Stationary solutions to a nonlocal fourth-order elliptic obstacle problem

Abstract

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.

1 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 Fig. 1. At rest, the cross-section of the elastic plate is $D:=(-L,L)$, $L>0$, and it is clamped at its boundary $(x,z)=(\pm L,0)$. The fixed ground plate has the same shape D and is located at $z=-H-d$. It is coated with an insulating layer

$$\begin{aligned} \Omega _1 := D \times (-H-d,-H) \end{aligned}$$

of thickness $d>0$ with a priori non-uniform dielectric permittivity $\sigma _1>0$ and cannot be penetrated by the elastic plate. As a consequence, the vertical displacement $u:{\bar{D}} \rightarrow {\mathbb {R}}$ of the elastic plate actually ranges in $[-H,\infty )$ and the contact region $\{ (x,-H)\ :\ x\in D, \ u(x)=-H\}$ between the insulating layer and the elastic plate might be non-empty. We assume also that the free space

$$\begin{aligned} \Omega _2(u):=\left\{ (x,z)\in D\times {\mathbb {R}}\,:\, -H< z < u(x)\right\} \end{aligned}$$

between the upper part of the insulating layer and the elastic plate has uniform permittivity $\sigma _2>0$, and we denote the electrostatic potential in the device

$$\begin{aligned} \Omega ({u}):=\left\{ (x,z)\in D\times {\mathbb {R}} \,:\, -H-d< z < u(x)\right\} =\Omega _1\cup \Omega _2( {u})\cup \Sigma (u) \end{aligned}$$

by $\psi _u$, where $\Sigma (u)$ is the interface

$$\begin{aligned} \Sigma (u):=\{(x,-H)\,:\, x\in D,\, u(x)>-H\}. \end{aligned}$$

Fig. 1

Geometry of $\Omega (u)$ for a state $u\in {\bar{S}}_0$ with non-empty (and disconnected) coincidence set ${\mathcal {C}}(u)$

According to the model derived in [13, Section 5], equilibrium configurations of the above described device are weak solutions $u\in H_D^2(D)$ to the fourth-order obstacle problem

$$\begin{aligned} \beta \partial _x^4 u-(\tau +\alpha \Vert \partial _x u\Vert _{L_2(D)}^2)\partial _x^2 u+\partial {\mathbb {I}}_{{\bar{S}}_0}(u) \ni -g(u) \;\;\text { in }\;\; D, \end{aligned}$$

(1.1)

where

$$\begin{aligned} H_D^2(D):= \left\{ v\in H^2(D)\,:\, v(\pm L) = \partial _x v(\pm L)=0 \right\} , \end{aligned}$$

and $\partial {\mathbb {I}}_{{\bar{S}}_0}(u)$ denotes the subdifferential in $H_D^2(D)$ of the indicator function ${\mathbb {I}}_{{\bar{S}}_0}$ of the closed convex subset

$$\begin{aligned} {\bar{S}}_0 := \left\{ v\in H_D^2(D)\,:\, v\ge -H \text { in } D \right\} \end{aligned}$$

of $H_D^2(D)$. We recall that, given $v\in {\bar{S}}_0$, the subdifferential $\partial {\mathbb {I}}_{{\bar{S}}_0}(v)$ is the subset of the dual space

$$\begin{aligned} H^{-2}(D):=\big (H_D^2(D)\big )' \end{aligned}$$

of $H_D^2(D)$ given by

$$\begin{aligned} \partial {\mathbb {I}}_{{\bar{S}}_0}(v) :=\left\{ \xi \in H^{-2}(D)\,:\, \langle \xi , v-w \rangle _{H_D^2} \ge 0, \, w\in {\bar{S}}_0\right\} , \end{aligned}$$

where $\langle \cdot , \cdot \rangle _{H_D^2}$ denotes the duality pairing between $H^{-2}(D)$ and $H_D^2(D)$. If $v\not \in {\bar{S}}_0$, then $\partial {\mathbb {I}}_{{\bar{S}}_0}(v):=\emptyset$. While $\partial {\mathbb {I}}_{{\bar{S}}_0}(u)$ 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 $\psi _u$ in the device. The latter solves the transmission problem

$$\begin{aligned} \mathrm {div}(\sigma \nabla \psi _u)&=0 \quad \text {in }\ \Omega (u), \end{aligned}$$

(1.2a)

$$\begin{aligned} \llbracket \psi _u \rrbracket = \llbracket \sigma \partial _z \psi _u \rrbracket&=0\quad \text {on }\ \Sigma (u), \end{aligned}$$

(1.2b)

$$\begin{aligned} \psi _u&=h_u\quad \text {on }\ \partial \Omega (u), \end{aligned}$$

(1.2c)

in the domain $\Omega (u)$, see Fig. 1. In (1.2), $\llbracket \cdot \rrbracket$ denotes the jump across the interface $\Sigma (u)$, the dielectric permittivity $\sigma$ is given by

$$\begin{aligned} \sigma (x,z) := \left\{ \begin{array}{lcl} \sigma _1(x,z) &{} \text { for } &{} (x,z)\in \Omega _1, \\ &{} &{} \\ \sigma _2 &{} \text { for } &{} (x,z)\in D\times (-H,\infty ), \end{array} \right. \end{aligned}$$

with

$$\begin{aligned} \sigma _1\in C^2\big ( {\overline{\Omega }}_1 \big ), \quad \min _{{\overline{\Omega }}_1} \sigma _1>0 ,\qquad \sigma _2\in (0,\infty ), \end{aligned}$$

(1.3a)

and the non-homogeneous Dirichlet boundary conditions $h_u$ are given by

$$\begin{aligned} h_u(x,z):= h(x,z,u(x)) = \left\{ \begin{array}{ll} h_1(x,z,u(x)), &{} (x,z)\in {\overline{\Omega }}_1, \\ h_2(x,z,u(x)), &{} (x,z)\in \overline{\Omega _2(u)}, \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} h_1: {\bar{D}}\times [-H-d,-H]\times [-H,\infty )\rightarrow [0,\infty ) \end{aligned}$$

(1.3b)

and

$$\begin{aligned} h_2: {\bar{D}}\times [-H,\infty )\times [-H,\infty )\rightarrow [0,\infty ) \end{aligned}$$

(1.3c)

are $C^2$-smooth functions satisfying

$$\begin{aligned} h_1(x,-H,w)&=h_2(x,-H,w),\quad (x,w)\in D\times [-H,\infty ), \end{aligned}$$

(1.3d)

$$\begin{aligned} \sigma _1(x,-H)\partial _z h_1(x,-H,w)&=\sigma _2\partial _z h_2(x,-H,w),\quad (x,w)\in D\times [-H,\infty ). \end{aligned}$$

(1.3e)

We note that (1.3d–1.3e) imply that $h_u$ satisfies

$$\begin{aligned} \llbracket h_u \rrbracket = \llbracket \sigma \partial _z h_u \rrbracket =0\quad \text {on }\ \Sigma (u) \end{aligned}$$

(1.4)

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

$$\begin{aligned} {\mathcal {C}}(u):=\{x\in D\,:\, u(x)=-H\} \end{aligned}$$

for $u\in {\bar{S}}_0$, the electrostatic force is given by

$$\begin{aligned} \begin{aligned} g(u)(x) :=&{\mathfrak {g}}(u)(x) -\frac{\sigma _2}{2} \left[ \big ((\partial _x h_2)_u\big )^2+ \big ((\partial _z h_2)_u+(\partial _w h_2)_u\big )^2 \right] (x, u(x))\\&+ \left[ \sigma _1(\partial _w h_1)_u\,\partial _z\psi _{u,1} \right] (x,-H-d) \end{aligned} \end{aligned}$$

(1.5a)

for $x\in D$, where

$$\begin{aligned}&{\mathfrak {g}}(u)(x)\nonumber \\&\quad :=\left\{ \begin{array}{ll} \displaystyle {\frac{\sigma _2}{2} \big (1+(\partial _x u(x))^2\big )\,\big [\partial _z\psi _{u,2}-(\partial _z h_2)_u-(\partial _w h_2)_u\big ]^2(x, u(x))} , &{} x\in D\setminus {\mathcal {C}}(u),\\ \displaystyle {\frac{\sigma _2}{2}\, \left[ \frac{\sigma _1}{\sigma _2}\partial _z\psi _{u,1}-(\partial _z h_2)_u-(\partial _w h_2)_u \right] ^2(x, -H)}, &{} x\in {\mathcal {C}}(u), \end{array}\right. \end{aligned}$$

(1.5b)

and $(\psi _{u,1},\psi _{u,2}) := \left( \psi _u \mathbf{1 }_{{\overline{\Omega }}_1} , \psi _u \mathbf{1 }_{\overline{\Omega _2(u)}}\right)$. 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 $\psi _u$ 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 in ${\bar{S}}_0$ of an energy functional E, which is actually the total energy of the device. Specifically,

$$\begin{aligned} E(u) := E_m(u) + E_e(u) \end{aligned}$$

(1.6a)

consists of the mechanical energy

$$\begin{aligned} E_m(u):=\frac{\beta }{2} \Vert \partial _x^2u\Vert _{L_2(D)}^2 +\left( \frac{\tau }{2} + \frac{\alpha }{4}\Vert \partial _x u\Vert _{L_2(D)}^2 \right) \Vert \partial _x u\Vert _{L_2(D)}^2 \end{aligned}$$

(1.6b)

and the electrostatic energy

$$\begin{aligned} E_e(u) := -\frac{1}{2} \int _{\Omega (u)} \sigma \vert \nabla \psi _u\vert ^2\,{\mathrm {d}}(x,z). \end{aligned}$$

(1.6c)

Then (1.1) subject to (1.5) is the Euler–Lagrange equation for minimizers of E in ${\bar{S}}_0$, 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 one minimizer on ${\bar{S}}_0$. This, however, requires additional assumptions ensuring that the electrostatic energy $E_e(u)$ does not grow faster than $\Vert u\Vert _{H^1(D)}^2$ 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

$$\begin{aligned} \vert \partial _x h_1(x,z,w)\vert +\vert \partial _z h_1(x,z,w)\vert \le \sqrt{m_1+m_2 w^2}, \quad \vert \partial _w h_1(x,z,w)\vert \le \sqrt{m_3}, \end{aligned}$$

(1.7a)

for $(x,z,w)\in {{\bar{D}}} \times [-H-d,-H] \times [-H,\infty )$ and

$$\begin{aligned} \vert \partial _x h_2(x,z,w)\vert +\vert \partial _z h_2(x,z,w)\vert \le \sqrt{\frac{m_1+m_2 w^2}{H+w}},\quad \vert \partial _w h_2(x,z,w)\vert \le \sqrt{\frac{m_3}{H+w}} \end{aligned}$$

(1.7b)

for $(x,z,w)\in {{\bar{D}}} \times [-H,\infty ) \times [-H,\infty )$. The existence result from [13] then reads:

Proposition 1.1

[13, Theorems 5.1 and 5.3] Let $\beta >0$ and $\tau ,\alpha \ge 0$. Assume that (1.3), (1.7), (1.8), and (1.9) hold, and that the ground plate and the elastic plate are kept at constant, but different, electrostatic potentials; that is, there is $V>0$ such that

$$\begin{aligned} h_1(x,-H-d,w)&=0,\quad (x,w)\in {\bar{D}}\times [-H,\infty ),\end{aligned}$$

(1.8)

$$\begin{aligned} h_2(x,w,w)&=V,\quad (x,w)\in {\bar{D}}\times [-H,\infty ). \end{aligned}$$

(1.9)

If

$$\begin{aligned} \max \{\alpha ,{\mathfrak {K}}\}>0 \end{aligned}$$

(1.10)

with

$$\begin{aligned} {\mathfrak {K}} :=\beta - 4L^2 \left[ (d+1) \max \{ \Vert \sigma _1\Vert _{L_\infty (\Omega _1)} , \sigma _2\} \left( 12 m_2L^2+2 m_3 \right) - \tau \right] _+ , \end{aligned}$$

then there is at least one solution $u\in {\bar{S}}_0$ to the variational inequality (1.1) in the following sense: for all $w\in {\bar{S}}_0$,

$$\begin{aligned}&\int _D \left\{ \beta \partial _x^2 u\,\partial _x^2 (w-u)+\big [\tau +\alpha \Vert \partial _x u\Vert _{L_2(D)}^2\big ]\partial _x u\, \partial _x(w-u)\right\} \,{\mathrm {d}}x\\&\quad \ge -\int _D g(u) (w-u)\, {\mathrm {d}}x \end{aligned}$$

where g(u) is given by (1.5) and $\psi _u$ is the solution to (1.2). Here, we interpret $\partial _x^4 u$ for $u\in {\bar{S}}_0$ as an element of $H^{-2}(D)$ by virtue of

$$\begin{aligned} \langle \partial _x^4 u,\phi \rangle _{H_D^2}:=\int _D \partial _x^2 u \partial _x^2\phi \,{\mathrm {d}}x,\quad \phi \in H_D^2(D). \end{aligned}$$

Moreover, this solution can be obtained as a minimizer of E on ${\bar{S}}_0$.

Obviously, a first step towards a full proof of Proposition 1.1 is to solve the transmission problem (1.2) for the electrostatic potential $\psi _u$ with sufficient regularity in order to give a meaning to the function g along with deriving suitable continuity properties. We refer to Sect. 2.1 for a detailed account on this issue (see in particular Lemmas 2.1 and 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 on ${\bar{S}}_0$. This property is ensured by assumption (1.10) (along with (1.7)). In particular, if $\alpha >0$, then the mechanical energy $E_m$ involves a super-quadratic term which allows a compensation of the negative contribution from the electrostatic energy $E_e$.

Remark 1.2

Note that (1.8) implies

$$\begin{aligned} \partial _w h_1(x,-H-d,w) =0,\quad (x,w)\in {\bar{D}}\times [-H,\infty ), \end{aligned}$$

while (1.9) implies

$$\begin{aligned} \partial _x h_2(x,w,w)=0,\quad (x,w)\in {\bar{D}}\times [-H,\infty ), \end{aligned}$$

and

$$\begin{aligned} \partial _z h_2(x,w,w) + \partial _w h_2(x,w,w)=0,\quad (x,w)\in {\bar{D}}\times [-H,\infty ), \end{aligned}$$

so that the formula (1.5) for g(u) simplifies and becomes $g(u) = {\mathfrak {g}}(u)$ 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 $\alpha >0$, we shall treat $\alpha$ 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

$$\begin{aligned} \partial _w h_1(x,-H-d,w) =0,\quad (x,w)\in {\bar{D}}\times [-H,\infty ), \end{aligned}$$

(1.11a)

and that there is a number $K>0$ such that

$$\begin{aligned} \vert \partial _x h_2(x,w,w)\vert + \vert \partial _z h_2(x,w,w)+\partial _w h_2(x,w,w)\vert \le K,\end{aligned}$$

(1.11b)

for $(x,w)\in {{\bar{D}}}\times [-H,\infty )$. 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

$$\begin{aligned} \begin{aligned} g(u)(x) =&{\mathfrak {g}}(u)(x) -\frac{\sigma _2}{2} \left[ \big ((\partial _x h_2)_u\big )^2+ \big ((\partial _z h_2)_u+(\partial _w h_2)_u\big )^2 \right] (x, u(x)). \end{aligned} \end{aligned}$$

(1.12)

With these assumptions we can now formulate the main result of the present paper.

Theorem 1.3

Let $\beta >0$, $\tau \ge 0$, and $\alpha = 0$. Assume that (1.3), (1.7), and (1.11) hold. Then there is at least one solution $u\in {\bar{S}}_0$ to the variational inequality (1.1) in the sense of Proposition 1.1. More precisely, the functional E is bounded from below on ${\bar{S}}_0$ and has a minimizer on ${\bar{S}}_0$ which is a weak solution to (1.1).

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 Sect. 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 Sect. 2 and relies on an idea introduced previously in a related work [10].

Remark 1.4

Theorem 1.3 remains valid when $\alpha >0$ and then only requires the assumptions (1.3) and (1.7). Indeed, the existence of a minimizer of E on ${\bar{S}}_0$ 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.

Corollary 1.5

Suppose (1.3), (1.8), and (1.9). If the coincidence set ${\mathcal {C}}(u)\subset D$ of a solution $u\in {\bar{S}}_0$ to (1.1) is non-empty, then it is an interval.

2 Proof of Theorem 1.3 and Corollary 1.5

2.1 Auxiliary results

Let us emphasize that the function g(u) defined in (1.5) involves gradient traces of the electrostatic potential $\psi _u$, the latter solving the transmission problem (1.2) posed on the non-smooth domain $\Omega (u)$ which possesses corners. In addition, $\Omega _2(u)$ need not be connected, but may consist of several components with non-Lipschitz boundaries (see Fig. 1), so that traces have first to be given a meaning. While the existence of a unique variational solution $\psi _u\in h_u+H_0^1(\Omega (u))$ to (1.2) readily follows from the Lax–Milgram theorem, the required further regularity for $\psi _u$ in order to make sense of its gradient traces is thus far from being obvious. Moreover, $\psi _u$ (and hence g(u)) depends non-locally on u so that continuity properties with respect to the plate deformation u are non-trivial.

Nevertheless, relying on shape optimization methods and Gamma convergence techniques the following result regarding the existence of a solution to the transmission problem (1.2) and its regularity is shown in [13].

Lemma 2.1

Suppose (1.3). For each $u\in {\bar{S}}_0$, there is a unique variational solution $\psi _u \in h_u + H^1_0(\Omega (u))$ to (1.2). Moreover,

$$\begin{aligned} \psi _{u,1} = \psi _u \mathbf{1 }_{{\overline{\Omega }}_1} \in H^2(\Omega _1),\qquad \psi _{u,2} = \psi _u \mathbf{1 }_{\overline{\Omega _2(u)}} \in H^2(\Omega _2(u)), \end{aligned}$$

and $\psi _{u}$ is a strong solution to the transmission problem (1.2) satisfying $\sigma \partial _z \psi _u\in H^1(\Omega (u))$. Also,

$$\begin{aligned} \inf _{\partial \Omega (u)} h_u \le \psi _u(x,z) \le \sup _{\partial \Omega (u)} h_u, \quad (x,z)\in \overline{\Omega (u)}. \end{aligned}$$

Proof

The existence, uniqueness, and regularity of the variational solution $\psi _u$ to (1.2) follows from [13, Theorem 1.1], while the upper and lower bounds for $\psi _u$ are consequences of the weak maximum principle [16, Chapter 7, Exercice 2.2], since $\sigma \in L_\infty (\Omega (u))$. $\square$

The regularity of $\psi _u$ provided by Lemma 2.1 in particular guarantees that g(u) defined in (1.5) is meaningful for $u\in {\bar{S}}_0$. As for the continuity of g(u) with respect to $u\in {\bar{S}}_0$ we recall:

Lemma 2.2

Suppose (1.3).

(a):

The mapping$g:{\bar{S}}_0\rightarrow L_2(D)$is well-defined, continuous, and bounded on bounded sets, the set${\bar{S}}_0$being endowed with the topology of$H^2(D)$.

(b):

Let$(u_j)_{j\ge 1}$be a sequence of functions in${\bar{S}}_0$such that$u_j\rightharpoonup u$in$H^2(D)$for some$u\in {\bar{S}}_0$. Then

$$\begin{aligned} \lim _{j\rightarrow \infty } \Vert g(u_j)-g(u)\Vert _{ L_2(D)} = 0 \quad \text { and }\quad \lim _{j\rightarrow \infty } E_e(u_j) = E_e(u). \end{aligned}$$

Proof

Part (a) follows from [13, Theorem 1.4], [13, Corollary 3.14 & Lemma 3.16], and the continuity of the trace operator from $H^1(\Omega _1)$ to $L_p(D\times \{-H\})$ for all $p\in [1,\infty )$.

Part (b) follows from [13, Proposition 3.17 & Corollary 3.12]. $\square$

2.2 Minimizers for a regularized energy

In the following we let $\beta >0$ and $\tau \ge 0$ and assume throughout that (1.3), (1.7), and (1.11) hold. We put

$$\begin{aligned} {\bar{\sigma }}:=\max \left\{ \Vert \sigma _1\Vert _{L_\infty (\Omega _1)},\,\sigma _2\right\} . \end{aligned}$$

In order to prove Theorem 1.3 it suffices to find a minimizer of the energy functional E on ${\bar{S}}_0$ since any such minimizer satisfies (1.1) according to [13, Theorem 5.3] (note that [13, Theorem 5.3] obviously remains true without imposing [13, Assumption (5.2a)]). However, as mentioned previously, the coercivity of the energy functional E is a priori unclear when dropping assumption (1.10). For this reason we introduce for $k \ge 0$ the regularized functional

$$\begin{aligned} {\mathcal {E}}_k(u):= E(u) + \frac{A}{2} \Vert (u-k)_+\Vert _{L_2(D)}^2,\quad u\in {\bar{S}}_0, \end{aligned}$$

(2.1)

where E(u) is defined in (1.6) and the constant A given by

$$\begin{aligned} A:=8 (d+1) {\bar{\sigma }} \left( \frac{3m_2}{2}+\frac{m_3^2 (d+1) {\bar{\sigma }}}{\beta }\right) \end{aligned}$$

with constants $m_j$ introduced in (1.7). We shall now prove, for each $k>0$, the existence of a minimizer $u_k$ of ${\mathcal {E}}_k$ on ${\bar{S}}_0$ and subsequently derive an a priori bound on such minimizers, so that the additional regularizing term drops out in ${\mathcal {E}}_k$. We first show the coercivity of the functional ${\mathcal {E}}_k$.

Lemma 2.3

Given $k\ge H$, there is a constant $c(k)>0$ such that

$$\begin{aligned} {\mathcal {E}}_k(u)\ge \frac{\beta }{4}\Vert \partial _x^2 u\Vert _{L_2(D)}^2+\frac{A}{4}\Vert (u-k)_+\Vert _{L_2(D)}^2- c(k),\quad u\in {\bar{S}}_0. \end{aligned}$$

Proof

Let $u\in {\bar{S}}_0$. The variational characterization of $\psi _u$, see [13, Lemma 3.2], readily gives

$$\begin{aligned} \begin{aligned} -E_e(u)&=\frac{1}{2}\int _{\Omega (u)} \sigma \vert \nabla \psi _u\vert ^2\, {\mathrm {d}}(x,z) \le \frac{1}{2}\int _{\Omega (u)} \sigma \vert \nabla h_u\vert ^2\, {\mathrm {d}}(x,z) . \end{aligned} \end{aligned}$$

Thus, invoking (1.3), Young’s inequality, and the definition of $\Omega (u)$ we derive

$$\begin{aligned} -E_e(u)&\le \int _{\Omega (u)} \sigma \left[ ( \partial _x h(x,z,u(x))^2+(\partial _w h(x,z,u(x))^2(\partial _x u)^2\right] \, {\mathrm {d}}(x,z) \nonumber \\&\quad +\frac{1}{2}\int _{\Omega (u)} \sigma (\partial _z h(x,z,u(x))^2\, {\mathrm {d}}(x,z) \nonumber \\&\le (d+1) {\bar{\sigma }} \left\{ \frac{3}{2} m_1 \vert D\vert +\frac{3}{2} m_2\Vert u\Vert _{L_2(D)}^2+m_3\Vert \partial _x u\Vert _{L_2(D)}^2\right\} . \end{aligned}$$

(2.2)

Next, since $u\in {\bar{S}}_0\subset H_D^2(D)$ implies

$$\begin{aligned} \int _D \vert \partial _x u\vert ^2\,{\mathrm {d}}x=-\int _D u\partial _x^2u\,{\mathrm {d}}x\le \Vert u\Vert _{L_2(D)} \Vert \partial _x^2u\Vert _{L_2(D)}, \end{aligned}$$

we deduce from Young’s inequality

$$\begin{aligned} \begin{aligned} -E_e(u)&\le (d+1) {\bar{\sigma }} \left\{ \frac{3}{2} m_1 \vert D\vert +\frac{3}{2} m_2\Vert u\Vert _{L_2(D)}^2+m_3\Vert u\Vert _{L_2(D)} \Vert \partial _x^2u\Vert _{L_2(D)}\right\} \\&\le (d+1) {\bar{\sigma }} \left\{ \frac{3}{2} m_1 \vert D\vert +\left( \frac{3m_2}{2}+\frac{m_3^2 (d+1) {\bar{\sigma }} }{\beta }\right) \Vert u\Vert _{L_2(D)}^2\right\} \\&\qquad +\frac{\beta }{4} \Vert \partial _x^2u\Vert _{L_2(D)}^2. \end{aligned} \end{aligned}$$

Finally, note that

$$\begin{aligned} \Vert u\Vert _{L_2(D)}^2&= \int _D u^2 \mathbf{1 }_{[-H,k]}(u)\,{\mathrm {d}}x + \int _D u^2 \mathbf{1 }_{(k,\infty )}(u)\, {\mathrm {d}}x \\&\le k^2 \int _D \mathbf{1 }_{[-H,k]}(u)\,{\mathrm {d}}x +2 \int _D (u-k)^2 \mathbf{1 }_{(k,\infty )}(u)\, {\mathrm {d}}x + 2 k^2 \int _D \mathbf{1 }_{(k,\infty )}(u)\, {\mathrm {d}}x \\&\le 2 k^2 |D| + 2 \Vert (u-k)_+\Vert _{L_2(D)}^2 . \end{aligned}$$

Hence, taking the previous two inequalities into account, the definition of ${\mathcal {E}}_k(u)$ entails

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_k(u)&\ge \frac{\beta }{4} \Vert \partial _x^2u\Vert _{L_2(D)}^2 + \frac{A}{2}\Vert (u-k)_+\Vert _{L_2(D)}^2 - \frac{3}{2} (d+1) {\bar{\sigma }} m_1 \vert D\vert \\&\qquad - \frac{A}{8} \left( 2\Vert (u-k)_+\Vert _{L_2(D)}^2+ 2k^2 |D| \right) \\&\ge \frac{\beta }{4} \Vert \partial _x^2u\Vert _{L_2(D)}^2 + \frac{A}{4} \Vert (u-k)_+\Vert _{L_2(D)}^2 - c(k) \end{aligned} \end{aligned}$$

as claimed. $\square$

The just established coercivity now easily yields the existence of a minimizer of ${\mathcal {E}}_k$ on ${\bar{S}}_0$.

Proposition 2.4

For each $k\ge H$, the functional ${\mathcal {E}}_k$ has at least one minimizer $u_k\in {\bar{S}}_0$ on ${\bar{S}}_0$; that is,

$$\begin{aligned} {\mathcal {E}}_k(u_k)=\min _{{\bar{S}}_0} {\mathcal {E}}_k. \end{aligned}$$

(2.3)

Moreover, $u_k \in {\bar{S}}_0$ is a weak solution to the variational inequality

$$\begin{aligned} \beta \partial _x^4 u_k-\tau \partial _x^2 u_k+ A (u_k-k)_++\partial {\mathbb {I}}_{{\bar{S}}_0}(u_k) \ni -g(u_k) \;\;\text { in }\;\; D. \end{aligned}$$

(2.4)

Proof

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 $u_k\in {\bar{S}}_0$ of ${\mathcal {E}}_k$ on ${\bar{S}}_0$. In particular,

$$\begin{aligned} 0\le \liminf _{s\rightarrow 0^+} \frac{1}{ s}\big ({\mathcal {E}}_k(u_k+s(w-u_k))-{\mathcal {E}}_k(u_k)\big ) \end{aligned}$$

for any fixed

$$\begin{aligned} w \in S_0 := \left\{ v\in H_D^2(D)\,:\, v> -H \text { in } D \right\} \subset {\bar{S}}_0. \end{aligned}$$

It was shown in [13, Theorem 1.4] that (since $u_k+s(w-u_k)\in S_0$ for $s\in (0,1)$)

$$\begin{aligned} \lim _{s\rightarrow 0^+} \frac{1}{s}\big (E_e(u_k+s(w-u_k))-E_e(u_k)\big )= \int _D g(u_k) (w-u_k)\, {\mathrm {d}}x. \end{aligned}$$

From the definition of ${\mathcal {E}}_k$ we then obtain by gathering the two limits that

$$\begin{aligned}&\int _D \left\{ \beta \partial _x^2 u_k\,\partial _x^2 (w-u_k) + \tau \partial _x u_k\, \partial _x (w-u_k) + A(u_k-k)_+ (w-u_k)\right\} \,{\mathrm {d}}x \\&\quad \ge - \int _D g(u_k) (w-u_k)\, {\mathrm {d}}x \end{aligned}$$

for all $w\in S_0$. Since $S_0$ is dense in ${\bar{S}}_0$, this inequality also holds for any $w\in {\bar{S}}_0$, and we thus have shown that $u_k$ satisfies the variational formulation of (2.4). $\square$

2.3 A priori bounds

We shall now show that $u_k$ is a priori bounded for k large enough (making the additional term in ${\mathcal {E}}_k$ 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 $\beta \partial _x^4-\tau \partial _x^2$ with clamped boundary conditions [4, 8, 11, 17].

Lemma 2.5

Let $G_0\ge 0$ and recall that $\beta >0$ and $\tau \ge 0$. For an interval $I:=(a,b)\subset (-L, L)$, let $S_I \in C^4([a,b])$ denote the unique solution to the boundary-value problem

$$\begin{aligned} \beta S_I'''' - \tau S_I'' = G_0\ , \qquad x\in (a,b)\ , \end{aligned}$$

(2.5)

supplemented with one of the boundary conditions:

$$\begin{aligned}&S_I(a) + H = S_I'(a)= S_I(b) + H = S_I'(b) = 0 \quad \text{ if } \quad -L<a<b<L\ , \end{aligned}$$

(2.6)

$$\begin{aligned}&S_I(-L) = S_I'(-L)= S_I(b) + H = S_I'(b) = 0 \quad \text{ if } \quad -L=a<b<L\ , \end{aligned}$$

(2.7)

$$\begin{aligned}&S_I(a)+H = S_I'(a)= S_I(L) = S_I'(L) = 0 \quad \text{ if } \quad -L<a<b=L\ , \end{aligned}$$

(2.8)

$$\begin{aligned}&S_I(-L) = S_I'(-L)= S_I(L) = S_I'(L) = 0 \quad \text{ if } \quad -L=a<b=L\ . \end{aligned}$$

(2.9)

Then, there is $\kappa _0>0$ depending only on $G_0$, $\beta$, L, H, and $\tau$ (but not on $I=(a,b)$) such that

$$\begin{aligned} \left| S_I(x) \right| \le \kappa _0\ , \quad x\in [a,b]\, . \end{aligned}$$

Proof

This result has already been observed in [10, Lemma A.1] and we include its proof here 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 $y\in [0,1]$ and note that P solves the boundary-value problem

$$\begin{aligned} \begin{aligned}&\beta P'''' - \tau (b-a)^2 P'' = (b-a)^4 G_0\ , \qquad y\in (0,1)\ , \\&P(0) = P'(0) = P(1) = P'(1) = 0\ . \end{aligned} \end{aligned}$$

(2.10)

Since $G_0\ge 0$ we deduce that $P\ge 0$ in (0, 1) from a version of Boggio’s comparison principle [4, 8, 11, 17] . Testing (2.10) by P we get

$$\begin{aligned} \beta \Vert P''\Vert _{L_2(0,1)}^2 + \tau (b-a)^2 \Vert P'\Vert _{L_2(0,1)}^2 = (b-a)^4 G_0 \int _0^1 P(y)\ {\mathrm {d}}y \ . \end{aligned}$$

Since

$$\begin{aligned} |P(y)| = \left| \int _0^y (y-y_*) P''(y_*) \ {\mathrm {d}}y_* \right| \le \Vert P''\Vert _{L_2(0,1)}\ , \end{aligned}$$

we infer from the above inequalities that

$$\begin{aligned} \beta \Vert P\Vert _{L_\infty (0,1)}^2 \le \beta \Vert P''\Vert _{L_2(0,1)}^2 \le (b-a)^4 G_0 \Vert P\Vert _{L_\infty (0,1)} \le 16L^4 G_0 \Vert P\Vert _{L_\infty (0,1)}\ . \end{aligned}$$

Consequently, $0\le P \le 16L^4 G_0/ \beta$ in [0, 1], hence $-H \le S_I \le 16L^4 G_0/ \beta - H$ in [a, b].

Case 2:$-L=a<b<L$. Define $Q(y):= y^2(y^2 +2(H-1)y +1- 3H)$ for $y\in [0,1]$ and note that $Q(0)=Q'(0) = Q(1)+ H = Q'(1)= 0$. Set $P(y) := S_I(-L+(b+L)y)-Q(y)$ for $y\in [0,1]$. Then, due to (2.5) and (2.7), P solves the boundary-value problem

$$\begin{aligned} \begin{aligned}&\beta P'''' - \tau (b+L)^2 P'' = (b+L)^4 G_0 - \beta Q'''' + \tau (b+L)^2 Q'',\,\qquad y\in (0,1),\\& P(0) = P'(0) = P(1) = P'(1) = 0\ . \end{aligned} \end{aligned}$$

(2.11)

The arguments of Case 1 give

$$\begin{aligned}&\beta \Vert P\Vert _{L_\infty (0,1)}^2 \\&\quad \le \beta \Vert P''\Vert _{L_2(0,1)}^2 \\&\quad \le \left[ (b+L)^4 G_0+ \beta \Vert Q''''\Vert _{L_\infty (0,1)}+ \tau (b+L)^2\Vert Q''\Vert _{L_\infty (0,1)}\right] \Vert P\Vert _{L_\infty (0,1)}\\&\quad \le \left[ (b+L)^4 G_0 + 24 \beta +14 \tau (H+1) (b+L)^2 \right] \Vert P\Vert _{L_\infty (0,1)} \\&\quad \le \left[ 16L^4 G_0 + 24 \beta + 56 \tau (H+1) L^2 \right] \Vert P\Vert _{L_\infty (0,1)}\ , \end{aligned}$$

since $Q''''=24$, $|Q''|\le 14 (H+1)$, and $b<L$. Therefore,

$$\begin{aligned} \begin{aligned} \Vert S_I \Vert _{L_\infty (-L,b)}&\le \Vert P\Vert _{L_\infty (0,1)} + \Vert Q\Vert _{L_\infty (0,1)} \\&\le \frac{16L^4 G_0 +24 \beta + 56 \tau (H+1) L^2}{\beta } + \Vert Q\Vert _{L_\infty (0,1)}\ . \end{aligned} \end{aligned}$$

Case 3:$-L<a<b=L$. Define $P(y) := S_I(a+y(L-a))-Q(1-y)$ for $y\in [0,1]$, where Q is as in Case 2. Arguing as in the previous case we obtain the same bound for $\Vert S_I\Vert _{L_\infty (a,L)}$.

Case 4:$-L=a<b=L$. Define $P(y) := S_I(-L+2Ly)$ for $y\in [0,1]$. Then, by (2.5) and (2.9), P solves the boundary-value problem

$$\begin{aligned} \begin{aligned}&\beta P'''' - 4\tau L^2 P'' = 16L^4 G_0\ , \qquad y\in (0,1)\ , \\&P(0) = P'(0) = P(1) = P'(1) = 0\ . \end{aligned} \end{aligned}$$

As in Case 1 we deduce that $0\le S_I \le 16L^4 G_0/\beta$ in $[-L,L]$. $\square$

The previous lemma now implies the desired a priori bounds on the minimizers $u_k$.

Proposition 2.6

There is $\kappa _0 \ge H>0$ depending only on K introduced in (1.11b) such that, for all $k\ge H$, the minimizer $u_k\in {\bar{S}}_0$ of ${\mathcal {E}}_k$ on ${\bar{S}}_0$ constructed in Proposition 2.4 satisfies $\Vert u_k\Vert _{L_\infty (D)}\le \kappa _0$.

Proof

Let $k\ge H$. We first note that, since ${\mathfrak {g}}(u_k)\ge 0$ in D by (1.5b), it easily follows from (1.11b) and (1.12) that

$$\begin{aligned} g(u_k)(x)\ge -G_0:= -\sigma _2 K^2, \quad x\in D. \end{aligned}$$

(2.12)

Next, since $u_k\in C({\bar{D}})$ with $u_k(\pm L)=\,0$, the set $\{ x \in D\,:\, u_k(x)>-H\}$ 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 $(I_j)_{j\in J}$. Consider a fixed $j\in J$ and let $S_{I_j}$ denote the solution to (2.5) in $I_j$ subject to the associated boundary conditions on $\partial I_j$ listed in (2.6–2.9), which vary according to whether ${\bar{I}}_j\subset D$ or not. Then Lemma 2.5 yields a constant $\kappa _0>H$ (independent of $j\in J$) such that

$$\begin{aligned} \Vert S_{I_j}\Vert _{L_\infty (I_j)}\le \kappa _0. \end{aligned}$$

(2.13)

Note that, by definition of $I_j$, the function $u_k$ restricted to $I_j$ satisfies the same boundary conditions on $\partial I_j$ as $S_{I_j}$. Hence, for $z:=u_k-S_{I_j}\in H^2(I_j)$ we have $z=\partial _x z=0$ on $\partial I_j$. Moreover, if $\theta \in {\mathcal {D}}(I_j)$, then $u_k\pm \delta \theta \in {\bar{S}}_0$ for $\delta >0$ small enough since $u_k>-H$ in the support of $\theta$. Invoking the weak formulation of (2.4) we derive

$$\begin{aligned} \pm \delta \int _{I_j} \left\{ \beta \partial _x^2 u_k\,\partial _x^2 \theta +\tau \partial _x u_k\, \partial _x\theta +A(u_k-k)_+ \theta \right\} \,{\mathrm {d}}x \ge \mp \delta \int _{I_j} g(u_k) \theta \, {\mathrm {d}}x , \end{aligned}$$

hence

$$\begin{aligned} \int _{I_j} \left\{ \beta \partial _x^2 u_k\,\partial _x^2 \theta +\tau \partial _x u_k\, \partial _x\theta +A(u_k-k)_+ \theta \right\} \,{\mathrm {d}}x =-\int _{I_j} g(u_k) \theta \, {\mathrm {d}}x . \end{aligned}$$

Thus, we conclude that $z=u_k-S_{I_j} \in H^2(I_j)$ weakly solves the boundary value problem

$$\begin{aligned} \beta \partial _x^4 z-\tau \partial _x^2 z&= -G_0-g(u_k)-A (u_k-k)_+\;\;\text { in }\;\; I_j, \end{aligned}$$

(2.14a)

$$\begin{aligned} z&=\partial _x z=0 \;\;\text { on }\;\; \partial I_j. \end{aligned}$$

(2.14b)

Now, since $g(u_k)+A (u_k-k)_+\in L_2(I_j)$ by Lemma 2.2(a), classical elliptic regularity theory [7] entails that $z=u_k-S_{I_j}\in H^4(I_j)$ 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 $\Vert u_k\Vert _{L_\infty (I_j)}\le \kappa _0$. Since $\kappa _0$ is independent of $j\in J$ and $(I_j)_{j\in J}$ covers $\{ x \in D\,:\, u_k(x)>-H\}$, the assertion follows. $\square$

2.4 Proof of Theorem 1.3

We are now in a position to finish off the proof of Theorem 1.3. Indeed, if $u_k\in {\bar{S}}_0$ is the minimizer of the functional ${\mathcal {E}}_k$ on ${\bar{S}}_0$ provided by Proposition 2.4, then $-H\le u_k\le \kappa _0$ in D due to Proposition 2.6. Consequently, for $k\ge \kappa _0$ we have

$$\begin{aligned} E(u_k)={\mathcal {E}}_{ \kappa _0}(u_k)={\mathcal {E}}_k(u_k)\le {\mathcal {E}}_k(v) = E(v) + \frac{A}{2} \Vert (v-k)_+\Vert _{L_2(D)}^2 ,\quad v\in {\bar{S}}_0. \end{aligned}$$

(2.15)

Now, since $0\in {\bar{S}}_0$ it follows from Lemma 2.3 that, for $k\ge \kappa _0$,

$$\begin{aligned} \frac{\beta }{4}\Vert \partial _x^2 u_k\Vert _{L_2(D)}^2 \le {\mathcal {E}}_{\kappa _0}(u_k)+ c(\kappa _0)\le {\mathcal {E}}_{ \kappa _0}(0)+ c(\kappa _0)=E(0)+ c(\kappa _0). \end{aligned}$$

Thus, $(u_k)_{k\ge \kappa _0}$ 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 $u_*\in {\bar{S}}_0$. 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

$$\begin{aligned} E(u_*)\le E(v),\quad v\in {\bar{S}}_0, \end{aligned}$$

recalling that the continuous embedding of $H^1(D)$ in $C({\bar{D}})$ readily implies that

$$\begin{aligned} \lim _{k\rightarrow \infty } \Vert (v-k)_+\Vert _{L_2(D)} = 0,\quad v\in {\bar{S}}_0. \end{aligned}$$

Therefore, $u_*\in {\bar{S}}_0$ minimizes E on ${\bar{S}}_0$. 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 $(u_k)_{k\ge \kappa _0}$ to $u_*$ to pass to the limit $k\rightarrow \infty$ in (2.4), observing that $(g(u_k))_{k\ge \kappa _0}$ converges to g(u) in $L_2(D)$ by Lemma 2.2 (b) and that $(u_k-k)_+=0$ for $k\ge \kappa _0$. This proves Theorem 1.3. $\square$

2.5 Proof of Corollary 1.5

Suppose (1.3), (1.8), and (1.9) and let $u\in {\bar{S}}_0$ be any solution to the variational inequality (1.1). Note that (1.8) and (1.9) imply $g(u) = {\mathfrak {g}}(u)$ in (1.5a). In particular, the function g(u) is non-negative. Assume now for contradiction that the coincidence set ${\mathcal {C}}(u)$ is not an interval. Then there are $-L<a<b<L$ with $u(a)+H=\partial _x u(a)=u(b)+H=\partial _x u(b)=0$ and $u>-H$ in $I:=(a,b)$. We may then argue as in the proof of Proposition 2.6 to conclude that $z:=u+H\in H^4(I)\cap H_D^2(I)$ is a strong solution to the boundary value problem

$$\begin{aligned} \beta \partial _x^4 z-\tau \partial _x^2 z&= -g(u)\;\;\text { in }\;\; I, \end{aligned}$$

(2.16a)

$$\begin{aligned} z&=\partial _x z=0 \;\;\text { on }\;\; \partial I. \end{aligned}$$

(2.16b)

Another application of a version of Boggio’s comparison principle [4, 8, 11, 17] implies $z=u+H \le 0$ in I. But this contradicts $u>-H$ in I. $\square$