Dynamics of a free boundary problem with curvature modeling electrostatic MEMS

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Dynamics of a free boundary problem with curvature modeling electrostatic MEMS

Joachim Escher, Philippe Laurencot, Christoph Walker

To cite this version:

Joachim Escher, Philippe Laurencot, Christoph Walker. Dynamics of a free boundary problem with curvature modeling electrostatic MEMS. Transactions of the American Mathematical Society, Amer- ican Mathematical Society, 2015, 367, pp.5693–5719. �hal-00794036�

ELECTROSTATIC MEMS

JOACHIM ESCHER, PHILIPPE LAURENC¸ OT, AND CHRISTOPH WALKER

Abstract. The dynamics of a free boundary problem for electrostatically actuated microelectrome- chanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference across the device. More precisely, the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane. The gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the mem- brane deformation. The main feature of the model considered herein is that, unlike most previous research, the small deformation assumption is dropped, and curvature for the deformation of the membrane is taken into account what leads to a quasilinear parabolic equation. The free boundary problem is shown to be well-posed, locally in time for arbitrary voltage values and globally in time for small voltages values. Furthermore, existence of asymptotically stable steady-state configura- tions is proved in case of small voltage values as well as non-existence of steady-states if the applied voltage difference is large. Finally, convergence of solutions of the free boundary problem to the solutions of the well-established small aspect ratio model is shown.

1. Introduction

The focus of this paper is the analysis of a model describing the dynamics of an electrostat- ically actuated microelectromechanical system (MEMS) when the deformation of the devices are not assumed to be small. More precisely, consider an elastic plate held at potentialV and suspended above a fixed ground plate held at zero potential. The potential difference between the two plates generates a Coulomb force and causes a deformation of the membrane, thereby converting electrostatic energy into mechanical energy, a feature used in the design of several MEMS-based devices such as micropumps or microswitches [26]. An ubiquitous phenomenon observed in such devices is the so called “pull-in” instability: a threshold value of the applied voltageVabove which the elastic response of the membrane cannot balance the Coulomb force and the deformable membrane smashes into the fixed plate. Since this effect might either be use- ful or, in contrast, could damage the device, its understanding is of utmost practical importance and several mathematical models have been set up for its investigation [7, 21, 26].

In the following subsection we give a brief description of an idealized device as depicted in Figure 1, where the state of the device is characterized by the electrostatic potential in the region between the two plates and the deformation of the membrane which is not assumed to be small from the outset, cf. [5].

1.1. The Model. To derive the model for electrostatic MEMS with curvature we proceed sim- ilarly to [5, 7, 22]. We consider a rectangular thin elastic membrane that is suspended above a rigid plate. The (x, ˆˆ y, ˆz)-coordinate system is chosen such that the ground plate of dimen- sion [−^L,L]×[0,l] in (x, ˆˆ y)-direction is located at ˆz = −H, while the undeflected membrane

Date: February 24, 2013.

2010Mathematics Subject Classification. 35R35, 35M33, 35Q74, 35B25, 74M05.

Key words and phrases. MEMS, free boundary problem, curvature, well-posedness, asymptotic stability, small aspect ratio limit.

1

with the same dimension [−^L,L]×[0,l] in(x, ˆˆ y)-direction is located at ˆz = 0. The membrane is held fixed along the edges in ˆy-direction while the edges in ˆx-direction are free. Assuming homogeneity in ˆy-direction, the membrane may thus be considered as an elastic strip and the

ˆ

y-direction is omitted in the sequel. The mechanical deflection of the membrane is caused by a voltage difference that is applied across the device. The membrane is held at potentialV while the rigid plate is grounded. We denote the deflection of the membrane at position ˆxand time ˆtby

ˆ

u=uˆ(t, ˆ^ˆ x)>−^Hand the electrostatic potential at position(x, ˆˆ z)and time ˆtby ˆψ=ψ^ˆ(t, ˆ^ˆ x, ˆz). We do not indicate the time variable ˆtfor the time being. The electrostatic potential ˆψis harmonic, i.e.

∆ψˆ =0 in Ωˆ(uˆ) (1.1)

and satisfies the boundary conditions

ψˆ(x,ˆ −^H) =0 , ψˆ(x, ˆˆ u(xˆ)) =V, xˆ∈(−^L,L), (1.2) where

ˆ

Ω(uˆ):={(x, ˆˆ z);−^L<_xˆ<_L,−^H<_zˆ<_uˆ(xˆ)}

is the region between the ground plate and the membrane. The total energy of the system constitutes of the electric potential and the elastic energy and reads

E(uˆ) =Ee(uˆ) +Es(uˆ).

Theelectrostatic energy Ee in dependence of the deflection ˆuis given by Ee(uˆ) =−^ǫ₂⁰

Z L

−L Z u(ˆ x)ˆ

−H |∇^ψˆ(x, ˆˆ z)|^{2d ˆzd ˆx}

withǫ₀being the permittivity of free space while theelastic energy Esonly retains the contribution due to stretching (in particular, bending is neglected) and is proportional to the tensionTand to the change of surface area of the membrane, i.e.

Es(uˆ) =T Z _L

−L

q

1+ (∂xˆuˆ(xˆ))²−¹

d ˆx . Introducing the dimensionless variables

x= ^xˆ

L , z= ^zˆ

H , u= ^uˆ

H , ψ= ^ψˆ V

and denoting the aspect ratio of the device byε= H/L, we may write the total energy in these variables in the form

E(u) =TL Z 1

−1

q

1+ε²(∂xu(x))²−¹

dx

−^ǫ0V

2

2ε Z

Ω(u)

ε²|^∂xψ(x,z)|²+|^∂zψ(x,z)|^2d(x,z),

(1.3)

with

Ω(u):={(x,z)∈(−^{1, 1})×(−^1,∞) : −¹<_z<_u(x)} ^, so that, formally, the corresponding Euler-Lagrange equations are

0=ε²∂x ∂xu p1+ε²(∂xu)²

!

−^ε2λε2|^∂xψ(x,u(x))|²+|^∂zψ(x,u(x))|^{2 (1.4)} forx∈(−^{1, 1}), where we have set

λ= ^ǫ0V

2

TLε³ .

We now take again time into account and derive the dynamics of the dimensionless deflection u = u(t,^ˆ x) by means of Newton’s second law. Letting ρ and δ denote the mass density per unit volume of the membrane and the membrane thickness, respectively, the sum over all forces equalsρδ∂²_tˆu. The elastic and electrostatic forces, given by the right hand side of equation (1.4), are combined with a damping force of the form−^a∂ˆtubeing linearly proportional to the velocity.

This yields

ρ δ ∂²_ˆtu+a∂tˆu=ε²∂x ∂xu p1+ε²(∂xu)²

!

−^ε2λε2|^∂xψ(x,u(x))|²+|^∂zψ(x,u(x))|^{2 .} Finally, scaling time based on the strength of damping according to t = tε^{ˆ 2}/a and setting γ:=

√ρδε

a , we derive for the dimensionless deflection the evolution problem γ²∂²_tu+∂tu = ∂x ∂xu

p1+ε²(∂xu)²

!

− ^λε2|^∂xψ(x,u(x))|²+|^∂zψ(x,u(x))|^{2 (1.5)} for t > _{0 and x} ∈ ^{I :}= (−^{1, 1}). Instead of considering this hyperbolic equation, however, we assume in this paper that viscous or damping forces dominate over inertial forces, i.e. we assume that γ ≪ 1 and thus neglect the second order time derivative term in (1.5). The membrane displacementu=u(t,x)∈(−^1,∞)then evolves according to

∂tu−^∂x p ^∂xu 1+ε²(∂xu)²

!

=−^{λ ε2}|^∂xψ(t,x,u(x))|²+|^∂zψ(t,x,u(x))|^{2 , (1.6)} fort>_{0 andx}∈ ^Iwith clamped boundary conditions

u(t,±¹) =0 , t>_{0 , (1.7)}

and initial condition

u(0,x) =u⁰(x), x∈ ^{I. (1.8)}

In dimensionless variables, equations (1.1)-(1.2) read

ε²∂²_xψ+∂²_zψ=0 , (x,z)∈^Ω(u(t)), t>_{0 , (1.9)} subject to the boundary conditions (extended continuously to the lateral boundary)

ψ(t,x,z) = ¹+z

1+u(t,x) ^, (x,z)∈^∂Ω(u(t)), t>_{0 . (1.10)} In the following we shall focus our attention on (1.6)-(1.10), its situation being depicted in Fig- ure 1.

1.2. Simplified Models. Besides assuming that damping forces dominate over inertial forces and thus reducing equation (1.5) to (1.6), other simplifications of the model above have been considered as well in the literature. For instance, restricting attention to small deformations of the membrane yields a linearized stretching term ∂²_xu in (1.6). The corresponding semilinear evolution problem with (1.6) being replaced by

∂tu−^∂2xu=−^{λ ε2}|^∂xψ(t,x,u(x))|²+|^∂zψ(t,x,u(x))|^{2 , x}∈ ^{I, t}>_{0 , (1.11)} is investigated in [6]. It is shown therein that the problem (1.7)-(1.11) is well-posed locally in time. Moreover, solutions exist globally for small voltage values λ 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, as the parameterεapproaches zero, the solutions are shown to converge toward the solutions of the so-called small aspect ratio model,

Figure1. Idealized electrostatic MEMS device.

see (1.13) below. Indeed, lettingε=0 (and applying a potentialVwithV²∼^ε3as suggested in [5]), one can solve (1.9)-(1.10) explicitly for the potentialψ=ψ0, that is,

ψ0(t,x,z) = ¹+z

1+u0(t,x) ^, (t,x,z)∈[0,∞)×^I×(−^{1, 0}), (1.12) and the displacementu=u0satisfies

∂tu0−^∂2xu0 = −(1+^λu₀)^{2, x}∈ ^{I, t}∈(0,∞), u0(t,±¹) = 0 , t∈(0,∞),

u₀(0,x) = u⁰(x), x∈ ^I.

(1.13)

In the limit ε →0, the free boundary problem is thus reduced to the singular semilinear heat equation (1.13) which has been studied thoroughly in recent years, see [7] for a survey as well as e.g. [8, 9, 13, 14, 15, 16, 18, 22, 25]. It is noteworthy to remark that the picture regarding pull-in voltage for the small aspect ratio model (1.13) is rather complete.

Let us point out that [6] is apparently the first mathematical analysis of the parabolic free boundary problem (1.7)-(1.11) while the corresponding elliptic (i.e. steady-state) free boundary problem is investigated in [20]. Moreover, we shall emphasize that the inclusion of non-small deformations is a feature of great physical relevance and, even though the results presented herein are reminiscent of the ones in [6], the quasilinear structure of (1.6) is by no means a trivial mathematical extension of (1.11).

1.3. Main Results. To state our findings on (1.6)-(1.10), we introduce the spaces

W_q,D^2α(I):=







{^u∈^Wq^2α(I);u(±¹) =0} ^{for 2α}∈¹q, 2i , W_q^2α(I) for 0≤^2α< ¹

q .

(1.14)

We shall prove the following result regarding local and global existence of solutions:

Theorem 1.1(Well-Posedness). Let q∈ (2,∞),ε >0, and consider an initial value u⁰∈ ^Wq,D² (I) such that u⁰(x)>−^{1for x}∈ I. Then, the following are true:

(i) For each voltage value λ > 0, there is a unique maximal solution(u,ψ) to(1.6)-(1.10)on the maximal interval of existence[0,T_m^ε)in the sense that

u∈^C1 [0,T_m^ε),Lq(I)∩^C [0,T_m^ε),W_q,D² (I) satisfies(1.6)-(1.8)together with

u(t,x)>−^{1 ,} (t,x)∈[0,T_m^ε)×^I,

andψ(t)∈^W2² Ω(u(t))solves(1.9)-(1.10)onΩ(u(t))for each t∈[0,T_m^ε).

(ii) If for eachτ>_{0there isκ}(τ)∈(0, 1)such that u(t)≥ −¹+κ(τ)andk^u(t)kW_q²(I)≤^κ(τ)⁻¹ for t∈[0,T_m^ε)∩[0,τ], then the solution exists globally, that is, T_m^ε =^∞.

(iii) If u⁰(x) ≤ ^{0 for x} ∈ ^{I, then u}(t,x) ≤^{0 for}(t,x) ∈ [0,T_m^ε)×^{I. If u0} = u⁰(x)is even with respect to x∈ I, then, for all t∈[0,T_m^ε), u=u(t,x)andψ=ψ(t,x,z)are even with respect to x∈^{I as well.}

(iv) Givenκ ∈(0, 1), there areλ_∗(κ)>_{0and r}(κ)>₀such that T_m^ε =^∞ with u(t,x)≥ −¹+κ for (t,x) ∈ [0,∞)×I provided that λ ∈ (0,λ_∗(κ)) and k^u0k_Wq²_(I) ≤ ^r(κ). In that case, u enjoys the following additional regularity properties:

u∈ ^BUCρ([0,∞),W_q,D^2−ρ(I))∩^L∞([0,∞),W_q,D² (I)) for someρ>_0small.

Note that part (iv) of Theorem 1.1 provides uniform estimates onu in theW_q²(I)-norm and ensures thatunever touches down on -1, not even in infinite time. In contrast to the semilinear case considered in [6], the global existence result for the quasilinear equation (1.6) requires ini- tially a small deformation, see also Remark 3.3 below. The proof of Theorem 1.1 is the content of Section 3. It is based on interpreting (1.6) as a abstract quasilinear Cauchy problem which allows us to employ the powerful theory of evolution operators developed in [4]. Let us emphasize at this point that the regularity properties of the right-hand side of equation (1.6) established in [6] are not sufficient to handle the quasilinear character of the curvature operator and we conse- quently have to derive Lipschitz properties of the right-hand side of (1.6) in weaker topologies than in [6]. This is the purpose of Section 2.

Regarding existence and asymptotic stability of steady-state solutions to (1.6)-(1.10) we have a similar result as in [20, Thm. 1] and [6, Thm.1.3].

Theorem 1.2(Asymptotically Stable Steady-State Solutions). Let q∈(2,∞)andε>_0.

(i) Letκ∈(0, 1). There areδ=δ(κ)>₀and an analytic function [λ7→^Uλ]:[0,δ)→^Wq,D² (I)

such that(Uλ,Ψ_λ)is the unique steady-state to(1.6)-(1.10)satisfyingk^UλkW_q,D² (I)≤^1/κwith Uλ≥ −¹+κandΨ_λ∈^W2²(^Ω(Uλ))whenλ∈(0,δ). The steady-state possesses the additional

regularity

U_λ∈^C2+α [−^{1, 1}],

Ψ_λ∈^W2² Ω(U_λ)∩^{C Ω}(U_λ)∩^{C2+α Ω}(U_λ)∪^Γ(U_λ), (1.15) whereα∈[0, 1)is arbitrary andΓ(U_λ)denotes the boundary ofΩ(U_λ)without corners. More- over, U_λis negative, convex, and even with U0=0andΨ_λ =^Ψ_λ(x,z)is even with respect to x∈^I.

(ii) Let λ ∈ (0,δ). There are ω0,m,R > ₀ such that for each initial value u⁰∈^Wq,D² (I) with k^u0−^UλkW_q,D² <_m,(1.6)-(1.10)has a unique global solution(u,ψ)with

u∈^C1 [0,∞),Lq(I)∩^C [0,∞),W_q,D² (I), ψ(t)∈^W2² Ω(u(t)), t≥^{0 ,} and

u(t,x)>−^{1 ,} (t,x)∈[0,∞)×^I. Moreover,

k^u(t)−^UλkW_q,D² (I)+k^∂tu(t)kLq(I)≤^Re−ω0tk^u0−^UλkW_q,D² (I), t≥^{0 . (1.16)} Part (ii) of Theorem 1.2 shows local exponential stability of the steady-states derived in part (i).

We also point out that the potentialψconverges exponentially toΨ_λin theW₂²-norm ast→^∞, see Remark 4.1 for a precise statement. The proof of Theorem 1.2 is given in Section 4 and relies on the Implicit Function Theorem for part (i) and the Principle of Linearized Stability for part (ii).

Clearly, Theorem 1.2 is just a local result with respect toλvalues. However, we next show that there is an upper threshold forλabove which no steady-state solution exists. This is expected on physical grounds and is related to the “pull-in” instability already mentioned in the introduction.

Theorem 1.3 (Non-Existence of Steady-State Solutions). Let q ∈ (2,∞) and ε > 0. There is λ¯(ε) > ₀ such that, ifλ ≥ ^λ¯(ε), then there is no steady-state solution(u,ψ)to(1.6)-(1.10)such that u∈^Wq,D² (I),ψ∈^W2²(^Ω(u)), and u(x)>−^{1for x}∈ I. In addition,λ¯(ε)→^2asε→^0.

Similar results have already been obtained for related models, including the small aspect ratio model [5, 7] and for the stationary free boundary problem corresponding to (1.7)-(1.11), see [20].

The proof of Theorem 1.3 relies on a lower bound on∂zψ(x,u(x))established in the latter paper and is given in Section 5.

The final issue we address is the connection between the free boundary problem (1.6)-(1.10) and the small aspect ratio limit (1.13). More precisely, we show the following convergence result:

Theorem 1.4 (Small Aspect Ratio Limit). Let λ > _{0, q} ∈ (2,∞), and let u⁰ ∈ ^W_q,D² (I) with

−¹ < _u⁰(x) ≤ ^{0 for x} ∈ ^{I. Forε} > ₀ we denote the unique solution to(1.6)-(1.10) on the maximal interval of existence[0,T_m^ε)by (uε,ψε). There areτ >_0,ε0 >_{0, andκ0}∈ (0, 1)depending only on q and u⁰such that T_m^ε ≥^{τwith uε}(t)≥ −¹+κ₀andk^uε(t)kW_q²(I)≤^κ−0¹for all(t,ε)∈[0,τ]×(0,ε₀). Moreover, asε→^0,

uε −→^u0 in C^1−θ [0,τ],W_q^2θ(I), 0<_θ<_{1 ,} and

ψε(t)1Ω(uε(t))−→^ψ0(t)1Ω(u0(t)) in L₂ I×(−^{1, 0}), t∈[0,τ], (1.17) where

u₀∈^C1 [0,τ],Lq(I)∩^C [0,τ],W_q,D² (I)

is the unique solution to the small aspect ratio equation (1.13)and ψ0 is the potential given in(1.12).

Furthermore, there is Λ(u⁰) > ₀ such that the results above hold true for each τ > ₀ provided that λ∈(0,Λ(u⁰)).

The proof is given in Section 6. A similar result has been established in [20, Thm. 2] for the stationary problem and in [6, Thm.1.4] for the semilinear parabolic version (1.11). As in the latter paper, the crucial step is to derive theε-independent lower boundτ >_{0 onTm}^ε_{, which is} not guaranteed by the analysis leading to Theorem 1.1. The proof of Theorem 1.4 uses several properties of (1.9)-(1.10) with respect to theε-dependence shown in [6].

2. On theEllipticEquation(1.9)-(1.10)

We shall first derive properties of solutions to the elliptic equation (1.9)-(1.10) in dependence of a given (free) boundary. To do so, we transform the free boundary problem (1.9)-(1.10) to the fixed rectangleΩ:=I×(0, 1). More precisely, letq∈(2,∞)be fixed and consider an arbitrary functionv∈^Wq,D² (I)taking values in(−^1,∞). We then define a diffeomorphismTv:Ω(v)→^Ω by setting

Tv(x,z):=

x, 1+z 1+v(x)

, (x,z)∈^Ω(v), (2.1) withΩ(v):={(x,z)∈ ^I×(−^1,∞); 1<_z<_v(x)}. Clearly, its inverse is

T_v⁻¹(x,η) = x,(1+v(x))η−^1, (x,η)∈^{Ω, (2.2)} and the Laplace operator from (1.9) is transformed to thev-dependent differential operator

Lvw :=ε²∂²_xw−^{2ε2η ∂xv}(x)

1+v(x) ^∂x∂ηw+¹+ε²η²(∂xv(x))² (1+v(x))^{2 ∂}

2ηw

+ε²η

"

2

∂xv(x) 1+v(x)

2

− ^∂

2xv(x) 1+v(x)

#

∂ηw.

An alternative formulation of the boundary value problem (1.9)-(1.10) is then Lu(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)} forφ=ψ◦^T_u(t)⁻¹. With this notation, the quasilinear evolution equation (1.6) forubecomes

∂tu−^∂x ∂xu p1+ε²(∂xu)²

!

=−^λ

1+ε²(∂xu)² (1+u)²

|^∂ηφ(·^{, 1})|^{2, x}∈^{I , t}>_{0 , (2.5)} where we have used∂xφ(t,x, 1) =0 forx∈ ^Iandt>_{0 due toφ}(t,x, 1) =1 by (2.4). The inves- tigation of the dynamics of (2.5) involves the properties of its nonlinear right hand side as well as the properties of the quasilinear curvature term. We shall see that these two features of (2.5) are somewhat opposite as the treatment of the former requires a functional analytic setting in W_q²(I)to handle the second order terms ofLu(t)in (2.3), while a slightly weaker setting has to be chosen to guarantee H ¨older continuity ofuwith respect to time which is required in quasilinear evolution equations (see Remark 3.3 for further details). To account for these features of (2.5) we have to refine the Lipschitz property of the right-hand side of (2.5) derived in [6] as stated in (2.8) below.

Defining forκ ∈(0, 1)the open subset Sq(κ):=

u∈^Wq,D² (I);k^ukW_q,D² (I)<_{1/κ and} −¹+κ<_u(x)forx∈ ^I

ofW_q,D² (I)(defined in (1.14)) with closure Sq(κ) =

u∈^Wq,D² (I);k^uk_W²

q,D(I)≤^{1/κ and} −¹+κ≤^u(x)forx∈ ^I

,

the crucial properties of the nonlinear right-hand side of (2.5) are collected in the following proposition:

Proposition 2.1. Let q∈(2,∞),κ ∈ (0, 1), andε>0. For each v∈^Sq(κ)there is a unique solution φv∈^W2²(^Ω)to

L^vφv(x,η) =0 , (x,η)∈^{Ω, (2.6)} φv(x,η) =η, (x,η)∈^{∂Ω. (2.7)} Ifv is defined by˜ v˜(x) := v(−^x)for x ∈ ^{I, then φ}v˜(x,η) =φv(−^x,η)for(x,η) ∈ ^Ω. Moreover, for 2σ∈[0, 1/2), the mapping

gε:Sq(κ)−→^W2,D^2σ (I), v7−→ ¹+ε²(∂xv)²

(1+v)² |^∂ηφv(·^{, 1})|²

is analytic and bounded with gε(0) =1. Finally, ifξ ∈ [0, 1/2)and ν ∈ [0,(1−^2ξ)/2), then there exists a constant c1(κ,ε)>_{0such that}

k^gε(v)−^gε(w)kW_2,D^ν (I)≤^c1(κ,ε)k^v−^wk_W²−^ξ

q,D(I), v,w∈^Sq(κ). (2.8) According to [6, Prop. 2.1] we actually only have to prove (2.8). Notice that this global Lipschitz property is in the weaker topology ofW_q,D^2−ξ(I) instead ofW_q,D² (I) and improves [6, Prop. 2.1] where it was established for ξ = 0. The property (2.8) will be a consequence of a sequence of lemmas. For the remainder of this section we fixε>_0,κ∈(0, 1), andq∈(2,∞).

In the following, ifα>_{1/2 we letW2,D}^α (^Ω)denote the subspace of elements inW₂^α(^Ω)whose boundary trace is zero, and if 0≤^α<_{1/2 we setW}^α

2,D(^Ω):=W₂^α(^Ω). We equipW_2,D¹ (^Ω)with the norm

k^Φk_W2,D¹ ₍Ω):=k^∂xΦk²_L2(Ω)+k^∂ηΦk²_L2(Ω)

1/2

, and introduce the notation

W_2,D^−θ(^Ω):= (W_2,D^θ (^Ω))^′, 0≤^θ≤^{1 .}

Lemma 2.2. For each v ∈ ^Sq(κ) and F ∈ ^W_2,D⁻¹(^Ω) there is a unique solutionΦ ∈ ^W_2,D¹ (^Ω)to the boundary value problem

−L^vΦ =F inΩ, (2.9)

Φ =0 on∂Ω, (2.10)

and there is a constant c₂(κ,ε)>_{0such that}

k^ΦkW_2,D¹ (Ω)≤^c2(κ,ε)k^Fk_W⁻¹

2,D(Ω). (2.11)

Furthermore, if F∈^L2(^Ω), thenΦ∈^W2,D² (^Ω)and

k^ΦkW_2,D² (Ω)≤^c2(κ,ε)k^FkL₂(Ω). (2.12) Proof. According to [12, Def. 1.3.2.3, Eq. (1,3,2,3)], we may write anyF ∈^W2,D⁻¹(^Ω)in the form F= f0+∂xf1+∂ηf2 with (f0,f1,f2) ∈ ^L2(^Ω)³. Consequently, [10, Thm. 8.3] ensures that the

boundary value problem (2.9)-(2.10) has a unique solutionΦ∈^W2,D¹ (^Ω). Furthermore, takingΦ as a test function in the weak formulation of (2.9)-(2.10) gives

h^F,Φi= Z

Ω

ε²(∂xΦ)²−^2ε2η₁^∂xv

+v∂xΦ∂ηΦ−^{ε2η ∂}x

∂xv 1+v

Φ∂ηΦ−^ε2₁^∂xv

+v Φ∂xΦ

d(x,η) +

Z

Ω

"

1+ε²η²(∂xv)²

(1+v)² (∂ηΦ)²+2ε²η ∂xv

1+v 2

Φ∂ηΦ

#

d(x,η)

+ Z

Ωε²η

"

∂x

∂xv 1+v

− ∂xv

1+v 2#

Φ∂ηΦd(x,η)

= Z

Ω

ε²(∂xΦ)²−^2ε2η₁^∂xv

+v∂xΦ∂ηΦ+ ¹+ε²η²(∂xv)²

(1+v)² (∂ηΦ)²

d(x,η)

− Z

Ωε² ∂xv 1+v

∂xΦ−^η₁^∂xv +v∂ηΦ

Φd(x,η) and thus

Z

Ω

"

ε²

∂xΦ −^η₁^∂+^xvv∂ηΦ 2

+ ∂ηΦ

1+v 2#

d(x,η)

≤^ε2

∂xv 1+v

_L

∞(I)

∂xΦ−^η₁^∂+^xvv∂ηΦ _L

2(Ω) k^ΦkL₂(Ω) + k^Fk_W−¹

2,D(Ω)k^ΦkW_2,D¹ (Ω).

(2.13)

Note then that by definition ofSq(κ)and Sobolev’s embedding, there isc₃>0 such that 1+v(x)≥^{κ, x} ∈^I, k^vkC¹([−1,1])≤ ^c_κ^{3 (2.14)} for allv∈^Sq(κ). Also, ifζ= (ζ1,ζ2)∈^R2, Young’s inequality ensures that, for(x,η)∈^Ω,

ε²ζ²₁ ≤^2ε2

ζ1−^{η ∂xv}(x) 1+v(x)^ζ2

2

+2ε²η²

∂xv(x) 1+v(x)

2

ζ²₂

≤²¹+2ε²k^∂xvk²∞

ε²

ζ1−^{η ∂xv}(x) 1+v(x)^ζ2

2

+¹ 2

ζ2

1+v(x) 2!

. Therefore, introducing

ν(κ,ε):= ¹ 2 min

( ε²κ²

κ²+2ε²c²₃, κ² (κ+c3)²

) , we infer from (2.14) that

ν(κ,ε) ζ²₁+ζ²₂

≤^ε2

ζ₁−^{η ∂xv}(x) 1+v(x)^ζ2

2

+ ζ₂

1+v(x) 2

. (2.15)

Consequently, (2.13), (2.14), and (2.15) give ε²

∂xΦ − ^η₁^∂+^xvv ∂ηΦ

2 L₂(Ω)

+

∂ηΦ 1+v

2 L₂(Ω)

≤^ε2_κ^c3₂

∂xΦ − ^η₁^∂+^xvv∂ηΦ _L

2(Ω)k^ΦkL₂(Ω)

+k^Fk_W−1 2,D(Ω)

pν(κ,ε) ^ε

2

∂xΦ− ^η₁^∂xv +v∂ηΦ

2 L₂(Ω)

+

∂ηΦ 1+v

2 L₂(Ω)

!1/2

,

whence, using again (2.15),

k^ΦkW_2,D¹ (Ω) ≤ p ^ε ν(κ,ε)

c₃

κ²k^ΦkL₂(Ω)+k^Fk_W−1 2,D(Ω)

ν(κ,ε) ^{. (2.16)}

We now proceed as in [10, Lem. 9.17] and argue by contradiction to show (2.11) (see also [6, Lem. 6.2] for a similar argument in a slightly different functional setting). The last statement of

Lemma 2.2 is proved in [6, Lem. 6.2].

Now, introducing

fv(x,η):=Lvη=ε²η

"

2

∂xv(x) 1+v(x)

2

− ^∂

2xv(x) 1+v(x)

#

, (x,η)∈^{Ω, (2.17)} forv∈^Sq(κ)given, we readily deduce that fv∈^L2(^Ω)with

k^fvkL₂(Ω)≤^c4(κ,ε). (2.18) Consequently, Lemma 2.2 provides a unique solutionΦ_v∈^W2,D² (^Ω)to

− L^vΦv(x,η) = fv, (x,η)∈^Ω, Φ_v(x,η) =0 , (x,η)∈^∂Ω. Clearly, defining

φv(x,η):=^Φv(x,η) +η, (x,η)∈^{Ω¯ , (2.19)} gives then the unique solutionφv∈^W₂²(^Ω)to (2.6)-(2.7). To prove a Lipschitz dependence ofφv

onv∈^Sq(κ), we introduce a bounded linear operator

A(v)∈ L ^W2,D¹ (^Ω),W_2,D⁻¹(^Ω)∩ L ^W2,D² (^Ω),L2(^Ω) by setting

A(v)^Φ:=−L^{vΦ, Φ}∈^W2,D¹ (^Ω).

Note thatA(v)is invertible according to Lemma 2.2 and thatΦ_v = A(v)⁻¹fv. For the inverse A(v)⁻¹we have:

Lemma 2.3. Givenθ∈[0, 1]\ {^1/2}, there is a constant c₅(κ,ε)>_{0such that} kA(v)⁻¹k_L(W^θ−1

2,D(Ω),W_2,D^θ+1(Ω)) ≤ ^c5(κ,ε), v∈^Sq(κ).

Proof. Due to Lemma 2.2,A(v)⁻¹belongs for eachv∈^Sq(κ)to bothL(W_2,D⁻¹(^Ω),W_2,D¹ (^Ω))and L(L2(^Ω),W_2,D² (^Ω))with

kA(v)⁻¹k_L(W−¹

2,D(Ω),W_2,D¹ (Ω))+ kA(v)⁻¹k_L(L2(Ω),W_2,D² (Ω)) ≤ ^2c2(κ,ε). Hence, using complex interpolation, we derive forθ∈[0, 1],

kA(v)⁻¹k_{L [W}−1

2,D(Ω),L₂(Ω)]_θ,[W_2,D¹ (Ω),W_2,D² (Ω)]_θ ≤ ^2c2(κ,ε),

and it then remains to characterize the interpolation spaces forθ ∈ [0, 1]\ {^1/2}. For this we first invoke [28, Thm. 1.1.11] to obtain that

[W_2,D⁻¹(^Ω),L2(^Ω)]_θ = [(W_2,D¹ (^Ω))^′,(L2(^Ω))^′]_θ = [W_2,D¹ (^Ω),L2(^Ω)]^′_θ

with equivalent norms and hence, using [12, Cor. 1.4.4.5] and [19, Prop. 2.11 & Prop. 3.3] to characterize the last interpolation space,

[W_2,D⁻¹(^Ω),L2(^Ω)]θ= (W_2,D^1−θ(^Ω))^′=W_2,D^θ−1(^Ω).

Finally, since the embedding

[W_2,D¹ (^Ω),W_2,D² (^Ω)]_θ֒→^W_2,D^θ+1(^Ω)

is obviously continuous by [28, Thm. 4.3.2/2] and definition of interpolation, the assertion fol-

lows.

Next, we show that A(v) is Lipschitz continuous with respect to v in a suitable topology.

More precisely:

Lemma 2.4. Givenξ∈[0,(q−¹)/q)andα∈(ξ, 1), there exists c₆(κ,ε)>_{0such that} kA(v)− A(w)k_L(W2,D² ₍Ω),W_2,D^−α(Ω))≤^c6(κ,ε)k^v−^wk_W²−ξ

q (I), v,w∈^Sq(κ).

Proof. Considerv,w∈^Sq(κ)andΦ∈^W2,D² (^Ω). ThenA(v)^Φ∈ ^L2(^Ω)֒→^W_2,D^−α(^Ω)and so, for anyϕ∈^W_2,D^α (^Ω),

Z

Ω

A(v)− A(w)^Φ ϕd(x,η) =2ε² Z

Ωη ∂xv

1+v −₁^∂xw +w

∂x∂ηΦϕd(x,η)

− Z

Ω

1+ε²η²(∂xv)²

(1+v)² −¹+ε²η²(∂xw)² (1+w)²

∂²_ηΦϕd(x,η)

−^2ε2 Z

Ωη

"

∂xv 1+v

2

− ∂xw

1+w 2#

∂ηΦϕd(x,η) +ε²

Z

Ωη ∂²_xv

1+v− ^∂

2xw 1+w

∂ηΦϕd(x,η)

= :J1+J2+J3+J4.

Sinceξ∈[0,(q−¹)/q)it follows from the continuous embeddingW_q^2−ξ(I)֒→^W∞¹(I)that

|^J1| ≤^2ε2

∂xv

1+v−₁^∂+^xww _L

∞(I)k^∂x∂ηΦkL₂(Ω)k^ϕkL₂(Ω)

≤^ε2c(κ)k^v−^wk_W²−^ξ

q (I)k^ΦkW_2,D² (Ω)k^ϕkW_2,D^α (Ω). Similarly, we have

|^J2|+|^J3| ≤(1+ε²)c(κ) k^v−^wk_W²−^ξ

q (I) k^ΦkW_2,D² (Ω)k^ϕkW_2,D^α (Ω).

Finally, we infer from the embedding W₂¹(^Ω) ֒→ ^L2q/(q−2)(^Ω) and the fact that (W_q^ξ_′(I))^′ = Wq^−ξ(I)sinceξ∈[0,(q−¹)/q), see, e.g., [3, Eq. (5.14)], that

|^J4| ≤^ε2 Z

Ω∂²_x(v−^w)^{η ∂ηΦϕ} 1+v d(x,η)

+ε²k^∂2xwkLq(I)

1

1+v −₁ ¹ +w

_L

∞(I)k^∂ηΦkL_2q/(q−2)(Ω)k^ϕkL₂(Ω)

≤^ε2k^∂2x(v−^w)k_W−ξ q (I)

1 1+v

Z ₁

0 η ∂ηΦ(·^,η)ϕ(·^,η)dη _Wξ

q′(I)

+ε²c(κ)k^v−^wk_W²−ξ

q (I)k^∂ηΦk_W¹

2,D(Ω)k^ϕkW_2,D^α (Ω).

Now, choosingβ∈[ξ,α), pointwise multiplication is continuous as mappings W_q²(I)·^W₂^β(I)֒→^W_q^ξ′(I), C²(^Ω¯)·^W2¹(^Ω)·^W2^α(^Ω)֒→^W₂^β(^Ω)