Reinforced Limit of a MEMS Model with Heterogeneous Dielectric Properties

A MEMS model with an insulating layer is considered and its reinforced limit is derived by means of a Gamma convergence approach when the thickness of the layer tends to zero. The limiting model inherits the dielectric properties of the insulating layer.


Introduction
Idealized microelectromechanical systems (MEMS) consist of two dielectric plates: a rigid ground plate above which an elastic plate is suspended. The latter is electrostatically actuated by a Coulomb force which is induced across the device by holding the two plates at different voltages. In this set-up there is thus a competition between attractive electrostatic forces and restoring mechanical forces due to the elasticity of Partially   the plate. When the two plates are not prevented from touching each other, a contact of the plates commonly leads to an instability of the device-also known in the literature as "pull-in instability"-which is revealed as a singularity in the corresponding mathematical equations, e.g., see [9,16] and the references therein. In contrast, when the ground plate is coated with an insulating layer preventing a direct contact of the plates, see Fig. 1, a touchdown of the elastic plate on this layer does not result in an instability as the device may continue to operate without interruption (though it still leads to a peculiar situation from a mathematical point of view). Different mathematical models describing this setting including an insulating layer were introduced [2,4,10,12,13,17]. The basic assumption in all these models is that the state of the device is fully described by the vertical deflection of the elastic plate and the electrostatic potential in the device. According to [4,10,12,13] the dynamics of the former is governed by an evolution equation while that of the latter is governed by an elliptic equation in a time-varying domain enclosed by the two plates. Due to the heterogeneity of the dielectric properties of the device, this elliptic equation is actually a transmission problem (see (1.1) below) on the non-smooth time-dependent domain with a transmission condition at the interface separating the insulating layer and the free space. The analysis of such a model turns out to be quite involved [10,Section 5]. Therefore, several simpler and more tractable models were derived on the assumption of a vanishing aspect ratio of the device [2,4,10,12,13]. Thanks to this approximation the electrostatic potential can be computed explicitly in terms of the deflection of the elastic plate, and the model thus reduces to a single equation for the deflection.
The aim of the present work is to derive an intermediate model by letting only the thickness of the insulating layer go to zero (instead of the aspect ratio of the device). Our starting point is the model analyzed in [10] in which we introduce an appropriate scaling of the dielectric permittivity in dependence on the layer's thickness (see (2.1a) below) and use a Gamma convergence approach to study the limiting behavior. The specific choice of the scaling is required in order to keep relevant information of the dielectric heterogeneity of the device and can be interpreted as a reinforced limit from a mathematical point of view [1].
To be more precise, we recall the model stated in [10,Section 5]. Let D ⊂ R n with n ≥ 1 be a bounded C 2 -domain representing the (identical) horizontal crosssection of the two plates (actually, only the cases n ∈ {1, 2} are physically relevant for applications to MEMS, the ground plate being D × (−H − d, −H ) and thus a two or three dimensional object). The dielectric layer of thickness δ > 0 on top of the ground plate located at z = −H − δ with H > 0 is then given by The deflection of the elastic plate from its rest position at z = 0 is described by a function u : is the free space between the elastic plate and the top of the dielectric layer. We let denote the interface separating free space and dielectric layer and put If the elastic plate and the insulating layer remain separate, that is, if u > −H in D, then (u) coincides with In contrast, a touchdown of the elastic plate on the insulating layer corresponds to a non-empty coincidence set and a different geometry as the free space (u) then has several connected components. It is worth pointing out that in this case -independent of the smoothness of the function u -these components may not be Lipschitz domains, a feature which requires some special care in the mathematical analysis.
The different situations with empty and non-empty coincidence sets are depicted in Fig. 1.
In the model considered in [10,Section 5], the deflection u of the elastic plate is governed by an evolution equation involving contributions from mechanical and electrostatic forces, the latter depending on the electrostatic potential denoted by ψ in the following. However, for the derivation of the limiting problem for the electrostatic potential, the evolution of u does not play any role. We thus consider throughout this paper a fixed geometry (u); that is, we consider the function u :D → [−H , ∞) with u = 0 on ∂ D describing the deflection of the elastic plate as given and fixed. We refer to [10,Section 5] for the full model. Given such a function u, the electrostatic potential ψ = ψ u,δ satisfies the transmission problem div(σ δ ∇ψ) = 0 in δ (u) , (1.1c) and the corresponding electrostatic energy of the device with geometry δ (u) is Here, σ δ is the permittivity of the device, which is different in the insulating layer and free space, and h u,δ is a given suitable function describing the boundary values of the electrostatic potential. By · we denote the jump of a function across the interface (u). The Lax-Milgram theorem provides the existence of a unique electrostatic potential ψ u,δ = χ u,δ + h u,δ solving (1.1) in a variational sense, and the function χ u,δ ∈ H 1 0 ( δ (u)) is the minimizer of the Dirichlet integral among functions ϑ ∈ H 1 0 ( δ (u)), see Proposition 3.2 below. In the following we shall derive the limiting model obtained from (1.1) as δ → 0 when imposing suitable assumptions on the function h u,δ defining the boundary values of the potential (see (2.2) below) and on the permittivity σ δ (see (2.1) below), so that information on the dielectric heterogeneity is inherited. As for the permittivity we assume that it is constant (normalized to 1) in (u) and a reinforced limit σ δ = O(δ) in R δ . We then shall follow [1] to compute the Gamma limit with respect to the L 2topology of the family of functionals (G δ ) δ∈(0,1) as δ → 0, which turns out to be the functional with h u and h u defined below in (2.6) and in (2.8), respectively, see Theorem 3.1. Let us emphasize here that an utmost challenging feature of the limiting problem is that (u) need not be a Lipschitz set as it may have cusps when the coincidence set C(u) is nonempty. Therefore, the usual trace theorem is not available and a meaningful definition of G requires a suitable definition of a trace on (D \ C(u)) × {−H } for functions in (a subset of) H 1 ( (u)), see Lemmas 2.1 and 2.2. Once this issue is settled, the existence of a minimizer χ u of G in a suitable subset of H 1 ( (u)) is shown by classical arguments, see Proposition 3.3. The derivation of the corresponding Euler-Lagrange equation offers further challenges again related to the non-smoothness of (u). Indeed, a formal computation reveals that ψ u = χ u + h u solves Laplace's equation on (u) with a Robin boundary condition along the interface (u) and a Dirichlet condition on the other boundary parts; that is, However, a rigorous computation relies on Gauß' theorem which requires some geometric condition on the boundary of (u) and the existence of boundary traces for ∇χ u , see [7]. 1 Due to the Robin boundary condition the resulting model is consistent in the sense that touching plates again do not lead to a singularity in the equations.
is a Lipschitz domain and the derivation of (1.2) only requires that χ u belongs to H 2 ( (u)). However, this property is not guaranteed by classical elliptic regularity theory since (u) is only Lipschitz. In the special case that D is a one-dimensional interval and under appropriate choices of h u and h u , we provide a rigorous justification of (1.2) in Theorem 3.5.
In Sect. 2 we first list the precise assumptions that we impose on the permittivity σ δ and on the function h u,δ defining the boundary values of the electrostatic potential. Moreover, since, as pointed out above, the set (u) may not be Lipschitz for deformations u with non-empty coincidence set C(u) and thus standard trace theorems are not valid, we derive in Sect. 2 also boundary trace theorems in weighted spaces for functions in H 1 ( (u)). Section 3 is dedicated to the computation of the Gamma limit of (G δ ) δ∈(0,1) as δ → 0, which is the main result of this paper, see Theorem 3.1. Moreover, we derive in Sect. 3 the limiting equations (1.2).
From now on, the function u is fixed and assumed to satisfy

Assumptions and Auxiliary Results
In this section we state the precise assumptions imposed on the permittivity σ δ and the function h u,δ defining the boundary values of the electrostatic potential. We also provide some auxiliary results regarding boundary traces for functions defined on the possibly non-Lipschitz set (u).

Assumptions on ı and h u,ı
To inherit in the limit δ → 0 the information of the permittivity from the insulating layer, we specifically assume that the permittivity scales with the layer's thickness; that is, we assume that the permittivity of the device is given in the form Regarding the boundary values of the electrostatic potential given in (1.1c) we fix two In the following, we shall also use the abbreviations and Furthermore, we set

Traces in H 1 (Ä(u))
As pointed out already in the introduction, the region (u) need not be Lipschitz (besides not being connected) when the elastic plate touches the insulating layer; that is, when C(u) = ∅. That there is still a meaningful definition of boundary traces on D \ C(u) for functions in H 1 ( (u)) in this case is the content of the subsequent result. We follow [14], exploiting the special geometry of (u) to show that traces are well-defined in weighted spaces.
(b) The proof being similar to that of (a), we omit it here.
For simplicity, we use the notation for ϑ ∈ H 1 ( (u)). Next, we introduce H 1 B ( (u)) as the closure in H 1 ( (u)) of the set Next, we introduce H 1 B ( (u)) as the closure in H 1 ( (u)) of the set Since ϑ(x, u(x)) = ϑ(x, −H ) = 0 for x ∈ C(u) and ϑ ∈ C 1 B ( (u)), we agree upon setting ϑ(x, u(x)) = ϑ(x, −H ) := 0 for all x ∈ C(u) and ϑ ∈ H 1 B ( (u)) in the reminder of this paper. For functions in H 1 B ( (u)) we derive a Poincaré inequality and improve the information on the trace along from Lemma 2.1: 12) and the trace ϑ → ϑ(·, −H ) yields a bounded linear operator from H 1 it follows from Hölder's inequality that, for x / ∈ C(u) and z ∈ (−H , u(x)), Consequently, using again Hölder's inequality gives Hence, and we complete the proof of (2.12) by a density argument. Next, consider again ϑ ∈ C 1 B (u) and x / ∈ C(u). We infer from (2.14) with z = −H that Since ϑ(x, −H ) = 0 for x ∈ C(u), we use once more Hölder's inequality to obtain which shows (2.13) for ϑ ∈ C 1 B (u) . We again complete the proof by a density argument.

The Reinforced Limit
As announced in the introduction we shall derive the limiting equations of (1.1) as δ → 0 when assuming the reinforced limit (2.1) on the permittivity. For this we first compute the Gamma limit of the functionals (G δ ) δ∈(0,1) and then study the behavior of the corresponding minimizers. Define for δ ∈ (0, 1)

The Gamma Limit of the Electrostatic Energy
with h u,δ given in (2.5). Also, for ϑ ∈ H 1 B ( (u)), we set with h u and h u defined in (2.6) and (2.8), respectively, and Then the main result of the present paper is the following convergence.

Minimizers
Now that we have shown the Gamma convergence of the functionals (G δ ) δ∈(0,1) towards G, we can deduce useful information on the relation between their minimizers. We first recall from the Lax-Milgram theorem (also see [10, Proposition 3.1, Lemma 3.2]) the following result regarding the solvability of the transmission problem (1.1):
As for the functional G we note:

Proposition 3.3 Suppose (1.3) and (2.2). There is a unique minimizer
Proof It readily follows from (2.1b), the Poincaré inequality (2.12), and the Lax-Milgram theorem that there is a unique minimizer we obtain the claimed estimate by taking ϑ ≡ 0 in the previous inequality.
As a consequence of Theorem 3.1 and Propositions 3.2 and 3.3, we obtain the convergence of the minimizers of the functionals.

The Limiting Model
Finally, we shall derive the analogue to Eqs. (1.1) satisfied by ψ u := χ u + h u for which we suppose that χ u ∈ H 1 B ( (u)) ∩ H 2 ( (u)) and that Gauß' theorem applies for (u) and ∇χ u (which requires some geometric condition on the boundary of (u) and the existence of boundary traces for ∇χ u , see [7]). Let Therefore, since φ vanishes on ∂ (u) \ , Consequently, in this case ψ u = χ u + h u ∈ H 2 ( (u)) solves Laplace's equation with mixed boundary conditions of Dirichlet and Robin type as announced in (1.2). The corresponding electrostatic energy is that is, where h u is defined in (2.8). By Corollary 3.4 we have E e,δ (u) → E e (u) as δ → 0.
For the special case that D is an interval in R and h b is of the form ∞)), the previous computation can be rigorously justified.  C(u)). It is given as ψ u = χ u + h u with χ u ∈ H 1 B ( (u)) being the unique minimizer of G on H 1 B ( (u)). Remark 3. 6 We point out once more that, since (u) need not be Lipschitz, the stated L 2 -regularity of ∂ z ψ u (·, −H ) does not follow from the H 2 -regularity of ψ u by a standard trace theorem. In fact, Lemma 2.1 only ensures that ∂ z ψ u (·, −H ) belongs to the weighted space L 2 (D \ C(u), (H + u)dx). That ∂ z ψ u (·, −H ) additionally belongs to L 2 (D \ C(u)) follows a posteriori from (1.2c), as shown in the proof below.

Proof of Theorem 3.5
Since D is a one-dimensional interval, H 2 (D) is embedded in C(D), which implies (1.3a) as well as (1.3b) by [11,Exercise 10.26]. It follows from [3,8] that the unique minimizer χ u ∈ H 1 B ( (u)) of G on H 1 B ( (u)) belongs to H 2 ( (u)). Thanks to (1.3b) and [11,Theorem 10.29], see also [15, II.Theorem 3.1], there is a sequence (χ u, j ) j≥1 in C ∞ (u) such that lim j→∞ χ u, j − χ u H 2 ( (u)) = 0. (3.23) Now, for j ≥ 1 and φ ∈ C 1 B (u) , we infer from [7, Folgerung 7.5] and the regularity of χ u, j , φ, and h that Gauß' theorem can be applied in each connected component of (u), as there are at most two singular points. Therefore, we obtain since φ vanishes on ∂ (u) \ (u). Now, thanks to (3.23), it is straightforward to pass to the limit j → ∞ in the two integrals over (u). Moreover, by Lemma 2. Thanks to the above analysis, the identity (3.21) holds true for any test function φ ∈ C 1
The analysis of the complete MEMS model coupling (1.2) to an equation for u is performed in a forthcoming research [8] for n = 1.