2. Some properties of the electrostatic energy and potential

DOI:10.1051/cocv/2015012 www.esaim-cocv.org

A VARIATIONAL APPROACH TO A STATIONARY FREE BOUNDARY PROBLEM MODELING MEMS

Philippe Laurenc ¸ot

and Christoph Walker

Abstract. A variational approach is employed to find stationary solutions to a free boundary prob- lem modeling an idealized electrostatically actuated MEMS device made of an elastic plate coated with a thin dielectric film and suspended above a rigid ground plate. The model couples a non-local fourth-order equation for the elastic plate deflection to the harmonic electrostatic potential in the free domain between the elastic and the ground plate. The corresponding energy is non-coercive reflecting an inherent singularity related to a possible touchdown of the elastic plate. Stationary solutions are constructed using a constrained minimization problem. A by-product is the existence of at least two stationary solutions for some values of the applied voltage.

Mathematics Subject Classification. 35J35, 35R35, 35Q74.

Received August 31, 2014. Revised January 20, 2015.

Published online March 8, 2016.

1. Introduction

Microelectromechanical systems (MEMS) play a key rˆole in many electronic devices nowadays and include micro-pumps, optical micro-switches, and sensors, to name but a few [19]. Idealized electrostatically actuated MEMS consist of an elastic plate lying above a fixed ground plate and held clamped along its boundary. A Coulomb force induced by the application of a voltage difference across the device deflects the elastic plate. It is known from applications that a stable configuration is only obtained for voltage differences below a certain critical threshold as above this value the elastic plate may “pull in” on the ground plate.

In a simplified and re-scaled geometry when presupposing zero variation in transversal direction (see Fig.1), the stationary problem can be described as finding the plate deflection u = u(x) ∈ (−1,∞) on the interval I:= (−1,1) according to

β∂_x⁴u(x)−

τ+a∂_xu²_L2(I)

∂_x²u(x) =−λ

ε²|∂_xψ(x, u(x))|²+|∂_zψ(x, u(x))|²

, x∈I, (1.1)

u(±1) =∂_xu(±1) = 0, (1.2)

Keywords and phrases.MEMS, free boundary, stationary solutions, multiplicity, constrained minimization.

∗Partially supported by the French-German PROCOPE project 30718Z.

1 Institut de Math´ematiques de Toulouse, UMR 5219, Universit´e de Toulouse, CNRS, 31062 Toulouse cedex 9, France.

[email protected]

2 Leibniz Universit¨at Hannover, Institut f¨ur Angewandte Mathematik, Welfengarten 1, 30167 Hannover, Germany.

[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2016

PH. LAURENC¸ OT AND CH. WALKER

Figure 1.Idealized electrostatic MEMS device.

along with the electrostatic potentialψ=ψ(x, z) satisfying

ε²∂_x²ψ+∂_z²ψ= 0, (x, z)∈Ω(u), (1.3)

ψ(x, z) = 1 +z

1 +u(x), (x, z)∈∂Ω(u), (1.4)

in the region

Ω(u) :={(x, z)∈I×R : −1< z < u(x)}

between the two plates. In equation (1.1), the fourth-order termβ∂_x⁴uwithβ >0 reflects plate bending while the linear second-order termτ ∂_x²uwith τ≥0 and the non-local second-order term a∂xu²_L2(I)∂_x²uwitha≥0 and

∂_xu²_L2(I):=

₁

−1|∂_xu|²dx

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 with parameterλ >0 proportional to the square of the applied voltage difference and the device’s aspect ratioε >0. The boundary conditions (1.2) mean that the elastic plate is clamped. According to (1.3)−(1.4), the electrostatic potential is harmonic in the regionΩ(u) enclosed by the two plates with value 1 on the elastic plate and value 0 on the ground plate. We refer the readere.g.to [6,16,19] and the references therein for more details on the derivation of the model.

A crucial feature of the model is the singularity arising in the term∂_zψ(x, u(x)) of (1.1) whenu(x) =−1 (due toψ(x,−1) = 0 andψ(x, u(x)) = 1),i.e.when the elastic plate touches down on the ground plate. The strength of this instability is in some sense tuned by the parameterλand it is thus expected that solutions to (1.1)−(1.4) only exist for small values ofλbelow a certain threshold. Obviously, the stable operating conditions of MEMS devices and hence the existence of stationary solutions are of utmost importance in applications. Questions related to the pull-in threshold were the focus of a very active research in the recent past, however, almost exclusively dedicated to the simplifiedsmall gap modelobtained by formally settingε= 0 in (1.1)−(1.4). This reduces the problem to a singular nonlinear eigenvalue problem foruof the form

β∂_x⁴u(x)−

τ+a∂_xu²_L2(I)

∂_x²u(x) =−λ 1

(1 +u(x))², x∈I, (1.5)

subject to the boundary conditions (1.2) with explicitly given electrostatic potential ψ(x, z) = 1 +z

1 +u(x)·

For detailed results on the small gap model we refer the reader to [6,10,14,17] and the references therein in which also higher dimensional counterparts are investigated. Roughly speaking, in the one-dimensional (and two- dimensional radially symmetric) fourth-order small gap model with clamped boundary conditions anda= 0 it is known [14] that there is a thresholdλ_∗>0 such that there are (at least) two solutions to (1.5) forλ∈(0, λ_∗), one solution forλ=λ_∗, and no solution forλ > λ_∗.

A similar result might also be expected for the free boundary problem (1.1)−(1.4) withε >0. A first step in this direction was made in Theorem 1.7 in [13], where the following result was shown fora= 0:

Proposition 1.1. Let a= 0.

(i) There is λ_s(ε) > 0 such that for each λ ∈ (0, λ_s(ε)) there exists a solution (U_λ, Ψ_λ) to (1.1)−(1.4) with U_λ∈H⁴(I)satisfying−1< U_λ<0 inI andΨ_λ∈H²(Ω(U_λ)). The mappingλ→(λ, U_λ)defines a smooth curve in R×H⁴(I)withU_λ−→0in H⁴(I)asλ→0.

(ii) There areε_∗>0andλ_c: (0, ε_∗)→(0,∞)such that there is no solution(u, ψ)to(1.1)−(1.4)forε∈(0, ε_∗) andλ > λ_c(ε).

Actually, (U_λ, Ψ_λ) forλ∈(0, λ_s) is an asymptotically stable steady state for the corresponding dynamic problem.

The proof of part (i) of Proposition1.1 is based on the Implicit Function Theorem and readily extends to the casea >0. For part (ii) one may employ a nonlinear variant of the eigenfunction method involving a positive eigenfunction in H⁴(I) associated to the fourth-order operator β∂_x⁴−τ ∂_x² subject to the clamped boundary condition (1.2). For further use we now state the extension of Proposition1.1 (i) toa >0.

Theorem 1.2. Let a ≥ 0. There is λ_s(a, ε) > 0 such that for each λ ∈ (0, λ_s(a, ε)) there exists a solution (U_λ, Ψ_λ) to (1.1)−(1.4) with U_λ ∈ H⁴(I) satisfying −1 < U_λ <0 in I and Ψ_λ ∈ H²(Ω(U_λ)). The mapping λ→(λ, U_λ)defines a smooth curve in R×H⁴(I) withU_λ−→0in H⁴(I)asλ→0.

Theorem 1.2 in particular ensures the existence of stationary solutions for small values of λ. However, it leaves open the question whether multiple solutions exist for such values ofλwhich is a remarkable feature of the simplified small gap model as pointed out above. The purpose of the present paper is to give (partially) an affirmative answer. More precisely, we shall prove herein:

Theorem 1.3. There is a one-parameter family (λ_ρ, u_ρ, ψ_ρ)_ρ>2 withλ_ρ >0,u_ρ∈H⁴(I), andψ_ρ∈H²(Ω(u_ρ)) such that (u_ρ, ψ_ρ)is a solution to (1.1)−(1.4)with λ=λ_ρ for each ρ >2. Bothu_ρ=u_ρ(x)and ψ_ρ =ψ_ρ(x, z) are even with respect to x∈I and−1< u_ρ<0 in I. Moreover, λ_ρ→0 asρ→ ∞andu_ρ=U_λρ for allρ >2 sufficiently large.

Theorem 1.3provides multiple solutions to (1.1)−(1.4) for small values ofλand is derived by a variational approach. It relies on the observation that (1.1) is the Euler-Lagrange equation of the total energyE given by E(u) :=Em(u)−λEe(u) with mechanical energy

Em(u) := β

2∂_x²u²_L2(I)+1 2

τ+a

2∂_xu²_L2(I)

∂_xu²_L2(I) and electrostatic energy

Ee(u) :=

Ω(u)

ε²|∂xψ_u|²+|∂zψ_u|²

d(x, z),

where the electrostatic potentialψ_u is the solution to (1.3)−(1.4) associated to the given (sufficiently smooth) deflectionu. Note thatEis the sum of terms with different signs. The possible pull-in instability thus manifests in

PH. LAURENC¸ OT AND CH. WALKER

the non-coercivity of the energyE, and due to this a plain minimization of the total energy is not appropriate.

In fact, it follows from Lemma 2.9 (ii) that E is not bounded from below for λ > 0 and we therefore take an alternative route and minimize the mechanical energy Em constrained to (certain) deflectionsuwith fixed electrostatic energyEe(u) =ρ. Each minimizeru_ρ of this constrained minimization problem together with the corresponding electrostatic potential ψ_ρ := ψ_uρ then yields a solution to (1.1)−(1.4) for the corresponding Lagrange multiplierλ=λ_ρ. Though lacking a continuity property with respect to ρ >2, the observation that Ee(U_λ)→2 asλ→0 while λ_ρ→0 forEe(u_ρ) =ρ→ ∞yields multiplicity of solutions to (1.1)−(1.4) for small values ofλin the sense that there is at least a sequenceλ_j→0 of voltage values for which there are two different solutions (u_j, ψ_j) (i.e. ρ =j in Thm. 1.3) and (U_λj, Ψ_λj) (i.e. λ =λ_j in Thm. 1.2). Note that, by taking a different sequenceρ_j→ ∞withρ_j =j, we obtain different solutions (u_ρj, ψ_ρj) – since the electrostatic energies differ – but with possibly equal voltage values. We conjecture that, as in the simplified small gap model, the solutions constructed in Theorem1.3actually lie on a smooth curve.

To prove Theorem1.3we first solve in Section2the elliptic problem (1.3)−(1.4) for the electrostatic potential ψ=ψ_u for a given deflectionuand investigate then its dependence and that of the corresponding electrostatic energyEe(u) with respect tou. Some technical details needed regarding continuity and differentiability properties ofEeand the right-hand side of (1.1) are postponed to Section4. The constrained minimization problem leading to Theorem1.3is studied in Section3.

2. Some properties of the electrostatic energy and potential

We first focus on the elliptic problem (1.3)–(1.4) and investigate its solvability and properties of the corre- sponding electrostatic energy.

We shall use the following notation. To account for the clamped boundary conditions (1.2) we introduce, for s≥0 andp≥2,

W_p,D^s (I) :=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

{v∈W_p^s(I) ; v(±1) =∂_xv(±1) = 0}, s > 3 2, {v∈W_p^s(I) ; v(±1) = 0}, 1

2 < s <3 2,

W_p^s(I), s < 1

2,

and writeH_D^s(I) :=W_2,D^s (I). Similarly,H_D¹(Ω(u)) :={v∈H¹(Ω(u)) ; v= 0 on ∂Ω(u)}. Fors≥1 we set S^s:={u∈H_D^s(I) : u >−1 on I}, K^s:={u∈H_D^s(I) : −1< u≤0 on I},

and givenu∈S¹ we define

b_u(x, z) :=

⎧⎪

⎪⎨

⎪⎪

⎩ 1 +z

1 +u(x) for (x, z)∈Ω(u), 1 for (x, z)∈Ω(0)\Ω(u),

(2.1)

withΩ(0) =I×(−1,0). Note that, ifu∈ K¹, then the functionb_ubelongs toH¹(Ω(0))∩C(Ω(0)) which allows us to defineB_u∈H⁻¹(Ω(0)) (i.e.the dual space ofH_D¹(Ω(0))) by setting

Bu, ϑ:=−

Ω(0)

ε²∂_xb_u∂_xϑ+∂_zb_u∂_zϑ

d(x, z), ϑ∈H_D¹(Ω(0)). (2.2)

2.1. Electrostatic potential

We first recall the existence and properties of weak solutions to (1.3)–(1.4) for u ∈ K¹ which follow from Theorem 8.3 in [7] and the Lax–Milgram theorem.

Lemma 2.1. Givenu∈S¹, there is a unique weak solutionψ_u∈H¹(Ω(u))to(1.3)−(1.4)such thatψ_u−b_u∈ H_D¹(Ω(u)). If, in addition, u∈ K¹, thenψ_u−b_u satisfies the variational inequality

Ω(u)

ε²|∂_x(ψ_u−b_u)|²+|∂_z(ψ_u−b_u)|²

d(x, z)−2B_u, ψ_u−b_u

≤

Ω(u)

ε²|∂xϑ|²+|∂zϑ|²

d(x, z)−2Bu, ϑ (2.3)

for allϑ∈H_D¹(Ω(u)).

Replacingϑ∈H_D¹(Ω(u)) in (2.3) byξ−b_u, whereξis an arbitrary function inH¹(Ω(u)) satisfyingξ−b_u∈ H_D¹(Ω(u)), one easily obtains the following consequence:

Lemma 2.2. Let u∈ K¹. For allξ∈H¹(Ω(u))such thatξ−b_u∈H_D¹(Ω(u))there holds

Ω(u)

ε²|∂xψ_u|²+|∂zψ_u|²

d(x, z)≤

Ω(u)

ε²|∂xξ|²+|∂zξ|²

d(x, z). (2.4)

We collect additional properties ofψ_u in the next result whenuis assumed to be more regular.

Proposition 2.3. Let α ∈ [0,1/2). If u ∈ S^2−α, then the weak solution ψ_u to (1.3)−(1.4) belongs to H^2−α(Ω(u)). In addition, if u∈ K^2−α, then

1 +z ≤ ψ_u(x, z)≤1, (x, z)∈Ω(u), (2.5)

∂_xψ_u(x, u(x)) =−∂_zψ_u(x, u(x))∂_xu(x), x∈I, (2.6)

∂_zψ_u(x, u(x))≥0, x∈I. (2.7)

Proof. Thatψ_u∈H^2−α(Ω(u)) foru∈S^2−α follows from Corollary4.2proved in Section4. Next, ifu∈ K^2−α, then owing to the non-positivity of u, the functions (x, z) → 1 +z and (x, z) → 1 are a subsolution and a supersolution to (1.3)−(1.4), respectively, and (2.5) follows from the comparison principle. To obtain (2.6), we simply differentiate the boundary conditionψ_u(x, u(x)) = 1, x ∈ I, with respect to x. Finally, (2.7) is a straightforward consequence of the boundary conditionψ_u(x, u(x)) = 1,x∈I, and (2.5).

Thanks to the continuity of the normal trace of the gradient fromH^2−α(Ω(u)) toH^(1−2α)/2(I) forα∈[0,1/2) (see Thm. 1.5.2.1 in [8]), the regularity of the solutionψ_u∈H^2−α(Ω(u)) to (1.3)–(1.4) foru∈S^2−αprovided by Proposition2.3gives a meaning to the right-hand side of (1.1). We introduce the functiong by

g(u)(x) :=ε²|∂xψ_u(x, u(x))|²+|∂zψ_u(x, u(x))|², x∈I, u∈S^2−α, (2.8) and observe:

Proposition 2.4. If α∈[0,1/2), then g∈C(S^2−α, H^σ(I))for allσ∈[0,1/2).

Proof. This is proved in Corollary4.2.

Remark 2.5. The fact thatgis a locally bounded map fromS²inH^σ(I) for allσ∈[0,1/2) is already noticed in Lemma 5.6 in [13].

PH. LAURENC¸ OT AND CH. WALKER

2.2. Electrostatic energy

We now study the properties of the electrostatic energy Ee(u) =

Ω(u)

ε²|∂xψ_u|²+|∂zψ_u|²

d(x, z), u∈S¹, (2.9)

whereψ_u∈H¹(Ω(u)) is provided by Lemma2.1. Alternatively, we may write foru∈ K¹ E_e(u) =

Ω(u)

ε²|∂_x(ψ_u−b_u)|²+|∂_z(ψ_u−b_u)|² d(x, z)

−2Bu, ψ_u−b_u+ ₁

−1

1 + ε²

3|∂xu|² dx

1 +u· (2.10)

We first establish a monotonicity property ofEe similar to Remark 4.7.14 in [11].

Proposition 2.6. Consider two functions u₁ andu₂ inK¹ such thatu₁≤u₂. Then E_e(u₂)≤ E_e(u₁).

Proof. Considerξ∈H¹(Ω(u₁)) such thatξ−b_u1 ∈H_D¹(Ω(u₁)) and define ξ(x, z) :=˜

⎧⎨

⎩

ξ(x, z) for (x, z)∈Ω(u₁),

1 for (x, z)∈Ω(u₂)\Ω(u₁).

Note that this definition is meaningful sinceΩ(u₁)⊂Ω(u₂). Since b_u1(x, u₁(x)) =b_u2(x, u₂(x)) = 1 for x∈I, the previous construction guarantees that ˜ξ∈H¹(Ω(u₂)) with

ξ˜−b_u2∈H_D¹(Ω(u₂)) and ∇ξ˜=1_Ω(u1)∇ξ. (2.11) We now infer from Lemma2.2and (2.11) that

E_e(u₂)≤

Ω(u2)

ε²|∂_xξ|˜²+|∂_zξ|˜²

d(x, z)

=

Ω(u1)

ε²|∂xξ|²+|∂zξ|²

d(x, z).

The above inequality being valid for allξ∈H¹(Ω(u₁)) satisfyingξ−b_u1 ∈H_D¹(Ω(u₁)), in particular forξ=ψ_u1,

we conclude thatE_e(u₂)≤ E_e(u₁).

We next turn to continuity and Fr´echet differentiability of the functionalEe.

Proposition 2.7. If α∈[0,1/2), then Ee∈C(K¹)∩C¹(S^2−α) with∂_uEe(u) =−g(u)for u∈S^2−α. Proof.

Step 1: Continuity. Let (u_n)_n≥1 be a sequence in K¹ and u ∈ K¹ such that u_n −→ u in H¹(I). We first observe that, for alln≥1,ψ_un−b_un∈H_D¹(Ω(u_n)) is a weak solution to

ε²∂_x²(ψ_un−b_un) +∂_z²(ψ_un−b_un) =−B_un, (x, z)∈Ω(u_n), (2.12) while the convergence of (u_n)_n≥1towarduinH¹(I) entails that

n→∞lim B_un−B_uH⁻¹_(Ω(0))= 0, (2.13)

where Ω(0) = I ×(−1,0). Next, denoting the Hausdorff distance between open subsets of Ω(0) by d_H (see Sect. 2.2.3 in [11] for instance), we realize that

d_H(Ω(u_n), Ω(u))≤ u_n−u_L∞(I), and deduce from the continuous embedding ofH¹(I) in L_∞(I) that

n→∞lim d_H(Ω(u_n), Ω(u)) = 0. (2.14)

Since Ω(0)\ Ω(u_n) has a single connected component for all n ≥ 1, it follows from (2.12), (2.13), (2.14), Theorem 4.1 in [20], and (Cor. 3.2.6 in [11]) that

ψ_un−b_un−→ψ_u−b_u in H_D¹(Ω(0)). (2.15) Therefore, since

n→∞lim ₁

−1

1 +ε²

3|∂_xu_n|² dx

1 +u_n = ₁

−1

1 + ε²

3|∂_xu|² dx

1 +u

thanks to the continuous embedding ofH¹(I) inL_∞(I), we may pass to the limit asn→ ∞in (2.10) foru_n and use (2.13) and (2.15) to complete the proof.

Step 2: Differentiability. Consider u ∈ S^2−α and v ∈ H_D^2−α(I). Owing to the continuous embedding of H^2−α(I) in L_∞(I), u+sv still belongs to S^2−α for s∈R small enough and the map s→ E_e(u+sv) is thus well-defined in a neighborhood of s = 0. We then argue as in the proof of Proposition 2.2 in [13], with the help of a shape optimization approach (by transforming the electrostatic energyEeto a fixed domain by means of (4.1); see [11], for instance) to show that this map is differentiable at s= 0 with

d

dsEe(u+sv)

s=0=− ₁

−1g(u)vdx.

Consequently,Eeis Gˆateaux-differentiable with derivative∂_uEe(u)∈ L

H_D^2−α(I),R

. Sinceg∈C(S^2−α, L₂(I)) by Proposition2.4, the Gˆateaux-derivative∂_uEeis continuous as a mapping fromS^2−αtoL

H_D^2−α(I),R . The

claim follows from Proposition 4.8 in [21].

We next derive additional properties of E_e and, in particular, the following lower and upper bounds which have been established in Lemma 7 in [3] and Lemma 5.4 in [13], respectively.

Lemma 2.8. Foru∈ K¹, 2≤

₁

−1

dx

1 +u(x) ≤ Ee(u)≤ ₁

−1

1 +ε²|∂xu(x)|² dx 1 +u(x)·

Proof. We recall the proof for the sake of completeness. We first deduce from (1.4) and the Cauchy−Schwarz inequality that, forx∈I,

1

1 +u(x)= (ψ_u(x, u(x))−ψ_u(x,−1))²

1 +u(x) = 1

1 +u(x)

_u(x)

−1 ∂_zψ_u(x, z) dz ₂

≤ _u(x)

−1 (∂_zψ_u(x, z))² dz.

PH. LAURENC¸ OT AND CH. WALKER

Integrating the above inequality with respect tox∈I readily gives the first inequality of Lemma2.8. We next infer from Lemma2.2withξ=b_u, the latter being defined in (2.1), that

E_e(u)≤

Ω(u)

ε²|∂_xb_u|²+|∂_zb_u|² d(x, z)

≤

Ω(u)

ε² (1 +z)²

(1 +u(x))⁴|∂_xu(x)|²+ 1 (1 +u(x))²

d(x, z),

from which the second inequality of Lemma2.8follows.

Finally, we recall the existence of a non-positive eigenfunction of the linear operator β∂_x⁴ − τ ∂_x² ∈ L(H_D⁴(I), L₂(I)) along with some of its properties.

Lemma 2.9.

(i) The linear operatorβ∂⁴_x−τ ∂²_x∈ L(H_D⁴(I), L₂(I))has a non-positive eigenfunctionϕ₁∈H_D⁴(I)∩C^∞([−1,1]) associated to a positive eigenvalueμ₁. Moreover,ϕ₁ is even and it can be chosen such thatϕ₁<0 inI with min_[−1,1]ϕ₁=−1.

(ii) Given ρ∈(2,∞), there is η_ρ∈(0,1) such thatE_e(η_ρϕ₁) =ρandη_ρ →0as ρ→2.

Proof. Part (i) follows from Theorem 4.7 in [15], which is a consequence of the version of Boggio’s principle [2]

established in [9,15,18]. As for part (ii), note thatηϕ₁∈ K¹ forη∈[0,1) and J(η) :=E_e(ηϕ₁)≥

₁

−1

dx

1 +ηϕ₁(x), η∈[0,1), (2.16)

by Lemma2.8. We infer from Propositions2.6 and2.7 thatJ is a non-decreasing and continuous function on [0,1) with J(0) = 2. In addition, ϕ₁ reaches necessarily its minimum −1 at some x₀ ∈ I and thus satisfies ϕ₁(x₀) =−1 and∂_xϕ₁(x₀) = 0. Therefore,

0≤1 +ϕ₁(x)≤ ∂_x²ϕ_1L∞(I)|x−x₀|² as x→x₀,

which implies that (1 +ϕ₁)⁻¹ ∈ L₁(I). This property along with (2.16) entails that J(η) → ∞ as η → 1.

Recalling the continuity ofJ, we have thus shown that [2,∞) equals the range ofJ. The existence ofη_ρfor each ρ∈(2,∞) such thatE_e(η_ρϕ₁) =ρnow follows. Thatη_ρ→0 asρ→2 is a consequence of the fact that (2.16)

impliesJ(η) = 2 if and only ifη= 0.

3. A minimization problem with constraint

Recall that, foru∈H_D²(I), the mechanical energyE_m is given by Em(u) = β

2∂_x²u²_L2(I)+1 2

τ+a

2∂_xu²_L2(I)

∂_xu²_L2(I). Our goal is now to minimizeEmon the set

A_ρ:=

u∈ K² ; u is even andE_e(u) =ρ

for a givenρ∈(2,∞). Note that Aρ is non-empty as it containsη_ρϕ₁ according to Lemma2.9. We set μ(ρ) := inf

u∈AρE_m(u)≥0 and first collect some properties of the functionρ→μ(ρ).

Proposition 3.1. The function μis non-decreasing on(2,∞)with

ρ→2limμ(ρ) = 0 and μ_∞:= lim

ρ→∞μ(ρ)<∞.

Proof. Letρ∈(2,∞). Sinceη_ρϕ₁∈ Aρ andη²_ρ<1, a straightforward computation gives 0≤μ(ρ)≤ E_m(η_ρϕ₁)≤η²_ρE_m(ϕ₁).

SinceEm(ϕ₁) is finite,η_ρ∈(0,1), and η_ρ→0 asρ→2 by Lemma 2.9, we readily obtain

ρ→2limμ(ρ) = 0 and 0≤μ(ρ)≤ E_m(ϕ₁). (3.1) Let us now check the monotonicity ofμ. To this end, fix 2< ρ₁< ρ₂andv∈ A_ρ2. For allt∈[0,1], the function tvbelongs toK², and Propositions2.6and2.7imply that the functionh: [0,1]→R, defined byh(t) :=E_e(tv), is continuous and non-decreasing withh(0) = 2 andh(1) =ρ₂. Sinceρ₁∈(2, ρ₂), there ist₁∈(0,1) such that h(t₁) =ρ₁, that is,t₁v∈ Aρ1. Consequently,

μ(ρ₁)≤ E_m(t₁v)≤ E_m(v).

As v was arbitrarily chosen in Aρ2, the above inequality allows us to conclude that μ(ρ₁)≤μ(ρ₂). Thus,μ is a non-decreasing function on (2,∞) which is bounded from above byE_m(ϕ₁) according to (3.1). It then has a

finite limitμ_∞∈[0,E_m(ϕ₁)] asρ→ ∞.

We next show the existence ofu_ρ∈ A_ρsuch that

Em(u_ρ) =μ(ρ), (3.2)

that is,u_ρ is a minimizer of E_m inA_ρ.

Proposition 3.2. For eachρ∈(2,∞), there is at least one solutionu_ρ∈ Aρto the minimization problem (3.2).

The first step of the proof of Proposition3.2is a pointwise lower bound for functions inAρ.

Lemma 3.3. Given ρ >2andv∈ A_ρ, assume that there isK≥2/ρsuch that ∂_x²v_L2(I)≤K. Then

[−1,1]min v≥ 1 ρ³K² −1.

Proof. Thanks to the continuous embedding of H_D²(I) in C¹([−1,1]), the function v reaches its minimum m at some point x_m∈ [−1,1]. Since Ee(v) =ρ > 2 and v ∈ K², we realize that v ≡0 and m∈ (−1,0) so that x_m∈I. Therefore,∂_xv(x_m) = 0 and we may assume thatx_m∈[0,1) sincev is even. Using Taylor’s expansion and H¨older’s inequality, we find, forx∈I,

v(x) =m− _x

xm

(y−x)∂²_xv(y) dy≤m+|x−√x_m|^3/2

3 ∂_x²v_L2(I)

≤m+K|x−x_m|^3/2. (3.3)

Next, sincev∈ A_ρ, we infer from Lemma 2.8and (3.3) that ρ=Ee(v)≥

₁

−1

dx 1 +v(x)= 2

₁

0

dx 1 +v(x) ≥2

₁

0

dx

1 +m+K|x−x_m|^3/2· (3.4)

PH. LAURENC¸ OT AND CH. WALKER

Ifx_m∈[1/2,1), thenx_m−(ρK)⁻²>0, and it follows from (3.4) that ρ≥2

_xm

xm−(ρK)⁻²

dx

1 +m+K|x−x_m|^3/2 ≥ 2(ρK)⁻² 1 +m+K(ρK)⁻³,

hencem≥ρ⁻³K⁻²−1 as claimed. Ifx_m∈[0,1/2), thenx_m+ (ρK)⁻²<1, and we deduce from (3.4) that ρ≥2

_xm+(ρK)⁻²

xm

dx

1 +m+K|x−x_m|^3/2 ≥ 2(ρK)⁻² 1 +m+K(ρK)⁻³,

and the same computation as in the previous case completes the proof.

Proof of Proposition3.2. Let (u_k)_k≥1 be a minimizing sequence ofEmin Aρ satisfying μ(ρ)≤ Em(u_k)≤μ(ρ) + 1

k· (3.5)

A first consequence of Proposition3.1and (3.5) is that∂_x²u_k²_L2(I)≤2(1 +μ_∞)/βfor allk≥1. Together with Lemma 3.3(withK= (2/ρ) + 2

(1 +μ_∞)/β) this property ensures 0≥u_k(x)≥ β

8ρ(β+ (1 +μ_∞)ρ²)−1, x∈[−1,1], k≥1. (3.6) Also, owing to (3.1), (3.5), and Poincar´e’s inequality, the sequence (u_k)_k≥1 is bounded inH_D²(I) and thus rela- tively compact inC¹([−1,1]). Consequently, there areu∈H_D²(I) and a subsequence of (u_k)_k≥1(not relabeled) such that

u_k −→u in C¹([−1,1]),

u_k u in H_D²(I). (3.7)

Combining (3.6) and (3.7) we conclude that

0≥u(x)≥ β

8ρ(β+ (1 +μ_∞)ρ²)−1, x∈[−1,1], henceu∈ K². We then infer from Proposition2.7that

Ee(u) = lim

k→∞Ee(u_k) =ρ, and sou∈ A_ρ. Since

E_m(u)≤lim inf

k→∞ E_m(u_k)≤μ(ρ)

by (3.5) and (3.7), we deduce thatE_m(u) =μ(ρ) so thatuis a minimizer ofE_min A_ρ. Theorem 3.4. Consider ρ∈(2,∞) and letu∈ Aρ be an arbitrary minimizer ofEm in Aρ. Thenu∈H_D⁴(I) and there isλ_u>0 such that

β∂⁴_xu(x)−

τ+a∂_xu²_L2(I)

∂_x²u(x) =−λ_u

ε²|∂_xψ_u(x, u(x))|²+|∂_zψ_u(x, u(x))|²

(3.8) for x ∈ I, where ψ_u ∈ H²(Ω(u)) denotes the associated solution to (1.3)−(1.4) given by Lemma 2.1 and Proposition 2.3. Furthermore,

0< λ_u≤ 8μ_∞√

β+ε²√ μ_∞

√β(ρ−2)² · (3.9)

Proof. Letu∈ A_ρ⊂ K²be a minimizer ofE_m. Recall from Proposition2.7that the derivative ofE_eis given by

∂uEe(u), ϑ=− ₁

−1g(u)ϑdx, ϑ∈H_D²(I), withg(u)∈L₂(I) while clearly

∂_uEm(u), ϑ= ₁

−1

β∂_x²u ∂_x²ϑ+

τ+a∂_xu²_L2(I)

∂_xu ∂_xϑ

dx, ϑ∈H_D²(I).

Sinceusolves (3.2) andg(u) is non-negative, (4.14. Prop. 1 in [22]) implies that there is a Lagrange multiplier λ_u∈Rsuch that

∂_uE_m(u), ϑ=λ_u∂_uE_e(u), ϑ, ϑ∈H_D²(I). (3.10) We may then combine (3.10) and classical elliptic regularity to conclude thatu∈H_D⁴(I) solves (3.8) in a strong sense. In addition, takingϑ=uin (3.10) gives

β∂_x²u²_L2(I)+τ∂_xu²_L2(I)+a∂_xu⁴_L2(I)=−λ_{u 1}

−1

ug(u) dx, (3.11)

henceλ_u>0 sinceg(u) is non-negative anduis non-positive and different from zero.

We are left with the upper bound (3.9) on λ_u. On the one hand, multiplying (1.3) by (1 +u)ψ_u−(1 +z), integrating overΩ(u), and using

(1 +u(x))ψ_u(x, z)−(1 +z) = 0, (x, z)∈∂Ω(u), we obtain from Green’s formula that

0 =

Ω(u)

ε²∂_xψ_u∂_x((1 +u)ψ_u) +∂_zψ_u((1 +u)∂_zψ_u−1) d(x, z)

=

Ω(u)

(1 +u)

ε²|∂_xψ_u|²+|∂_zψ_u|²

+ε²ψ_u∂_xψ_u ∂_xu

d(x, z)−2,

whence

Ω(u)

u

ε²|∂xψ_u|²+|∂zψ_u|²

+ε²ψ_u∂_xψ_u ∂_xu

d(x, z) = 2− Ee(u). (3.12) On the other hand, we multiply (1.3) byuψ_uand integrate overΩ(u). Using again Green’s formula along with the values ofuandψ_u on the boundary ofΩ(u), we find

0 =−

Ω(u)

ε²∂_xψ_u∂_x(uψ_u) +∂_zψ_u(u∂_zψ_u) d(x, z)

−ε² ₁

−1u(x)∂_xu(x)∂_xψ_u(x, u(x)) dx+ ₁

−1u(x)∂_zψ_u(x, u(x)) dx

=−

Ω(u)

u

ε²|∂_xψ_u|²+|∂_zψ_u|²

+ε²ψ_u∂_xψ_u∂_xu) d(x, z) +

₁

−1u(x)

∂_zψ_u(x, u(x))−ε²∂_xu(x)∂_xψ_u(x, u(x)) dx.

Combining (3.12) with the above identity and (2.6) we end up with

− ₁

−1u(x)

1 +ε²|∂xu(x)|²

∂_zψ_u(x, u(x)) dx=Ee(u)−2. (3.13)

PH. LAURENC¸ OT AND CH. WALKER

Now it follows from (2.6), (3.2), (3.11), (3.13), Jensen’s inequality, the bounds −1 < u ≤ 0, and the non- negativity (2.7) ofx→∂_zψ_u(x, u(x)) that

4μ(ρ)≥β∂_x²u²_L2(I)+τ∂_xu²_L2(I)+a∂_xu⁴_L2(I)

=−λ_{u 1}

−1u(x)

1 +ε²|∂_xu(x)|²

|∂_zψ_u(x, u(x))|² dx

≥λ_{u 1}

−1|u(x)|

1 +ε²|∂_xu(x)|²

∂_zψ_u(x, u(x)) dx 2

₁

−1|u(x)|

1 +ε²|∂xu(x)|² dx

≥λ_u (Ee(u)−2)² 2 +ε²∂_xu²_L2(I)· We finally observe thatE_e(u) =ρasu∈ A_ρ while

∂xu²_L2(I)=− ₁

−1u ∂_x²udx≤ ₁

−1|∂_x²u|dx≤√

2∂_x²u_L2(I)≤2

μ(ρ) β , sinceu∈ K² solves (3.2). Therefore,

4μ(ρ)≥λ_u

√β(ρ−2)² 2

√

β+ε² μ(ρ)

,

which gives (3.9) after using Proposition3.1.

Proof of Theorem 1.3. Clearly, Proposition 3.2 and Theorem3.4 imply that for each ρ >2 there are λ_ρ >0, u_ρ ∈H_D⁴(I), andψ_ρ ∈H²(Ω(u_ρ)) such that (u_ρ, ψ_ρ) is a solution to (1.1)−(1.4) with λ=λ_ρ. We recall that λ→(λ, U_λ) defines a smooth curve inR×H⁴(I) starting at (0,0) according to Theorem1.2so thatEe(U_λ)→2 asλ→0 due to Proposition2.7. Consequently, sinceEe(u_ρ) =ρandλ_ρ→0 asρ→ ∞, we realize thatu_ρ=U_λρ for large ρ. Finally, since u_ρ is even and uniquely determinesψ_ρ, it readily follows that ψ_ρ =ψ_ρ(x, z) is even

with respect tox∈I.

4. Regularity of solutions to (1.3) – (1.4)

In this section we provide the technical proofs of Propositions2.3and2.4that were postponed. That is, we shall improve the regularity of the weak solution ψ_u to (1.3)−(1.4) given in Lemma 2.1 for smoother deflec- tionsu and prove continuity properties of the function g defined in (2.8). In order to do so we introduce the transformation

T_u(x, z) :=

x, 1 +z 1 +u(x)

, (x, z)∈Ω(u),

mapping Ω(u) onto the fixed rectangleΩ:=I×(0,1). We then transform the elliptic problem (1.3)–(1.4) for ψ_u in the variables (x, z)∈Ω(u) to the elliptic problem

L_uΦ_u=f_u in Ω, Φ_u= 0 on ∂Ω, (4.1)

forΦ_u(x, η) =ψ_u◦T_u⁻¹(x, η)−η in the variables (x, η) =T_u(x, z)∈Ω, where the operatorLu is given by Luw :=ε² ∂_x²w−2ε²η ∂_xu(x)

1 +u(x) ∂_x∂_ηw+1 +ε²η²(∂_xu(x))² (1 +u(x))² ∂²_ηw +ε² η

2

∂_xu(x) 1 +u(x)

₂

− ∂_x²u(x) 1 +u(x)

∂_ηw (4.2)