Convergence of energy minimizers of a MEMS model in the reinforced limit

Energy minimizers to a MEMS model with an insulating layer are shown to converge in its reinforced limit to the minimizer of the limiting model as the thickness of the layer tends to zero. The proof relies on the identification of the $\Gamma$-limit of the energy in this limit.


Introduction
A microelectromechanical system (MEMS), such as an electrostatic actuator, consists of an elastic plate, which is coated with a thin dielectric layer, clamped on its boundary, and suspended above a rigid ground plate. The latter is also coated with a dielectric layer but with positive thickness δ > 0, see Figures 1.1 and 1.2. Applying a voltage difference between the two plates generates a Coulomb force accross the device and induces a deformation of the elastic plate, thereby changing the geometry of the device and converting electrostatic energy to mechanical energy through a balance between electrostatic and mechanical forces [2,3,8,22]. Assuming that the physical state of the MEMS device is fully described by the vertical deflection u of the elastic plate and the electrostatic potential ψ inside the device, a mathematical model is derived in [17]. It characterizes equilibrium configurations of the device as critical points of the total energy which is the sum of the mechanical and electrostatic energies, with an additional constraint stemming from the property that the elastic plate cannot penetrate the layer covering the ground plate. Specifically, ignoring variations in the transverse horizontal direction, we consider a two-dimensional MEMS in which the rigid ground plate and the undeflected elastic plate have the same one-dimensional shape D := (−L, L) with L > 0. The ground plate is located at height z = −H − δ, where H > 0, and is coated with a dielectric layer As for the electrostatic potential ψ, it is defined in the full device It is worth mentioning at this point that the geometry of the full device Ω δ (u) has different properties according to the minimal value of u. In the model derived in [17], equilibrium configurations of the above described MEMS device are critical points of the total energy given by E δ (u) := E m (u) + E e,δ (u) .
(1. 2) In (1.2), E m (u) is the mechanical energy with β > 0, a ≥ 0, and τ ≥ 0, and includes bending and external stretching effects of the elastic plate. The electrostatic energy is with σ δ denoting the permittivity of the device (see (2.1) below), and ψ = ψ u,δ is the electrostatic potential satisfying the transmission problem Here, · denotes the jump of a function across the interface Σ(u). The boundary values of the electrostatic potential are prescribed by a function h u,δ which satisfies the assumptions listed below in (3.1). A specific example, when σ does not depend on the vertical coordinate z, is , , (1.4) Since the elastic plate is clamped at the boundary and cannot penetrate the dielectric layer R δ , the set of admissible deflections is Equilibrium configurations of the MEMS device are then critical points u ∈S 0 of the total energy E δ . Their analysis involves the associated transmission problem (1.3) solved by the electrostatic potential ψ u,δ . A natural question is what happens when the thickness δ of the dielectric layer tends to zero, in particular, whether the reduced model derived in this limit retains the dielectric inhomogeneity of the device. When the dielectric permittivity σ δ of the device does not depend on δ, the influence of the dielectric layer is lost in the limit δ → 0, and the reduced model is obtained simply by setting δ = 0 in (1.2) and (1.3), discarding the jump condition (1.3b) which is then meaningless. Building upon the outcome of [1,5], it turns out that it is rather the reinforced limit, where the dielectric permittivity scales as δ in the layer R δ , which leads to a relevant reduced model. For a given deflection u ∈S 0 , the reinforced limit of the transmission problem (1.3) is identified in [14] by a Γ-convergence approach. More precisely, it is shown in [14] that the reinforced limit as δ → 0 of (1.3) is div(σ∇ψ u ) = 0 in Ω(u) , (1.5a) that is, in the reinforced limit the electrostatic potential ψ u solves Laplace's equation in Ω(u) with a Robin boundary condition along the interface Σ(u) and a Dirichlet condition on the other boundary parts. Here, σ := σ δ 1 Ω(u) is assumed to be independent of δ. The total energy is then given by where and h u is defined below in (3.1). The purpose of this research is to complete the outcome of [14] by identifying the reinforced limit of the full model and showing that, in this limit, if u * δ ∈S 0 is a minimizer of E δ inS 0 for each δ ∈ (0, 1), then the cluster points of (u * δ ) δ∈(0,1) in L 2 (D) are minimizers of the reduced total energy E inS 0 . The main tool we shall employ in the forthcoming analysis is the theory of Γ-convergence. We shall actually show that, under suitable assumptions on the dielectric permittivity σ δ and the boundary values in (1.3), the Γlimit in L 2 (D) of (E δ ) δ∈(0,1) is the reduced total energy E defined in (1.6).
Let us finally remark that, in this paper, we focus on the energy approach to take into account the influence of the thickness of a dielectric layer as first developed in [16] for a related model. We refer to [3,[19][20][21] for alternative approaches to model dielectric layers, all designed within the so-called small aspect ratio approximation. Recall that, in the latter, the electrostatic potential is given explicitly as a function of the deflection u and the model then reduces to a single equation for u. Such models have been extensively studied in the last decades in the mathematical literature since the pioneering works of [4,9,11,21], see the book [7], the survey [15] and the references therein.

Convergence of minimizers
As already mentioned, the reinforcement limit requires that the permittivity σ δ in the dielectric layer R δ scales with the layer's thickness; that is, the (scaled) permittivity of the device is given in the form With this specific form of σ, we can show that cluster points as δ → 0 of minimizers of the total energy E δ onS 0 are minimizers of the reduced total energy E. More precisely: Theorem 2.1. Suppose that the dielectric permittivity satisfies (2.1) and that the assumptions on the boundary values in (1.3c) are given by (3.1) below. For δ ∈ (0, 1) let u * δ ∈S 0 be any minimizer of E δ onS 0 with corresponding electrostatic potential ψ u * δ ,δ satisfying (1.3). Then and there are a subsequence δ j → 0 and a minimizer u * ∈S 0 of E onS 0 such that where ψ u * satisfies (1.5) (with u replaced by u * ).
As we shall see below, the main step in the proof of Theorem 2.1 is the Γ-convergence of the sequence (E δ ) δ∈(0,1) in L 2 (D) to E which is established in Section 4. We then combine this property with estimates on the minimizers of E δ onS 0 , which do not depend on δ ∈ (0, 1) and are derived in Sections 4.3-4.4 to complete the proof.
As for the reduced total energy E, the existence of minimizers of E onS 0 has already been established in [13, Theorem 2.3] by a direct approach, assuming additionally that

Assumptions and auxiliary results
This section is devoted to a precise definition of the boundary conditions (1.3c) and (1.5c), and includes as well useful properties of h u,δ on which we rely on in the sequel.
3.1. Boundary data. We fix two C 2 -functions and In order to guarantee the coercivity of the energy functional E δ we require that there is a constant m > 0 such that and . Moreover, we assume that and that there is K > 0 such that Given a function u :D → [−H, ∞) we shall also use the abbreviations Note that (3.1c)-(3.1d) imply that h u,δ satisfies the transmission conditions (1.3b): Simple computations show that the example provided in (1.4) satisfies (3.1) with
(ii) First, M is well-defined and finite owing to the continuous embedding of H 1 (D) in C(D) and the strong convergence of (u δ ) δ∈(0,1) in H 1 (D). Next, the stated convergences readily follow from the smoothness of h and h b , from the convergence of u δ → u in H 1 (D), and the continuous embedding of H 1 (D) in C(D).

Convergence of minimizers
Three steps are needed to prove Theorem 2.1: we begin by establishing in Section 4.1 the convergence of the electrostatic energy E e,δ as δ → 0, building upon the analysis performed in [14] for a reduced problem. This convergence, along with the weak lower semicontinuity of the mechanical energy E m , leads us to the Γ-convergence of E δ to E in L 2 (D), see Section 4.2. Such a property provides information on the relationship between minimizers for the cases δ > 0 and δ = 0, which we use in Sections 4.3-4.4 to complete the proof of Theorem 2.1.
4.2. Γ-convergence of the total energy. We now turn to the Γ-convergence of the total energy and first establish that the H 2 -norm of u is controlled by the total energy E δ (u) (defined in (1.2)) and the L 2 -norm of u, whatever the value of δ ∈ (0, 1).
The total energies (defined in (1.2) and (1.6)), being a priori defined only onS 0 , are extended to functionals on L 2 (D) by setting Then we can prove: Proof. (i) Recovery sequence. Concerning the construction of a recovery sequence it is sufficient to consider u ∈S 0 . Let us observe from [14,Corollary 3.4] that lim δ→0 E e,δ (u) = E e,0 (u) .

4.4.
Remaining arguments for the proof of Theorem 2.1: The case a = 0. To finish off the proof of Theorem 2.1, we are left with the case a = 0 for which the weak compactness of minimizers in H 2 (D) is harder to derive. Additional information on these minimizers is actually required and follows from the analysis performed in [17,18], using that they are critical points of the total energy. Lemma 4.4. There is a constant c 2 > 0 which does not depend on δ ∈ (0, 1) such that, if u is a minimizer of E δ onS 0 for some δ ∈ (0, 1), then u L∞(D) ≤ c 2 , δ ∈ (0, 1) .
Taking Lemma 4.4 for granted, we are in a position to complete the proof of Theorem 2.1 when a = 0.
We are left with proving Lemma 4.4, which relies on the same comparison argument as [18, Proposition 2.6] and uses in an essential way the Euler-Lagrange equation satisfied by minimizers of the total energy E δ .