Energy minimizers for an asymptotic MEMS model with heterogeneous dielectric properties

A model for a MEMS device, consisting of a fixed bottom plate and an elastic plate, is studied. It was derived in a previous work as a reinforced limit when the thickness of the insulating layer covering the bottom plate tends to zero. This asymptotic model inherits the dielectric properties of the insulating layer. It involves the electrostatic potential in the device and the deformation of the elastic plate defining the geometry of the device. The electrostatic potential is given by an elliptic equation with mixed boundary conditions in the possibly non-Lipschitz region between the two plates. The deformation of the elastic plate is supposed to be a critical point of an energy functional which, in turn, depends on the electrostatic potential due to the force exerted by the latter on the elastic plate. The energy functional is shown to have a minimizer giving the geometry of the device. Moreover, the corresponding Euler-Lagrange equation is computed and the maximal regularity of the electrostatic potential is established.


Introduction
The modeling and analysis of microelectromechanical systems (MEMS) has attracted a lot of interest in recent years, see, e.g., [10,11,21,22,30,31,35] and the references therein. Idealized devices often consist of a rigid dielectric ground plate above which an elastic dielectric plate is suspended. Applying a voltage difference between the two plates induces a competition between attractive electrostatic Coulomb forces and restoring mechanical forces, the latter resulting from the elasticity of the upper plate. When electrostatic forces dominate mechanical forces, the two plates may come into contact, a phenomenon usually referred to as pull-in instability or touchdown. From a mathematical point of view, this phenomenon may be accounted for in different ways. In fact, in most mathematical models considered so far in the MEMS literature, the pull-in instability is revealed as a singularity in the corresponding mathematical equations which coincides with a breakdown of the model, see [10,21,31] and the references therein. There is a close connection between the singular character of the touchdown and the fact that the modeling does not account for the thickness of the plates. Indeed, coating the ground plate with a thin insulating layer prevents a direct contact of the plates, so that a touchdown of the elastic plate on the insulating layer does not interrupt the operation of the device [6,17,24,25]. Due to the presence of this layer, the MEMS device features heterogeneous dielectric properties (with a jump of the permittivity at the interface separating the coated ground plate and the free space beneath the elastic plate) and the electrostatic potential solves a free boundary transmission problem in the non-smooth domain enclosed between the two plates [17]. The shape of the domain itself is given by a partial differential equation governing the deflection of the elastic plate from rest, which, in turn, involves the electrostatic force exerted on the latter. The mathematical treatment of such a model is rather complex, see [17,Section 5] and [18]. It is thus desirable to derive simpler and more tractable models. As the modeling involves two small spatial scales -the aspect ratio ε of the device and the thickness d of the insulating layer -a variety of reduced models may be obtained. For instance, the assumption of a vanishing aspect ratio of the device, when either the ratio d/ε has a positive finite limit [2,6,20,24,25] or converges to zero, see [10,30,31] and the references therein, leads to a model which no longer involves a free boundary. Indeed, in that case, the electrostatic potential can be computed explicitly in terms of the deflection of the elastic plate and the model reduces to a single equation for the deflection, with the drawback that some important information on the electrostatic potential may thus be lost.
For this reason an intermediate model is derived in [16] by letting only the thickness of the insulating layer d go to zero (keeping the aspect ratio of the device of order one). Assuming an appropriate scaling of the dielectric permittivity in dependence on the layer's thickness (in order to keep relevant information of the dielectric heterogeneity of the device) and using a Gamma convergence approach, the resulting energy, which is the building block of the model, is computed. The next step is the mathematical analysis of the thus derived model, in which stationary solutions correspond to critical points of the energy, while the dynamics is described by the gradient flow associated with the energy. The aim of the present work is to show the existence of a particular class of stationary solutions, which are additionally energy minimizers, and to identify the corresponding Euler-Lagrange equations.
Let us provide beforehand a more precise description of the MEMS configuration under study. We consider an idealized MEMS device composed of two rectangular two-dimensional dielectric plates: a fixed ground plate above which an elastic plate, with the same shape at rest, is suspended and clamped in only one direction while free in the other. We assume that the device is homogeneous in the free direction and that it is thus sufficient to consider only a cross-section of the device orthogonal to the free direction. The shape of the ground plate and that of the elastic plate at rest are then represented by D := (−L, L) ⊂ R, the ground plate being located at z = −H with H > 0 and covered with an infinitesimally thin dielectric layer (in consistency with the aforementioned limit). The vertical deflection of the elastic plate from its rest position at z x ∈ D, u(x) > −H} = D \ C(u) × {−H} be the part of the ground plate which is not in contact with the elastic plate. A touchdown of the elastic plate on the ground plate corresponds to a non-empty coincidence set, in which case Σ(u) is a strict subset of D × {−H}. Note that the free space Ω(u) then has a different geometry with at least two connected components, which may not be Lipschitz domains due to cusps (independent of the smoothness of the function u). In Figure 1 the different situations with empty and non-empty coincidence sets are depicted. As already mentioned, the building block of the model studied in this paper is the total energy E(u) of the device at a state u given by E(u) := E m (u) + E e (u) and derived in [16] in the limit of an infinitesimally small insulating layer. It consists of the mechanical energy E m (u) and the electrostatic energy E e (u). The former is given by with β > 0 and τ, α ≥ 0, taking into account bending and external-and self-stretching effects of the elastic plate. The electrostatic energy is where ψ u is the electrostatic potential in the device and solves the elliptic equation with mixed boundary conditions ∆ψ u = 0 in Ω(u) , (1.3a) The main results of this work are the existence of at least one minimizer of the total energy E and the derivation of the corresponding Euler-Lagrange equation. This requires, of course, first to study the well-posedness of the elliptic problem (1.3) subject to its mixed boundary conditions. A first step in that direction is to guarantee that the electrostatic energy E e is well-defined, which turns out to require some care. Indeed, it should be pointed out that Ω(u) is a non-smooth domain with corners and possibly features turning points, for instance when C(u) includes an interval, see Figure 1. Thus, Ω(u) might consist of several components no longer having a Lipschitz boundary, so that traces have first to be given a meaning. Once this matter is settled, the existence of a variational solution ψ u to (1.3) readily follows from the Lax-Milgram Theorem and the electrostatic energy is then well-defined. This paves the way to the proof of the existence of minimizers of the total energy by the direct method of calculus of variations but does not yet allow us to conclude. Indeed, since E involves two contributions with opposite signs, it might be unbounded from below. We overcome this difficulty by adding a penalization term to the total energy. This additional term can be removed afterwards, thanks to an a priori upper bound on the minimizers which follows from the corresponding Euler-Lagrange equation. However, it turns out that the derivation of the latter requires additional regularity of the electrostatic potential ψ u . Such a regularity is actually not obvious, as the highest expected smoothness of the boundary of Ω(u) is Lipschitz regularity (when the coincidence set C(u) is empty). Consequently, one needs to establish sufficient regularity for ψ u both for states u with empty and with non-empty coincidence sets C(u). In particular, this will ensure a well-defined normal trace of the gradient of ψ u on Σ(u) as required by (1.3c) and on the part of G(u) lying above Σ(u) as required by (2.6a) below. The above mentioned difficulties are actually not the only ones that we face in the forthcoming analysis. To name but a few, the electrostatic energy E e (u) features a nonlocal and intricate dependence upon the state u and appropriate continuity properties are needed in the minimizing procedure. This requires a thorough understanding of the dependence of ψ u on the state u, this dependence being due to the domain Ω(u) as well as the functions h u and h u . Also, due to the prescribed constraint u ≥ −H, the Euler-Lagrange equation solved by minimizers is in fact a variational inequality.

Main Results
Throughout this work we shall assume that As for the functions h u and h u appearing in (1.3) we shall assume in the following that (2.1c) Assumption (2.1c) allows us later to rewrite (1.3) as an elliptic equation with homogeneous boundary conditions. In the following, we shall use the notation introduced in (1.4).
A simple example of boundary functions (h, h) satisfying (2.1b) and (2.1c) may be derived from [17,Example 5.5] with the scaling from [16]: and h ≡ 0. Then (h, h) clearly satisfies (2.1b) and (2.1c), the former being a consequence of the regularity (2.1a) of σ. Note that h u (x, u(x)) = V, x ∈ D, for a given state u; that is, in this example the electrostatic potential is kept constant to the value V along the elastic plate, see (1.3b).
2.1. The Electrostatic Potential. We first turn to the existence of an electrostatic potential for a given state u. To have an appropriate functional setting for u we introducē and point out that C(u) = ∅ if and only if u belongs to the interior ofS; that is, u ∈ S, where Regarding the well-posedness of (1.3) we shall prove the following result.
Theorem 2.2. Suppose (2.1). For each u ∈S there exists a unique strong solution ψ u ∈ H 2 (Ω(u)) to (1.3). Moreover, given κ > 0 and r ∈ [2, ∞), there are c(κ) > 0 and c(r, κ) > 0 such that , taking into account the clamped boundary conditions (1.1). We shall now focus on the existence of energy minimizers onS 0 . We have already observed that the total energy E is the sum of two terms E m and E e with different signs. Hence, the coercivity of E is not obvious. However, if α > 0, the first order term in the mechanical energy E m is quartic and thus dominates the negative contribution coming from the electrostatic energy E e . This property allows us to follow the lines of [17,Section 5] to derive the coercivity of E based on the following growth assumption for h: there is a constant K > 0 such that This approach no longer works if α = 0 and the coercivity of E is not granted. To remedy this drawback, we shall use a regularized energy functional (see (6.1) below), which includes a penalization term ensuring its coercivity if, in addition to (2.3), we assume that and We complete the analysis when α = 0 by showing that minimizers of the regularized energy functional for a suitable choice of the penalization parameter give rise to a minimizer of E, establishing indirectly that E is bounded from below in that case as well. Consequently, in both cases we can prove the existence of at least one energy minimizer as stated in the next result.
At this point, no further qualitative information on energy minimizers u * is available, and a particularly interesting question, which is yet left unanswered by our analysis, is whether the coincidence set C(u * ) is empty or not. Another interesting open issue is the uniqueness of minimizers. The proof of Theorem 2.3 is given in Section 6 for α = 0 and in Section 7 for α > 0.

Euler-Lagrange Equation.
We next aim at deriving the Euler-Lagrange equation satisfied by minimizers of the total energy E. Recalling the prescribed constraint u ≥ −H for u ∈S 0 , we are dealing with an obstacle problem and the resulting equation is actually a variational inequality. For the precise statement we introduce, for a given u ∈S, the function g(u) : D → R by setting for x ∈ C(u). In fact, g(u) represents the electrostatic force exerted on the elastic plate and is computed as the differential (in a suitable sense) of the electrostatic energy E e (u) with respect to u. We emphasize here that the regularity properties of ψ u established in Theorem 2.2 are of utmost importance to guarantee that g(u) is well-defined on D \ C(u), since it features the trace of ∂ z ψ u on G(u). With this notation, we are able to identify the variational inequality solved (in a weak sense) by energy minimizers.
Theorem 2.4. Assume (2.1). Assume that u ∈S 0 is a minimizer of E onS 0 . Then g(u) ∈ L 2 (D) and u is an H 2 -weak solution to the variational inequality where ∂IS 0 denotes the subdifferential of the indicator function IS 0 of the closed convex subsetS 0 of H 2 (D); that is, At this point, we do not know whether minimizers of E inS 0 are the only solutions to (2.7), a question closely connected to the uniqueness issue for (2.7). It is, however, expected that the set of solutions to (2.7) exhibits a complex structure. Indeed, in the much simpler situation studied in [20], the minimizer may coexist with other steady states, depending on the boundary values of the electrostatic potential.
The proof of Theorem 2.4 is given in Section 6 for α = 0 and in Section 7 for α > 0. It relies on the computation of the shape derivative of the electrostatic energy E e (u), which is performed in Section 5. The subsequent sections are dedicated to the proofs of the results stated in this section.
Throughout the paper, we impose assumptions (2.1) and set (3.1) Let Ω(M ) := D × (−H, M ). Then Proof. Integrating with respect to y ∈ [−L, L] and taking into account the boundary condition v(±L) = 0, we obtain Hence, by Hölder's inequality we get Using this inequality and the fact that h and its derivatives up to second order are bounded onD× which yields (a). As for (b) we first note that (3.2) and the compact embedding of We shall now prove Theorem 2.2 and thus focus on (1.3), which is more conveniently considered with homogeneous boundary conditions. To this end, we introduce for a given and fixed function v ∈S. Due to assumption (2.1c), problem (1. Hence, the next result can be seen as a reformulation of Theorem 2.2 in terms of χ v . Then there exists a unique strong solution χ v ∈ H 2 (Ω(v)) to (3.7) and there is C(κ) > 0 depending only on σ and κ such that Moreover, for any r ∈ [2, ∞), there is C(κ) > 0 depending only on σ and κ such that The remainder of this section is devoted to the proof of Theorem 3.2.
3.1. Variational Solution to (3.7). We first establish the existence of a variational solution to (3.7). To this end, we introduce for v ∈S the space H 1 B (Ω(v)) as the closure in H 1 (Ω(v)) of the set , and shall then minimize the functional with respect to ϑ ∈ H 1 B (Ω(v)). Let us recall from [16, Lemma 2.2] that the trace ϑ(·, −H) ∈ L 2 (D) is well-defined for ϑ ∈ H 1 B (Ω(v)) (see also Lemma 3.7 below for a complete statement), while Lemma 3.1 ensures that h v ∈ H 1 (Ω(v)) and that h v (·, −H) and h v belong to L 2 (D). Thus, given as the unique minimizer of the functional G(v) on H 1 B (Ω(v)). Moreover, χ v is also the unique minimizer on ϑ∆h v d(x, z) .
Proof. As noted above, G(v) and G D (v) are both well-defined on H 1 B (Ω(v)). Moreover, owing to the Poincaré inequality established in [16,Lemma 2.2], the functional G(v) is coercive on H 1 B (Ω(v)). It thus readily follows from the Lax-Milgram Theorem that there is a unique minimizer ). Since each connected component of Ω(v) has at most two singular points, we infer from [15,Folgerung 7.5] that we may apply Gauß' Theorem on each connected component of Ω(v) and deduce from (2.1c) that ϑ∆h v d(x, z) Consequently, χ v is also the unique minimizer of the functional G D (v) on H 1 B (Ω(v)). For further use we state the following weak maximum principle.
Finally, a similar argument with ϑ * : in Ω(v) and completes the proof.
We now improve the regularity of χ v as stated in Theorem 3.2 and show that χ v belongs to H 2 (Ω(v)). Once this is shown, it then readily follows that χ v is a strong solution to (3.7) (see [16,Theorem 3.5]).
As pointed out previously, for a general v ∈S, the set Ω(v) may consist of several connected components without Lipschitz boundaries when the coincidence set C(v) is non-empty. The global H 2 (Ω(v))-regularity of χ v is thus clearly not obvious. The main idea is to write the open set D \C(v) as a countable union of disjoint open intervals (I j ) j∈J , see [1, IX.Proposition 1.8], and to establish the H 2 -regularity for χ v first locally on This local regularity is performed in Section 3.2. The global H 2 (Ω(v))-regularity is subsequently established in Section 3.3.

Local
We define the open set O I (v) by and split its boundary as these correspond to cuspidal boundary points, see Figure 2. Let f ∈ L 2 (O I (v)) be a fixed function. The aim is to investigate the auxiliary problem We shall show the existence and uniqueness of a variational solution ζ v := ζ I,v ∈ H 1 (O I (v)) to (3.18) and then prove its H 2 -regularity. The main difficulty encountered here is the just mentioned possible lack of Lipschitz regularity of O I (v). Indeed, the trace of functions in H 1 (O I (v)) on ∂O I (v) have no meaning yet in that case, and so (3.18b) and (3.18c) are not well-defined. We shall thus first give a precise meaning to traces for functions in H 1 (O I (v)).
then ζ I,v coincides -at least formally -with the restriction of χ v to I and we shall also deduce Theorem 3.2 from Theorem 3.9. We thus do not impose that v(a) = −H or v(b) = −H in (3.13), so as to be able to handle simultaneously the above mentioned different cases also depicted in Figure 2.
3.2.1. Traces. As already noticed in [27], one can take advantage of the particular geometry of O I (v), which lies between the graphs of two continuous functions, in order to define traces for functions in H 1 (O I (v)) along these graphs. More precisely, one can derive the following result [16, Lemma 2.1]. (a) There is a linear bounded operator There is a linear bounded operator For simplicity, for ϑ ∈ H 1 (O I (v)), we use the notation We next introduce the variational setting associated with (3.18) and define the space   21) and the trace operator is well-defined in L 2 (I) and, thus, so is the functional . We now derive the existence of a unique variational solution to (3.18), or, equivalently, of a unique minimizer of Proof. It readily follows from (2.8), Lemma 3.7, and the Lax-Milgram Theorem that there is a unique variational solution Taking ϑ ≡ 0 in the previous inequality, we deduce from (3.21) and Hölder's and Young's inequalities that . Combining the Poincaré inequality (3.21) and the above inequality completes the proof.
We next investigate the regularity of the variational solution ζ v to (3.18); that is, we establish a local version of Theorem 3.2.
Theorem 3.9. Consider a function v satisfying (3.13) and let κ > 0 be such that v H 2 (I) ≤ κ . (3.26) The variational solution ζ v = ζ I,v ∈ H 1 B (O I (v)) to (3.18) given by Lemma 3.8 belongs to H 2 (O I (v)), and there is C 1 (κ) > 0 depending only on σ and κ such that Moreover, there is C 2 (κ) > 0 depending only on σ and κ such that, for any r ∈ [2, ∞), Several difficulties are encountered in the proof of Theorem 3.9, due to the low regularity of the domain O I (v) which has a Lipschitz boundary if v(a) > −H and v(b) > −H but may have cusps otherwise, see Figure 2, and due to the mixed boundary conditions (3.18b) and (3.18c). As in [12,Section 3.3], to remedy these problems requires to construct suitable approximations of O I (v) and to pay special attention in the derivation of functional inequalities and estimates on the dependence of the constants on v and I. To be more precise, we shall begin with the case where v satisfies v ∈ W 3 ∞ (I) and min an assumption which is obviously stronger than (3.13 in Lemma 3.11. We then shall show that the last two integrals on the right-hand side of (3.30) are controlled by the H 2 -norm of ζ v with a sublinear dependence, a feature which will allow us to derive (3.27) when v satisfies (3.29). To this end, we shall use the embedding of the subspace ) and the continuity of the trace operator from After this preparation, we will be left with relaxing the assumption (3.29) to (3.13) and this will be achieved by an approximation argument, see Section 3.2.5. (3.29). Throughout this section, we assume that v satisfies (3.29) and fix M > 0 such that (3.32) We also denote positive constants depending only on σ by C and (C i ) i≥3 . The dependence upon additional parameters will be indicated explicitly.
We begin with the H 2 -regularity of the variational solution ζ v to (3.18), which follows from the analysis performed in [3][4][5].
Proof. We first recast the boundary value problem (3.18) in the framework of [5]. Owing to (3.29), the boundary of the domain O I (v) includes four W 3 ∞ -smooth edges (Γ i ) 1≤i≤4 given by and four vertices (S i ) 1≤i≤4 We set and note that D Γ = ∅ as required in [5].
Since v ∈ W 3 ∞ (I), the measure ω i of the angle at S i taken towards the interior of O I (v) satisfies For 1 ≤ i ≤ 4, we denote the outward unit normal vector field and the corresponding unit tangent vector field by ν i and τ i , respectively. According to the geometry of O I (v), We also define 34) and note that the measure Ψ i ∈ [0, π] of the angle between µ i and τ i , 1 ≤ i ≤ 4, is given by We also set We finally define the boundary operator Now, on the one hand, the regularity of σ implies that [5, Assumption (1.5)] is satisfied, while [5, Assumption (1.6)] obviously holds since N = ∅. On the other hand, we note that µ 1 (S 1 ) = −µ 2 (S 1 ) and . We then set ε 1 = −1 and ε 4 = 1. We are left with checking [5, Assumptions (2.3)-(2.4)] but this is obvious due to (3.36). We finally observe that for any m ∈ Z. We then infer from [5, Theorem 2.2] that ζ v has no singular part and thus belongs to We now investigate the quantitative dependence of the just established H 2 -regularity of ζ v on v and derive an H 2 -estimate, which is related to the regularity of v. To this end, we need the following identity.
The identity of Lemma 3.11 is reminiscent of [17,Lemma 3.5]. Its proof is rather technical and thus postponed to Appendix B.
The next step of the analysis is to show that the two integrals over I on the right-hand side of the identity stated in Lemma 3.11 can be controlled by the H 2 -norm of ζ v with a mild dependence on v. To this end, we need some auxiliary functional and trace inequalities which are established in Appendix C. With this in hand, we begin with an estimate of the last integral.
In particular, there is C 4 (M ) > 0 such that Proof. To lighten notation, we set O := O I (v) and introduce P : by Lemma 3.10 and σ ∈ C 2 (Ī), the function P belongs to H 1 (O) and satisfies (C.2) by (3.18b) and (3.18c). In addition, we observe that . Moreover, by (2.8) and Lemma 3.8, Collecting the previous estimates, we end up with from which (3.37) follows. We next deduce from (3.37) (with r = 4) and Hölder's inequality that , and the proof is complete.
We are now in a position to derive quantitative estimates in H 2 for ζ v , which only depends on the H 2 -norm of v, even though v is assumed to be more regular.
To complete the proof of Lemma 3.13, we simply notice that (3.18a) ensures that and deduce (3.39b) from (3.39a).
Summarizing, we have established the following result: Proposition 3.14.
Consider v ∈ H 2 (I) satisfying (3.29); that is, v ∈ W 3 ∞ (I) and min [a,b] v > −H , Then the elliptic boundary value problem (3.18) has a unique strong solution ζ v ∈ H 2 (O I (v)) which satisfies We emphasize that, though derived for functions v ∈ H 2 (I) satisfying the additional assumption (3.29), the estimates stated in Proposition 3.14 only depend on the H 2 -norm of v and, neither on its W 2 ∞ -norm, nor on the value of its minimum (provided that it stays above −H). The outcome of Proposition 3.14 is thus likely to extend to any configuration depicted in Figure 2 under the sole assumption (3.13) and this will be shown in the next section by an approximation argument.
3.2.5. H 2 -regularity: Proof of Theorem 3.9. We now prove the H 2 -regularity of ζ v as stated in Theorem 3.9. We thus assume that v satisfies (3.13); that is, x ∈ I , and fix κ > 0 such that v H 2 (I) ≤ κ. Owing to the density of C ∞ ([a, b]) in H 2 (I) and since v satisfies (3.13), we employ classical approximation arguments to construct a sequence (v n ) n≥1 of functions in C ∞ ([a, b]) with the following properties: A first consequence of (3.44a) and the continuous embedding of (3.45) According to (3.13) and (3.44b), the function v n satisfies (3.29) for each n ≥ 1 and, since O I (v) ⊂ O I (v n ), we infer from Proposition 3.14 that the strong solution ζ vn to (3.18) with v n instead of v (and f replaced by its trivial extension to O I (v n )) satisfies ). On the one hand, since both φ and ζ vn belong to Hence, by (3.48), On the other hand, since ζ vn ∈ H 1 B (O I (v n )) and v n ≥ v, it follows from Lemma A.1 and (3.46) that Hence, by (3.45), Combining the previous two limits, we deduce We next recall that ζ vn is the unique solution in H 1 (3.50) and use the convergences (3.48) and (3.49) to pass to the limit n → ∞ and conclude that φ ∈ H 1 B (O I (v)) satisfies the variational formulation of (3.18). Therefore, Lemma 3.8 guarantees that φ = ζ v . We have thus shown that ζ v ∈ H 2 (O I (v)) and it follows from (3.46) and (3.48) that (3.51) A further consequence of (3.20) and (3.48) is that (∂ x ζ vn (·, −H)) n≥1 converges to ∂ x ζ v (·, −H) in L 2 (I, (H + v)dx), which, together with the positivity of H + v in I, implies that (∂ x ζ vn (·, −H)) n≥1 converges to Combining this convergence with (3.46) and using Fatou's lemma to take the limit ε → 0 give Finally, by (3.19) and (3.46), Hence, by (3.48), Moreover, owing to Lemma A.1, (3.46), and the properties ζ vn ∈ H 1 B (O I (v n )) and v n ≥ v, and it follows from (3.45) that Since H + v > 0 in I, we may extract a further subsequence (not relabeled) such that (∂ z ζ vn (·, v n )) n≥1 converges a.e. in I to ∂ z ζ v (·, v). We then use Fatou's lemma to pass to the limit n → ∞ in (3.47) and conclude that thereby completing the proof of Theorem 3.9.

Global
Hence, Ω(v) is the disjoint union of the open domains O I j (v). Now recall from Proposition 3.3 that Furthermore, since ∆h v belongs to L 2 (Ω(v)) by Lemma 3.1, it follows from the definition of H 1 B (Ω(v)) that with constants C 1 (κ) and C 2 (κ) not depending on I j . Therefore, summing with respect to j ∈ J, we conclude that χ v ∈ H 2 (Ω(v)) and satisfies (3.9) and (3.10), since ∆h v L 2 (Ω(v)) ≤ c(κ) by Lemma 3.1. Therefore, as in [16, Theorem 3.5], we may use the version of Gauß' Theorem stated in [15,Folgerung 7.5] in the variational characterization of χ v featuring G(v) to deduce that χ v ∈ H 2 (Ω(v)) is indeed a strong solution to (3.7). This proves Theorem 3.2. Owing to (3.6) and Lemma 3.1, this also entails Theorem 2.2.

Continuity of χ v with Respect to v
In this section we derive continuity properties of χ v and its gradient trace ∂ z χ v (·, v) with respect to v ∈S. The latter will also yield the continuity of the function g defined in (2.6). Throughout this section we denote positive constants depending only on σ by C. The dependence upon additional parameters will be indicated explicitly.
Proof. The proof is very similar to that of [17,Proposition 3.11].
Also, from (4.6) and Lemma 3.1 we deduce that Gathering the outcome of the above analysis gives (4.3).
Moreover, the continuity of the trace from These two properties, along with (3.4) and (3.5), imply that that is, (ϑ n ) n≥1 is a recovery sequence for ϑ and the claim is proved.

4.2.
Continuity of ∂ z χ v (·, v) with Respect to v. Finally, in order to establish the continuity of the function g defined in (2.6) we need also to investigate the continuous dependence of the gradient trace ∂ z χ v (·, v) on v ∈S, the main difficulty arising when C(v) = ∅. In this regard we note: Then where ℓ(v) is given by Proof. Thanks to (4.19) and the continuous embedding of H 2 (D) in L ∞ (D), we may fix M > H (only depending on κ) such that Step 1. We first establish an estimate ensuring that there is no concentration of , so that it follows from the boundary conditions (3.18b) and (3.18c) that for a.a. x ∈ D \ C(v). Thus, for an arbitrary measurable subset E ⊂ D \ C(v), we infer from Hölder's inequality that Clearly, the same proof implies that, for any n ≥ 1 and arbitrary measurable subset Step 2. We next handle the behavior of ∂ z χ v (·, v) where v stays away from −H. To this end, let ε ∈ (0, H/2) and define Λ(ε) : A straightforward consequence of (4.23) and (4.24) is that Therefore, the function X n,ε , given by is well-defined. Let j ∈ J and n ≥ n ε . Since ∂ z χ v and ∂ z χ vn both belong to (3.14), it follows from (3.19), (4.21), and the definition of Λ(ε) that Summing the above inequality over j ∈ J and noticing that by the Cauchy-Schwarz inequality, (4.19), Theorem 3.2, we obtain We now infer from (4.14) and the above inequality that x ∈ Λ(ε) , n ≥ n ε . Using (4.24) and Hölder's and Young's inequalities, we obtain, for j ∈ J, Summing over j ∈ J and using (4.19) and Theorem 3.2 give Owing to (4.26), we may take the limit n → ∞ in the previous inequality and obtain lim sup Since Λ(ε) ⊂ Λ(δ) for all δ ∈ (0, ε), we infer from the above inequality that lim sup and we may pass to the limit δ → 0 to conclude that lim n→∞ Y n L 1 (Λ(ε)) = 0, ε ∈ (0, H/2). (4.27) Step 3. Finally, we infer from (4.19), (4.21), (4.22), and Theorem 3.2 that for n ≥ n ε by (4.23) and (4.24), we further obtain We now first let n → ∞ with the help of (4.27) and then take the limit ε → 0 to conclude that Finally, given r ∈ [1, ∞), we infer from Hölder's inequality, Lemma 3.1, (3.10), and (4.19) that and the assertion follows from (4.28).
Summarizing the outcome of this section, we have obtained continuity properties of the electrostatic energy E e and the function g introduced in (2.6). Proof. Let us first recall that, if (v n ) n≥1 is a sequence inS converging weakly in H 2 (D) to v ∈S, then there is κ > 0 such that (4.12) and (4.19) hold true. Consequently, we infer from Corollary 4.2 that thereby establishing the stated continuity of E e . Next, let v ∈S. Since ∂ x v = 0 a.e. in C(v), it follows from (2.6) and Proposition 4.3 that for x ∈ D. The stated continuity of g then readily follows from Proposition 4.3 and the C 1 -regularity of h and h (see also Lemma 3.1(b)).

Shape Derivative of the Electrostatic Energy
In this section we investigate differentiability properties of the electrostatic energy with respect to u ∈S, where ψ u is the strong solution to (1.3), see Theorem 2.2. Owing to the dependence of ψ u on the domain Ω(u) this resembles the computation of a shape derivative, a topic which has received considerable attention in recent years, see [8,14,32] and the references therein. Note that we may write alternatively E e (u) = −G(u)[ψ u − h u ], since χ u = ψ u − h u is the strong solution to (3.7) (with v = u) given by Theorem 3.2.
As might be expected, the switch between boundary conditions for ψ u when C(u) = ∅ generates additional difficulties and we begin with the differentiability of ψ u with respect to u ∈ S.
Lemma 5.1. Let u ∈ S be fixed and define, for v ∈ S, the transformation Θ v : Ω(u) → Ω(v) by Then there exists a neighborhood U of u in S such that the mapping (Ω(v)) solves (3.7), see Theorem 3.2, and S is endowed with the H 2 (D)-topology.
Proof. The proof follows the lines of [14,Theorem 5.3.2], a similar proof is given in [17,Lemma 4.1]. We thus only provide a very brief sketch here. Let u ∈ S and v ∈ S. Setting ξ v := χ v • Θ v and performing a change of variables (x,z) = Θ v (x, z), the weak formulation (3.12) satisfied by χ v (as a critical point of G(v)) can be written in the form for some Fréchet differentiable function (Ω(u))) ′ . One then uses the Implicit Function Theorem to derive that ξ v depends smoothly on v.
As a next step we establish the Fréchet differentiability of E e on the open set S. For u ∈ S recall that g(u) is given by (2.6a) since C(u) = ∅ in this case.
Proposition 5.2. Let S be endowed with the H 2 (D)-topology. Then the electrostatic energy E e : S → R is continuously Fréchet differentiable with for u ∈ S and ϑ ∈ H 2 (D) ∩ H 1 0 (D). Proof. In this proof we shall use the notation from Lemma 5.1. We fix u ∈ S and recall from Lemma 5.1 that the mapping v → ξ v = χ v • Θ v is continuously differentiable with respect to v in a neighborhood U of u in S and takes values in We introduce the functions Then, recalling that h and h are C 1 -functions in all their arguments by (2.1b), we conclude that the Fréchet derivative of E e at u applied to ϑ ∈ H 2 (D) ∩ H 1 0 (D) is given by Using J u = 1, j(u) = ∇χ u + ∇h u = ∇ψ u in Ω(u), and m(u) = ψ u − h u in D, we see that Since Using that Θ u is the identity on Ω(u), DΘ u = id, and that ξ u = χ u , we compute from the definition of j(v) that and in Ω(u) . Moreover, The above three identities yield Next we shall simplify the right-hand side of (5.3). Using Gauß' Theorem, the fact that ψ u is a strong solution to (1.3a), ϑ = 0 on ∂D, and the fact that ∂ v ξ v [ϑ]| v=u belongs to H 1 B (Ω(u)), the first integral on the right-hand side of (5.3) can be rewritten in the form Since, due to (1.3c), On account of (∂ z h) u = ∂ z ψ u − ∂ z χ u in Ω(u), the second integral on the right-hand side of (5.3) can be written as we obtain We write the second integral in (5.5) in the form and use integration by parts and (5.6) to get Therefore, we deduce from (5.5) that

Combining this identity with (5.3) and (5.4) yields
and hence, for x ∈ D, Inserting this identity into (5.7) gives according to (2.6a). Finally, the continuity of ∂ u E e : S → L H 2 (D) ∩ H 1 0 (D), R readily follows from Theorem 4.4.
We finally provide the differentiability property of E e on the closed setS. More precisely, we show that E e admits a directional derivative at a point u ∈S in any direction of −u + S, which is given by g(u) defined in (2.6). Recall that C(u) may be non-empty in this case.
Since u s ∈ S for s ∈ (0, 1), we obtain from Proposition 5.2 that d ds E e (u s ) = D g(u s )(w − u) dx for s ∈ (0, 1). Therefore, letting s → 0, we derive with the help of Theorem 4.4 that Now, Theorem 4.4 guarantees that E e (u s ) → E e (u) as s → 0, so that and we conclude from (5.8) that as claimed.
6. Proofs of Theorem 2.3 and Theorem 2.4 for α = 0 In this section we deal with the case α = 0 and recall that the total energy is then given by and electrostatic energy 6.1. Existence of a Minimizer of a Regularized Energy. As already noted in [17], the boundedness from below of the functional E is a priori unclear since α = 0. To cope with this issue, we work with the regularized functional given by for some constant c(k) > 0.
Proof. By (2.3), (2.8), and Proposition 3.3, and we further obtain with the help of Young's inequality that Using this estimate in the definition of E k (u) along with thereby completing the proof.
Due to the weak lower semicontinuity of E m in H 2 (D) and the continuity of E e with respect to the weak topology of H 2 (D) (see Theorem 4.4), Lemma 6.1 allows us to apply the direct method of the calculus of variations to derive the existence of a minimizer of E k inS 0 . Corollary 6.2. For each k ≥ H, the functional E k has at least one minimizer u k ∈S 0 ; that is, where ∂IS 0 is the subdifferential of the indicator function IS 0 of the closed convex subsetS 0 of H 2 (D); that is, for all w ∈S 0 .
Proof. Let k ≥ H be fixed. Consider a minimizer u ∈S 0 of E k onS 0 and fix w ∈ S 0 :=S 0 ∩ S. Owing to the convexity ofS 0 , the function u + s(w − u) = (1 − s)u + sw belongs to S 0 for all s ∈ (0, 1] and the minimizing property of u guarantees that Since u ∈S 0 ⊂S and w ∈ S 0 ⊂ S, Proposition 5.3 implies that for all w ∈ S 0 . Since S 0 is dense inS 0 and (u, g(u)) belongs to H 2 (D) × L 2 (D), this inequality also holds for any w ∈S 0 .
Step 1: Thanks to (2.3b) and (2.4a), we infer from Lemma 3.4 that |ψ u | ≤ K in Ω(u). Combining this bound with (2.3), (2.4), (2.6), and (2.8) readily gives (6.4) Step 2: Consider first j ∈ J with j ≥ 2 and let θ ∈ D(I j ). Since u > −H in the support of θ, the function u ± δθ belongs to S 0 for δ > 0 small enough. We thus infer from (6.3b) that Consequently, using the function S I j defined in Proposition D.1, we realize that u − S I j ∈ H 2 (I j ) is a weak solution to the boundary value problem the boundary conditions (6.5b) being a consequence of the definition of I j , j ≥ 2, the H 2 (D)-regularity of u, and the constraint u ≥ −H. Taking into account that g(u) + A(u − k) + ∈ L 2 (I j ) by Theorem 4.4, classical elliptic regularity theory implies that u − S I j ∈ H 4 (I j ) is a strong solution to (6.5). Since the right hand side of (6.5a) is non-positive due to (6.4), it now follows from a version of Boggio's comparison principle [7,13,19,29] that u − S I j < 0 in I j , so that u(x) ≤ κ 0 for x ∈Ī j and j ≥ 2 by Proposition D.1.
Step 3: We next handle the case j = 0 in which I 0 = (−L, a 0 ). We first argue as in the previous step to conclude that for all θ ∈ D(I 0 ) and that u(−L) = ∂ x u(−L) = u(a 0 ) + H = ∂ x u(a 0 ) = 0. Consequently, we infer from (6.6) and Proposition D.1 that u − S I 0 ∈ H 2 (I 0 ) is a weak solution to the boundary value problem We then argue as in Step 2 to establish that u − S I 0 < 0 in I 0 = (−L, a 0 ). Hence, u ≤ κ 0 in [−L, a 0 ] by Proposition D.1.
Step 4: For the case I 1 we proceed as in Step 3 using Proposition D.1 to deduce that u ≤ κ 0 in [b 0 , L]. This completes the proof.
6.3. Proof of Theorem 2.3 for α = 0. Let k ≥ H and consider a minimizer u k ∈S 0 of the functional E k onS 0 as provided by Corollary 6.2. Then, −H ≤ u k ≤ κ 0 in D according to Proposition 6.4. Therefore, if k ≥ κ 0 , then Now, it follows from Lemma 6.1 and the fact that 0 ∈S 0 that, for k ≥ κ 0 , Therefore, (u k ) k≥κ 0 is bounded in H 2 (D) and there is a subsequence of (u k ) k≥κ 0 (not relabeled) which converges weakly in H 2 (D) and strongly in H 1 (D) towards some u * ∈S 0 . Due to the weak lower semicontinuity of E m in H 2 (D) and the continuity of E e with respect to the weak topology of H 2 (D) (see Theorem 4.4), we readily infer from (6.7) that E(u * ) ≤ E(v) , v ∈S 0 , after taking into account that Consequently, u * ∈S 0 is a minimizer of E onS 0 . This proves Theorem 2.3.
6.4. Proof of Theorem 2.4 for α = 0. Let u ∈S 0 be any minimizer of E onS 0 . Proceeding as in the proof of Proposition 6.3, this implies that u ∈S 0 is an H 2 -weak solution to the variational inequality Consider now α > 0. In that case, the total energy is given by and electrostatic energy Observe that, since α > 0, the mechanical energy E m features a super-quadratic term in ∂ x u L 2 (D) which has the following far-reaching consequence.
Proof. As in the proof of Lemma 6.1, we deduce from (2.3), (2.8), and Proposition 3.3 that Therefore, since u L 2 (D) ≤ 2|D| ∂ x u L 2 (D) by Poincaré's inequality, it follows fromYoung's inequality that − c , and the proof is complete. Once Lemma 7.1 is established, the existence of a minimizer of E onS 0 follows from the weak lower semicontinuity of E m in H 2 (D) and the continuity of E e with respect to the weak topology of H 2 (D) (see Corollary 4.2) by the direct method of the calculus of variations, hence Theorem 2.3 for α > 0 (see also [17,Theorem 5.1]). As for the proof of Theorem 2.4 for α > 0, it is the same as that for α = 0, see Section 6.4.

Appendix A. A technical lemma
Lemma A.1. Let I and J be two bounded intervals in R, and let U be a bounded open subset of I × J. Consider ϑ ∈ H 1 (U ) and functions v ∈ C(Ī), w ∈ C(Ī), and ρ ∈ C(Ī), ρ ≥ 0, such that (a) x → ϑ(x, v(x)) and x → ϑ(x, w(x)) are well-defined and belong to L 2 (I, ρ dx); Proof. Owing to (b) we have, for a.a. x ∈ I, Integrating with respect to x ∈ I after multiplication by ρ(x) and using Hölder's inequality give and the proof is complete.

Appendix C. Some Functional Inequalities
Let I = (a, b) ⊂ D be an open interval and consider v ∈ W 3 ∞ (I) such that min [a,b] We derive in this section functional inequalities for functions in the subspace We begin with Poincaré and Sobolev inequalities and pay special attention to the dependence of the constants on v.
Since O I (v) is a two-dimensional domain, a classical consequence of Lemma C.1 is the continuous embedding of H 1 W S (O I (v)) in L r (O I (v)) for r ∈ [1, ∞). We stress here once more that our main concern is the precise dependence of the embedding constant on v. Proof. Step 1. Assume first that r ≥ 4. For n ≥ 1, we define the truncation T n by T n (s) := s for s ∈ [−n, n] and T n (s) := n sign(s) for s ∈ (−∞, −n) ∪ (n, ∞). Since T n is a Lipschitz continuous function on R with |T ′ n | ≤ 1 and vanishes at zero, the function T n (P ) r/2 also belongs to H 1 W S (O I (v)). We then infer from Lemma C.1, the bound |T ′ n | ≤ 1, and Hölder's inequality that Since the right-hand side of the above inequality does not depend on n ≥ 1, we may take the limit n → ∞ and deduce from Fatou's lemma that P ∈ L r (O I (v)) and satisfies the stated bound for r ≥ 4.
Case 3: −L < a < b = L. We set P (y) := S I (a + y(L − a)) − Q(1 − y) for y ∈ [0, 1] and proceed as in the previous case to derive the same bound for S I L∞(I) .
We then argue as in Case 1 to conclude that 0 ≤ S I ≤ 16L 4 G 0 /β in [−L, L].