ON A THREE-DIMENSIONAL FREE BOUNDARY PROBLEM MODELING ELECTROSTATIC MEMS

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ON A THREE-DIMENSIONAL FREE BOUNDARY PROBLEM MODELING ELECTROSTATIC MEMS

Philippe Laurencot, Christoph Walker

To cite this version:

Philippe Laurencot, Christoph Walker. ON A THREE-DIMENSIONAL FREE BOUNDARY PROB- LEM MODELING ELECTROSTATIC MEMS. Interfaces and Free Boundaries, European Mathemat- ical Society, 2016, 18, pp.393–411. �hal-01154864�

ELECTROSTATIC MEMS

PHILIPPE LAURENC¸ OT AND CHRISTOPH WALKER

ABSTRACT. We consider the dynamics of an electrostatically actuated thin elastic plate being clamped at its boundary above a rigid plate. The model includes the harmonic electrostatic potential in the three-dimensional time-varying region between the plates along with a fourth-order semilinear parabolic equation for the elastic plate deflection which is coupled to the square of the gradient trace of the electrostatic potential on this plate.

The strength of the coupling is tuned by a parameterλproportional to the square of the applied voltage. We prove that this free boundary problem is locally well-posed in time and that for small values ofλsolutions exist globally in time. We also derive the existence of a branch of asymptotically stable stationary solutions for small values ofλand non-existence of stationary solutions for large values thereof, the latter being restricted to a disc-shaped plate.

1. INTRODUCTION ANDMAINRESULTS

We focus on an idealized model for an electrostatically actuated microelectromechanical system (MEMS).

The device is built of a thin conducting elastic plate being clamped at its boundary above a rigid conducting plate. A Coulomb force is induced across the device by holding the ground and the elastic plates at different electric potentials which results in a deflection of the elastic plate and thus in a change in geometry of the device, see Figure 1. An ubiquitous feature of such MEMS devices is the occurrence of the so-called “pull- in” instability which manifests above a critical threshold for the voltage difference in a touchdown of the elastic plate on the rigid ground plate. Estimating this threshold value is of utmost interest in applications as it determines the stable operating regime for such a MEMS device. To set up a mathematical model we assume that the dynamics of the device can be fully described by the deflection of the elastic plate from its rest position (when no voltage difference exists) and the electrostatic potential in the varying region between the two plates. We further assume that the elastic plate in its rest position and the fixed ground plate can be described by a regionDinR². After a suitable scaling the rigid ground plate is located atz=−1and the rest position of the elastic plate is atz= 0. Ifu= u(t,x,y)fort≥ 0and(x,y)∈ Ddescribes the vertical displacement of the elastic plate from its rest position, thenuevolves in the damping dominated regime according to

∂tu+β∆²u− τ+ak∇uk²₂^∆u=−λ|∇εψ(t,x,y,u(t,x,y))|², (x,y)∈D, t>_{0 , (1.1)} with clamped boundary conditions

u=∂νu=0 , (x,y)∈∂D, t>_{0 , (1.2)} and initial condition

u(0,x,y) =u⁰(x,y), (x,y)∈D. (1.3) Here we put

∇εψ:= ε∂xψ,ε∂yψ,∂zψ ,

Date: May 25, 2015.

2010Mathematics Subject Classification. 35K91, 35R35, 35M33, 35Q74.

Key words and phrases. MEMS, free boundary problem, stationary solutions.

Partially supported by the French-German PROCOPE project 30718ZG..

1

FIGURE1. Schematic diagram of an idealized electrostatic MEMS device.

whereε > ₀ is the aspect ratio of the device, i.e. the ratio between vertical and horizontal dimensions, λ > ₀is proportional to the square of the applied voltage difference, andψ = ψ(t,x,y,z)denotes the dimensionless electrostatic potential. The latter satisfies a rescaled Laplace equation

ε²∂²_xψ+ε²∂²_yψ+∂²_zψ=0 , (x,y,z)∈^Ω(u(t)), t>_{0 , (1.4)} in the cylinder

Ω(u(t)):={(x,y,z)∈D×(−1,∞) : −1<_z<_u(t,x,y)}

between the rigid ground plate atz=−1and the deflected elastic plate. The boundary conditions forψare ψ(t,x,y,z) = ¹+z

1+u(t,x,y) ^, (x,y,z)∈∂Ω(u(t)), t>_{0 . (1.5)} In equation (1.1), the fourth-order termβ∆²uwithβ > ₀reflects plate bending while the linear second- order termτ∆uwithτ≥0and the non-local second-order termak∇uk²₂^∆uwitha≥0and

k∇uk²₂:= Z

D|∇u|²d(x,y)

account for external stretching and self-stretching forces generated by large oscillations, respectively. The right-hand side of (1.1) is due to the electrostatic forces exerted on the elastic plate and is tuned by the strength of the applied voltage difference which is accounted for by the parameterλ. The boundary con- ditions (1.2) mean that the elastic plate is clamped. According to (1.4)-(1.5), the electrostatic potential is harmonic in the regionΩ(u)enclosed by the two plates with value1on the elastic plate and value0on the ground plate. We refer the reader e.g. to [5, 9, 12, 20] and the references therein for more details on the derivation of the model.

Equations (1.1)-(1.5) feature a singularity which reflects the pull-in instability occurring when the elastic plate touches down on the ground plate. Indeed, whenureaches the value−1somewhere, the regionΩ(u) gets disconnected. Moreover, the imposed boundary conditions (1.5) imply that the vertical derivative

∂zψ(x,y,u(x,y))blows up at the touchdown point(x,y)and, in turn, the right-hand side of (1.1) becomes singular. Questions regarding (non-)existence of stationary solutions and of global solutions to the evolution problem as well as the qualitative behavior of the latter are strongly related. Due to the intricate coupling of the possibly singular equation (1.1) and the free boundary problem (1.4)-(1.5) in non-smooth domains, answers are, however, not easy to obtain.

It is thus not surprising that most mathematical research to date has been dedicated to various variants of the so-calledsmall gap model, a relevant approximation of (1.1)-(1.5) obtained by formally settingε=0 therein. This approximation allows one to compute the electrostatic potential

ψ(t,x,z) = ¹+z 1+u(t,x,y) explicitly in dependence ofu, the latter to be determined via

∂tu+β∆²u− τ+ak∇uk²₂^∆u=−λ 1

(1+u)^{2 ,} (x,y)∈D, t>_{0 ,}

subject to (1.2) and (1.3). We note that this small gap approximation is a singular equation but no longer a free boundary problem. We refer to [5, 7, 13, 17] and the references therein for more information on this case.

The free boundary problem (1.1)-(1.5) has been investigated in a series of papers by the authors [4, 10–

12, 15], though in a simpler geometry assumingDto be a rectangle and presupposing zero variation in the y-direction, see also [16] for the case of a non-constant permittivity profile. In this geometry the deflection u =u(t,x)is independent ofyandψis harmonic in a two-dimensional domain. In the present paper we remove this assumption and tackle for the first time the evolution problem with a three-dimensional domain Ω(u(t)), assuming only a convexity property onD. More precisely, we assume in the following that

Dis a bounded and convex domain inR²with aC²-smooth boundary. (1.6) A typical example forDis a disc. We shall see later on that condition (1.6) is used to obtain sufficiently smooth solutions to the elliptic problem (1.4)-(1.5) in order for the trace of∇εψto be well-defined on

∂Ω(u(t)), a fact which is not clear at first glance asΩ(u(t))is only a Lipschitz domain. Still it turns out that the regularity of the square of the gradient trace of this solution occurring on the right-hand side of (1.1) is weaker than in the case of a two-dimensional domainΩ(u(t))studied in [4, 10–12, 15] restricting us to study the fourth-order caseβ>₀only, see Remark 3.1 for more information.

The following result shows that (1.1)-(1.5) is locally well-posed in general and globally well-posed for smallaor small initial values provided thatλis small as well.

Theorem 1.1(Local and Global Well-Posedness). Suppose(1.6). Letε>_0,4ξ∈(7/3, 4), and consider an initial valueu⁰∈W₂^4ξ(D)^{such that}u⁰(x,y)>−1for(x,y)∈ Dandu⁰=∂νu⁰=0on∂D. Then, the following are true:

(i) For each voltage value λ > ₀, there is a unique solution(u,ψ) ^to(1.1)-(1.5)on the maximal interval of existence[0,Tm)in the sense that

u∈C [0,Tm),W₂^4ξ(D)∩C (0,Tm),W₂⁴(D)∩C¹ (0,Tm),L2(D) (1.7) satisfies(1.1)-(1.3)together with

u(t,x,y)>−1 , (t,x,y)∈[0,Tm)×D,

andψ(t)∈W₂² Ω(u(t))^solves(1.4)-(1.5)inΩ(u(t))^{for each}t∈[0,Tm)^. (ii) If, for eachT>0, there isκ(T)∈(0, 1)^{such that}

ku(t)k_W4ξ

2 (D)≤κ(T)⁻¹, u(t)≥ −1+κ(T) in D fort∈[0,Tm)∩[0,T], then the solution exists globally, that is,Tm=^∞.

(iii) Givenκ∈(0, 1/2), there existsm∗:=m∗(κ,ε)>₀such thatTm=^∞and ku(t)k_W4ξ

2 (D)≤κ⁻¹, u(t)≥ −1+κinD

fort≥0, provided thatλ+ak∇u⁰k²₂≤m∗and ku⁰k_W4ξ

2 (D)≤(2κ)⁻¹, u⁰≥ −1+2κinD.

(iv) If Dis a disc inR²andu⁰ = u⁰(x,y)is radially symmetric with respect to(x,y) ∈ D, then, for all t ∈ [0,Tm)^, u = u(t,x,y) ^andψ = ψ(t,x,y,z)are radially symmetric with respect to(x,y)∈D.

The global existence criterion stated in part (ii) of Theorem 1.1 involving a blow-up of some Sobolev- norm or occurrence of a touchdown is not yet optimal as the possible norm blow-up has rather mathematical than physical reasons. In the case of a two-dimensional domainΩ(u)this condition is superfluous as shown in [11]. Note that part (iii) of Theorem 1.1 provides uniform estimates onuand ensures, in particular, that unever touches down on1, not even in infinite time.

Regarding existence of stationary solutions to (1.1)-(1.5) for an arbitrary convex domainDwe have:

Theorem 1.2 (Stationary Solutions). Suppose (1.6). Then, given ε > _{0 and κ} ∈ (0, 1), there are δ:=δ(κ,ε)> ₀ and an analytic function [λ 7→ U_λ] : [0,δ) → W₂⁴(D) ^{such that} (U_λ,Ψ_λ) ^{is for} eachλ ∈ (0,δ)an asymptotically stable stationary solution to(1.1)-(1.5)withU_λ > −1+κ inDand Ψ_λ∈W₂²(^Ω(Uλ)).

We shall point out that while Theorem 1.2 ensures the existence of at least one stationary solution for a fixed, sufficiently small voltage valueλ, a recent result [15] yields a second one for (some of) these values in the two-dimensional case.

WhenDis a disc inR², additional information on stationary solutions can be retrieved, in particular a non-existence result whena=0. Roughly speaking, this additional information is provided by the fact that the operatorβ∆²−τ∆satisfies the maximum principle when restricted to radially symmetric functions (see Section 5 for more details).

Theorem 1.3. Assume thatDis a disc inR²and letε>_0.

(i) For anyκ∈(0, 1)andλ∈(0,δ(κ,ε)), the stationary solution(U_λ,Ψ_λ)to(1.1)-(1.5)constructed in Theorem 1.2 enjoys the following properties: the functionU_λis radially symmetric andU_λ<₀ inDwhileΨ_λis radially symmetric in the first two variables(x,y).

(ii) Assume thata= 0. There areε∗ >₀and a functionΛ: (0,ε∗) →(0,∞)such that there is no radially symmetric stationary solution(u,ψ)to(1.1)-(1.5)forλ>Λ(ε)andε∈(0,ε∗).

The outline of this paper is as follows. In Section 2 we first investigate the elliptic problem (1.4)-(1.5) for the electrostatic potential in dependence on a given deflection of the elastic plate. The main result of this section is Proposition 2.1 which implies that (1.1)-(1.5) can be rewritten as a semilinear equation for the deflectionuonly. The proof is rather involved and divided into several steps. Section 3 is then devoted to the proof of Theorem 1.1 while the proof of Theorem 1.2 is given in Section 4. Finally, in Section 5 we indicate how the proof of Theorem 1.3 can be carried out based on the corresponding proof for the two-dimensional case.

2. THEELECTROSTATICPOTENTIALEQUATION

We first focus on the free boundary problem (1.4)-(1.5) which we transform to the cylinder Ω:=D×(0, 1).

More precisely, let q ≥ 2 be fixed and consider an arbitrary function v ∈ W_q,B² (D)taking values in (−1,∞), where

W_p,B^α (D):=













{w∈W_p^α(D) : w=∂νw=0on∂D}, α>₁+1/p, {w∈W_p^α(D) : w=0on∂D}, α∈(1/p, 1+1/p), W_p^α(D), 0≤α<_1/p,

forp≥2. We define

Ω(v):={(x,y,z)∈D×(−1,∞) : −1<_z<_v(x,y)}

and consider the rescaled Laplace equation

ε²∂²_xψv+ε²∂²_yψv+∂²_zψv = 0 , (x,y,z)∈^Ω(v), (2.1) ψv(x,y,z) = ¹+z

1+v(x,y)^, (x,y,z)∈∂Ω(v). (2.2) Introducing the diffeomorphismTv:Ω(v)→^Ωgiven by

Tv(x,y,z):=

x,y, 1+z 1+v(x,y)

, (x,y,z)∈^Ω(v), (2.3) we note that its inverse is

T_v⁻¹(x,y,η) = x,y,(1+v(x,y))η−1

, (x,y,η)∈^Ω, (2.4)

and that the rescaled Laplace operator in (2.1) is transformed to thev-dependent differential operator Lvw :=ε²∂²_xw+ε²∂²_yw−2ε²η ∂xv(x,y)

1+v(x,y) ^∂x∂ηw−2ε²η ∂yv(x,y)

1+v(x,y) ^∂y∂ηw +¹+ε²η²|∇v(x,y)|²

(1+v(x,y))^{2 ∂}

2ηw+ε²η

2 |∇v(x,y)|²

(1+v(x,y))²− ^∆v(x,y) 1+v(x,y)

∂ηw. The boundary value problem (2.1)-(2.2) is then equivalent to

Lvφv

(x,y,η) =0 , (x,y,η)∈^Ω, (2.5)

φv(x,y,η) =η, (x,y,η)∈∂Ω, (2.6) via the transformationφv = ψv◦ T_v⁻¹. Observe that (2.5)-(2.6) is an elliptic equation with non-constant coefficients but in the fixed domainΩ.

We now aim at studying precisely the well-posedness of (2.5)-(2.6) as well as the regularity of its solu- tions. To this end, we introduce forκ ∈(0, 1)andp≥2the set

Sp(κ):=

w∈W_p,B² (D) : kwk_W2

p,B(D)<_{1/κ and} −1+κ<_w(x,y)for(x,y)∈D

, with

Sp(κ):=

w∈W²_p,B(D) : kwk_W2

p,B(D)≤1/κ and −1+κ≤w(x,y)for(x,y)∈D

being its closure inW_p,B² (D). The key result of this section reads:

Proposition 2.1. Suppose(1.6). Letε>_0,κ ∈ (0, 1), andq≥3. For eachv ∈ Sq(κ)there is a unique solutionφv∈W₂²(^Ω)to(2.5)-(2.6). Furthermore there isc₁(κ,ε)>_{0such that}

kφv₁−φv₂k_W2

2(Ω)≤ c₁(κ,ε)kv₁−v₂k_W2

q(D), v₁,v₂∈Sq(κ), (2.7) and the mapping

gε :Sq(κ)−→ L2(D), v7−→ ¹+ε²|∇v|²

(1+v)² |∂ηφv(·,·, 1)|² is analytic, globally Lipschitz continuous, and bounded.

Several steps are needed to prove Proposition 2.1. We begin with the analysis of the Dirichlet problem associated toLv.

Lemma 2.2. Suppose(1.6). Letε>_0,κ∈(0, 1), andq>2. For eachv∈Sq(κ)andF ∈L2(^Ω), there exists a unique solution

Φ∈W_2,B¹ (^Ω):={w∈W₂¹(^Ω) : w=0on∂Ω} to the boundary value problem

−LvΦ=F in Ω, Φ=0 on ∂Ω. (2.8)

Proof. Sinceq>2, the definition ofSq(κ)and Sobolev’s embedding theorem guarantee the existence of some constantc₀>₀depending only onqandDsuch that, forv∈Sq(κ),

1+v(x,y)≥κ, (x,y)∈D, and kvk_C1(D)¯ ≤ ^c0

κ . (2.9)

We now claim that, due to (2.9), the operator−Lvis elliptic with ellipticity constantν(κ,ε) > _{0 being} independent ofv∈Sq(κ). Indeed, for(x,y,η)∈^Ω, set

a11(v):=ε², a13(v) =a31(v):=−ε²η ∂xv

1+v

(x,y), a22(v):=ε², a23(v) =a32(v):=−ε²η

∂yv 1+v

(x,y), a₃₃(v):=

1+ε²η²|∇v|² (1+v)²

(x,y), b₁(v):=ε² ∂xv

1+v

(x,y), b2(v):=ε²

∂yv 1+v

(x,y), b3(v):=−ε²η

|∇v|² (1+v)²

(x,y), which allows us to write−Lvin divergence form:

−Lvw=−∂x a₁₁(v)∂xw+a₁₃(v)∂ηw

−∂y a₂₂(v)∂yw+a₂₃(v)∂ηw

−∂η a₃₁(v)∂xw+a32(v)∂yw+a33(v)∂ηw

−b1(v)∂xw−b2(v)∂yw−b3(v)∂ηw. Denoting the principal of−Lvby−L⁰_v, that is,

−L⁰_vw:=−∂x a11(v)∂xw+a13(v)∂ηw

−∂y a22(v)∂yw+a23(v)∂ηw

−∂η a31(v)∂xw+a32(v)∂yw+a33(v)∂ηw , and introducing the associated matrix

P :=











ε² 0 −^ε

2∂xv(x,y) 1+v(x,y)^η

0 ε² −^ε

2∂yv(x,y) 1+v(x,y)^η

−^ε

2∂xv(x,y) 1+v(x,y)^η −^ε

2∂yv(x,y) 1+v(x,y)^η

1+ε²η²|∇v(x,y)|² (1+v(x,y)²











we observe that, for(x,y,η)∈^Ω, the eigenvalues ofPareε²and µ± = ¹

2 t±^pt²−4d withtanddgiven by

t:=ε²+¹+ε²η²|∇v(x,y)|²

(1+v(x,y))^{2 , d:}= ^ε

2

(1+v(x,y))^{2 .}

By (2.9),

µ+ >_µ− ≥ ^d

t ≥ ^κε

2

(1+ε²)κ²+2ε²c²₀ >_{0 ,}

which implies that−Lvis elliptic with a positive ellipticity constant depending onκandεonly. Further- more, we infer from (2.9) and the definition ofSq(κ)that

3

∑

i,j=1

kaij(v)k_L∞(Ω)+

3

∑

i=1

kbi(v)k_L∞(Ω)≤c2(κ,ε) (2.10) for allv ∈ Sq(κ). It then follows from [6, Theorem 8.3] that, givenv ∈ Sq(κ) andF ∈ L2(^Ω), the boundary value problem (2.8) has a unique weak solutionΦ∈W_2,B¹ (^Ω). Next, for smoother functionsv, we make use of the convexity ofΩto gain more regularity on the solution to (2.8).

Lemma 2.3. Suppose(1.6). Letε > _0andκ ∈ (0, 1). For eachv ∈ S∞(κ)andF ∈ L2(^Ω), the weak solutionΦ∈W_2,B¹ (^Ω)to(2.8)belongs toW_2,B² (^Ω):=W₂²(^Ω)∩W_2,B¹ (^Ω).

Proof. ConsiderF ∈L₂(^Ω)and denote the corresponding weak solution to (2.8) byΦ∈W_2,B¹ (^Ω). The regularity ofΦandvensure that

G:=F+b₁(v)∂xΦ+b₂(v)∂yΦ+b₃(v)∂ηΦ∈L₂(^Ω).

SinceΩis convex anda_ij(v)∈W∞¹(^Ω)for1≤i,j≤3, we are in a position to apply [8, Theorem 3.2.1.2]

to conclude that there is a unique solutionΦb ∈W_2,B² (^Ω)to the boundary value problem

−L⁰_v^Φb =G in Ω, Φb =0 on ∂Ω. (2.11)

whereL⁰_vis the principal part of the operatorLv. It also follows from [6, Theorem 8.3] that (2.11) has a unique weak solution inW_2,B¹ (^Ω). Due to (2.8) and the definition ofG, the functionsΦandΦb are both weak solutions inW_2,B¹ (^Ω)to (2.11) and thusΦ=Φb ∈W_2,B² (^Ω). The next step is to adapt the analysis performed in the two-dimensional case in [15, Section 4] to derive an estimate on theW₂¹(^Ω)-norm of∂ηΦwhich is suitably uniform with respect tov. We begin with the following trace estimate.

Lemma 2.4. Suppose(1.6). Letp∈[2, 4]. There existsc3(p)>₀such that kw(·,·, 1)k^p_L

p(D)≤c3(p)kwk^(3p−4)/2_W1

2(Ω) kwk^(4−p)/2_L

2(Ω) , w∈W₂¹(^Ω). (2.12) Proof. Letw∈W₂¹(^Ω). Forη∈(0, 1)and(x,y)∈D, one has

|w(x,y, 1)|^p=|w(x,y,η)|^p+p Z ₁

η |w(x,y,z)|^p−2w(x,y,z)∂ηw(x,y,z)dz. Integrating the above identity with respect to(x,y)∈Dgives

kw(·,·, 1)k^p_L

p(D)≤ Z

D|w(x,y,η)|^pd(x,y) +p

Z

Ω|w(x,y,z)|^p−1|∂ηw(x,y,z)|d(x,y,z).

We next integrate with respect toη∈(0, 1)and use H¨older’s inequality to obtain kw(·,·, 1)k^p_L

p(D)≤ kwk^p_L

p(Ω)+pkwk^p−1_L

2(p−1)(Ω)k∂ηwk_L2(Ω)

≤ kwk2kwk^p−1_L

2(p−1)(Ω)+pkwk^p−1_L

2(p−1)(Ω)k∂ηwk_L2(Ω)

≤pkwk^p−1_L

2(p−1)(Ω)kwk_W1 2(Ω). We finally use the Gagliardo-Nirenberg inequality

kwk^p−1_L

2(p−1)(Ω)≤c(p)kwk^3(p−2)/2_W1

2(Ω) kwk^(4−p)/2_L

2(Ω)

to complete the proof.

Lemma 2.5. Suppose(1.6). Letε> ₀,κ ∈ (0, 1)^{, and}q>₂. For eachv ∈ Sq(κ)^andF ∈ L2(^Ω)^{, the} weak solutionΦ∈W_2,B¹ (^Ω)^to(2.8)belongs to the Hilbert spaceX(^Ω)^{defined by}

X(^Ω):=ⁿw∈W_2,B¹ (^Ω) : ∂ηw∈W₂¹(^Ω)^o , and there isc4(κ,ε)>_{0such that}

k^Φk_W1

2(Ω)+k∂ηΦk_W1

2(Ω)≤c4(κ,ε)k^Φk_L2(Ω)+kFk_L2(Ω)

. (2.13)

Proof. We first recall that, due to the continuous embedding of W_q²(D) inW∞¹(D), there is a positive constantc0>₀depending only onqandDsuch that, forv∈Sq(κ),

1+v(x,y)≥κ, (x,y)∈D, and kvk_C1(D)¯ ≤ ^c0

κ . (2.14)

ConsiderF∈ L2(^Ω)and denote the corresponding weak solution to (2.8) byΦ∈W_2,B¹ (^Ω). We begin with an estimate forΦinW₂¹(^Ω)and first infer from (2.8) and the Divergence Theorem that

Z

ΩFΦd(x,y,η) =− Z

Ω

ΦLvΦd(x,y,η)

=ε²kPxk²_L

2(Ω)+ε²kPyk²_L

2(Ω)+kPηk²_L

2(Ω)

−ε² Z

Ω

∂xv

1+vΦ∂xΦ+ ^∂yv

1+vΦ∂yΦ−η |∇v|² (1+v)^2Φ∂ηΦ

d(x,y,η) with

Px:=∂xΦ−η ∂xv

1+v∂ηΦ, Py:=∂yΦ−η ∂yv

1+v∂ηΦ, Pη:= ^∂ηΦ 1+v . Note that (2.14) ensures that

k∇^Φk²_L2(Ω)≤c(κ)^hkPxk²_L2(Ω)+kPyk²_L2(Ω)+kPηk²_L2(Ω)ⁱ . (2.15) Using (2.14) and Cauchy-Schwarz’ and Young’s inequalities, we obtain

ε²kPxk²_L

2(Ω)+ε²kPyk²_L

2(Ω)+kPηk²_L

2(Ω)

= Z

ΩFΦd(x,y,η) +ε² Z

Ω

∂xv

1+vΦPx+ ^∂yv 1+vΦPy

d(x,y,η)

≤ kFk_L2(Ω)k^Φk_L2(Ω)+ε²k^Φk_L2(Ω)

|∇v| 1+v _L

∞(D)

hkPxk_L2(Ω)+kPyk_L2(Ω)

i

≤ ^ε

2

2

hkPxk²_L2(Ω)+kPyk²_L2(Ω)ⁱ+c(κ,ε)^hkFk²_L2(Ω)+k^Φk²_L2(Ω)ⁱ ,

whence

ε²kPxk²_L2(Ω)+ε²kPyk²_L2(Ω)+kPηk²_L2(Ω)≤c(κ,ε)^hkFk²_L2(Ω)+k^Φk²_L2(Ω)ⁱ . Combining (2.15) with the above estimate gives

k^Φk²_W1

2(Ω)≤c(κ,ε)^hkFk²_L

2(Ω)+k^Φk²_L

2(Ω)

i

. (2.16)

We now turn to an estimate on∂ηΦinW₂¹(^Ω)which is established first for smooth functionsv, the constants appearing in the estimates depending only onqandκ. Indeed, assume first that, besides being in Sq(κ), the functionvalso belongs toS∞(κ^′)for someκ^′ ∈(0, 1). ThenΦ∈W_2,B² (^Ω)by Lemma 2.3 and, setting

ζx:=∂x∂ηΦ, ζy:=∂y∂ηΦ, ζη :=∂²_ηΦ, it follows from Lemma 2.6 below that

Z

Ω∂²_xΦ∂²_ηΦd(x,y,η) = Z

Ω|ζx|²d(x,y,η), Z

Ω∂²_yΦ∂²_ηΦd(x,y,η) = Z

Ω|ζy|²d(x,y,η).

(2.17)

We then infer from (2.8) and (2.17) that

− Z

ΩFζηd(x,y,η) = Z

Ω∂²_ηΦLvΦd(x,y,η)

=ε²kQxk²_L2(Ω)+ε²kQyk²_L2(Ω)+kQηk²_L2(Ω)

+ε² Z

Ωη

2|∇v|²−(1+v)^∆v (1+v)²

∂ηΦ∂²_ηΦd(x,y,η), where

Qx:=ζx−η ∂xv

1+vζη, Qy:=ζy−η ∂yv

1+vζη, Qη:= ^ζη 1+v . Owing to (2.14) we note that

k∇∂ηΦk²_L

2(Ω)≤c(κ)^hkQxk²_L

2(Ω)+kQyk²_L

2(Ω)+kQηk²_L

2(Ω)

i

. (2.18)

Since∂ηΦ∂²_ηΦ=∂η |∂ηΦ|²/2, using integration by parts to handle the last term on the right-hand side of the above identity leads us to

ε²kQxk²_L

2(Ω)+ε²kQyk²_L

2(Ω)+kQηk²_L

2(Ω)

= −

Z

ΩFζηd(x,y,η) +^ε

2

2 Z

Ω

2|∇v|²−(1+v)^∆v (1+v)²

|∂ηΦ|²d(x,y,η)

−^ε

2

2 Z

D

2|∇v|²−(1+v)^∆v (1+v)²

|∂ηΦ(x,y, 1)|²d(x,y). (2.19) We now estimate successively the three terms on the right-hand side of (2.19) and begin with the first one which is the easiest. Indeed, it follows from (2.14) and Cauchy-Schwarz’ and Young’s inequalities that

−

Z

ΩFζηd(x,y,η)

≤ k(1+v)Fk_L2(Ω)kQηk_L2(Ω)≤ ¹

4kQηk²_L2(Ω)+c(κ)kFk²_L2(Ω). (2.20)

Next, introducingq^′ :=q/(q−1)∈(1, 2), we infer from H¨older’s and Gagliardo-Nirenberg inequalities

that

Z Ω

2|∇v|²−(1+v)^∆v (1+v)²

|∂ηΦ|²d(x,y,η)

≤ ¹ κ²

h

2k∇vk_L∞(D)k∇vk_Lq(D)+k1+vk_L∞(D)k^∆vk_Lq(D)ⁱk∂ηΦk²_L

2q′(Ω)

≤c(κ)k∂ηΦk^3/q_W1

2(Ω)k∂ηΦk^(2q−3)/q_L

2(Ω)

=c(κ)k∂ηΦk²_L2(Ω)+k∇∂ηΦk²_L2(Ω)^3/2qk∂ηΦk^(2q−3)/q_L

2(Ω) . Using (2.18) and Young’s inequality we end up with

Z

Ω

2|∇v|²−(1+v)^∆v (1+v)²

|∂ηΦ|²d(x,y,η)

≤ ¹

2kQxk²_L2(Ω)+¹

2kQyk²_L2(Ω)+ ¹

2ε²kQηk²_L2(Ω)+c(κ,ε)k∂ηΦk²_L2(Ω). (2.21) Similarly, since2q^′∈(2, 4), H¨older’s and Young’s inequalities combined with (2.18) and Lemma 2.4 entail that

Z D

2|∇v|²−(1+v)^∆v (1+v)²

|∂ηΦ(x,y, 1)|²d(x,y)

≤ ¹ κ²

h

2k∇vk_L∞(D)k∇vk_Lq(D)+k1+vk_L∞(D)k^∆vk_Lq(D)ⁱk∂ηΦ(·,·, 1)k²_L

2q′(D)

≤c(κ)k∂ηΦk^(q−2)/q_L

2(Ω) k∂ηΦk^(q+2)/q_W1

2(Ω)

≤ ¹ 2kQxk²_L

2(Ω)+¹ 2kQyk²_L

2(Ω)+ ¹

2ε²kQηk²_L

2(Ω)+c(κ,ε)k∂ηΦk²_L

2(Ω). (2.22) Inserting (2.20)-(2.22) in (2.19) leads us to

ε²kQxk²_L2(Ω)+ε²kQyk²_L2(Ω)+kQηk²_L2(Ω)≤c(κ,ε)kFk²_L2(Ω)+k∂ηΦk²_L2(Ω) , from which we deduce, thanks to (2.16) and (2.18), that (2.13) holds true.

Since the estimate (2.13) does not depend on the regularity ofv, the fact that it extends to all functions

inSq(κ)follows by a classical approximation argument.

It remains to prove the auxiliary result used in (2.17) which is recalled in the following lemma.

Lemma 2.6. IfΦ∈W_2,B² (^Ω)^{, then} Z

Ω∂²_xΦ∂²_ηΦd(x,y,η) = Z

Ω|∂x∂ηΦ|²d(x,y,η), Z

Ω∂²_yΦ∂²_ηΦd(x,y,η) = Z

Ω|∂y∂ηΦ|²d(x,y,η).

Proof. SinceΩ=D×(0, 1)andDis a convex subset ofR², the projectionpr2(D)ofDonto they-axis as well as the sections

D_[y] :={x∈^R : (x,y)∈D}, y∈ pr2(D),

are intervals. Moreover, the Fubini-Tonelli Theorem (together with Nikodym’s characterization of Sobolev spaces via absolutely continuous functions, see [21, Theorem 2.1.4]) implies that for a.a. y ∈ pr2(D), the functionΦ(·,y,·)belongs toW₂²(D_[y]×(0, 1))withΦ(·,y,·) = 0on∂(D_[y]×(0, 1))sinceΦ ∈

W_2,B² (^Ω)⊂ C(^Ω). Thus,D_[y]×(0, 1)being a rectangle, we may apply [8, Lemma 4.3.1.2] to conclude

that _Z

D_[y]×(0,1)∂²_xΦ(x,y,η)∂²_ηΦ(x,y,η)d(x,η) = Z

D_[y]×(0,1)|∂x∂ηΦ(x,y,η)|²d(x,η) for a.a.y∈ pr2(D). Recalling thatpr2(^Ω) =pr2(D)is measurable and

Ω_[y]:={(x,η)∈^R×^R : (x,y,η)∈^Ω}=D_[y]×(0, 1),

the first assertion follows by integrating the above identity on pr₂(^Ω) with respect to y and using the

Fubini-Tonelli Theorem. The second assertion is analogous.

To recover the fullW₂²-regularity of the solution to (2.8) we need to have slightly smoother functionsv.

Proposition 2.7. Suppose(1.6). Letε>_0,κ ∈ (0, 1)^{, and}q ≥3. For eachv ∈ Sq(κ)^andF∈ L₂(^Ω)^, the weak solutionΦ∈W_2,B¹ (^Ω)^to(2.8)belongs toW_2,B² (^Ω)and there isc5(κ,ε)>₀such that

k^Φk_W2

2(Ω)≤c₅(κ,ε)kFk_L2(Ω). (2.23) Proof. ConsiderF∈L2(^Ω)and denote the corresponding weak solution to (2.8) byΦ. Introducing

J(x,y,η):=2ε²η ∂xv

1+v

(x,y)∂x∂ηΦ(x,y,η) +2ε²η ∂yv

1+v

(x,y)∂y∂ηΦ(x,y,η) +

1−¹+ε²η²|∇v|² (1+v)²

(x,y)∂²_ηΦ(x,y,η)

−ε²η

2 |∇v|²

(1+v)²− ^∆v 1+v

(x,y)∂ηΦ(x,y,η), it follows from (2.8) thatΦsolves

ε²∂²_xΦ+ε²∂²_yΦ+∂²_ηΦ= J in Ω, Φ=0 on ∂Ω.

Moreover, Lemma 2.5 and the continuous embeddings ofW_q²(D)inW∞¹(D)andW₂¹(^Ω)inL₆(^Ω)guar- antee thatJbelongs toL₂(^Ω)with

kJk²_L2(Ω)≤c(κ,ε)k^Φk²_L2(Ω)+kFk²_L2(Ω) .

We then infer from [8, Theorem 3.2.1.2] thatΦ∈W₂²(^Ω)and inspecting the proof of [8, Theorem 3.2.1.2]

along with [8, Theorem 3.1.3.1 & Lemma 3.2.1.1] ensures that k^Φk_W2

2(Ω)≤c(κ,ε)kJk_L2(Ω). We have thus shown thatΦ∈W₂²(^Ω)and satisfies

k^Φk_W2

2(Ω)≤c(κ,ε)k^Φk²_L2(Ω)+kFk²_L2(Ω) .

Finally, since the embeddings ofW₂²(^Ω)inW₂¹(^Ω)andW_q²(D)inC¹(D^¯)are compact, we may proceed

as in the proof of [4, Eq. (19)] to derive (2.23).

After these preliminary steps we are in a position to prove Proposition 2.1.

Proof of Proposition 2.1. Forv∈Sq(κ)and(x,y,η)∈^Ω, we set fv(x,y,η):=Lvη=ε²η

2 |∇v(x,y)|²

1+v(x,y)²− ^∆v(x,y) 1+v(x,y)

. SinceW_q²(D)is embedded inW∞¹(D), the function fvbelongs toL₂(^Ω)with

kfvk_L2(Ω)≤c6(κ,ε), (2.24)

and Proposition 2.7 ensures that there is a unique solutionΦ_v∈W_2,B² (^Ω)to

−LvΦ_v= fv in Ω, Φ_v=0 on ∂Ω, satisfying

k^Φvk_W2

2(Ω)≤c5(κ,ε)kfvk_L2(Ω). (2.25) Settingφv(x,y,η) =^Φv(x,y,η) +ηfor(x,y,η)∈^Ω, the functionφvobviously solves (2.5)-(2.6) while we deduce from (2.24) and (2.25) that

kφvk_W2

2(Ω)≤c₇(κ,ε). (2.26)

We next define a bounded linear operatorA(v)∈ L W_2,B² (^Ω),L₂(^Ω)by A(v)^Φ:=−LvΦ, Φ∈W_2,B² (^Ω).

Proposition 2.7 guarantees thatA(v)is invertible with inverseA(v)⁻¹∈ L L₂(^Ω),W_2,B² (^Ω))satisfying

A(v)⁻¹

L L₂(Ω),W_2,²_B(Ω))≤c5(κ,ε). (2.27) We then note that

kA(v₁)− A(v₂)k_L(W2

2,B(Ω),L₂(Ω))≤c₈(κ,ε)kv₁−v₂k_W2

q(D), v₁,v₂∈Sq(κ), (2.28) which follows from the definition ofLvand the continuity of pointwise multiplication

W_q¹(D)·W_q¹(D)֒→W_q¹(D)֒→L∞(D)

except for the terms involving∂²_xv_iand∂²_yv_i,i=1, 2, where continuity of pointwise multiplication Lq(D)·W₂¹(^Ω)֒→L₂(^Ω),

the latter being true thanks to the continuous embedding ofW₂¹(^Ω)inL6(^Ω)and the choiceq≥3. Similar arguments also show that

kfv₁−fv₂k_L

2(Ω)≤c9(κ,ε)kv₁−v2k_W2

q(D), v₁,v2∈Sq(κ). (2.29) Now, forv₁,v₂∈Sq(κ), we infer from (2.27) and (2.28) that

kA(v1)⁻¹− A(v2)⁻¹k_L(L2(Ω),W2

2,B(Ω))≤c10(κ,ε)kv1−v2k_W2 q(D),

which, combined with (2.24), (2.27), (2.29), and the observation thatφv=A(v)⁻¹fvimplies (2.7).

SinceW₂^1/2(D)embeds continuously inL₄(D), pointwise multiplication

W₂^1/2(D)·W₂^1/2(D)֒→L2(D) (2.30) is continuous and hence, invoking [19, Chapter 2, Theorem 5.5] (sinceΩ = D×(0, 1) is a bounded Lipschitz domain), the mapping

Sq(κ)→L2(D), v7→∂ηφv(·,·, 1)² (2.31) is bounded and globally Lipschitz continuous. Thanks to the continuity of the embedding ofW_q¹(D)in L∞(D), the mapping

Sq(κ)→W_q¹(D), v7→ ¹+ε²|∇v|²

(1+v)^{2 (2.32)}

is bounded and globally Lipschitz continuous with a Lipschitz constant depending only onκandε, and the Lipschitz continuity ofgε stated in Proposition 2.1 follows at once from those of the mappings in (2.31) and (2.32). Finally, to prove thatgεis analytic, we note thatSq(κ)is open inW_q,B² (D)and that the mappings

A:Sq(κ)→ L(W_2,B² (^Ω),L2(^Ω)) and [v7→ fv]:Sq(κ)→L2(^Ω)