H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2$$\end{document}-regularity for a two-dimensional transmission problem with geometric constraint

The H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2$$\end{document}-regularity of variational solutions to a two-dimensional transmission problem with geometric constraint is investigated, in particular when part of the interface becomes part of the outer boundary of the domain due to the saturation of the geometric constraint. In such a situation, the domain includes some non-Lipschitz subdomains with cusp points, but it is shown that this feature does not lead to a regularity breakdown. Moreover, continuous dependence of the solutions with respect to the domain is established.


Introduction
The H 2 -regularity of variational solutions to a two-dimensional transmission problem with geometric constraint is investigated, in particular when part of the interface becomes part of the outer boundary of the domain due to the geometric constraint, a situation in which the domain includes some non-Lipschitz subdomains with cusp points. Such a regularity is required in particular to guarantee that the variational solutions satisfy the strong formulation of the transmission problem. H 2 -regularity is, however, not true in general and known to depend heavily on the geometry and smoothness of the domain and the interfaces. In An extended version of this manuscript with the same title is available at https://arxiv.org/pdf/2103.07301. pdf. Some proofs being similar to [7] are only sketched herein but detailed proofs are supplied in the extended version for the sake of completeness. Geometry of (w) for a state w ∈S with non-empty coincidence set fact, when interfaces intersect the outer boundary of the domain, regularity of variational solutions to transmission problems in non-smooth domains is a challenging issue, even for transversal intersections, see [1-3, 5, 10, 11, 13] and the references therein. Motivated by the mathematical study of microelectromechanical systems (MEMS), we identify herein a class of two-dimensional domains possibly featuring cusps for which H 2 -regularity is true. We actually derive H 2 -estimates which hold uniformly with respect to suitable perturbations of the underlying domain. We point out that such quantitative estimates are not contained in the above mentioned literature, but they turn out to be instrumental for a thorough study of MEMS models [9].
To set up the geometric framework, let D := (−L, L) be a finite interval of R, L > 0, and let H > 0 and d > 0 be two positive parameters. Given a function u ∈ C(D, [−H , ∞)) with u(±L) = 0, we define the subdomain (u) of D × (−H , ∞) by for some positive constants σ 1 = σ 2 , and n (u) denotes the unit normal vector field to (u) (pointing into 2 (u)) given by In (1.2c), h u is a suitable function reflecting the boundary behavior of ψ u , see Section 2 for details. In addition, · denotes the (possible) jump across the interface (u); that is, whenever meaningful for a function f : (u) → R. Let us already mention that there are several features of the specific geometry of (u) which may hinder the H 2 -regularity of the solution ψ u to (1.2). Indeed, on the one hand, the interface (u) always intersects with the boundary ∂ (u) of (u) and it follows from [10] that this sole property prevents the H 2 -regularity of ψ u , unless σ and the angles between (u) and ∂ (u) at the intersection points satisfy some additional conditions. On the other hand, (u) and 2 (u) are at best Lipschitz domains, while 1 (u) may consist of non-Lipschitz domains with cusp points.
The particular geometry (u) = 1 (u) ∪ 2 (u) ∪ (u), in which the boundary value problem (1.2) is set, is encountered in the investigation of an idealized electrostatically actuated MEMS as already pointed out and described in detail in [6,14]. Such a device consists of an elastic plate of thickness d which is fixed at its boundary {±L} × (0, d) and suspended above a rigid conducting ground plate located at z = −H . The elastic plate is made up of a dielectric material and deformed by a Coulomb force induced by holding the ground plate and the top of the elastic plate at different electrostatic potentials. In this context, u represents the vertical deflection of the bottom of the elastic plate, so that the elastic plate is given by 2 (u), while 1 (u) denotes the free space between the elastic plate and the ground plate. An important feature of the model is that the elastic plate cannot penetrate the ground plate, resulting in the geometric constraint u ≥ −H . Still, a contact between the elastic plate and the ground plate -corresponding to a non-empty coincidence set C(u)is explicitly allowed. The dielectric properties of 1 (u) and 2 (u) are characterized by positive constants σ 1 and σ 2 , respectively. The electrostatic potential ψ u is then supposed to satisfy (1.2) and is completely determined by the deflection u. The state of the MEMS device is thus described by the deflection u, and equilibrium configurations of the device are obtained as critical points of the total energy which is the sum of the mechanical and electrostatic energies, the former being a functional of u while the latter is the Dirichlet integral of ψ u . Owing to the nonlocal dependence of ψ u on u, minimizing the total energy and deriving the associated Euler-Lagrange equation demand quite precise information on the regularity of the electrostatic potential ψ u for an arbitrary, but fixed function u and its continuous dependence thereon. This first step of provisioning the required information is the main purpose of the present research. In the companion paper [9], we use the results obtained herein to analyze the minimizing problem leading to the determination of u and compute the associated Euler-Lagrange equation.
Since the regularity of the variational solution ψ u to (1.2) is intimately connected with the regularity of the boundaries of (u), 1 (u), and 2 (u), let us first mention that (u) and 2 (u) are always Lipschitz domains and that the measures of the angles at their vertices do not exceed π, a feature which complies with the H 2 -regularity of ψ u away from the interface (u) [4]. This property is shared by 1 (u) when the coincidence set C(u) is empty, see Fig. 1, so that it is expected that ψ| i (u) belongs to H 2 ( i (u)), i = 1, 2, in that case. However, when C(u) is non-empty, the open set 1 (u) is no longer connected and the boundary of its connected components is no longer Lipschitz, but features cusp points. Moreover, there is an interplay between the transmission conditions (1.2b) and the boundary condition (1.2c) when C(u) = ∅. Whether ψ| i (u) still belongs to H 2 ( i (u)), i = 1, 2, in this situation is thus an interesting question, that we answer positively in our first result. For the precise statement, we introduce the functional setting we shall work with in the sequel. Specifically, we set . Clearly, the coincidence set C(u) is empty if and only if u ∈ S. In addition, the situation already alluded to, where C(u) is non-empty and 1 (u) is a disconnected open set in R 2 with a non-Lipschitz boundary, corresponds to functions u ∈S\S. Also, we include the constraint ± σ ∂ x u(±L) ≤ 0 in the definition of S andS to guarantee that the way (u) and ∂ (u) intersect does not prevent the H 2 -regularity of ψ u in smooth situations (i.e. u ∈ S ∩W 2 ∞ (D)), see [10].
(a) For each u ∈S, there is a unique variational solution ψ u ∈ h u + H 1 0 ( (u)) to (1.2). Moreover, ψ u,1 := ψ u | 1 (u) ∈ H 2 ( 1 (u)) and ψ u,2 := ψ u | 2 (u) ∈ H 2 ( 2 (u)), and ψ u is a strong solution to the transmission problem (1.2). (b) Given κ > 0, there is c(κ) > 0 such that, for every u ∈S satisfying u H 2 (D) ≤ κ, It is worth emphasizing that, for i ∈ {1, 2}, the restriction of ψ u to i (u) belongs to H 2 ( i (u)) for all u ∈S. In particular, there is no regularity breakdown when the coincidence set C(u) is non-empty. Moreover, the H 2 -regularity of ψ u is uniformly valid when u ranges in a bounded subset ofS. A similar observation is made in [7] for a different geometric setting when one of the two subsets does not depend on the function u. Identifying other non-smooth geometries for which H 2 -regularity of the variational solution to a transmission problem depends in a somewhat uniform way on some specific features of the domain is an interesting issue, which is worth a forthcoming investigation.

Remark 1.2 When the upper part 2 (v) is clamped at its lateral boundaries in the sense that
Theorem 1.1 applies whatever the values of σ 1 and σ 2 .
Theorem 1.1 is an immediate consequence of Proposition 4.9 below. Its proof begins with quantitative H 2 -estimates on ψ u depending only on u H 2 (D) for sufficiently smooth functions in S, the H 2 -regularity of ψ u being guaranteed by [10] in that case. Since the class of functions for which these estimates are valid is dense inS, we complete the proof with a compactness argument, the main difficulty to be faced being the dependence of (u) on u. More precisely, we begin with a variational approach to (1.2) and first show in Section 3 by classical arguments that, given u ∈S, the variational solution ψ u to (1.2) corresponds to the minimizer on h u + H 1 0 ( (u)) of the associated Dirichlet energy Thanks to this characterization, we use -convergence tools to show the H 1 -stability of ψ u with respect to u in Sect. 3.2. Section 4 is devoted to the study of the H 2 -regularity of ψ u which we first establish in Sect. 4.1 for smooth functions u ∈ S ∩ W 2 ∞ (D) (thus having an empty coincidence set), relying on the analysis performed in [10]. It is worth mentioning that the constraint involving σ in the definition of S comes into play here. For u ∈ S ∩ W 2 ∞ (D), we next derive quantitative H 2 -estimates on ψ u which only depend on u H 2 (D) as stated in Theorem 1.1 (b), see Sect. 4.2. The building block is an identity in the spirit of [4, Lemma 4.3.1.2] allowing us to interchange derivatives with respect to x and z in some integrals involving second-order derivatives, its proof being provided in Appendix 1. We then combine these estimates with the already proved H 1 -stability of variational solutions to (1.2) and use a compactness argument to extend the H 2 -regularity of ψ u to arbitrary functions u ∈S in Sect. 4.3. In this step, special care is required to cope with the variation of the functional spaces with u. In fact, as a side product of the proof of Theorem 1.1, we obtain qualitative information on the continuous dependence of ψ u with respect to u, which we collect in the next result.
In addition, if i ∈ {1, 2} and U i is an open subset of i (u) such thatŪ i is a compact subset of i (u), then Also, for any p ∈ [1, ∞), Clearly, the quantity M introduced in Theorem 1.3 is finite due to (1.3) and the continuous embedding of H 1 (D) in C(D).

Remark 1.4
An interesting issue is the extension of the above results to a three-dimensional setting, where D is a bounded domain of R 2 instead of an interval. There are, however, at least two difficulties to overcome, which are both of geometric nature. On the one hand, the coincidence set C(u) defined in (1.1) is no longer a countable union of open intervals when D is a two-dimensional domain and it might have a much more complicated structure. The former property plays an essential role in the proof of Proposition 4.9 (a) below. On the other hand, the -convergence argument involved in the proof of Proposition 3.3 strongly makes use of the two-dimensional geometry of (u). In fact, the literature on regularity of solutions to transmission problems in non-smooth three-dimensional domains when the interfaces intersect the outer boundary seems to be rather sparse and restricted to specific geometries. We refer to [1,3,5,11,13] for results in that direction.
Throughout the paper, c and (c k ) k≥1 denote positive constants depending only on L, H , d, σ 1 , and σ 2 . The dependence upon additional parameters will be indicated explicitly.

The boundary values
We state the precise assumptions on the function h v occurring in (1.2c). Roughly speaking, we assume that it is the trace on ∂ (v) of a function h v ∈ H 1 ( (v)) which is such that h| i (v) belongs to H 2 ( i (v)) for i = 1, 2 and satisfies the transmission conditions (1.2b), as well as suitable boundedness and continuity properties with respect to v.
Specifically, for every v ∈S, let and suppose that h v satisfies the transmission conditions Moreover, given v ∈S and a sequence (v n ) n≥1 inS satisfying Observe that the convergence of (v n ) n≥1 , the continuous embedding of From now on, we impose the conditions (2.1) throughout. We finish this short section by providing an example of h v satisfying the imposed conditions (2.1).
Then (2.1a)-(2.1e) are satisfied. In addition, In the context of a MEMS device alluded to in the introduction, these additional properties mean that the ground plate and the top of the elastic plate are kept at constant potential. For instance, ζ(r ) := V min{1, (r − 1) 2 /d 2 } for r > 1 and ζ ≡ 0 on (−∞, 1] will do.

Variational solution to (1.2)
In this section we investigate the properties of the variational solution ψ v to (1.2) for v ∈S and, in particular, its H 1 -stability.

A variational approach to (1.2)
Given v ∈S we introduce the set of admissible potentials The variational solution ψ v to the transmission problem (1.2) is then the minimizer of the functional J (v) on the set A(v): In addition, readily follows from the direct method of calculus of variations due to the lower semicontinuity and coercivity of J (v) on A(v), the latter being ensured by the assumption σ ≥ min{σ 1 , σ 2 } > 0 and Poincaré's inequality. The uniqueness of ψ v is guaranteed by the strict convexity of For further use, we report the following version of Poincaré's inequality for functions in Hence, after integration with respect to (x, z) over (v), from which we deduce the stated inequality.

H 1 -stability of Ã v
The purpose of this section is to study the continuity properties of the solution ψ v to (3.2) with respect to v. More precisely, we aim at establishing the following result.

4)
and set

5)
which is finite by (3.4

) and the continuous embedding of H 1 (D) in C(D). Then
To prove Proposition 3.3, we make use of a -convergence approach and argue as in [7, Section 3.2] with minor changes. We thus omit the proof here and refer to the extended version of this paper [8] for details.

H 2 -regularity
In the previous section we introduced the variational solution In Sect. 4.3 we extend these estimates to the general case v ∈S by means of a compactness argument.

H 2 -regularity for
Assuming that v is smoother with an empty coincidence set, see Fig. 1, the existence of a strong solution ψ v to (1.2) is a consequence of the analysis performed in [10].
and the transmission problem Besides [10], the proof of Proposition 4.1 requires the following auxiliary result.

.2)
Proof We set e x = (1, 0) and e z = (0, 1). Given θ ∈ C ∞ c (v) and j ∈ {x, z} we note that Proof of Proposition 4. 1 We check that the transmission problem (4.1) fits into the framework of [10]. Since v ∈ S ∩ W 2 ∞ (D) and v(±L) = 0, the boundaries of 1 (v) and 2 (v) are W 2 ∞ -smooth curvilinear polygons and the interface (v) meets the boundary ∂ (v) of (v) at the vertices A ± := (±L, 0). Moreover, at the vertex A ± , the measures ω ±,1 and ω ±,2 of the angles between −e z and (1, ∓∂ x v(±L)) and between (1, ∓∂ x v(±L)) and e z , respectively, satisfy ω ±,1 + ω ±,2 = π, as well as by definition of S. According to the analysis performed in [10], these conditions guarantee that the variational solution ψ v to (3.2) provided by Lemma 3.1 satisfies ) for i = 1, 2 and solves the transmission problem (1.2) in a strong sense. Next, owing to the just established H 2 -regularity of ψ v,1 and ψ v,2 , we may differentiate with respect to x the transmission condition ψ v (x, v(x)) = 0, x ∈ D, and find that The stated H 1 -regularity of ∂ x ψ v + ∂ x v∂ z ψ v then follows from Lemma 4.2 and the boundedness of ∂ x v and ∂ 2 x v. In the same vein, due to (1.2b), the regularity of v, and the identity

H 2 -Estimates on
The H 2 -regularity of ψ v being guaranteed by Proposition 4.1 for v ∈ S ∩ W 2 ∞ (D), the next step is to show that this property extends to any v ∈S. To this end, we shall now derive quantitative H 2 -estimates on ψ v , paying special attention to their dependence upon the regularity of v. As in [7], it turns out to be more convenient to study a non-homogeneous transmission problem with homogeneous Dirichlet boundary conditions instead of (4.1).
where ψ v ∈ H 1 ( (v)) is the unique solution to (4.1) provided by Proposition 4.1. Since ) for i = 1, 2, we readily infer from (2.1a) and (4.3) that We omit in the following the dependence of χ on v for ease of notation. According to (2.1a), (2.1b), and Proposition 4.1, χ solves the transmission problem For that purpose, we transform (4.5) to a transmission problem on the rectangle R := D × (0, 1 + d). More precisely, we introduce the transformation mapping 1 (v) onto the rectangle R 1 := D × (0, 1), and the transformation mapping 2 (v) onto the rectangle R 2 := D × (1, 1 + d). The interface separating R 1 and R 2 is It is worth pointing out here that Then, (4.4) implies For further use, we also introducê and derive the following fundamental identity for , which provides a connection between some integrals involving products of second-order derivatives of and is in the spirit of [  Consequently, since (∂ x 1 , ∂ x 2 ) lies in H 1 (R 1 ) × H 1 (R 2 ) by (4.8), we may argue as in the proof of Lemma 4.2 and deduce from (4.9) that

Lemma 4.3 Given
Moreover, by (4.8), Similarly, setting we derive from (4.8) that G i := G| R i ∈ H 1 (R i ) for i = 1, 2, while (4.5b), (4.6), (4.7), and (4.8) imply that, for x ∈ D, that is, G = 0 on 0 , and we argue as in the proof of Lemma 4.2 to conclude that In addition, by (4.8), Owing to (4.10), (4.11), and the H 1 -regularity of F and G, we are in a position to apply Lemma A.1 (see Appendix 1) with (4.12) Using the definitions of F and G, the identity (4.12) reads Noticing that the first terms on both sides of the above identity are the same and that the assertion follows, recalling that ∂ xσ = 0 in R 2 . We now translate the outcome of Lemma 4.3 in terms of the solution χ to (4.5).

Lemma 4.5 Let
Proof Let us first recall the regularity of stated in (4.8) which validates the subsequent computations. Using the transformations T 1 and T 2 introduced in (4.6) and (4.7), respectively, we obtain We use Lemma 4.3 to express the first integral on the right-hand side and get We then compute separately the integrals over R i , i = 1, 2, and begin with the contribution of R 1 . We complete the square to get Thanks to the identities and the property ∂ η 1 (±L, η) = 0 for η ∈ (0, 1) stemming from (4.8), we may perform integration by parts in the last two integrals on the right-hand side of the previous identity and obtain Transforming the above identity back to 1 (v) yields (4.14) Next, arguing in a similar way, Transforming this formula back to 2 (v) yields and we deduce from (4.13), (4.14), (4.15), and the above identity that It remains to simplify the last two integrals on the right-hand side of (4.16). To this end, we first recall that the regularity of χ allows us to differentiate with respect to x the transmission condition χ = 0 on (v) to deduce that while the second transmission condition in (4.5b) reads In particular, (4.17) and (4.18) imply that, on (v), Therefore, Hence, Consequently, (4.16) and (4.19) entail as claimed.
In order to estimate the boundary and the transmission terms in Lemma 4.5, we first report the following trace estimates.

Lemma 4.6
Given κ > 0 and α ∈ (0, 1/2], there is c(α, κ) > 0 such that, for any v ∈S satisfying v H 2 (D) ≤ κ and θ ∈ H 1 ( 2 (v)), Proof Let θ ∈ H 1 ( 2 (v)). Using the transformation T 2 defined in (4.7) which maps 2 and so that the continuous embedding of H 2 (D) in W 1 ∞ (D) and the assumed bound on v readily imply that (4.21) By complex interpolation, from which we deduce that Since α > 0, the trace maps H α+1/2 (R 2 ) continuously on H α (D × {1}), and we thus infer from (4.20) and (4.21) that Based on Lemma 4.6 we are in a position to estimate the boundary and transmission terms in the identity provided by Lemma 4.5.

(4.23)
Proof To prove (4.22), let us first note that H ζ −1/2 (D) embeds continuously into L 4 (D). We use the Cauchy-Schwarz inequality and Lemma 4.6 with α = ζ − 1/2 and deduce H 1 ( 2 (v)) . As for (4.23) we obtain analogously and (4.25) At this point, we use (4.17) and (4.18) to show that Consequently, so that Owing to (4.25) and the above inequality, we may then argue as in the proof of (4.24) to conclude that , as claimed in (4.23).
We now gather the previous findings to deduce the following crucial H 2 -estimate on the solution ψ v to (4.1) for v ∈ S ∩ W 2 ∞ (D), which only depends on the H 2 (D)-norm of v (but not on its W 2 ∞ (D)-norm).
There is a constant c 0 (κ) > 0 such that the solution ψ v to (4.1) satisfies and Since σ is constant on 1 (v) and on 2 (v), it readily follows from (4.5a) that we infer from Lemma 4.5 and the above two formulas that Using Lemma 4.7 with ζ = 7/8, along with the identity , we further obtain Hence, thanks to Young's inequality, Recalling that by (4.5) and that min{σ 1 , σ 2 } > 0, we conclude that (4.27) Owing to the continuous embedding of H 2 (D) in C(D), combining (4.27) and Lemma 3.2 leads us to the estimate . The bound (4.26a) then readily follows from the assumptions (2.1a) and (2.1c). Finally, (4.26a), together with (2.1a) and (2.1c), yields (4.26b).  . 3 Geometry of (w) for a state w ∈S with non-empty and disconnected coincidence set regularity of v assumed in the previous sections and also allow for a non-empty coincidence set.

H 2 -regularity and
and is a strong solution to the transmission problem (4.1). Moreover, there is c 1 (κ) > 0 such that The proof involves three steps: we first establish Proposition 4.9 (b) under the additional assumption Building upon this result, we take advantage of the density of S ∩ W 2 ∞ (D) inS and of the estimates derived in Proposition 4.8 to verify Proposition 4.9 (a) by a compactness argument. Combining the previous steps leads us finally to a complete proof of Proposition 4.9 (b). We thus start with the proof of Proposition 4.9 (b) when the solutions (ψ v n ) n≥1 to (4.1) associated with the sequence (v n ) n≥1 satisfies the above additional bound. We state this result as a separate lemma for definiteness.

Lemma 4.10
Let κ > 0 and v ∈S be such that v H 2 (D) ≤ κ and consider a sequence (v n ) n≥1 inS satisfying (4.29). Assume further that, for each n ≥ 1, (ψ v n ,1 , ψ v n ,2 ) belongs to H 2 ( 1 (v n )) × H 2 ( 2 (v n )) and that there is μ > 0 such that The proof is very close to that of [7, Proposition 3.13 & Corollary 3.14], so that we omit the details here and refer to the extended version of this paper [8] instead.

Proof of Proposition 4.9 (a)
Let v ∈S be such that v H 2 (D) ≤ κ. We may choose a sequence Owing to (4.32) and the regularity property v n ∈ S ∩ W 2 ∞ (D), n ≥ 1, Proposition 3.3 guarantees that (ψ v n ,1 , ψ v n ,2 ) belongs to H 2 ( 1 (v n )) × H 2 ( 2 (v n )) and (ψ v n ) n≥1 satisfies (4.30) with μ = c 0 (2κ). We then infer from Lemma 4.10 that (ψ v,1 , ψ v,2 ) belongs to Combining the above bound with (2.1d) and Lemma 4.10 gives (4.28). Checking that ψ v is a strong solution to (4.1) is then done as in [7,Corollary 3.14], see also the extended version of this paper [8] for a complete proof. We supplement the H 2 -weak continuity of ψ v with respect to v reported in Proposition 4.9 with the continuity of the traces of ∇ψ v,2 on the upper and lower boundaries of 2 (v).

Proposition 4.11
Let κ > 0 and v ∈S be such that v H 2 (D) ≤ κ and consider a sequence (v n ) n≥1 inS satisfying (4.29). Then, for p ∈ [1, ∞), ∇ψ v n ,2 (·, v n ) → ∇ψ v,2 (·, v) in L p (D, R 2 ) , (4.33) ∇ψ v n ,2 (·, v n + d) → ∇ψ v,2 (·, v + d) in L p (D, R 2 ) , (4.34) and ∇ψ v,2 (·, v) L p (D,R 2 ) + ∇ψ v,2 (·, v + d) L p (D,R 2 ) ≤ c( p, κ) . Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Next, after integrating by parts, Since V (x, 0) = V (x, 1 + d) = 0 for x ∈ D by (A.1a) and the second and fourth terms cancel each other out, we obtain