) = 0 , t ∈ (0, ∞) , u 0 (0, x) = u 0 (x) ,

x ∈ I .

(1.7)

Several mathematical results have been obtained for (1.7), including a characterization of the critical value of λ which corresponds to the value beyond which no steady-state exists as well as a possible space dependence of the permittivity of the membrane, see, e.g., [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF][START_REF] Ghoussoub | On the partial differential equations of electrostatic MEMS devices: stationary case[END_REF][START_REF] Lin | Nonlinear non-local elliptic equation modelling electrostatic actuation[END_REF][START_REF] Pelesko | Modeling MEMS and NEMS[END_REF] for the stationary problem and [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF][START_REF] Flores | Analysis of the dynamics and touchdown in a model of electrostatic MEMS[END_REF][START_REF] Ghoussoub | On the partial differential equations of electrostatic MEMS devices. II. Dynamic case[END_REF][START_REF] Guo | A nonlocal quenching problem arising in a micro-electro mechanical system[END_REF][START_REF] Guo | Global solutions of singular parabolic equations arising from electrostatic MEMS[END_REF][START_REF] Guo | On the partial differential equations of electrostatic MEMS devices. III. Refined touchdown behavior[END_REF][START_REF] Hui | The existence and dynamic properties of a parabolic nonlocal MEMS equation[END_REF][START_REF] Pelesko | Modeling MEMS and NEMS[END_REF] for the evolution problem. Inertial effects are taken into account in [START_REF] Guo | Dynamical solutions of singular wave equations modeling electrostatic MEMS[END_REF][START_REF] Kavallaris | A hyperbolic non-local problem modelling MEMS technology[END_REF].

To the best of our knowledge, the first analytical research without assumption of a small aspect ratio and thus dedicated to the original free boundary problem (1.1)-(1.5) is [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF], where the existence of steady-states has been established for small voltage values λ and a non-existence result for steady-states is obtained for large values of λ.

Here, we address the evolution problem. A rough summary of our results reads as follows: We prove the local well-posedness of (1.1)-(1.5) for all voltage values and show that the solutions exist globally in time provided the voltage value is sufficiently small. In contrast to the stationary case [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF] it turns out that a W 2 ∞ (I)-setting is no longer suitable for the u-component of (1.1)- (1.5). This is due to the fact that the heat semigroup does not enjoy suitable properties in L ∞ (I). Instead, we are therefore lead to work in the framework of W 2 q (I)-spaces for q < ∞, which generates additional difficulties as now ∂ 2

x u may become unbounded. For small voltage values we further prove that there is a locally asymptotically stable steady-state. For high voltage values we prove that global existence of solutions does not hold. In addition, we analyze the behavior of the solutions as the small aspect ratio ε → 0, showing convergence towards (1.7) as expected from a formal analysis.

To state precisely our results we introduce for q ∈ [2, ∞) and κ ∈ (0, 1) the set

S q (κ) := u ∈ W 2 q,D (I) ; u W 2 q,D (I) < 1/κ and -1 + κ < u(x) for x ∈ I ,
where W 2α q,D (I) := {u ∈ W 2α q (I) ; u(±1) = 0} for 2α ∈ (1/q, 2] and W 2α q,D (I) := W 2α q (I) for 0 ≤ 2α < 1/q. The local existence result now reads: Theorem 1.1 (Local Well-Posedness). Let q ∈ (2, ∞), ε > 0, and consider an initial value u 0 ∈ W 2 q,D (I) such that u 0 (x) > -1 for x ∈ I. Then, the following are true:

(i) For each voltage value λ > 0, there is a unique maximal solution (u, ψ) to (1.1)-(1.5) on the maximal interval of existence [0, T ε m ) in the sense that u ∈ C 1 [0, T ε m ), L q (I) ∩ C [0, T ε m ), W 2 q,D (I) satisfies (1.1)-(1.3) together with u(t, x) > -1 , (t, x) ∈ [0, T ε m ) × I , and ψ(t) ∈ W 2 q Ω(u(t)) solves (1.4)-(1.5) on Ω(u(t)) for each t ∈ [0, T ε m ). (ii) If for each τ > 0 there is κ(τ) ∈ (0, 1) such that u(t) ∈ S q (κ(τ)) for t ∈ [0, T ε m ) ∩ [0, τ], then the solution exists globally, that is, T ε m = ∞. (iii) If u 0 (x) ≤ 0 for x ∈ I, then u(t, x) ≤ 0 for (t, x) ∈ [0, T ε m ) × I. If u 0 = u 0 (x) is even with respect to x ∈ I, then, for all t ∈ [0, T ε m ), u = u(t, x
) and ψ = ψ(t, x, z) are even with respect to x ∈ I as well.

The proof of Theorem 1.1 is performed as follows. We first transform the Laplace equation (1.4) to a fixed rectangle which results in an elliptic boundary value problem with non-constant coefficients depending on u and its derivatives up to order 2. Solving this elliptic equation (for a given u) allows us to interpret the full free boundary problem as a nonlocal semilinear heat equation for u (see (2.5)). We then employ a fixed point argument to solve this evolution problem. Since the nonlinearity in the u-equation depends on the trace of the gradient of the potential, two ingredients are essential: precise estimates based on the regularizing effects of the heat semigroup and elaborated investigations of the properties of the solution to the transformed elliptic problem. The proof is given in Section 2.

We now address global existence issues. From a physical viewpoint a "pull-in" instability occurs for high voltage values. Accordingly, for large values of λ solutions cease to exist globally while solutions corresponding to small λ values exist globally in time. More precisely, we have:

Theorem 1.2 (Global Existence). Let q ∈ (2, ∞), ε > 0, λ > 0, and let u 0 ∈ W 2 q,D (I) satisfy -1 < u 0 (x) ≤ 0 for x ∈ I. Let (u, ψ) be the corresponding solution to (1.1)-(1.5) on the maximal interval of existence [0, T ε m ). (i) Given κ ∈ (0, 1) there exists λ * := λ * (κ, ε) > 0 and κ 0 := κ 0 (κ, ε) > 0 such that T ε m = ∞ and u(t) ∈ S q (κ 0 ) for t ≥ 0 provided that u 0 ∈ S q (κ) and λ ∈ (0, λ * ). (ii) There is λ * (ε) > 0 depending only on ε such that T ε m < ∞ provided λ > λ * (ε).
Note that part (i) of Theorem 1.2 provides uniform estimates on u in the W 2 q (I)-norm and ensures that u never touches down on -1, not even in infinite time. Its proof is contained in Section 2 and it is a consequence of the above mentioned fixed point argument. The second part of Theorem 1.2 is proven in Section 3 by constructing a suitable strict Lyapunov functional. Let us mention that similar results as stated in Theorem 1.2 are known to hold for the small aspect ratio model (1.7), see [START_REF] Flores | Analysis of the dynamics and touchdown in a model of electrostatic MEMS[END_REF][START_REF] Ghoussoub | On the partial differential equations of electrostatic MEMS devices. II. Dynamic case[END_REF]. However, the nonlocal features of (1.1)-(1.5) prevents one from using similar techniques and we thus have to develop an alternative approach. Also, there is a qualitative difference of the interpretation of the finiteness of T ε m in Theorem 1.2(ii). Indeed, according to Theorem 1.1, T ε m < ∞ implies that the W 2 q (I)-norm of u blows up or u touches down on -1 in finite time. This is in clear contrast to the small aspect ratio model (1.7) for which touchdown is the only mechanism for a finite time singularity. The difference stems from the fact that in (1.7) the nonlinearity is of zero order while for the free boundary problem (1.1)-(1.5) the nonlocal nonlinearity is rather of order "3/2" in the L q -sense (see Proposition 2.1). Nevertheless, we strongly believe that finite time touchdown occurs in the present model as well when T ε m is finite. We next turn to stability of steady-states. This is a delicate issue since it is expected in analogy to what is known for the small aspect ratio model [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF][START_REF] Pelesko | Modeling MEMS and NEMS[END_REF] that there are two steady states for small λ values. In [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF] it was shown that there is at least one steady-state to (1.1)-(1.5) for small values of λ (and none for large λ). We shall refine this result here and prove that, provided λ is small, this steady-state is unique with a first component in the set S q (κ) and locally asymptotically stable.

Theorem 1.3 (Asymptotic Stability). Let q ∈ (2, ∞), ε > 0, and κ ∈ (0, 1).

(i) There are δ = δ(κ) > 0 and an analytic function [λ → U λ ] : [0, δ) → W 2 q,D (I) such that (U λ , Ψ λ ) is for each λ ∈ (0, δ) the unique steady-state to (1.1)-(1.5) with U λ ∈ S q (κ) and Ψ λ ∈ W 2 2 (Ω(U λ )). Moreover, U λ is negative, convex, and even for λ ∈ (0, δ) and U 0 = 0. (ii) Let λ ∈ (0, δ). There are ω 0 , r, R > 0 such that for each initial value u 0 ∈ W 2 q,D (I) with u 0 -U λ W 2 q,D < r, the solution (u, ψ) to (1.1)-(1.5) exists globally in time and

u(t) -U λ W 2 q,D (I) + ∂ t u(t) L q (I) ≤ Re -ω 0 t u 0 -U λ W 2 q,D (I) , t ≥ 0 . (1.8)
The first part of Theorem 1.3 is a consequence of the Implicit Function Theorem while the second part follows from the Principle of Linearized Stability, and the proofs are given in Section 4. We shall point out that Theorem 1.3 provides uniqueness of steady-states with first components in S q (κ) for fixed λ small. A result in this spirit is also shown in [START_REF] Ghoussoub | On the partial differential equations of electrostatic MEMS devices: stationary case[END_REF]Thm.5.6]. But, as pointed out before, for the small aspect ratio model (1.7) it is known that below the critical threshold there are exactly two steady-states. If this would turn out to be true for the free boundary problem as well, that is, if there would be another smooth branch of steady-states emanating from λ = 0, say, V λ = U λ , then the fact that S q (κ 1 ) ⊂ S q (κ 2 ) for 0 < κ 2 < κ 1 < 1 would imply that δ(κ) ց 0 as κ ց 0 in Theorem 1.3. Obviously, V λ / ∈ S q (κ) for λ < δ(κ) and thus, as λ ց 0, the minimum of V λ has to approach -1 or the W 2 q -norm of V λ has to blow up1 . We also note that ψ converges exponentially to Ψ λ in the W 2 2 -norm as t → ∞, see Corollary 4.1 for a precise statement. Finally, both components of the steady-state enjoy more regularity than stated, see [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF]Cor.10].

More insight in the connection between the free boundary model and its small aspect ratio limit is offered in the next theorem. Indeed, we show that the solution (u, ψ) = (u ε , ψ ε ) to (1.1)-(1.5) provided by Theorem 1.1 converges to the solution (u 0 , ψ 0 ) of the small aspect ratio model (1.6), (1.7) as ε → 0. This gives a rigorous justification of the formal derivation.

Theorem 1.4 (Small Aspect Ratio Limit). Let λ > 0, q ∈ (2, ∞), κ ∈ (0, 1), and let u 0 ∈ S q (κ) with u 0 (x) ≤ 0 for x ∈ I. For ε > 0 let (u ε , ψ ε ) be the unique solution to (1.1)-(1.5) on the maximal interval of existence [0, T ε m ).
There are τ > 0, ε 0 > 0, and κ 0 ∈ (0, 1) depending only on q and κ such that T ε m ≥ τ and u ε (t) ∈ S q (κ 0 ) for all (t, ε) ∈ [0, τ] × (0, ε 0 ). Moreover, the small aspect ratio equation (1.7) has a unique solution

u 0 ∈ C 1 [0, τ], L q (I) ∩ C [0, τ], W 2
q,D (I) satisfying u 0 (t) ∈ S q (κ 0 ) for all t ∈ [0, τ] and such that the convergences

u ε -→ u 0 in C 1-θ [0, τ], W 2θ q (I) , 0 < θ < 1 , and ψ ε (t)1 Ω(u ε (t)) -→ ψ 0 (t)1 Ω(u 0 (t)) in L 2 I × (-1, 0) , t ∈ [0, τ] , (1.9)
hold as ε → 0, where ψ 0 is the potential given in (1.6). Furthermore, there is Λ(κ) > 0 such that the results above hold true for each τ > 0 provided that λ ∈ (0, Λ(κ)).

A similar result has been established for the stationary problem in [21, Theorem 2] and the proof of Theorem 1.4 is performed along the same lines provided one ensures an ε-independent lower bound τ > 0 on T ε m . In addition, in [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF] we took advantage of the fact that a W 2 ∞ (I)-bound is available for solutions to the stationary problem. We refine the arguments here by showing that a W 2 q (I)-bound is sufficient for q > 2.

2. Local and Global Well-Posedness: Proof of Theorem 1.1 and Theorem 1.2(i)

The starting point for the proof of Theorem 1.1 is to transform the free boundary problem (1.4)-(1.5) to the fixed rectangle Ω := I × (0, 1). More precisely, let q > 2 be fixed and consider an arbitrary function v ∈ W 2 q,D (I) taking values in (-1, ∞). We then define a diffeomorphism T v := Ω(v) → Ω by setting

T v (x, z) := x, 1 + z 1 + v(x) , (x, z) ∈ Ω(v) (2.1)
with

Ω(v) = {(x, z) ∈ I × (-1, ∞) ; -1 < z < v(x)}. Clearly, its inverse is T -1 v (x, η) = x, (1 + v(x))η -1 , (x, η) ∈ Ω , (2.2)
and the Laplace operator is transformed to the v-dependent differential operator

L v w := ε 2 ∂ 2 x w -2ε 2 η ∂ x v(x) 1 + v(x) ∂ x ∂ η w + 1 + ε 2 η 2 (∂ x v(x)) 2 (1 + v(x)) 2 ∂ 2 η w + ε 2 η 2 ∂ x v(x) 1 + v(x) 2 - ∂ 2 x v(x) 1 + v(x) ∂ η w .
The boundary value problem (1.4)-(1.5) is then obviously equivalent to

L u(t) φ (t, x, η) = 0 , (x, η) ∈ Ω , t > 0 , (2.3) φ(t, x, η) = η , (x, η) ∈ ∂Ω , t > 0 , (2.4
)

for φ = ψ • T -1 u(t)
. With this notation, the evolution equation (1.1) for u becomes

∂ t u -∂ 2 x u = -λ 1 + ε 2 (∂ x u) 2 (1 + u) 2 |∂ η φ(•, 1)| 2 , x ∈ I , t > 0 , (2.5)
after noticing that we have ∂ x φ(t, x, 1) = 0 for x ∈ I and t > 0 due to φ(t, x, 1) = 1 by (2.4). To set the stage for the proof of Theorem 1.1 we first observe:

Proposition 2.1. Let κ ∈ (0, 1) and ε > 0. For each v ∈ S q (κ) there is a unique solution

φ v ∈ W 2 2 (Ω) to L v φ v (x, η) = 0 , (x, η) ∈ Ω , (2.6
)

φ v (x, η) = η , (x, η) ∈ ∂Ω . (2.7)
In addition, defining ṽ by ṽ(x) := v(-x) for x ∈ I, we have

φ ṽ(x, η) = φ v (-x, η) for (x, η) ∈ Ω.
Moreover, for 2σ ∈ [0, 1/2), the mapping

g ε : S q (κ) -→ W 2σ 2,D (I) , v -→ 1 + ε 2 (∂ x v) 2 (1 + v) 2 |∂ η φ v (•, 1)| 2
is analytic, globally Lipschitz continuous, and bounded with g ε (0) = 1.

The proof of Proposition 2.1 shares some common steps with that of [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF]Lem. 5 & 6], but requires further developments, in particular establishing the Lipschitz continuity of g ε which was not needed in [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF]. We first derive suitable properties of the operator L v for v in the closure

S q (κ) = u ∈ W 2 q,D (I) ; u W 2 q,D (I) ≤ 1/κ and -1 + κ ≤ u(x) for x ∈ I of S q (κ), which we gather in the next lemma. Lemma 2.2. Let κ ∈ (0, 1) and ε > 0. For each v ∈ S q (κ) and F ∈ L 2 (Ω), there is a unique solution Φ ∈ W 2 2,D (Ω) to the boundary value problem -L v Φ = F in Ω , (2.8) Φ = 0 on ∂Ω .
(2.9)

Moreover, there is a constant c 1 (κ, ε) > 0 depending only on q, κ, and ε such that

Φ W 2 2 (Ω) ≤ c 1 (κ, ε) F L 2 (Ω) .
(2.10)

Proof. First note that the definition of S q (κ) and Sobolev's embedding theorem guarantee the existence of some constant c 0 > 0 depending only on q such that, for v ∈ S q (κ),

1 + v(x) ≥ κ , x ∈ I , and v C 1 ([-1,1]) ≤ c 0 κ . (2.11)
It follows from the proof of [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF]Lem. 5] that, due to (2.11), the operator -L v is elliptic with ellipticity constant ν(κ, ε) > 0 being independent of v ∈ S q (κ). Moreover, writing -L v in divergence form,

-L v w = -∂ x a 11 (v) ∂ x w + a 12 (v) ∂ η w -∂ η a 21 (v) ∂ x w + a 22 (v) ∂ η w + b 1 (v) ∂ x w + b 2 (v) ∂ η w , with a 11 (v) := ε 2 , a 22 (v) := 1 + ε 2 η 2 |∂ x v(x)| 2 (1 + v(x)) 2 , a 12 (v) := -ε 2 η ∂ x v(x) 1 + v(x) , a 21 (v) := a 12 (v) , b 1 (v) := ε 2 ∂ x v(x) 1 + v(x) , b 2 (v) := -ε 2 η ∂ x v(x) 1 + v(x) 2 ,
we see from (2.11) and the definition of S q (κ)

that 2 ∑ i,j=1 a ij (v) W 1 q (Ω) + 2 ∑ i=1 b i (v) L ∞ (Ω) ≤ c 2 (κ, ε) (2.12)
for all v ∈ S q (κ). Moreover, the embedding of

W 1 q (I) in C([-1, 1]
) ensures that a ij (v) belongs to C(Ω) for 1 ≤ i, j ≤ 2 and v ∈ S q (κ). It then follows from [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Thm. 8.3] that, given v ∈ S q (κ) and F ∈ L 2 (Ω), the boundary value problem (2.8)-(2.9) has a unique weak solution Φ ∈ W 1 2,D (Ω). Furthermore, the regularity of Φ and (2.12) ensure that

G := F -b 1 (v) ∂ x Φ -b 2 (v) ∂ η Φ belongs to L 2 (Ω)
, and we are in a position to apply [20, Chapt. 3, Thm. 9.1] to conclude that Φ is actually the unique solution in W 2 2,D (Ω) to the boundary value problem

-L 0 v Φ = G in Ω , Φ = 0 on ∂Ω , where L 0 v denotes the principal part of the operator L v , that is, -L 0 v w := -∂ x a 11 (v) ∂ x w + a 12 (v) ∂ η w -∂ η a 21 (v) ∂ x w + a 22 (v) ∂ η w .
In addition, it follows from [20, Chapt. 3, Thm. 10.1] that there is a constant c 3 (κ, ε) > 0 depending only on q, κ, and ε such that

Φ W 2 2 (Ω) ≤ c 3 (κ, ε) Φ L 2 (Ω) + G L 2 (Ω) .
Combining the previous inequality with (2.12) and the inequality

∂ x Φ 2 L 2 (Ω) + ∂ η Φ 2 L 2 (Ω) = - Ω Φ ∂ 2 x Φ + ∂ 2 η Φ d(x, η) ≤ δ 2 Φ 2 W 2 2 (Ω) + 1 4δ 2 Φ 2 L 2 (Ω)
which is valid for all δ > 0, we are led to

Φ W 2 2 (Ω) ≤ c 3 (κ, ε) Φ L 2 (Ω) + F L 2 (Ω) + c 2 (κ, ε) ∂ x Φ L 2 (Ω) + ∂ η Φ L 2 (Ω) ≤ c 3 (κ, ε) Φ L 2 (Ω) + F L 2 (Ω) + 2c 2 (κ, ε) δ Φ W 2 2 (Ω) + 1 2δ Φ L 2 (Ω)
whence, after choosing δ sufficiently small,

Φ W 2 2 (Ω) ≤ c 4 (κ, ε) Φ L 2 (Ω) + F L 2 (Ω) . (2.13)
We finally prove (2.10) and argue as in the proof of [10, Lemma 9.17]. Assume for contradiction that (2.10) is not true. Then, for each n ≥ 1, there are

v n ∈ S q (κ), Φ n ∈ W 2 2,D (Ω), Φ n ≡ 0, and F n ∈ L 2 (Ω) such that -L v n Φ n = F n in Ω and Φ n W 2 2 (Ω) ≥ n F n L 2 (Ω)
.

(2.14)

Setting Φ n := Φ n / Φ n L 2 (Ω)
and

F n := F n / Φ n L 2 (Ω)
, we realize that (2.13) and (2.14) imply

-L v n Φ n = F n in Ω , Φ n ∈ W 2 2,D (Ω) , (2.15) Φ n L 2 (Ω) = 1 , (2.16)
and

n F n L 2 (Ω) ≤ Φ n W 2 2 (Ω) ≤ c 4 (κ, ε) Φ n L 2 (Ω) + F n L 2 (Ω) = c 4 (κ, ε) 1 + F n L 2 (Ω) .
Consequently, we have for n ≥ 2c 4 (κ, ε),

n F n L 2 (Ω) ≤ 2c 4 (κ, ε) and Φ n W 2 2 (Ω) ≤ 1 + 2 n c 4 (κ, ε) . (2.17)
Since W 2 2 (Ω) and W 2 q (I) are compactly embedded in W 1 2 (Ω) and C 1 ([-1, 1]), respectively, we infer from (2.17) and the boundedness of S q (κ) in W 2 q (I) that there are

(Φ, v) ∈ W 2 2,D (Ω) × W 2 q,D (I) and a subsequence of (Φ n , v n ) n (not relabeled) such that (Φ n , v n ) ⇀ (Φ, v) in W 2 2,D (Ω) × W 2 q,D (I) , (2.18)
(Φ n , v n ) -→ (Φ, v) in W 1 2,D (Ω) × C 1 ([-1, 1]) . (2.19)
It readily follows from (2.18) and (2.19) that v ∈ S q (κ) and Proof of Proposition 2.1. For v ∈ S q (κ) and (x, η) ∈ Ω, we set

a ij (v n ), b i (v n ) -→ a ij (v), b i (v) in C([-1, 1]) (2.20) for all 1 ≤ i, j ≤ 2. Since F n -→ 0 in L 2 (Ω)
f v (x, η) := L v η = ε 2 η 2 ∂ x v(x) 1 + v(x) 2 - ∂ 2 x v(x) 1 + v(x)
.

Since q > 2, the function f v belongs to L 2 (Ω) with

f v L 2 (Ω) ≤ c 5 (κ, ε) , (2.21)
and Lemma 2.2 ensures that there is a unique solution

Φ v ∈ W 2 2,D (Ω) to -L v Φ v = f v in Ω , (2.22) Φ v = 0 on ∂Ω , (2.23) satisfying Φ v W 2 2 (Ω) ≤ c 1 (κ, ε) f v L 2 (Ω) . (2.24) Letting φ v (x, η) = Φ v (x, η) + η for (x, η)
∈ Ω, the function φ v obviously solves (2.6)-(2.7), and from (2.21) and (2.24) we obtain

φ v W 2 2 (Ω) ≤ c 6 (κ, ε) . (2.25)
In addition, if v ∈ S q (κ) and ṽ denotes the function defined by ṽ(x) := v(-x) for x ∈ I, we obviously have ṽ ∈ S q (κ) and the properties of L ṽ ensure that (x, η) -→ φ v (-x, η) solves (2.22)-(2.23) with ṽ instead of v. The uniqueness of the solution to (2.22)-(2.23) then readily implies that φ ṽ(x, η) = φ v (-x, η) for (x, η) ∈ Ω and thus that

g ε ( ṽ)(x) = g ε (v)(-x) , x ∈ I . (2.26)
Next, given v ∈ S q (κ), we define a bounded linear operator

A(v) ∈ L W 2 2,D (Ω), L 2 (Ω) by A(v)Φ := -L v Φ , Φ ∈ W 2 2,D (Ω) . Lemma 2.2 guarantees that A(v) is invertible with inverse A(v) -1 ∈ L L 2 (Ω), W 2 2,D (Ω)) satis- fying A(v) -1 L L 2 (Ω),W 2 2,D (Ω)) ≤ c 1 (κ, ε) . (2.27)
We then note that

A(v 1 ) -A(v 2 ) L(W 2 2,D (Ω),L 2 (Ω)) ≤ c 7 (κ, ε) v 1 -v 2 W 2 q (I) , v 1 , v 2 ∈ S q (κ) , (2.28)
which follows from the definition of L v and the continuity of pointwise multiplication

W 1 q (I) • W 1 q (I) ֒→ W 1 q (I) ֒→ L ∞ (I)
except for the terms involving ∂ 2 x v i , i = 1, 2, where continuity of pointwise multiplication

L q (Ω) • W 1 2 (Ω) ֒→ L 2 (Ω) is used. Now, for v 1 , v 2 ∈ S q (κ
), we infer from (2.27) and (2.28) that

A(v 1 ) -1 -A(v 2 ) -1 L(L 2 (Ω),W 2 2,D (Ω)) ≤ A(v 1 ) -1 L(L 2 (Ω),W 2 2,D (Ω)) A(v 2 ) -A(v 1 ) L(W 2 2,D (Ω),L 2 (Ω)) A(v 2 ) -1 L(L 2 (Ω),W 2 2,D (Ω)) ≤c 1 (κ, ε) 2 c 7 (κ, ε) v 1 -v 2 W 2 q (I)
, which, combined with (2.21), the observation that 0 ∈ S q (κ) and

f v 1 -f v 2 L 2 (Ω) ≤ c 8 (κ, ε) v 1 -v 2 W 2 q (I) , v 1 , v 2 ∈ S q (κ) , ensures that φ v 1 -φ v 2 W 2 2 (Ω) = Φ v 1 -Φ v 2 W 2 2 (Ω) = A(v 1 ) -1 f v 1 -A(v 2 ) -1 f v 2 W 2 2,D (Ω) ≤ c 1 (κ, ε) 2 c 7 (κ, ε) v 1 -v 2 W 2 q (I) f v 1 L 2 (Ω) + c 8 (κ, ε) v 1 -v 2 W 2 q (I) A(v 2 ) -1 L(L 2 (Ω),W 2 2,D (Ω)) ≤ c 9 (κ, ε) v 1 -v 2 W 2 q (I) (2.29)
for v 1 , v 2 ∈ S q (κ). We may then invoke [24, Chapt. 2, Thm. 5.4] and the continuity of pointwise multiplication [START_REF] Amann | Multiplication in Sobolev and Besov spaces[END_REF]Thm. 4.1] to conclude that the mapping

W 1/2 2 (I) • W 1/2 2 (I) ֒→ W 2σ 1 2 (I) for 2σ 1 < 1/2 according to
S q (κ) → W 2σ 1 2 (I) , v → ∂ η φ v (•, 1) 2 (2.30)
is globally Lipschitz continuous. Thanks to the continuity of the embedding of W 2 q (I) in W 1 ∞ (I), the mapping

S q (κ) → W 1 q (I) , v → 1 + ε 2 (∂ x v) 2 (1 + v) 2 (2.31)
is globally Lipschitz continuous with a Lipschitz constant depending only on κ and ε, and the Lipschitz continuity of g ε stated in Proposition 2.1 follows at once from (2.30), (2.31), and continuity of pointwise multiplication

W 1 q (I) • W 2σ 1 2 (I) ֒→ W 2σ 2 (I) = W 2σ 2,D (I) , where 2σ < 2σ 1 < 1/2, see again [1, Thm. 4.1]. Finally, to prove that g ε is analytic, we note that S q (κ) is open in W 2
q,D (I) and that the mappings

A : S q (κ) → L(W 2 2,D (Ω), L 2 (Ω)) and [v → f v ] : S q (κ) → L 2 (Ω) are analytic.
The analyticity of the inversion map ℓ → ℓ -1 for bounded operators implies that also the mapping

[v → φ v ] : S q (κ) → W 2 2
(Ω) is analytic, and the assertion follows as above from the results on pointwise multiplication .

Let p ∈ (1, ∞). We define A p ∈ L(W 2 p,D (I), L p (I)) by A p v := -∂ 2 x v for v ∈ W 2 p,D (I). Since A r ⊂ A p for r ≥ p, we suppress the subscript in the following and write A := A p . Note then that (2.5) subject to the boundary condition (1.2) and the initial condition (1.3) may be recast as an abstract parameter-dependent Cauchy problem

u + Au = -λg ε (u) , t > 0 , u(0) = u 0 , (2.32)
where we recall that the function g ε was defined in Proposition 2.1. To prove Theorem 1.1 it then suffices to focus on (2.32). For that purpose, let {e -tA ; t ≥ 0} denote the heat semigroup on L p (I) corresponding to -A. In order to state suitable regularizing properties we recall that we have set W (i

) If 0 ≤ α ≤ β ≤ 1 with 2α, 2β = 1/p, then e -tA L(W 2α p,D (I),W 2β
p,D (I)) ≤ Me -ωt t α-β , t > 0 , for some number M ≥ 1 depending on p, α, and β.

(ii) If 0 ≤ α ≤ β ≤ 1 with 2α = 1/p and 2β = 1/r, then e -tA L(W 2α p,D (I),W 2β r,D (I)) ≤ Me -ωt t α-β-1 2 ( 1 p -1 r ) , t > 0 ,
for some number M ≥ 1 depending on p, r, α, and β.

Proof. (i) Since -A ∈ L(W 2 p,D (I), L p (I)) is the generator of the analytic semigroup {e -tA ; t ≥ 0} on L p (I) with a negative spectral bound, it follows from [3, Chapt. V, Thm.2.1.3] that there are ω > 0 and M ≥ 1 such that

e -tA L(E α ,E β ) ≤ Me -ωt t α-β , t > 0 ,
where, for θ ∈ [0, 1],

E θ := (L p (I), W 2 p,D (I)) θ with (•, •) θ chosen as real interpolation functor (•, •) θ,p if 2θ = 1 and as complex interpolation functor [•, •] 1/2 if 2θ = 1. Since E θ = W 2θ
p,D (I) with equivalent norms for 2θ ∈ [0, 2] \ {1/p} by [START_REF] Grisvard | Équations différentielles abstraites[END_REF][START_REF] Seeley | Interpolation in L p with boundary conditions[END_REF], assertion (i) follows.

(ii) From Sobolev's embedding we have

W 2 p,D (I) ֒→ W 2θ r,D (I) for 2θ = 2 -(1/p -1/r) = 1/r, whence e -tA L(W 2α p,D (I),W 2β r,D (I)) ≤ c e -t 2 A L(W 2θ r,D (I),W 2β r,D (I)) e -t 2 A L(W 2α p,D (I),W 2 p,D (I)) , t > 0 ,
and so assertion (ii) follows from (i).

We are now in a position to prove Theorem 1.1 and Theorem 1.2(i).

Proof of Theorem 1.1 and Theorem 1.2(i).

Let λ > 0, q ∈ (2, ∞), ε > 0, and consider u 0 ∈ W 2 q,D (I) with u 0 (x) > -1 for x ∈ I. Clearly, there is κ ∈ (0, 1/2) such that

u 0 ∈ S q (2κ) . (2.33)
We now fix 1 2 -1 q < 2σ < 1 2 with 2σ = 1/q and put κ 0 := κ/M, where M ≥ 1 is such that

e -tA L(W 2 q,D (I)) + t -σ+1+ 1 2 ( 1 2 -1 q ) e -tA L(W 2σ 2,D (I),W 2 q,D (I)) ≤ Me -ωt , t ≥ 0 (2.34)
with ω > 0 according to Lemma 2.3. By Proposition 2.1 there is c 10 (κ, ε) > 0 such that

g ε (v 1 ) -g ε (v 2 ) W 2σ 2,D (I) ≤ c 10 (κ, ε) v 1 -v 2 W 2 q,D (I) , v 1 , v 2 ∈ S q (κ 0 ) . ( 2

.35)

Since 0 ∈ S q (κ 0 ) and g ε (0) = 1, we deduce from (2.35) that

g ε (v) W 2σ 2,D (I) ≤ 1 + c 10 (κ, ε) κ 0 = c 11 (κ, ε) , v ∈ S q (κ 0 ) . (2.36) Now, for τ > 0, define V τ := C([0, τ], S q (κ 0 )) and F(v)(t) := e -tA u 0 -λ t 0 e -(t-s)A g ε v(s) ds for 0 ≤ t ≤ τ and v ∈ V τ . Consider v 1 , v 2 ∈ V τ and t ∈ [0, τ]. Then, introducing I(τ) := τ 0 e -ωs s σ-1-1 2 ( 1 2 -1 q ) ds ≤ I(∞) := ∞ 0 e -ωs s σ-1-1 2 ( 1 2 -1 q ) ds , (2.37)
which is finite thanks to the positivity of ω and the choice of σ, it readily follows from (2.33), (2.34), and (2.36) that

F(v 1 )(t) W 2 q,D (I) ≤ M u 0 W 2 q,D (I) + λ M t 0 e -ω(t-s) (t -s) σ-1-1 2 ( 1 2 -1 q ) g ε (v 1 (s)) W 2σ 2,D (I) ds ≤ M 2κ + λ M c 11 (κ, ε) I(τ) , (2.38)
and from (2.34) and (2.35) that

F(v 1 )(t) -F(v 2 )(t) W 2 q,D (I) ≤ λ M c 10 (κ, ε) I(τ) v 1 -v 2 C([0,τ],W 2 q,D (I)) .
(2.39) Moreover, since W 2 q,D (I) embeds in L ∞ (I) with embedding constant 2 and u 0 ≥ 2κ -1, we deduce from the positivity of the heat semigroup, (2.34), and (2.36) that

F(v 1 )(t) ≥ -1 + 2κ -2 λ t 0 e -(t-s)A g ε (v 1 (s)) W 2 q,D (I) ds ≥ -1 + 2κ -2 λ M t 0 e -ω(t-s) (t -s) σ-1-1 2 ( 1 2 -1 q ) g ε (v 1 (s)) W 2σ 2,D (I) ds ≥ -1 + 2κ -2 λ M c 11 (κ, ε) I(τ) .
(2.40)

We finally note that F(v 1 )(t) ≤ 0 if u 0 ≤ 0 since g ε (v 1 ) ≥ 0. Consequently, due to (2.38)-(2.40) and the fact that I(τ) → 0 as τ → 0, there is τ 0 := τ 0 (λ, κ, ε, q, σ) > 0 sufficiently small such that F defines a contraction from V τ 0 into itself. This shows that there is a unique maximal solution

u ∈ C 1 [0, T ε m ), L q (I) ∩ C [0, T ε m ), W 2 q,D (I) ∩ C (0, T ε m ), W 2+2σ 2,D (I) to (2.32) for some T ε m ∈ (τ 0 , ∞], satisfying u(t, x) > -1 , (t, x) ∈ [0, T ε m ) × I , and, in addition, u(t, x) ≤ 0 , (t, x) ∈ [0, T ε m ) × I if u 0 (x) ≤ 0 , x ∈ I . Moreover, if for each τ > 0 there is κ(τ) ∈ (0, 1) such that u(t) ∈ S q (κ(τ)) for t ∈ [0, T ε m ) ∩ [0, τ], then necessarily T ε m = ∞.
This proves the statements (i) and (ii) of Theorem 1.1 after observing that ψ(t) := φ u(t) • T u(t) belongs to W 2 q Ω(u(t)) and solves (1.4)-(1.5) for each t ∈ [0, T ε m ), where the transformation T u was introduced in (2.1).

As for the statement (i) of Theorem 1.2, we choose λ * := λ * (κ, ε, q, σ) > 0 such that (recall (2.37))

λ * M max {c 10 (κ, ε), c 11 (κ, ε)} I(∞) ≤ 1 2 < 1 2κ 0 and 2 λ * M c 11 (κ, ε) I(∞) ≤ κ 0 . Letting λ ≤ λ * ,
it readily follows that, for each τ > 0, the mapping F defines a contraction from C([0, τ], S q (κ 0 )) into itself. This implies that T ε m = ∞ in this case and that u(t) ∈ S q (κ 0 ) for t ≥ 0. To prove statement (iv) of Theorem 1.1 suppose that u 0 is even on I and let u be the corresponding maximal solution to (2.32) with maximal existence time T ε m ∈ (0, ∞]. Introducing the function ũ defined by ũ(t, x) = u(t, -x) for (t, x) ∈ [0, T ε m ) × I, we deduce from Proposition 2.1 and the evenness of u 0 that ũ also solves (2.32), so that ũ actually coincides with u. Thus u(t, .) is even on I for all t ∈ [0, T ε m ) and the proof of Theorem 1.1 is complete.

We end this section with some useful properties of the component ψ of solutions to (1.1)-(1.5).

Proposition 2.4. Let q ∈ (2, ∞), ε > 0, λ > 0, and consider an initial value u 0 ∈ W 2 q,D (I) such that u 0 (x) > -1 for x ∈ I. Denoting the corresponding maximal solution to (1.1)-(1.5) by (u, ψ), we have for all t ∈ [0,

T ε m ) 1 + z -max u 0 + ≤ ψ(t, x, z) ≤ 1 , (x, z) ∈ Ω(u(t)) , (2.41) ∂ z ψ(t, x, u(t, x)) ≥ 0 , x ∈ I , (2.42) ∂ x ψ(t, x, u(t, x)) = -∂ x u(t, x) ∂ z ψ(t, x, u(t, x)) , x ∈ I . (2.43)
Proof. It readily follows from the positivity of λ that the constant max (u 0 ) + is a supersolution to (1.1)-(1.3), so that u ≤ max

(u 0 ) + in [0, T ε m ) × [-1, 1]. This property entails that (x, z) -→ 1 + z -max u 0
+ is a subsolution to (1.4)-(1.5) and the comparison principle gives the lower bound in (2.41). The upper bound in (2.41) also follows from the comparison principle as the constant 1 is clearly a supersolution to (1.4)- (1.5). This implies in particular that ψ(t) reaches its maximum value on the graph of u(t) and thus that (2.42) holds true. Finally, (2.43) is an obvious consequence of (1.5).

On Nonexistence of Global Solutions: Proof of Theorem 1.2(ii)

We now prove that there are no global solutions for large λ values as stated in Theorem 1.2(ii) (note that part (i) of this theorem was shown in the previous section). For this we first need some preparations. Let q ∈ (2, ∞), ε > 0, λ > 0, and consider an initial value u 0 ∈ W 2 q,D (I) such that -1 < u 0 (x) ≤ 0 for x ∈ I. By Theorem 1.1, there is a unique solution (u, ψ) to (1.1)-(1.5) defined on the maximal interval of existence [0,

T ε m ) for some T ε m ∈ (0, ∞] and satysfying u ∈ C 1 [0, T ε m ), L q (I) ∩ C [0, T ε m ), W 2 q,D (I) together with -1 < u(t, x) ≤ 0 , (t, x) ∈ [0, T ε m ) × I , (3.1)
and

ψ(t) ∈ W 2 q Ω(u(t)) solves (1.4)-(1.5) on Ω(u(t)) for each t ∈ [0, T ε m ).
Our aim is to show that, if λ is sufficiently large, the maximal existence time T ε m is finite. To this end, define ζ 1 (x) := π cos (πx/2)/4 for x ∈ [-1, 1] and µ 1 := π 2 /4. Then, µ 1 is the principal eigenvalue of the L 2 (I)-realization of -∂ 2

x and -∂ 2

x ζ 1 = µ 1 ζ 1 in I , ζ 1 (±1) = 0 , ζ 1 L 1 (I) = 1 . (3.2)
A classical technique to show that solutions only exist on a finite time interval is to study the evolution of

E 0 (t) := 1 -1 ζ 1 (x) u(t, x) dx , t ∈ [0, T ε m ) ,
and show that E 0 reaches -1 in finite time, a feature contradicting (3.1). Such an approach has been used successfully for the small aspect ratio model (1.7) [START_REF] Flores | Analysis of the dynamics and touchdown in a model of electrostatic MEMS[END_REF][START_REF] Guo | Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties[END_REF] and the stationary version of (1.1)-(1.5) [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF], the proof of the latter relying also heavily on the convexity of u. But, such a convexity property is not known for the evolution problem (1.1)-(1.5) (neither it is for (1.7)) and studying the time evolution of E 0 does not seem to work. However, as we shall see below, the study of the time evolution of

E α (t) := 1 -1 ζ 1 (x) u + α 2 u 2 (t, x) dx , t ∈ [0, T ε m ) , (3.3)
for a suitable choice of α ∈ (0, 1) leads us to the expected result. Performing that study requires to connect the behavior of ψ to that of u and we devote the next two results to this issue. We first start with an easy consequence of the boundary conditions (1.5).

Lemma 3.1. For t ∈ [0, T ε m ) and p ∈ [1, ∞), we have 4p (p + 1) 2 1 -1 ζ 1 (x) 1 + u(t, x) dx ≤ p Ω(u(t)) ζ 1 (x) ψ(t, x, z) p-1 |∂ z ψ(t, x, z)| 2 d(x, z) . (3.4) Proof. For (t, x) ∈ [0, T ε m ) × I and p ∈ [1, ∞)
, it follows from (1.5) and the Cauchy-Schwarz inequality that

1 =ψ(t, x, u(t, x)) (p+1)/2 -ψ(t, x, -1) (p+1)/2 = p + 1 2 u(t,x) -1 ψ(t, x, z) (p-1)/2 ∂ z ψ(t, x, z) dz ≤ p + 1 2 u(t,x) -1 ψ(t, x, z) p-1 |∂ z ψ(t, x, z)| 2 dz 1/2 1 + u(t, x) , hence 4 (p + 1) 2 1 1 + u(t, x) ≤ u(t,x) -1 ψ(t, x, z) p-1 |∂ z ψ(t, x, z)| 2 dz .
Owing to the nonnegativity of ζ 1 , the estimate (3.4) follows from the above inequality after multiplying both sides by pζ 1 (x) and integrating over I with respect to x.

The next lemma is a consequence of (

.

Lemma 3.2. For t ∈ [0, T ε m ) and p ∈ [1, ∞), we have 1 -1 ζ 1 (x) 1 + ε 2 |∂ x u(t, x)| 2 ∂ z ψ(t, x, u(t, x)) dx = Ω(u(t)) ζ 1 (x) pε 2 ψ p-1 |∂ x ψ| 2 + p ψ p-1 |∂ z ψ| 2 + µ 1 ε 2 p + 1 ψ p+1 (t, x, z) d(x, z) - µ 1 ε 2 (p + 1)(p + 2) - µ 1 ε 2 p + 1 1 -1 ζ 1 (x) u(t, x) dx . (3.5)
Proof. We multiply (1.4) by ζ 1 ψ p and integrate over Ω(u). Integrating by parts and using the boundary conditions for ψ and ζ 1 and (2.43), we obtain 0 = -

Ω(u) pε 2 ζ 1 ψ p-1 |∂ x ψ| 2 + ε 2 ∂ x ζ 1 ψ p ∂ x ψ + p ζ 1 ψ p-1 |∂ z ψ| 2 d(x, z) + 1 -1 ζ 1 (x) -ε 2 ∂ x u(x) ∂ x ψ(x, u(x)) + ∂ z ψ(x, u(x)) dx = 1 -1 ζ 1 (x) 1 + ε 2 |∂ x u(x)| 2 ∂ z ψ(x, u(x)) dx -p Ω(u) ζ 1 ψ p-1 ε 2 |∂ x ψ| 2 + |∂ z ψ| 2 d(x, z) - ε 2 p + 1 Ω(u) ∂ x ζ 1 ∂ x ψ p+1 d(x, z) . Since - Ω(u) ∂ x ζ 1 ∂ x ψ p+1 d(x, z) = Ω(u) ∂ 2 x ζ 1 ψ p+1 d(x, z) + (∂ x ζ 1 (-1) -∂ x ζ 1 (1)) 0 -1 (1 + z) p+1 dz + 1 -1 ∂ x ζ 1 (x) ∂ x u(x) dx = -µ 1 Ω(u) ζ 1 ψ p+1 d(x, z) + µ 1 p + 2 - 1 -1 ∂ 2 x ζ 1 (x) u(x) dx = -µ 1 Ω(u) ζ 1 ψ p+1 d(x, z) + µ 1 p + 2 + µ 1 1 -1 ζ 1 (x) u(x) dx
by (1.5) and (3.2), we end up with (3.5).

Proof of Theorem 1.2 (ii).

Let α ∈ (0, 1) to be determined later. We first note that (3.1) implies that the function E α , defined in (3.3), satisfies

-1 ≤ α -2 2 ≤ E α (t) ≤ 0 , t ∈ [0, T ε m ) . ( 3.6)
We next multiply (1.1) by ζ 1 (1 + α u), integrate over I and use (3.2) and (2.43) to obtain

dE α dt + µ 1 E α + α 1 -1 ζ 1 |∂ x u| 2 dx = 1 -1 ζ 1 (1 + α u) ∂ t u -∂ 2 x u dx = -λ 1 -1 ζ 1 (x) (1 + α u(x)) 1 + ε 2 |∂ x u(x)| 2 |∂ z ψ(x, u(x))| 2 dx .
Since ζ 1 (x) ≥ 0 and 1 + α u(x) ≥ 1α by (3.1), we further obtain

dE α dt + µ 1 E α + α 1 -1 ζ 1 |∂ x u| 2 dx ≤ -λ(1 -α) R (3.7) with R(t) := 1 -1 ζ 1 (x) 1 + ε 2 |∂ x u(t, x)| 2 |∂ z ψ(t, x, u(t, x))| 2 dx , t ∈ [0, T ε m ) .
We now look for a lower bound for R. To this end we observe that R reminds of the left-hand side of (3.5) while (3.4) provides a lower bound of the right-hand side of (3.5). More precisely, let β > 0 and p ≥ 1 be two positive real numbers to be determined later. It follows from Young's inequality that

1 -1 ζ 1 (x) 1 + ε 2 |∂ x u(x)| 2 ∂ z ψ(x, u(x)) dx ≤β R + 1 4β 1 -1 ζ 1 (x) 1 + ε 2 |∂ x u(x)| 2 dx , that is, R ≥ 1 β 1 -1 ζ 1 (x) 1 + ε 2 |∂ x u(x)| 2 ∂ z ψ(x, u(x)) dx - 1 4β 2 1 + ε 2 1 -1 ζ 1 (x) |∂ x u(x)| 2 dx .
We now infer from Lemma 3.2, Lemma 3.1, (3.2), and the non-positivity of

ζ 1 u that R ≥ 1 β p Ω(u) ζ 1 ψ p-1 |∂ z ψ| 2 d(x, z) - µ 1 ε 2 (p + 1)(p + 2) - µ 1 ε 2 p + 1 1 -1 ζ 1 u dx - 1 4β 2 1 + ε 2 1 -1 ζ 1 |∂ x u| 2 dx ≥ 1 β 4p (p + 1) 2 1 -1 ζ 1 1 + u dx - µ 1 ε 2 (p + 1)(p + 2) - 1 4β 2 1 + ε 2 1 -1 ζ 1 |∂ x u| 2 dx ≥ 1 βp 1 -1 ζ 1 1 + u + α u 2 /2 dx - µ 1 ε 2 βp 2 - 1 4β 2 - ε 2 4β 2 1 -1 ζ 1 |∂ x u| 2 dx .
Finally, since y → (1 + y) -1 is convex and ζ 1 L 1 (I) = 1, we use Jensen's inequality as in [START_REF] Flores | Analysis of the dynamics and touchdown in a model of electrostatic MEMS[END_REF] and get

R ≥

1 βp

1 1 + E α - µ 1 ε 2 βp 2 - 1 4β 2 - ε 2 4β 2 1 -1 ζ 1 |∂ x u| 2 dx .
Inserting this estimate in (3.7) and using (3.6) give

dE α dt -µ 1 + α 1 -1 ζ 1 |∂ x u| 2 dx ≤ - λ(1 -α) βp 1 1 + E α - µ 1 ε 2 p - p 4β - pε 2 4β 1 -1 ζ 1 |∂ x u| 2 dx , whence dE α dt + α - λ(1 -α)ε 2 4β 2 1 -1 ζ 1 |∂ x u| 2 dx ≤ µ 1 + λ(1 -α) βp µ 1 ε 2 p + p 4β - 1 1 + E α .
At this point, the role of the additional parameter α becomes clear as it allows us to control the λ-dependent term involving ∂ x u. We thus choose

α = λε 2 4β 2 + λε 2 ∈ (0, 1) , so that α = λ(1 -α)ε 2 4β 2 ,
and obtain the following differential inequality for E α :

dE α dt ≤ F (E α ) := µ 1 + 4λβ (4β 2 + λε 2 )p µ 1 ε 2 p + p 4β - 1 1 + E α . (3.8)
Since F is an increasing function on (-1, ∞), it readily follows from the non-positivity of E α and (3.8) that, if F (0) < 0, then E α (t) ≤ E α (0) ≤ 0 and dE α (t)/dt ≤ F (0) < 0 for all t ∈ [0, T ε m ). Integrating this differential inequality and using (3.6), we conclude that -1 ≤ F (0)t for all t ∈ [0, T ε m ) and thus that T ε m ≤ -1/F (0) < ∞ as claimed. We are then left with showing that we can find parameters β > 0 and p ≥ 1 such that F (0) < 0 for λ large enough. To this end we choose β = √ λ/2 > 0 and p = 1

+ 2µ 1 ε 2 ≥ 1 so that α = ε 2 /(1 + ε 2 ) and F (0) ≤ µ 1 + 2 √ λ 1 + ε 2 1 2 + 1 + 2µ 1 ε 2 2 √ λ -1 ≤ µ 1 + √ λ 1 + ε 2 1 + 2µ 1 ε 2 √ λ -1 . Therefore, if √ λ > 4µ 1 (1 + ε 2 ), we have F (0) ≤ µ 1 - √ λ 2(1 + ε 2 ) < 0 ,
and the proof of Theorem 1.2(ii) is complete.

4. Asymptotic Stability: Proof of Theorem 1.3

Let q ∈ (2, ∞), ε > 0, and κ ∈ (0, 1) be fixed. We first prove Theorem 1.3(i). Recall that, choosing σ ∈ (1/2 -1/q, 1/2) so that W 2σ 2 (I) ֒→ L q (I), Proposition 2.1 states that g ε : S q (κ) → L q (I) is analytic. Therefore, since the generator of the heat semigroup -A := -A q ∈ L(W 2 q,D (I), L q (I)) is invertible, we obtain that the mapping

F : R × S q (κ) → W 2 q,D (I) , (λ, v) → v -λA -1 g ε (v) is analytic with F(0, 0) = 0 and D v F(0, 0) = id W 2 q,D
. Now, the Implicit Function Theorem ensures the existence of δ > 0 and an analytic function

[λ → U λ ] : [0, δ) → W 2 q,D (I) such that F(λ, U λ ) = 0 for λ ∈ [0, δ). Denoting the solution to (1.4)-(1.5) with U λ instead of u by Ψ λ ∈ W 2 2
(Ω(U λ )), the pair (U λ , Ψ λ ) is for each λ ∈ (0, δ) the unique steady-state to (1.1)-(1.5) with U λ in S q (κ) and U 0 = 0. Since U λ is convex and satisfies the Dirichlet conditions U λ (±1) = 0 we clearly have U λ ≤ 0 for λ ∈ (0, δ). That U λ is even follows from uniqueness and [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF]Thm.1]. This proves Theorem 1.3 (i).

To prove part (ii) of Theorem 1.3, we use the Principle of Linearized Stability. For this, we fix λ ∈ (0, δ) and introduce the linearization of g ε ,

B λ := λD v g ε (U λ ) ∈ L(W 2
q,D (I), L q (I)) . Since B λ L(W 2 q,D (I),L q (I)) → 0 as λ → 0, it follows from [3, I.Cor.1.4.3] that -A -B λ is the generator of an analytic semigroup on L q (I) and there is ω 1 > 0 such that the complex half plane [Re z ≥ -ω 1 ] belongs to the resolvent set of -A -B λ provided that λ is sufficiently small. Now write v = u -U λ and consider the linearization of (2.32),

v + (A + B λ )v = G λ (v) , t > 0 , v(0) = v 0 , (4.1)
where G λ ∈ C 2 (O λ , L q (I)) is defined on some open zero neighborhood O λ in W 2 q,D (I) such that U λ + O λ ⊂ S q (κ) and given by

G λ (v) := -λ g ε (U λ + v) -g ε (U λ ) -D v g ε (U λ )v .
Since -(A + B λ ) is the generator of an analytic semigroup on L q (I) with a negative spectral bound as observed above, we may apply [23, Thm.9.1.1] and conclude statement (ii) of Theorem 1.3 by making δ > 0 smaller, if necessary.

A straightforward consequence of (1.8) and (2.29) is: Corollary 4.1. Under the assumptions of Theorem 1.3(ii)

there is R 1 > 0 such that φ u(t) -φ U λ W 2 2 (Ω) ≤ R 1 e -ω 0 t u 0 -U λ W 2 q,D (I) , t ≥ 0 , where φ v is defined in Proposition 2.1.
5. Small Aspect Ratio Limit: Proof of Theorem 1.4

We shall now prove Theorem 1.4. Fix λ > 0, q ∈ (2, ∞), κ ∈ (0, 1), and let u 0 ∈ S q (κ) with u 0 (x) ≤ 0 for x ∈ I. For ε > 0 we denote the unique solution to (1.1)-(1.5) by (u ε , ψ ε ) which is defined on the maximal interval of existence [0, T ε m ). In the following, (K i ) i≥1 and K denote positive constants depending only on q and κ, but not on ε > 0 sufficiently small. Set κ 0 := κ/(2M) < κ, where M ≥ 1 is the constant defined in (2.34). Owing to the continuity properties of u ε , we have

τ ε := sup t ∈ [0, T ε m ) : u ε (s) ∈ S q (κ 0 ) for all s ∈ [0, t] > 0 . (5.1)
Thanks to the continuity of the embedding of W 2 q (I) in W 1 ∞ (I), there is a positive constant K 1 such that, for all ε > 0,

-1 + κ 0 ≤ u ε (t, x) ≤ 0 , (t, x) ∈ [0, τ ε ] × [-1, 1] , (5.2
)

u ε (t) W 2 q (I) + u ε (t) W 1 ∞ (I) ≤ K 1 , t ∈ [0, τ ε ] . (5.3)
As a consequence of (5.3) there is ε 0 > 0 depending only q and κ such that

ε 2 0 ∂ x u ε (t) 2 L ∞ (I) ≤ 1 2 , (t, ε) ∈ [0, τ ε ] × (0, ε 0 ] . ( 5.4)
For ε ∈ (0, ε 0 ), we set

φ ε (t) := φ u ε (t) = ψ ε (t) • T -1 u ε (t) , t ∈ [0, τ ε ] ,
with T -1 u ε (t) given by (2.2) and

Φ ε (t, x, η) := φ ε (t, x, η) -η , (t, x, η) ∈ [0, τ ε ] × Ω .
The cornerstone of the proof of Theorem 1.4 is to derive appropriate estimates on Φ ε , showing that it converges to zero as ε → 0. To this end, we further develop and improve the analysis performed in [21, Section 3] and establish the following bounds:

Lemma 5.1. There exists a positive constant K 2 such that, for ε ∈ (0, ε 0 ) and t ∈ [0, τ ε ],

∂ x Φ ε (t) L 2 (Ω) + 1 ε Φ ε (t) L 2 (Ω) + ∂ η Φ ε (t) L 2 (Ω) ≤ K 2 , (5.5)
1 ε ∂ x ∂ η Φ ε (t) L 2 (Ω) + 1 ε 2 ∂ 2 η Φ ε (t) L 2 (Ω) ≤ K 2 , (5.6)
1 ε ∂ η Φ ε (t, •, 1) W 1/2 2 (I) ≤ K 2 .
(5.7)

Proof. Fix ε ∈ (0, ε 0 ) and t ∈ [0, τ ε ]. It first follows from (2.41) that Φ ε (t) L ∞ (Ω) ≤ 1 , (5.8)
while (5.2) and (5.3) entail that the function

f ε (t, x, η) := f u ε (t) (x, η) = ε 2 η 2 ∂ x u ε 1 + u ε 2 - ∂ 2 x u ε 1 + u ε (t, x) , (t, x, η) ∈ [0, τ ε ] × Ω , satisfies f ε (t) L q (Ω) ≤ ε 2 2 κ 2 0 ∂ x u ε (t) L ∞ (I) ∂ x u ε (t) L q (I) + 1 κ 0 ∂ 2 x u ε (t) L q (I) ≤ 2K 2 1 κ 2 0 + K 1 κ 0 ε 2 .
Therefore, by H ölder's inequality,

f ε (t) L p (Ω) ≤ 2 (q-p)/qp f ε (t) L q (Ω) ≤ K 3 ε 2 (5.9)
for p ∈ [1, q]. From now on, the time t plays no particular role anymore and is thus omitted in the notation. We multiply (2.22) by Φ ε , integrate over Ω, and proceed as in [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF]Lemma 11] to obtain

Ω f ε 1 -∂ η Φ ε Φ ε d(x, η) = ε 2 Ω ∂ x Φ ε -η ∂ x u ε 1 + u ε ∂ η Φ ε 2 d(x, η) + Ω |∂ η Φ ε | 2 (1 + u ε ) 2 d(x, η)
To estimate the right-hand side of the above identity from below, we use the elementary inequality (rs) 2 ≥ (r 2 /2)s 2 , (5.2), and (5.4) to obtain Ω) . Next, thanks to (5.8), (5.9), and H ölder's inequality, we can estimate the left-hand side of the above inequality and obtain

Ω f ε 1 -∂ η Φ ε Φ ε d(x, η) ≥ ε 2 2 ∂ x Φ ε 2 L 2 (Ω) -ε 2 ∂ x u ε 2 L ∞ (I) Ω ∂ η Φ ε 2 (1 + u ε ) 2 d(x, η) + Ω |∂ η Φ ε | 2 (1 + u ε ) 2 d(x, η) ≥ ε 2 2 ∂ x Φ ε 2 L 2 (Ω) + 1 2 Ω |∂ η Φ ε | 2 (1 + u ε ) 2 d(x, η) ≥ ε 2 2 ∂ x Φ ε 2 L 2 (Ω) + 1 2 ∂ η Φ ε 2 L 2 (
ε 2 ∂ x Φ ε 2 L 2 (Ω) + ∂ η Φ ε 2 L 2 (Ω) ≤ 2 f ε L 2 (Ω) 1 -∂ η Φ ε L 2 (Ω) Φ ε L ∞ (Ω) ≤ 2K 3 ε 2 1 + ∂ η Φ ε L 2 (Ω) ≤ 2K 3 ε 2 + 1 2 ∂ η Φ ε 2 L 2 (Ω) + 2K 2 3 ε 4 , whence ε 2 ∂ x Φ ε 2 L 2 (Ω) + ∂ η Φ ε 2 L 2 (Ω) ≤ K 4 ε 2 . (5.10) Since Φ ε (x, 1) = 0 for x ∈ I, we have Φ ε L 2 (Ω) ≤ √ 2 ∂ η Φ ε L 2 (
Ω) and (5.5) readily follows from this inequality and (5.10).

We next establish (5.6). For that purpose, we set

ζ ε := ∂ 2 η Φ ε , ω ε := ∂ x ∂ η Φ ε ,
Ω f ε 1 -∂ η Φ ε ζ ε d(x, η) = Ω ζ 2 ε (1 + u ε ) 2 + ε 2 ω ε -η ∂ x u ε 1 + u ε ζ ε 2 d(x, η) .
Using once more the inequality (rs) 2 ≥ (r 2 /2)s 2 and (5.4) to estimate the right-hand side of the above inequality from below, we find ) , we infer from H ölder's inequality, (5.9), and the previous inequality that (recall that q > 2)

Ω f ε 1 -∂ η Φ ε ζ ε d(x, η) ≥ Ω ζ 2 ε (1 + u ε ) 2 + ε 2 2 ω 2 ε -ε 2 η 2 |∂ x u ε | 2 (1 + u ε ) 2 ζ 2 ε d(x, η) ≥ Ω 1 2 ζ 2 ε (1 + u ε ) 2 + ε 2 2 ω 2 ε d(x, η) ≥ 1 2 ζ ε 2 L 2 (Ω) + ε 2 ω ε 2 L 2 (Ω) . Introducing Q ε := ζ ε 2 L 2 (Ω) + ε 2 ω ε 2 L 2 (Ω
Q 2 ε ≤ 2 f ε L q (Ω) 1 -∂ η Φ ε L 2q/(q-2) (Ω) ζ ε L 2 (Ω) ≤ Kε 2 1 + ∂ η Φ ε L 2q/(q-2) (Ω) Q ε ,
that is,

Q ε ≤ Kε 2 1 + ∂ η Φ ε L 2q/(q-2) (Ω) .
(5.11)

At this point, we infer from the Gagliardo-Nirenberg inequality [START_REF] Nirenberg | On elliptic partial differential equations[END_REF] and (5.10),that

∂ η Φ ε L 2q/(q-2) (Ω) ≤ K ∂ η Φ ε 2/q W 1 2 (Ω) ∂ η Φ ε (q-2)/q L 2 (Ω) ≤ K ∂ η Φ ε 2 L 2 (Ω) + ω ε 2 L 2 (Ω) + ζ ε 2 L 2 (Ω)
1/q ε (q-2)/q ≤ Kε (q-4)/q ε 4 + ε 2 ω ε 2

L 2 (Ω) + ε 2 ζ ε 2 L 2 (Ω)
1/q ≤ Kε (q-4)/q ε 4/q + Q 2/q ε ≤ K ε + ε (q-4)/q Q 2/q ε . Inserting this estimate in (5.11) leads us to

Q ε ≤ Kε 2 1 + ε + ε (q-4)/q Q 2/q ε ≤ Kε 2 + Kε (3q-4)/q Q 2/q ε ≤ Kε 2 + 2 q Q ε + Kε (3q-4)/(q-2) , whence Q ε ≤ Kε 2 1 + ε q/(q-2) ≤ Kε 2 ,
and the proof of (5.6) is complete. As a consequence of (5.5) and (5.6), we have ∂ η Φ ε W 1 2 (Ω) ≤ Kε and the properties of the trace operator readily give (5.7), see [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]Thm. 1.5.1.3].

A first consequence of Lemma 5.1 is that τ ε (and thus also T ε m ) does not collapse to zero as ε → 0, so that the solutions (u ε , ψ ε ) ε∈(0,ε 0 ) to (1.1)-(1.5) have a common interval of existence.

Lemma 5.2.

(i) There is τ > 0 depending only on q, λ, and κ such that τ ε ≥ τ for all ε ∈ (0, ε 0 ). (ii) There is Λ := Λ(κ) > 0 such that τ ε = T ε m = ∞ for all ε ∈ (0, ε 0 ) provided λ ∈ (0, Λ). Proof. Owing to (5.2), (5.3) and (5.7), we may argue as at the end of the proof of Proposition 2.1 to conclude that, fixing 2σ ∈ (1/2 -1/q, 1/2), there is K 5 > 0 such that

g ε (u ε (t)) W 2σ 2 (I) ≤ K 5 , t ∈ [0, τ ε ] . (5.12)
As in the proof of (2.38) and (2.40), we infer from (2.34), (5.12), the fact that u 0 ∈ S q (κ), and the Variation-of-Constant formula that, for t ∈ [0, τ ε ], σ-1-1 2 ( 1 2 -1 q ) g ε (u ε (s)) W 2σ 2,D (I) ds ≤1κ + 2 λ M K 5 I(t) , (5.14) where I(t) is defined in (2.37). Since I(t) → 0 as t → 0, there exists τ > 0 which depends only on q and κ such that I(t) < 1 λκK [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF] and I(t) < (2M -1)κ 4λM 2 K 5 for all t ∈ [0, τ] . (5.15) Thanks to this choice, we readily deduce from (5.13) and (5.14) that u ε (t) W 2 q,D (I) ≤ 1/κ 0 and u ε (t) ≤ 1κ 0 for all t ∈ [0, τ] ∩ [0, τ ε ]. In other words, u ε (t) ∈ S q (κ 0 ) for all t ∈ [0, τ] ∩ [0, τ ε ] and the definition of τ ε implies that τ ε ≥ τ. Finally, setting Λ(κ) := min 1 κK 5 I(∞) , (2M -1)κ 4M 2 K 5 I(∞) , and taking λ ∈ (0, Λ(κ)), it follows that (5.15) holds for any τ > 0 which entails that τ ε ≥ τ for any τ > 0.

u ε (t) W 2 q,D (I) ≤ M u 0 W 2 q,D (I) + λ M t 0 e -ω(t-s) (t -s) σ-1-1 2 ( 1 2 -1 q ) g ε (u ε (s)) W
Proof of Theorem 1.4. A straightforward consequence of (5.7) and the continuous embedding of W

≤ K (1 + λ) .
Consequently, the family (u ε ) ε∈(0,ε 0 ) is bounded in C 1 ([0, τ], L q (I)) ∩ C([0, τ], W 2 q (I)) and thus, given θ ∈ ((q + 1)/2q, 1), we may extract a sequence (ε k ) k≥1 of positive real numbers, ε k → 0, such that u ε k -→ u 0 in C 1-θ ([0, τ], W 2θ q (I)) (5.18) for some function u 0 ∈ C 1-θ ([0, τ], W 2θ q (I)). Since θ > (q + 1)/2q, we deduce from (5.18) and the continuous embedding of W 2θ q (I) in W 1 ∞ (I) that .19)

u ε k -→ u 0 in C([0, τ], W 1 ∞ (I)) . ( 5

Lemma 2 . 3 .

 23 for 2β ∈ (0, 1/p) andW 2β p,D (I) = {u ∈ W 2β p (I) ; u(±1) = 0} for 2β ∈ (1/p, 2]. Then we have: Let 1 < p ≤ r < ∞.There exists ω > 0 such that the following hold.

e

  in addition the embedding of W 2 q,D (I) in L ∞ (I) with constant 2,u ε (t) L ∞ (I) ≤1κ + 2 λ t 0 e -(t-s)A g ε (u ε (s)) -ω(t-s) (ts)

  by (2.17), the convergences (2.18),(2.19), and (2.20) allow us to pass to the limit as n → ∞ in the weak formulation of (2.15) and conclude thatΦ ∈ W 2 2,D (Ω) is a weak solution to -L v Φ = 0 in Ω. Using again [10, Thm. 8.3], this implies Φ ≡ 0 contradicting Φ L 2 (Ω) =1 as follows from (2.16) and (2.19).

  and multiply (2.3) by ζ ε . After integrating over Ω, we proceed as in [21,Lemma 11] with the help of[START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] Lem. 4.3.1.2 & 4.3.1.3] to deduce that

  1/2 2 (I) in L 2q (I) is that -∂ 2 x u ε = -λ g ε (u ε (t)) , x ∈ I , t ∈ [0, τ] ,(5.17)with g ε defined in Proposition 2.1 and τ in Lemma 5.2, it readily follows from (5.2), (5.3), Lemma 5.1, and the continuous embedding of W 1/2 2 (I) in L 2q (I) that, for t ∈ [0, τ],∂ t u ε (t) L q (I) ≤ ∂ 2 x u ε (t)

	lim ε→0	sup t∈[0,τ]	∂ η φ ε (t, •, 1)	2 -1	L q (I)	= 0 .	(5.16)
	Since
	∂ t u ε L q (I)	+ λ	1 + ε 2 |∂ x u ε (t)| 2 (1 + u ε (t)) 2	L ∞ (I)	∂ η φ ε (t, •, 1)	2 L 2q (I)
	≤ K 1 +	9λK 8κ 2 0	1 + ∂ η Φ ε (t, •, 1)	2 W 1/2 2 (I)

For the case of the small aspect ratio model, V λ approaches the V-shaped function x → |x| -1 as λ ց 0.

Acknowledgments

Part of this research was done while J.E. and Ch.W. were visiting the Institut de Mathématiques de Toulouse, Université Paul Sabatier. The financial support and kind hospitality is gratefully acknowledged.

A first consequence of (5.2) and (5.19) is that -1 (5.16) and (5.19) ensures that

Recalling (5.17), classical stability properties of the linear heat equation entails that u 0 is a solution to the small aspect ratio equation (1.7) and it is clearly the unique solution to (1.7) which belongs to S q (κ 0 ) for all t ∈ [0, τ]. This implies in particular that not only a subsequence but the whole family (u ε ) ε∈(0,ε 0 ) converges towards u 0 in C 1-θ ([0, τ], W 2θ q (I)), θ ∈ (0, 1), as ε → 0. Finally, the remaining assertion (1.9) follows easily from Lemma 5.1 as shown in the proof of [START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF]Thm. 2]. This completes the proof of Theorem 1.4.


Reza Akbarinia

Emmanuel Castanier

Rémi Coletta

Fady Draidi

Esther Pacitti

Didier Parigot

Guillaume Verger

Patrick Valduriez

P2PShare: a Social-based P2P Data Sharing System 1

Keywords: Pair à Pair, Recommendation, Base de Données Probabiliste 1

P2PShare est un système Pair à Pair (P2P) pour le partage à grande échelle de données probabilistes qui s'appuie sur un système de recommandation basée sur le contenu et l'expertise de chaque utilisateur. Il est conçu pour gérer des données probabilistes et déterministes dans un environnement P2P. Il fournit un environnement flexible pour l'intégration de sources hétérogènes, et prend en compte les aspects sociaux pour découvrir des résultats de haute qualité pour les requêtes en privilégiant les données des amis (ou des amis d'amis), qui sont les experts sur les sujets liés à la requête. Nous avons mis en place un prototype de P2PShare à l'aide du logiciel Shared-Data Overlay Network (SON), une plate-forme de développement open source pour des réseaux P2P utilisant les services web, JXTA et OSGi. Dans cet article, nous décrivons les principaux services de P2PShare, par exemple, la diffusion des sujets d'intérêt entre les amis, les requêtes sur les contenus et les requêtes probabilistes sur les jeux de données.

Introduction

P2PShare is a P2P system for large-scale probabilistic data sharing, particularly in scientific communities. It takes into account heterogeneous data, and leverages content-based and expert-based recommendation for discovering the data relevant to queries. It is composed of three main components:

• ProbDB (http://probdb.gforge.inria.fr) [1 1] is a probabilistic database system built on top of a classical DBMS.

Instead of directly modifying the DBMS and adding "native" primitives to it, we have chosen to implement ProbDB on top of the DBMS, and thus to be able to change the underlying DBMS with a slight programming effort.

•

WebSmatch (http://websmatch.gforge.inria.fr) [1 2] is a flexible environment for Web data integration, based on real, end-to-end data integration scenarios (e.g. over public data or scientific data). WebSmatch supports the full process of importing, refining and integrating data sources and uses third party tools for high quality visualization.

•

P2Prec (http://p2prec.gforge.inria.fr) [1 3][1 4] is a social-based P2P recommendation system for large-scale content sharing that leverages content-based and social-based recommendation. The main idea is to recommend high quality documents related to query topics, by exploiting friendship networks.

P2PShare is organized as a P2P social network and combines the services of the three above components. The scientific data are stored at peers, without any central administration. Using the recommendation service of P2Prec, a user (a peer) is able to build its list of expert friends, and discover the relevant data provided by its friends. WebSmatch is used for integrating heterogeneous datasets, and ProbDB is utilized for storing and querying the probabilistic or deterministic datasets shared in the network.

We have implemented a prototype of P2PShare using SON2 , an open source development platform for P2P networks. With SON, we can easily provide a highly modular architecture for integration of various data types. As P2Prec was already built on top of SON [1 4], we integrated WebSmatch and ProbDB to the architecture.

In the rest of this paper, we first describe an application scenario, and then present P2PShare's architecture. Finally, we introduce the demonstration.

Application Scenario

As a motivating application for P2PShare, we consider a social network of scientists interested in sharing the scientific data (documents, annotations, datasets, etc.) they store locally in their databases. The scientists typically store different kinds of data including: papers and reports they have written, articles they have downloaded, datasets they have used or produced for scientific experiments, etc. In addition, they annotate the documents they have with tags, ratings, etc. The goal of this social network is that each user (peer) can discover data provided by others and send queries to them. For instance, one may want to know which peers are expert in a topic and get all the scientific data highly rated by those experts on that topic. Or another peer may look for the best datasets used by others for some kinds of experiments.

Figure 1 shows an example of the different relations needed to store the scientific data for this social network. The relation Documents (doc_id, file_name) stores the id and address of documents (e.g. papers, reports, and articles) which are stored as files. The relation Datasets (dataset_id, location) stores the id and location of the datasets. The datasets may contain probabilistic or deterministic (ordinary) data. In the Datasets table, the location attribute will be the name of the database that keeps the data. The relationship between the documents and datasets can be specified explicitly by the researchers or be inferred by the system (from the existing relationships). This relationship is shown in the relation Related_Datasets (doc_id, dataset_id, probability).

Figure 1. Relations between documents and datasets

The confidence on the relationship of documents and datasets is stated in the form of a probability, by using the probability attribute. If the probability is set to 1, it means that a researcher has explicitly specified the relationship between the document and the dataset. If the system infers the relationship, then the probability is set according to the semantic distance between the document and the dataset. The semantic distance is computed by WebSmatch by using the combination of several techniques, and normalizing their outputs. The closer the document and the dataset, the lower is the distance value. By using the probability attribute, P2PShare can answer important probabilistic queries issued by researchers, for example: given a document d and a probability p, return all datasets that are related to d with a probability higher than p. To answer such queries, P2PShare takes advantage of probabilistic query processing in ProbDB. In order to locate the data that are related to a query in the P2P network, P2PShare uses the recommendations given by P2PRec.

Prototype Architecture

P2PShare has been developed on top of the SON platform. SON is based on a set of basic concepts for developing and deploying multiple services (e.g. directory, query, summary or recommendation) in a simple and effective way.

It is an open source development platform for P2P networks using web services, JXTA and OSGi. SON combines three powerful paradigms: components, SOA and P2P. To provide weak coupling between system entities, components communicate by asynchronous message passing. To scale up, we rely on a decentralized organization based on a DHT for publishing and discovering services or data. In terms of communication, the infrastructure is based on JXTA virtual communication pipes, a technology that has been extensively used within the Grid community.

Using SON, the development of a P2P application is done through the design and implementation of a set of components. Each component includes a non-functional code that provides the component services and a code component that provides the component logic (business code). The complex aspects of asynchronous distributed programming (non-functional code) are separated from code components. The Component Generator (CG) automatically generates this non-functional code from a description of services (provided or required services) for each component. This CG component is not present at the execution of the SON infrastructure. SON is implemented in Java on top of OSGi components that provide all basic services for the lifecycle of our components, in particular, We developed P2PShare as a SON application with three components (see Figure 2): the WebSmatch component to extract the keywords of a document, the ProbDB component for the management of probabilistic databases and the P2Prec component for the recommendation process.

With P2Prec we exploit the social (friendship) connections between participants to support data location in dynamic groups of participants. The main idea is to recommend high quality documents related to query topics hold by useful friends (of friends) of the users, by exploiting the friendship network. To exploit friendship links, we rely on Friend-Of-A-Friend (FOAF)3 descriptions. Our recommendation model relies on a distributed graph, where each node represents a user (peer) labeled with the contents it shares and its topics of interests. The topics each peer is interested in are automatically calculated by analyzing the documents the peer holds. Peers become relevant for a topic if they hold a minimum number of highly rated documents on this topic. A peer p becomes useful to a peer q, if the number of their common topics of interest is higher than a threshold. To disseminate information about relevant peers, P2Prec relies on gossip algorithms that provide scalability, robustness, simplicity and load balancing.

ProbDB is responsible for processing queries over probabilistic data. It is built on top of a classical Database management system. When a query is received by ProbDB, it is analyzed and probabilistic keywords are extracted.

Then classical (non probabilistic) sub-queries are sent to the DBMS that process them and returns intermediate results. Afterwards, probabilistic functions are applied to the intermediate results, and the final results are returned to the user. In its current version, ProbDB is built on PosgreSQL, but can easily be adapted to work on other DBMSs, for instance MySQL.

Each peer can contain different schemas for describing its database. To be able to reformulate queries, P2PShare uses the schema matching done by WebSmatch. Schema matching is hard as we must deal with both structural heterogeneity (differences in formats, keys, etc.) and semantic heterogeneity (synonyms, homonyms, hypernyms, ambiguous names, etc.) among the metadata. The approach of WebSmatch is not to develop new matching techniques, but rather to combine the state of the art techniques. In its current version, WebSmatch supports dozens of string metrics (from the Second Strings project4 ), dictionary measures from Wordnet5 , Information Retrieval techniques (using the Lucent library6 ) but also some structure and instance-based measures. WebSmatch has two phases: a learning phase that produces a dedicated schema matcher and a matching phase in which the dedicated matcher is used over new input schemas. The learning process can use preferences in terms of precision and recall and some expert-provided correspondences.

Demonstration

To demonstrate the P2PShare systems, we use it for implementing the social network of scientists that we described in Section 2. We install P2PShare on each peer of the social network. Each peer will be started with an initial state containing a list of documents, list of datasets, and a list of friends. For the list of documents, we use the Ohsumed document corpus. It is a set of 348566 references from MEDLINE, the on-line medical information database, consisting of a large set of medical publications over a five-year period (1987)(1988)(1989)(1990)(1991). We show how the application works, from the global initialization to the utilization by an end-user.

Initialization. As the demonstration starts, an entry point, and several virtual peers (e.g. 30 peers) are created. Each peer is given an initial friend-of-a-friend (FOAF) file, which determines its friends and some information about them. When connecting a new peer to the network, we show how it gets initial information in its FOAF file in two cases: (1) it has already joined the network in the past (i.e. it knows other peers);

(2) it connects to the network for the first time.

Figure 2 .

 2 Figure 2. P2PShare Architecture

Work partially funded by the DataRing project of the French ANR.

http://www-sop.inria.fr/teams/zenith/SON

http://www.foaf-project.org

http://secondstring.sourceforge.net/

http://wordnet.princeton.edu/

http://lucene.apache.org/

Gossiping. The gossip service is at the heart of P2PShare, and is transparent to the end-user. While peers exchange gossiping messages, the system recommends new friendships to users. For the sake of the demonstration, we developed an interface showing what is internally happening during gossiping. The interface (see Figure 3) shows the current friends of the user, the gossip messages sent and received by the peer, the gossip local-view that permits to find friends, etc. We show how the gossip mechanism notifies the user that other users share the same interests, and asks to add them to her list of friends.

Recommendation-based querying processing. One of the main objectives of P2PShare is to answer queries issued by users looking for documents and datasets. The user is able to send a query for getting documents recommendations from her friends. Each friend may recommend documents depending on the similarity in terms of keywords and the document rates. Figure 3 shows the result returned to the user for an issued query. We show the results of the query for a user who has been in the network for a long time compared to a new user, and compare the accuracy and the number of answers she gets.

Probabilistic querying. Users can also issue queries over the probabilistic data of the systems by using the ProbDB query service. The issued queries are sent to the peers who may be able to partially answer it. When a peer receives a probabilistic query from the ProbDB query service, it processes the whole query and answers directly to the initiator of the query. Depending on the nature of the query, it may need to transfer the query to other pertinent peers and wait for them to answer. Each peer needs to reformulate a global query to its local schema. In order to do so, the WebSmatch component matches the schemas (the tables) of the global query with the local schemas of the user.


Ibrahim Amadou
email: ibrahim.amadou@insa-lyon.fr

Anis Ouni
email: anis.ouni@insa-lyon.fr

Hervé Rivano
email: herve.rivano@inria.fr

Fabrice Valois
email: fabrice.valois@insa-lyon.fr

Energy-efficient Relay Selection Schemes for Contention-based Geographic Forwarding

has emerged recently as an energy-efficient communication protocol to improve the wireless sensor networks lifetime. By exploiting the wireless broadcast advantage and node spatial diversity, it mitigates the effect of varying channel condition and node unavailability. The main contributions are based on classical relay selection scheme such as most forward within radius. However, this relay selection scheme performs poorly under real radio environments because it tends to forward packets on unreliable link, which leads to high expected number of retransmissions in each hop and an important number of duplicated packets. In this paper, we investigate the performance of several relay selection schemes. First, extensive simulations are proposed to evaluate their performance locally in terms of packet delivery ratio and duplicated packets. Then, we extend the work in multihop wireless networks and evaluate their performance in term of energy. To highlight their gain, we propose a smart solution obtained by linear programming. Based on the intelligent relay selection scheme, we develop a new solution, Furthest Forward within Reliable neighbors (F F Re), which tries to save more energy than the most efficient relay selection schemes.

I. INTRODUCTION Contention-based Geographic Forwarding protocols (CGF)

were firstly developed to eliminate the proactive beaconing drawbacks of geographic routing protocols in wireless sensor networks (WSNs). Unlike the well known geographic routing such as GPSR [START_REF] Karp | Gpsr: Greedy perimeter stateless routing for wireless networks[END_REF], they do not use pre-requisite knowledge of the neighborhood. Local informations are known reactively through distributed contention process. Basically, the core idea of CGF routing protocols is to combine greedy forwarding with recovery forwarding. In greedy forwarding, packets are broadcast in the neighborhood and then, the next hop is selected distributively inside a dedicated area called forwarding area by a contention scheme using an additional delay. This delay is computed by each node according to the expected forward progress or distance or remaining energy. Recovery forwarding is used when there is no forwarder candidate (hole problem [START_REF] Chen | A survey of void handling techniques for geographic routing in wireless networks[END_REF]). The difference among the existing CGF protocols comes from either the relay selection scheme (priority policy [START_REF] Xue | Cost efficiency of anycast-based forwarding in duty-cycled wsns with lossy channel[END_REF]) or the distribution contention function or the forwarding area design or the recovery strategy. Prior studies mainly focus on networks with high density in order to perform in greedy mode, assume an ideal disk channel model 1 1 Radio links are perfect within a given communication range and focus on finding the shortest path from a source to a destination. However, while assuming high density network deployment may be acceptable in wireless sensor networks, it is now clear that the perfect radio links assumption is unlikely to be valid with realistic physical layer model. In fact, wireless links are more unstable due to varying link qualities and unreliable than wired networks. These behaviors lead to poor performance when explicit neighbor discovery is used during the forwarding process, a large number of retransmissions and high duplicated packets, which can increase network congestion.

In this paper, we investigate the performance in terms of packet delivery ratio and duplicated packets of the existing relay selection schemes for CGF or geographic routing protocols under a realistic wireless channel model and considering only the 1-hop neighborhood. Then, we focus on multi-hop wireless networks using these previous CGF forwarding strategies. To highlight the performance in term of energy consumption. Results are compared with an optimal solution given by a linear programming. Based on these studies, we develop a new solution Furthest Forward within Reliable neighbors (F F R e ), which tries to minimize both the per-hop expected number of retransmissions and the number of hops in order to save energy.

The rest of the paper is organized as follows: Section II presents some existing relay selection schemes and some well known proposed relay selection schemes under realistic wireless channel model. Section III introduces our assumption and our radio channel model. Section IV presents a performance evaluation by simulation while the section V presents analytically the extension in case of multi-hop wireless networks. Analytical results are presented in Section VI. Section VII presents our proposal relay selection scheme and evaluate its gain. Section VIII concludes this work.

II. RELATED WORK

Unlike the recent works such as [START_REF] Douglas | A high-throughput path metric for multi-hop wireless routing[END_REF], [START_REF] Helmy | Energy-efficient forwarding strategies for geographic routing in lossy wireless sensor networks[END_REF], [START_REF] Woo | Taming the underlying challenges of reliable multihop routing in sensor networks[END_REF]- [START_REF] Zúñiga | An analysis of unreliability and asymmetry in low-power wireless links[END_REF], that try to improve the existing routing protocol relay selection policies in order to take into account the realistic wireless channel model or to improve the network capacity or the energy efficiency by keeping the shortest path metric approach, in this work, our aim is to study both the reliability and energy efficiency of the different relay selection schemes for greedy geographic forwarding. Most Forward within Radius (MFR)/Most Forwarding Progress (MFP) [START_REF] Takagi | Optimal transmission ranges for randomly distributed packet radio terminals[END_REF], Nearest Forwarding Progress (NFP)/Nearest Forward within Radius (NFR) [START_REF] Hou | Transmission range control in multihop packet radio networks[END_REF] and Random Progress Method (RMP) [START_REF] Nelson | The spatial capacity of a slotted aloha multihop packet radio network with capture[END_REF] are some of the earliest position-based routing protocol that were proposed for packet radio networks and were lately adopted for wireless ad hoc and sensor networks such as GPSR [START_REF] Karp | Gpsr: Greedy perimeter stateless routing for wireless networks[END_REF] or BLR [START_REF] Valois | Igf: A state-free robust communication protocol for wireless sensor networks[END_REF]. When MFR/MFP is used, at each hop, the node forwards its packets to the neighbor that maximizes the euclidean distance or the progress 2 . NFR/NFP was proposed in order to maximize throughput in wireless networks by selecting its relay node among the nearest neighbors in terms of the euclidean distance or the progress. While RMP was proposed in order to introduce a load balance during the forwarding. Compass or Angular [START_REF] Kranakis | Compass routing on geometric networks[END_REF] was proposed in 1999s for ad hoc networks. Angular forwarding consists to forward the packets to the neighbor that has the smallest angle. It means that the selected forwarder is the neighbor which has lower angle between the destination node, the source node and itself.

III. ASSUMPTIONS AND RADIO MODEL

In this section, we present the assumptions and radio link model used in this work.

A. Assumptions

We consider a WSN composed of sensor nodes. The nodes are randomly deployed with uniform distribution on 2D plane. We assume that sensors are aware of their own locations and know the location and the identity of sink node in the plane. There is no assumption about the properties of radio channel: channel may be either symmetric or asymmetric depending on radio environment. Radio communication range is unknown parameter, it depends on transmission power of each sensor and also of the radio environment. Given the wireless broadcast advantage, a packet sent by a node is assumed to be received by its neighbors. Packets are assumed to be acknowledged hop by hop either explicitly as in [?] or implicitly as in [START_REF] Heissenbüttel | Blr: Beacon-less routing algorithm for mobile ad-hoc networks[END_REF]. Finally, we consider that there is no hole in the network in order to guarantee the use of greedy forwarding strategy.

B. Radio channel model

For both simulation and analysis undertaken in this study, we consider a lognormal shadowing model, which is useful when dealing with more realistic radio environment. In our analysis, as in [START_REF] Jiu-Qiang | An analytical model for link evaluation of wireless sensor networks[END_REF] and based on DSSS-OQPSK modulation of CC2420 chip for MicaZ nodes, the probability of successfully receiving packet with length l bits can be expressed as :

p f (γ(d)) = 1 -Q 2 B R bit 10 γ(d) 10 l ( 1
)
2 it is defined as the projection of the distance traveled over the last hop from the forwarder to the final destination D Where γ(d) = P t -P L(d 0 ) -10η log( d d0 ) + X σ -P n is signal-to-noise ratio at the transmitter-receiver distance d. P t , P L(d 0 ), X σ and P n are respectively the transmission power in dBm, the path loss at a reference distance d 0 in dBm, the shadow fading component, with X σ ∼ N (0, σ) and the noise power in dBm. The Q (.) is the Q-function. R bit is the radio bit rate while B is the noise bandwidth. Let us now define p r (γ(d)) as probability of successfully receiving acknowledgment from receiver after the transmission of data packet. This acknowledgment can be either a data packet or an explicit acknowledgement packet ACK. We define

p u = p f (γ(d)) × p r (γ(d))
as the probability of successful and acknowledged transmission in 1-hop. This means that both the unicast data and acknowledgment packets are successfully received. Let us consider the overall probability p follows a geometrical distribution with parameter p u , which is given by:

p = [ i-1 j=1 (1 -p u )] × p u . (2)
The expected number of retransmissions is then defined by N T x = ∞ i=1 i × p and can be expressed as:

N T x = ∞ i=1 i × p u × (1 -p u ) i-1 = 1 p u (3)

IV. RELAY SELECTION SCHEMES IN 1-HOP NEIGHBORHOOD

The purpose of this study is to evaluate the effect of realistic radio model on performance of relay selection schemes in the 1-hop neighborhood of the source node. We extensively evaluate the performance of relay selection schemes through WSNET [START_REF]Wsnet's web site[END_REF] simulations. We evaluated their performance according to the following metrics:

• Average Packet Delivery Ratio: It defines the average ratio of total number of packets successfully relayed by the next hop in the neighborhood of the source node to total number of packets generated by the source node.

Packet is assumed to be successfully relayed if and only if the source node is aware of its relaying. It determines the reliability of relay selection scheme. More reliable is the relay selection scheme, better its performance is in terms of the number of duplicated packets and network congestion. It is useful when dealing with asymmetric links. • Average Duplicated packet: It provides the average number of times the packet is relayed after its first retransmission. The results are presented with a confidence interval of 95%. Simulation parameters are summarized in the table I.

Results and discussions

Fig. 1 and Fig. 2 display the average packet delivery ratio and average number of duplicated packets of the six reviewed relay selection schemes when the source node degree increases. From the two figures, we can make out that whatever the The reason is due to the use of the reliable short wireless links while the remaining other relay selection schemes select their next hop among nodes that have the unreliable long wireless links. This is due to the fact that at each hop they attempt to minimize the expected number of hops by maximizing the forwarding distance. The difference between NFR and NFP is mainly due to the fact that NFP uses the progress, which is the product of the distance with the cosinus of angle instead of the distance used by NFR. When the cosinus is very low, whatever the distance, their product is low. Therefore, NFP selects its next hop among nodes that present a lower cosinus of the angle and longer distance. This is likely to lead a significant performance degradation when the degree increases. This explains the difference between these two nearest forwarding schemes. The opposite reasoning can be applied to MFP and MFR when the progress is used instead of the distance. It explains their difference. While Angular and RMP present the similar performance with a small difference in term of the number of duplicated packets. This can be mainly explained through the similarity between their relay nodes distribution in the vinicity of the source node.

A. Network model and notations

The static wireless sensor network is modeled as a directed graph G(V, E), where V and E are respectively the set of nodes and links. The set of nodes V is decomposed into one sink and a set of sensors that are respectively denoted by S and V s . Given a link (u, v) ∈ E, u is the sending node and v is the receiving node. Each link is characterized by the capacity C(u, v) and expected number of retransmissions N r (u, v). Both the link capacity and expected number of retransmissions depend on the modulation and coding scheme. The expected number of retransmissions expresses the radio link properties, where link with smaller expected number of retransmissions has higher link quality or is a reliable link (more details are presented in subsection III-B).

B. Energy consumption model

An important observation in the case of wireless sensor network is that the radio is the sensor node's component, which causes a significant energy dissipation [START_REF] Raghunathan | Energy-aware wireless microsensor networks[END_REF]. In this work, we consider an energy consumption model based on the node operation modes: transmission, reception and idle. A node u in transmission mode spends a transmitting cost E t (u), while each node v able to receive the packet ((u, v) ∈ E), spends a receiving cost E r (v). We assume that each node consumes a fixed cost which corresponds to the idle state. This energy consumption model is described in Eq.(4). f (r, u, v) is a binary variable which presents an unit of flow generated by the sensor r and routed through link (u, v).

∀u ∈ V E(u) = X r∈Vs ( X v∈Γ + (u) Nr(u, v) * Et(u) * f (r, u, v) + X v∈Γ -(u) X w∈Γ + (v) Nr(v, w) * Er(u) * f (r, v, w)) (4)

C. Routing problem

Each sensor u ∈ V s can generate and inject a quantity of traffic d(u) into the network. This traffic is routed from the source node to the sink S through several hops without loss. This route is calculated based on Eq.( 5), which presents the conservation flow constraint. It guarantees that the unit of flow f generated by a source node and entering a relay node is equal to the amount of its outflow. It also guarantees that the flow exiting at the source node is equal to the flow entering at the destination node. The link capacity constraints [START_REF] Heissenbüttel | Blr: Beacon-less routing algorithm for mobile ad-hoc networks[END_REF] impose that the total flow on the link (u, v) does not exceed its capacity.

∀r ∈ Vs, v ∈ V X (v,w) f (r, v, w) - X (w,v) f (r, w, v) = 8 > < > : d(r), if v = r -d(r), if v = S 0 if v ∈ Vs \ {r} (5) ∀(u, v) ∈ E X r∈Vs f (r, u, v) ≤ C(u, v) (6)

D. Linear programming formulation

The life time of a sensor is mainly related to its total energy consumption during its operation. Hence, minimizing the energy consumption of sensors allows to maximize the life time of each sensor in the network. Our linear program computes the optimal route that minimizes the maximum consumption energy of all sensors, i.e., min max u∈Vs {E(u)}. This is a min-max optimization problem, respecting a number of constraints: flow conservation, capacity constraints and energy model consumption (Eq.( 4)-Eq.( 6)). We can then formulate this problem as follows:

min f,E max u∈Vs E(u)
subject to Eq. ( 4)-Eq. [START_REF] Heissenbüttel | Blr: Beacon-less routing algorithm for mobile ad-hoc networks[END_REF] Note that this linear program can calculate several routes which give the same optimal solution (min-max value). An optimal route is the one that contains the "min-max" consumption node that consumes more than each other nodes involved in the route. Hence, various routes with different number of hops are possible. To solve this problem, we choose among all routes the one that minimizes the total energy consumption (min u E(u)).

Table II summarizes all the linear programming parameters and notations.

VI. ANALYTICAL RESULTS AND DISCUSSION

In this section, we extensively evaluate the energy consumption of these relay selection schemes in the case of the multihop wireless sensor network.

E, V

Set of links and nodes

(u, v) Radio link C(u, v) Capacity of the link (u, v) f (r, u, v)
Flow of sensor r and routed through link (u, v) Et(u)

Energy consumption of transmitter u Er(v)

Energy consumption of receiver v E ( u)

Total Energy consumption of node u Nr(u, v)

Expected number of retransmissions of link (u, v) d(u)

Traffic generated by node u

Γ + (u)/Γ -(u)
Set of outgoing/ingoing links of node u

TABLE II: LP Model Notations

A. Scenarios and Model Parameters

We have developed and tested the linear programming formulation using AMPL/CPLEX [START_REF] Fourer | A modeling language for mathematical programming[END_REF], [START_REF] Ilog | Solver cplex[END_REF], to evaluate the performance in term of energy consumption of the different relay selection schemes. We consider a sensor network where 100 nodes are randomly deployed with uniform distribution on a network of dimension 100m * 100m. For each pair of nodes u and v, when the transmission is from u to v, we use equation [START_REF]Wsnet's web site[END_REF] to compute the packet reception rate over links (u, v) and (v, u). Then, based on these values, we compute the expected number of retransmissions. We consider the energy dissipated during the transmission as twice of the energy dissipated during the reception. Some of analytical evaluation parameters have already been presented in the Table I.

B. Analytical results

Fig. 3 and Fig. 4 present the maximum energy cost of the node and total energy consumption for the different relay selection schemes. Two different scenarios are studied: the first one with single source node and second one with several source nodes (concurrent flows).

Scenario 1: Fig. 3 shows that NFR and NFP are more energy efficient than the remaining other schemes. These results can be explained by the fact that NFR and NFP in some scenarios, use the reliable wireless links (i.e. they are characterized by the least expected number of retransmissions) to route the traffic through several hops. Therefore, the impact of the number of transmissions and receptions at each hop is less important than those based on the most forward, RMP and Angular, which use a longer unreliable wireless links with several number of retransmissions. Note that these results are very similar to those obtained in the section IV, except for RMP scheme.

Scenario 2: The objective of this study is twofold. The first one is to study their energy efficiency. While the second one is to study their behavior with an optimal solution obtained by the linear programming. Fig. 4 shows that whatever the number of the source nodes, NFR is better than the remaining other schemes. The results also show that its performance is closer to the optimal solution performance, thanks to the reliable links. By analyzing both the results and Fig. 5, we show that to minimize the maximum energy consumption, the optimal solution uses disjoint paths 3 between the source nodes and sink node. Moreover, it forwards its packets in network area where both the node density and expected number of retransmissions are low in order to minimize the energy cost of retransmissions and receptions at each hop and attempts to minimize the path lengths. Based on the previous concluding remarks about the optimal solution behavior, we propose in this section a new relay selection scheme, F F R e , which can outperfom the performance of NFR in terms of the energy efficiency and number of hops. Unlike the optimal solution approach which does not forward its packets based on the destination position area, F F R e is based on position-based routing and uses the reliable links and attempts to minimize the number of hops between the source nodes and sink node. F F R e classifies the relay 3 Two paths are disjoint if they have no common relay nodes along the path To highlight the performance of our proposal, we objectively compare its performance with NFR. Fig. 7 and Fig. 8 display the total energy consumption and average number of hops respectively. Results show the energy efficiency and number of hops efficiency of F F R e . They also show that F F R e converges to the optimal solution (see Fig. 7).

VIII. CONCLUSIONS

In this paper, we investigate the performance of the different relay selection schemes for contention-based geographic forwarding under realistic radio channel model, where wireless links are unstable and unreliable. In this case, links may present varying link qualities during the time. Through extensive simulations, we first evaluate their performance in terms of packet delivery ratio and duplicated packets. We show that relay selection scheme such as Nearest Forward within Radius (NFR), which selects the relay of each packet in the nearest Then, we extend the previous work in multi-hop wireless network by analytically evaluate their energy efficiency. A linear programming was developed to calculate an optimal path which minimizes the maximum node cost and the total energy consumption. Based on these results, we also show that NFR outperforms the remaining other schemes. We propose the Furthest Forward within Reliable neighbors (F F R e ), which attempts to minimize both the expected number of transmissions per-hop and number of hops. The performance evaluation shows that F F R e allows to save more energy than NFR by minimizing the number of hops from the source nodes and the destination node.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Average packet delivery ratio vs Source node degree.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Maximum energy consumption for each strategy.

Fig. 5 :Fig. 6 :

 56 Fig. 5: Optimal paths: the linear program adapts the routing paths depending to the number of source nodes.

Fig. 8 :

 8 Fig. 7: Total energy consumption

TABLE I :

 I Simulation parameters degree, NFR is the most suitable relay selection scheme in terms of packet delivery ratio and the least number of duplicated packets.