Stationary solutions to a nonlocal fourth-order elliptic obstacle problem

Existence of stationary solutions to a nonlocal fourth-order elliptic obstacle problem arising from the modelling of microelectromechanical systems with heterogeneous dielectric properties is shown. The underlying variational structure of the model is exploited to construct these solutions as minimizers of a suitably regularized energy, which allows us to weaken considerably the assumptions on the model used in a previous article.


INTRODUCTION
Idealized electrostatically actuated microelectromechanical systems (MEMS) are made up of an elastic conducting plate which is clamped on its boundary and suspended above a rigid conducting ground plate. Their dynamics results from the competition between mechanical and electrostatic forces in which the elastic plate is deformed by a Coulomb force induced by holding the two plates at different electrostatic potentials. When the electrostatic forces dominate the mechanical ones, the elastic plate comes into contact with the ground plate, thereby generating a short circuit and leading to the occurrence of a touchdown singularity in the related mathematical models, see [5,6,9,12,18,19] and the references therein. However, covering the ground plate with a thin insulating layer prevents a direct contact of the two plates and, from a mathematical point of view, features a constraint of obstacle-type which hinders the touchdown singularity. Different models have been developed to take into account the influence of the coating layer deposited on the ground plate, most of them relying on the so-called small aspect ratio approximation and describing the state of the MEMS device by the sole deformation of the elastic plate [2,3,14,15,20]. A more elaborate model is derived in [13,Section 5], in which the state of the device is not only given by the deformation of the elastic plate, but also by the electrostatic potential in the region between the two plates.
To give a more precise account, we restrict ourselves to a two-dimensional setting, neglecting variations in the transverse horizontal direction, so that the geometry of the device under study herein is the following, see Figure 1. At rest, the cross-section of the elastic plate is D := (−L, L), L > 0, and it is clamped at its boundary (x, z) = (±L, 0). The fixed ground plate has the same shape D and is located at z = −H − d. It is coated with an insulating layer of thickness d > 0 with a priori non-uniform dielectric permittivity σ 1 > 0 and which cannot be penetrated by the elastic plate. As a consequence, the vertical displacement u :D → R of the elastic plate actually ranges in [−H, ∞) and the contact region Geometry of Ω(u) for a state u ∈S 0 with non-empty (and disconnected) coincidence set C(u).
between the insulating layer and the elastic plate might be non-empty. We assume also that the free space between the upper part of the insulating layer and the elastic plate has uniform permittivity σ 2 > 0, and we denote the electrostatic potential in the device According to the model derived in [13,Section 5], equilibrium configurations of the above described device are weak solutions u ∈ H 2 D (D) to the fourth-order obstacle problem β∂ 4 : v(±L) = ∂ x v(±L) = 0 , and ∂IS 0 (u) denotes the subdifferential in H 2 D (D) of the indicator function IS 0 of the closed convex subsetS where ·, · H 2 D denotes the duality pairing between H −2 (D) and H 2 D (D). If v ∈S 0 , then ∂IS 0 (v) := ∅. While ∂IS 0 (u) accounts for the non-penetrability of the insulating layer, the fourth-and second-order terms in (1.1) represent forces due to plate bending and plate stretching, respectively. These forces are balanced by the electrostatic force g(u) acting on the elastic plate, which is derived in [13] and involves the electrostatic potential ψ u in the device. The latter solves the transmission problem in the domain Ω(u), see Figure 1. In (1.2), · denotes the jump across the interface Σ(u), the dielectric permittivity σ is given by and the non-homogeneous Dirichlet boundary conditions h u are given by and (1.3e) We note that (1.3d)-(1.3e) imply that h u satisfies and thus complies with the transmission conditions (1.2b), see [13,Example 5.5] for an example of functions h 1 and h 2 satisfying the above assumptions. With these assumptions, the electrostatic force g(u) is computed in [13, Theorems 1.2 and 1.3]. It has a different expression at contact points between the plates and at points where the elastic plate is strictly above the insulating layer. Specifically, introducing the coincidence set for u ∈S 0 , the electrostatic force is given by (1.5b) and (ψ u,1 , ψ u,2 ) := ψ u 1 Ω 1 , ψ u 1 Ω 2 (u) . It is readily seen from (1.5) that g(u) features a nonlinear and nonlocal dependence on u, the latter being due to the terms involving ψ u in (1.5).
The investigation of the solvability of (1.1)-(1.2) is initiated in [13], exploiting the variational structure underlying the derivation of (1.1)-(1.2) which implies that solutions to (1.1) are critical points inS 0 of an energy functional E, which is actually the total energy of the device. Specifically, (1.6b) and the electrostatic energy Then (1.1) subject to (1.5) is the Euler-Lagrange equation for minimizers of E inS 0 , see [13], and g(u) defined in (1.5) corresponds to the (directional) derivative of E e (u) with respect to u, see [13,Theorem 1.4]. The existence of solutions to (1.1) is established in [13,Section 5] by showing that the energy functional has at least a minimizer onS 0 . This, however, requires additional assumptions ensuring that the electrostatic energy E e (u) does not grow faster than u 2 H 1 (D) as well as the coercivity of the energy functional E. More precisely, to guarantee the former (see (2.2) below) we assume that there are constants m i > 0, i = 1, 2, 3, such that and then there is at least one solution u ∈S 0 to the variational inequality (1.1) in the following sense: Moreover, this solution can be obtained as a minimizer of E onS 0 .
Obviously, a first step towards a full proof of Proposition 1.1 is to solve the transmission problem (1.2) for the electrostatic potential ψ u with sufficient regularity in order to give a meaning to the function g along with deriving suitable continuity properties. We refer to Section 2.1 for a detailed account on this issue (see in particular Lemma 2.1 and Lemma 2.2 where these results are recalled). A second essential step in the proof of Proposition 1.1 consists of deriving the coercivity of the energy functional E onS 0 . This property is ensured by assumption (1.10) (along with (1.7)). In particular, if α > 0, then the mechanical energy E m involves a super-quadratic term which allows a compensation of the negative contribution from the electrostatic energy E e .
so that the formula (1.5) for g(u) simplifies and becomes g(u) = g(u) in (1.5a). In particular, the function g(u) is non-negative.
The aim of the present work is to establish the existence of a solution to (1.1) under considerably weaker assumptions. In particular, we shall get rid of the technical and somewhat artificial assumption (1.10). Since (1.10) is obviously satisfied when α > 0, we shall treat α as zero in the following computations. Moreover, we no longer need a sign for the function g(u) and can slightly weaken assumption (1.9). Indeed, we only require that and that there is a number K > 0 such that Clearly, (1.8)-(1.9) imply (1.11). Also note that, due to (1.11a), the last term in the definition of g(u) in (1.5a) vanishes, i.e. g(u) reduces to With these assumptions we can now formulate the main result of the present paper. Since we no longer impose assumption (1.10) in Theorem 1.3, the boundedness from below of the functional E is a priori unclear, due to the negative contribution from the electrostatic energy E e . We thus shall work with regularized coercive functionals instead (see (2.1) below) and use comparison principle arguments to derive a priori bounds on minimizers of the regularized functionals, see Section 2 below. We shall then prove that cluster points of these minimizers are actually minimizers of the original functional E. The full proof of Theorem 1.3 is given in Section 2 and relies on an idea introduced previously in a related work [10].  3) and (1.7). Indeed, the existence of a minimizer of E onS 0 is shown as in the proof of [13,Theorem 5.1] and this minimizer is a weak solution to (1.1) as a consequence of [13,Theorem 5.3].
Finally, we provide an additional property of weak solutions to (1.1) when the potentials applied on the elastic plate and the ground plate are constant. 2.1. Auxiliary Results. Let us emphasize that the function g(u) defined in (1.5) involves gradient traces of the electrostatic potential ψ u , the latter solving the transmission problem (1.2) posed on the non-smooth domain Ω(u) which possesses corners. In addition, Ω 2 (u) need not be connected, but may consist of several components with non-Lipschitz boundaries (see Figure 1), so that traces have first to be given a meaning. While the existence of a unique variational solution ψ u ∈ h u + H 1 0 (Ω(u)) to (1.2) readily follows from the Lax-Milgram theorem, the required further regularity for ψ u in order to make sense of its gradient traces is thus far from being obvious. Moreover, ψ u (and hence g(u)) depends non-locally on u so that continuity properties with respect to the plate deformation u is non-trivial.
Nevertheless, relying on shape optimization methods and Gamma convergence techniques the following result regarding the existence to a solution of the transmission problem (1.2) and its regularity is shown in [13].
The regularity of ψ u provided by Lemma 2.1 in particular guarantees that g(u) defined in (1.5) is meaningful for u ∈S 0 . As for the continuity of g(u) with respect to u ∈S 0 we recall: where E(u) is defined in (1.6) and the constant A given by with constants m j introduced in (1.7). We shall now prove, for each k > 0, the existence of a minimizer u k of E k onS 0 and subsequently derive an a priori bound on such minimizers, so that the additional regularizing term drops out in E k . We first show the coercivity of the functional E k .
we deduce from Young's inequality Finally, note that . Hence, taking the previous two inequalities into account, the definition of E k (u) entails as claimed.
The just established coercivity now easily yields the existence of a minimizer of E k onS 0 .
Proposition 2.4. For each k ≥ H, the functional E k has at least one minimizer u k ∈S 0 onS 0 ; that is, Moreover, u k ∈S 0 is a weak solution to the variational inequality Clearly, E m defined in (1.6b) is weakly lower semicontinuous on H 2 (D) while E e is continuous with respect to the weak topology of H 2 (D) due to Lemma 2.2. Thanks to Lemma 2.3, the direct method of the calculus of variations now easily yields the existence of a minimizer u k ∈S 0 of E k onS 0 . In particular, From the definition of E k we then obtain by gathering the two limits that for all w ∈ S 0 . Since S 0 is dense inS 0 , this inequality also holds for any w ∈S 0 , and we thus have shown that u k satisfies the variational formulation of (2.4).

A Priori Bounds.
We shall now show that u k is a priori bounded for k large enough (making the additional term in E k superfluous). To this aim we need an a priori bound for the solution to the fourth-order boundary value problem (2.5) subject to suitable Dirichlet boundary conditions as stated below. The bound relies on the maximum principle for the fourth order operator β∂ 4 x − τ ∂ 2 x with clamped boundary conditions [4,8,11,17]. Then, there is κ 0 > 0 depending only on G 0 , β, L, H, and τ (but not on I = (a, b)) such that Proof. This result has already been observed in [10, Lemma A.1] and we include its proof only for the sake of completeness. Note that (2.5) subject to one of the boundary conditions (2.6)-(2.9) indeed admits a unique solution S I .
Case 1: −L < a < b < L. Set P (y) := S I (a + (b − a)y) + H for y ∈ [0, 1] and note that P solves the boundary-value problem Since G 0 ≥ 0 we deduce that P ≥ 0 in (0, 1) from a version of Boggio's comparison principle [4,8,11,17] . Testing (2.10) by P we get we infer from the above inequalities that y ∈ (0, 1) , The arguments of Case 1 give The previous lemma now implies the desired a priori bounds on the minimizers u k . Proof. Let k ≥ H. We first note that, since g(u k ) ≥ 0 in D by (1.5b), it easily follows from (1.11b) and (1.12) that (2.12) Next, since u k ∈ C(D) with u k (±L) = 0, the set {x ∈ D : u k (x) > −H} is a non-empty open subset of D. Owing to [1, IX.Proposition 1.8] we can thus write it as a countable union of open intervals (I j ) j∈J . Consider a fixed j ∈ J and let S I j denote the solution to (2.5) in I j subject to the associated boundary conditions on ∂I j listed in (2.6)-(2.9), which vary according to whether I j ⊂ D or not. Then Lemma 2.5 yields a constant κ 0 > H (independent of j ∈ J) such that Note that, by definition of I j , the function u k restricted to I j satisfies the same boundary conditions on ∂I j as S I j . Hence, for z := u k − S I j ∈ H 2 (I j ) we have z = ∂ x z = 0 on ∂I j . Moreover, if θ ∈ D(I j ), then u k ± δθ ∈S 0 for δ > 0 small enough since u k > −H in the support of θ.
Invoking the weak formulation of (2.4) we derive Thus, we conclude that z = u k − S I j ∈ H 2 (I j ) weakly solves the boundary value problem (2.14b) Now, since g(u k )+A(u k −k) + ∈ L 2 (I j ) by Lemma 2.2 (a), classical elliptic regularity theory [7] entails that z = u k − S I j ∈ H 4 (I j ) is a strong solution to (2.14). Furthermore, it follows from (2.12) and the non-negativity of A(u k − k) + that the right-hand side of (2.14a) is a non-positive function. Boggio's comparison principle [4,8,11,17] then implies that z = u k − S I j < 0 in I j . Consequently, we infer from (2.13) that u k L∞(I j ) ≤ κ 0 . Since κ 0 is independent of j ∈ J and (I j ) j∈J covers {x ∈ D : u k (x) > −H}, the assertion follows.
Thus, (u k ) k≥κ 0 is bounded in H 2 (D) so that there is a subsequence (not relabeled) converging weakly in H 2 (D) and strongly in H 1 (D) towards some u * ∈S 0 . Since E m is weakly lower semicontinuous on H 2 (D) and since E e is continuous with respect to the weak topology of H 2 (D) owing to Lemma 2.2 (b), we obtain from (2.15) that recalling that the continuous embedding of H 1 (D) in C(D) readily implies that Therefore, u * ∈S 0 minimizes E onS 0 . Now [13,Theorem 5.3] entails that u * satisfies the variational inequality (1.1). Alternatively, one can use the weak convergence in H 2 (D) and the strong convergence in H 1 (D) of (u k ) k≥κ 0 to u * to pass to the limit k → ∞ in (2.4), observing that (g(u k )) k≥κ 0 converges to g(u) in L 2 (D) by Lemma 2.2 (b) and that (u k − k) + = 0 for k ≥ κ 0 . This proves Theorem 1.3. (2.16b) Another application of a version of Boggio's comparison principle [4,8,11,17] implies z = u + H ≤ 0 in I. But this contradicts u > −H in I.