## Abstract

The nonlinear non-local elliptic equation governing the deflection of charged plates in electrostatic actuators is studied under the pinned and the clamped boundary conditions. Results concerning the existence, construction and approximation, and behaviour of classical and singular solutions with respect to the variation of physical parameters of the equation in various situations are presented.

## 1. Introduction

In 1959, Feynman delivered a speech entitled ‘There's Plenty of Room at the Bottom’ in which he described the coming technology that would make things of very small scales, including much condensed information storage, powerful microscopes, miniature computers, infinitesimal machinery, and pointed out the possible ways to make them work based on the fundamental principles of physics, chemistry and biology (Feynman 1992). Approximately 23 years later, Feynman revisited the pictures he depicted in his earlier speech, elaborated in greater detail on infinitesimal machinery and described the mechanisms of small machines and computers based on techniques and ideas ranging from electrostatic actuation to quantum computation at the atomic-electron levels (Feynman 1993). In these speeches, Feynman advocated, anticipated and inspired an important area of contemporary technology known as microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS), which came into fast advancement after Nathanson *et al*. (1967) who produced the resonant gate transistor. The nature of MEMS and NEMS (small scales, multiphysics, etc.) suggests that mathematical analysis and numerical simulation of the problems arising in the modelling, optimization and design of MEMS and NEMS devices often play a crucial role in understanding these devices (Ye *et al*. 1998; Hung & Senturia 1999; Elata *et al*. 2003; Pelesko & Bernstein 2003; Chen *et al*. 2004; Chatterjee & Aluru 2005; see also Wang & Hadaegh 1996; Legtenberg *et al*. 1997; Castaner & Senturia 1999; Younis *et al*. 2003).

In MEMS and NEMS devices, an important method is called the electrostatic actuation, which is based on an electrostatic-controlled tunable capacitor and widely used in microresonators, switches, micromirrors, accelerometers, etc. In fact, almost every kind of MEMS and NEMS systems has one or more electrostatic actuation-based devices in routine operation. Although electrostatic actuation technology is being vigorously developed and improved, its physical principle has only two simple components: (i) the electrostatic Coulomb law that gives the force between two charged objects (e.g. two plates) and (ii) the elastic deformation relation, which gives the force that responds to the electrostatic force. The Coulomb force satisfies the inverse square law with respect to the distance of the two charged objects, which is a function of the deformation variable. On the other hand, by continuum mechanics, the simplest elastic force depends on the Laplace or bi-Laplace of the deformation variable. Hence, we are led to a nonlinear elliptic equation with an inverse square type nonlinearity, which has not been well studied as a mathematical problem. Interestingly, when circuit series capacitance is considered, the nature of electric charge distribution leads moreover to the presence of a non-local term and the equation becomes a singular integro-differential equation. Our purpose of this paper is to establish some basic existence and construction results for this important equation and its various variations. Some of our results may be directly used in numerical computation of the solutions and others may provide some qualitative understanding of the problems.

Here is an outline of the rest of the paper. In §2, we review the problem and the governing equation. In §3, we consider the solution of the equation under the pinned boundary condition. It will be seen that the validity of the maximum principle allows us to obtain classical solutions when the applied voltage is below a critical value. In §4, we consider radially symmetric solutions and use them to estimate the critical values of the parameters. In §5, we consider the solution of the equation under the clamped boundary condition. In this situation, the maximum principle is not valid and it is only possible to obtain a unique small-amplitude solution when the parameters are in some suitable ranges. In §6, we summarize our results.

## 2. Electrostatic actuation and nonlinear equations

The Coulomb law states that the electrostatic force *F* between two charges *q*_{1} and *q*_{2} placed at a distance *r* apart is given in normalized units by *F*=*q*_{1}*q*_{2}/*r*^{2}. Suppose the two charges are uniformly distributed over two parallel plates subject to a capacitive influence of capacitance *C* and a fixed electric voltage *V*. Then we can rewrite *q*_{1} and *q*_{2} as *q*_{1}=*q*=*CV*=−*q*_{2} and *F* satisfies *F*=−*C*^{2}*V*^{2}/*r*^{2}. Imagine one of the plates stretches a small vertical distance d*r*. Then the work done is *F* d*r* which decreases the electric potential stored in the capacitor. As a consequence, the electric potential *W* can be expressed as(2.1)In the general situation, when the electric charges are not uniformly distributed as a result of varying distance *r*=*L*+*u*(*x*), where *L*>0 is the distance between the two plates in the absence of plate deformation and *u*(*x*) is the plate deformation variable, we need to replace equation (2.1) by(2.2)where *Ω* is a bounded domain in ^{2} and *a*>0 is a constant related to permittivity.

On the other hand, in the presence of elastic deformation characterized by *u*≠0, it is well known that the elastic energy collects contributions from two sectors, i.e. the stretching energy sector given by(2.3)where *T*>0 is the tension constant and the bending energy sector given by(2.4)where *D*=2*h*^{3}*Y*/3(1−*ν*^{2}) in which *h* is the plate thickness, *Y* is the Young modulus and *ν* is the Poisson ratio.

Consequently, the total energy *E*=*P*+*Q*+*W* may be represented as(2.5)so that its Euler–Lagrange equation is(2.6)

In the zero plate thickness limit *D*=0, the problem reduces to a second-order equation, which is a special case of a problem that concerns the determination of the equilibrium state of two neighbouring charged liquid drops suspended over two circular rings that is governed by the equation (Taylor 1968, 1971; Ackerberg 1969)(2.7)where *α*, *β*>0 are constants reflecting fluid-mechanical and electrostatic properties of the drops, respectively, so that *α*=0 corresponds to the situation where uncharged drops are flat. Note also that the equation Δ*v*=*v*^{−p} (0<*p*≤1) has also been studied in the literature to some extent (see Meadows (2004) and the references therein).

In more realistic situations, the capacitance *C* of the actuator depends on the deformation variable *u* according to the relation (Pelesko & Bernstein 2003)(2.8)Besides, as a result of the circuit series capacitance *C*_{f} and the sensitivity of the actuator capacitance to the elastic deformation variable *u*, the voltage drop *V* at the actuator can no longer be kept at the constant supply voltage *V*_{s}, but is instead given by the series circuit formula *V*=*V*_{s}/(1+*C*/*C*_{f}). Therefore, in view of equation (2.8), *V* depends on the deformation variable *u* according to(2.9)where *Χ*=1/*C*_{f} counts the influence of the circuit series capacitance and we arrive at the non-local equation (Pelesko & Bernstein 2003)(2.10)where *λ*>0 is a constant which is proportional to the supply voltage squared, .

## 3. Solution under pinned boundary condition

We first consider equation (2.10) subject to the pinned boundary condition(3.1)which is also called the Navier boundary condition. Physically, this situation gives rise to a device which is ideally hinged along all its edges so that it is free to rotate and does not experience any torque or bending moment about its edges. In the case when *D*=0, the boundary condition is simply the Dirichlet boundary condition(3.2)

It is easily seen in view of the maximum principle that a classical solution of equation (2.10) subject to equation (3.1) must satisfy *u*(*x*)<0 and Δ*u*(*x*)>0 for *x*∈*Ω*.

We begin by considering the situation that the influence of the circuit series capacitance is negligible, *Χ*=0. Thus, equation (2.10) becomes(3.3)where we add a constant *α*≥0 in order to accommodate the equation (2.7) when *D*=0. Of course, a classical solution of equation (3.3) also satisfies *L*+*u*(*x*)>0, and other properties *u*<0 and Δ*u*>0 in *Ω* still hold. We shall see that it is not hard to construct a classical solution of equation (3.3).

We first show that when *λ*>0 is sufficiently large, equation (3.3) has no classical solution. The physical reason is that as the supply voltage *V*_{s} becomes sufficiently large, the charged plates undergo a deflection so that they collapse onto each other, *r*=*L*+*u*=0, and the system fails to operate. Such a situation is called a ‘pull-in’ situation and the level of *V*_{s} when pull-in takes place is called the pull-in voltage and the corresponding *λ* value is called the pull-in value, *λ*_{c}.

We use *λ*_{1} to denote the first eigenvalue of the operator −Δ subject to equation (3.2) and *U*_{1} an associated eigenfunction satisfying *U*_{1}>0 in *Ω*. Then *T*Δ*U*_{1}−*D*Δ^{2}*U*_{1}=−*λ*_{1}(*T*+*Dλ*_{1})*U*_{1}. If *u* is a classical solution of equation (3.3), we add *λ*_{1}(*T*+*Dλ*_{1})*u* to both sides of equation (3.3), multiply the resulting relation by *U*_{1} and integrate it over *Ω*. We obtain(3.4)Consider the function *f*(*u*)=*λ*_{1}(*T*+*Dλ*_{1})*u*+*λ*/(*L*+*u*)^{2} for −*L*<*u*≤0. Then *f*(0)=*λ*/*L*>0 and *f*(*u*)→∞ as *u*→−*L*^{+}. We see that we can find a condition so that(3.5)which implies that equation (3.4) cannot hold and equation (3.3) has no solution. Indeed, the only critical point of *f*(*u*) is(3.6)If *u*_{0}≥0, then *f*′(*u*)≤0 for all −*L*<*u*≤0 and *f*(*u*)≥*f*(0)>0 (−*L*<*u*≤0). Therefore, the only situation to be checked is when *u*_{0}<0. We have(3.7)Requiring *f*(*u*_{0})≥0 in equation (3.7), we arrive at the non-existence condition(3.8)Similarly, we have another non-existence condition expressed in terms of *α*,(3.9)

To obtain a solution of equation (3.3), we assume that it has a subsolution *u*_{*} which satisfies −*L*<*u*_{*}≤0 in *Ω* and the inequalities(3.10)With such a function *u*_{*}, we can use the maximum principle to show straightforwardly that the iterative scheme(3.11)gives rise to a monotone sequence {*u*_{n}} satisfying(3.12)In particular, the limit *u*=lim *u*_{n} is a solution of equation (3.3) and *u*_{*}≤*u*. Therefore, *u* is also the maximal solution of equation (3.3).

We now show that when both *α* and *λ* are small enough, equation (3.3) has a subsolution. To see this, we use *λ*_{1} and *U*_{1} to denote the first eigenpair of −Δ over a large bounded domain *Ω*_{1} containing so that *U*_{1}=0 on ∂*Ω*_{1} and *U*_{1}>0 in *Ω*_{1}. Let *ϵ*>0 be a small number so that *ϵU*_{1}<*L*/2. Then (*T*Δ−*D*Δ^{2})(−*ϵU*_{1})=*λ*_{1}(*T*+*Dλ*_{1})*ϵU*_{1}, which can be made greater than *α*+*λ*/(*L*−*ϵU*_{1})^{2} over when *α* and *λ* are sufficiently small. Of course, −*ϵU*_{1}<0 and Δ(−*ϵU*_{1})=*ϵλ*_{1}*U*_{1}>0 on ∂*Ω*. Therefore, −*ϵU*_{1} is a subsolution of equation (3.3) satisfying −*L*<*ϵU*_{1}<0. As a consequence, a solution of equation (3.3) exists, which can be constructed by the iterative scheme equation (3.11).

For given *α*≥0, we define(3.13)We show that *Λ*^{α} is either empty or a nontrivial bounded interval. In fact, if *Λ*^{α} is not empty and *λ*_{0}∈*Λ*, we can easily prove that (0, *λ*_{0}]⊂*Λ*^{α}. Indeed, let be a solution of equation (3.3) at *λ*=*λ*_{0}. Then, for any 0<*λ*<*λ*_{0}, we see that *u*_{0} is a subsolution of equation (3.3) at *λ*. Therefore it follows that *λ*∈*Λ*^{α} and the claim is verified.

As a by-product, we have seen in the above that if *λ*′, *λ*″∈*Λ*^{α} and *λ*′<*λ*″, then the corresponding maximal solutions *u*_{λ′} and *u*_{λ″} satisfy(3.14)Physically, this is a natural relation because higher supply voltage results in greater elastic deformation or deflection.

If *Λ*^{α}≠, then(3.15)which determines the pull-in voltage. We now study the limit of *u*_{λ} as .

For simplicity, we first consider the situation when *D*=0 and see that, after we rescale the tension constant *T* to unity to save some notation, *u*_{λ} satisfies(3.16)The monotonicity condition (3.14) implies that there is a well-defined function *U* so that(3.17)

In order to find the meaning of *U*, we again use (*λ*_{1}, *U*_{1}) to denote a first eigenpair of −Δ under the boundary condition (3.2) for which *U*_{1} satisfies 0<*U*_{1}≤1. Multiplying equation (3.16) by *U*_{1} and integrating, we have(3.18)Since *U*_{1} only vanishes at ∂*Ω*, equation (3.18) gives us the uniform bound(3.19)for any compact subset *K* of *Ω*. For any subdomain *Ω*_{0}⊂⊂*Ω* (i.e. ), insert another subdomain *Ω*_{1} satisfying *Ω*_{0}⊂⊂*Ω*_{1}⊂⊂*Ω*. Let *η* be a cut-off function satisfying *η*=1 on *Ω*_{0}, supp(*η*)⊂*Ω*_{1} and 0≤*η*≤1. Multiplying equation (3.16) by *η*^{2}*u*_{λ} and integrating, we have(3.20)Therefore, in view of the Schwartz inequality, we have(3.21)In particular(3.22)Combining equations (3.19) and (3.22), we arrive at the uniform bound(3.23)Consequently, we obtain that, for any *Ω*_{0}⊂⊂*Ω*, there holds(3.24)As a corollary, this shows that *U* is a distributional solution of the equation (3.16) at the critical (pull-in) ,(3.25)so that and −*L*≤*U*<0 in *Ω*. In other words, is attainable (i.e. ) for distributional solutions in any dimensions.

Note that the Hausdorff dimension of the zero set *Σ* of a *W*^{1,p}(*Ω*) solution *v*=*L*+*U* of the critical equation (3.25) has been studied in Jiang & Lin (2004) where *Σ* is defined by(3.26)In the special case, when we consider a one-dimensional situation, the distributional solution of equation (3.25) is continuous and *Σ* is the zero set of *L*+*U* in the classical sense. Using the result of Jiang & Lin (2004), we must have *Σ*=. In other words, *U*>−*L* everywhere and we obtain a classical solution at as well. Therefore, in dimension one, we have a complete description of the set *Λ*^{α},(3.27)

Another special case is when *Ω* is a disc or ball in ^{N}. Assume *Ω*=*B*_{R} (a ball of radius *R* centred at the origin). Using the symmetry theorem of Gidas *et al*. (1979), we know that the solution *u*_{λ} of equation (3.16) is radially symmetric and strictly increasing along any radial direction. Hence the function *U*(*x*) defined in equation (3.17) is also radially symmetric, thus continuous except at the origin, and non-decreasing along any radial direction. Using the upper bound equation (3.19), we see that *U* can only attain the value −*L* at the origin. In other words, when the domain *Ω* is radially symmetric, the pull-in collapse of the device at the pull-in voltage can only happen at the centre of the domain, although we have learned that in one-dimensional situation, the pull-in collapse does not happen at the pull-in voltage at all.

In order to derive similar results for the equation (3.3), we set *v*=*L*+*u*. Hence 0<*v*<*L* and the maximum principle implies that *Tv*−*D*Δ*v*<*TL* in *Ω*⊂^{N}. Now define(3.28)Then *w* satisfies the inequality(3.29)In other words, *w* is a bounded positive subharmonic function in *Ω* whose gradient necessarily has *L*^{2}-local bounds. In fact, multiplying equation (3.29) by *η*^{2}*w* where *η* is a smooth cut-off function used earlier and integrating by parts, we have(3.30)Using the Schwartz inequality in equation (3.30), we see that the left-hand side of equation (3.30) is bounded from above by a constant depending only on the cut-off function *η* and the uniform bound of *w* over *Ω*. Hence the claim follows. Therefore, we have(3.31)

On the other hand, as a consequence of equation (3.4), we also note that by replacing equation (3.19), we have(3.32)for any compact subset *K* of *Ω*.

We are now ready to obtain a high-order interior estimate for the classical solution *u* of equation (3.3) constructed earlier below the pull-in threshold . For this purpose, we multiply equation (3.3) by *η*^{4}*u* and integrate it twice by parts to get(3.33)Applying equations (3.31) and (3.32), and the Schwartz inequality in equation (3.33), we obtain the uniform interior bound(3.34)

As before, we use *u*_{λ} to denote the classical solution obtained at *λ*>0 which is below the pull-in threshold value . Then equations (3.31) and (3.34), and standard elliptic theory (Gilbarg & Trudinger 1977) imply that, for any subdomain *Ω*_{0}⊂⊂*Ω*, we have the uniform bound(3.35)In particular, there is a function such that(3.36)Consequently, *U* is a weak solution of the equation (3.3) at the critical parameter ,(3.37)and in the sense of weak solutions, the critical value is again attainable. Besides, when *N*=1 (a one-dimensional device again), the result of Jiang & Lin (2004) implies that the zero set of the function *v*=*L*+*U* is empty. Hence, in this situation, the solution of equation (3.37) is a classical solution satisfying −*L*<*U*<0 in *Ω* as in the second-order (*D*=0) situation and the pull-in phenomenon does not happen at the pull-in voltage.

We now turn to the non-local equation (2.10). Let be the critical parameter obtained for the equation (3.3). We show that a sufficient condition so that the existence of a classical solution to equation (2.10) is ensured reads(3.38)In fact, we can consider the closed convex set defined by(3.39)where *u*_{0} is the (maximum) classical solution of the equation(3.40)whose existence is guaranteed by the condition (3.38). For any *w*∈, since −*L*<*u*_{0}<*w*≤0, we see that *u*_{0} satisfies(3.41)Therefore *u*_{0} is a subsolution of the equation(3.42)subject to the boundary condition (3.1) (or (3.2) if *D*=0). As a consequence of our earlier discussion, equation (3.42) has a unique maximal solution, say *u*, satisfying *u*_{0}≤*u*<0 in *Ω*. In particular, we have thus defined a map *M*:→ with *M*(*w*)=*u*. Since the functions in *M*() are uniformly bounded away from −*L*, we can show by *L*^{2}-estimates that *M*() is a bounded subset of *W*^{4,2}(*Ω*) (or *W*^{2,2}(*Ω*) if *D*=0). Hence *M* is a completely continuous map. As a result, *M* has a fixed point in , which is of course a classical solution of equation (2.10).

Let the solution of equation (2.10) obtained above be denoted by *u*_{Χ} (which may not be unique). Suppose now . We can take *u*_{0} in equation (3.39) to be the classical solution of equation (3.40) with setting *Χ*=0. Then our discussion shows that the family {*u*_{Χ}}_{Χ>0} is bounded in *W*^{4,2}(*Ω*) (or *W*^{2,2}(*Ω*) if *D*=0). Therefore, in the sense of subsequence, we see that(3.43)This is a naturally expected result which says that we recover the equation (3.3) from (2.10) when we neglect the circuit series capacitance by setting *Χ*=0.

The solution obtained for the equation (2.10) or (3.3) is denoted by *u*_{α,λ}. We have seen that when *α*, *λ* are sufficiently small, *u*_{α,λ} are uniformly bounded away by the classical solutions of equation (3.3) corresponding to the larger values of *α* and *λ*. As a consequence, when we take *α*, *λ*→0, we see immediately through the well-known elliptic estimates that *u*_{α,λ}→0 in any standard function space topology over *Ω*, which says correctly that when the applied voltage is switched off, the elastic deflection vanishes in the special case where *α*=0.

## 4. Symmetric solutions and applications

Using the method of moving planes as in Gidas *et al*. (1979), we can show that when *Ω*=*B*_{R} (the ball in ^{N} of radius *R*>0 and centred at the origin), the classical solutions of equation (3.3) or (2.10) subject to the pinned boundary condition (3.1) are radially symmetric about the origin and strictly increasing along any radial direction (we omit the details here, but simply note that the problem may be reformulated as a cooperative second-order system (Troy 1981) so that a symmetry proof follows). As seen in §3, such a property has interesting applications and one of them is the conclusion that the pull-in collapse can only occur at the centre of the domain, i.e. the origin, owing to the uniform boundedness condition stated in equation (3.32). In this section, we show how to use radially symmetric solutions to get lower estimates for the pull-in threshold .

Let *Ω*=*B*_{R} in equation (3.3). We first consider the easier situation when *D*=0 and try a solution of the form(4.1)where *r*=|*x*|. The boundary condition *u*=0 at *r*=*R* implies *A*=*LR*^{−k} which automatically gives us the desired condition −*L*<*u*<0 for *r*<*R*. Assuming *α*=0 and inserting equation (4.1), we have(4.2)Thus *k*=2/3, *N*≥2, and the parameter *λ* is determined by(4.3)To see the meaning of this solution given in equation (4.1), we set *u*=*u*_{R}≡*L*(−1+[*r*/*R*]^{2/3}). We have |∇*u*_{R}|^{2}=4*LR*^{−4/3}*r*^{−2/3}/9. Therefore, *u*_{R}∈*W*^{1,p}(*B*_{R}) for any *p*<3*N*. Besides, |∂_{i}∂_{j}*u*_{R}|=*O*(*r*^{−4/3}) for *r* small, which implies ∂_{i}∂_{j}*u*_{R}∈*L*^{p}(*B*_{R}) for *p*<3*N*/4. Note that the classical solutions we obtained below the pull-in threshold in §3 are all smooth. So *u*_{R} with *λ* given in equation (4.3) is a new (non-classical) solution at the limit *D*=0 and *α*=0.

For 0<*λ*<*λ*_{R}, consider the function(4.4)Since lim_{t→1}*f*(*t*)=*λ*_{R}/*λ*>1, we see that *f*(*t*)>1 uniformly in *B*_{R} when *t* is close to 1. Therefore, when *t* is close to 1, we have(4.5)In other words, *tu*_{R} is a subsolution of the equation (3.3) when *D*=0 and *α*=0. Therefore, there is a classical solution *u* satisfying *tu*_{R}<*u*<0 in *B*_{R} which implies for *Ω*=*B*_{R}.

We can apply the above result to general domains. For this purpose, we observe that we can always use a classical solution of the equation over a larger domain to serve as a subsolution of the equation over a smaller domain. In other words, if we use to denote the dependence of the pull-in threshold on the domain *Ω*, we have the monotonicity relation when *Ω*_{1}⊂*Ω*_{2}. Consequently, if *Ω* can be contained in a ball of radius *R*, then the critical value of the equation (3.3) with *D*=0 and *α*=0 over *Ω* satisfies the lower estimate(4.6)

Similarly, we can also get an exact radial solution of equation (2.10) when *D*=0 using the same ansatz equation (4.1). Of course, we still have *k*=2/3 and *N*≥2. However, the parameter *λ* is now replaced by(4.7)where *s*_{N} is the surface area of the unit sphere in ^{N}.

We now turn to another limiting case of equation (3.3) when *T*=0 and *α*=0. With *Ω*=*B*_{R} and *u* given in equation (4.1), we have *k*=4/3, *N*≥3 and(4.8)Note that unfortunately the most interesting situation in which *N*=2 is excluded. The exact solution thus obtained is again not a classical solution, but can be used to derive a lower estimate for the critical threshold as before. If *Ω* is contained in a ball of radius *R*, then the critical value for the equation (3.3) with *T*=0 and *α*=0 has the lower bound given in equation (4.8).

Likewise, for the non-local equation (2.10) with *T*=0, we have(4.9)

We must point out that the exact radial solution just obtained for the equations (3.3) with *T*=0 and *α*=0; and (2.10) with *T*=0, i.e.(4.10)does not satisfy the full Navier boundary condition (4.1). Instead, we have(4.11)everywhere, which suggests that we can use it as a subsolution for the full equation (3.3) even when *α*>0. Indeed, equation (4.11) implies that, if(4.12)then *u*_{R} satisfies the inequality(4.13)where *u*_{R}=0 and Δ*u*_{R}>0 on ∂*B*_{R} and *λ*_{R} is defined by equation (4.8). Hence, *tu*_{R} is a classical subsolution of equation (3.3) (equation (3.10)) for 0<*λ*<*λ*_{R} and *t* (0<*t*<1) close to 1 and, consequently, the existence of a classical solution of equation (3.3) follows. Hence . In particular, if *Ω* is an arbitrary domain which is contained in a ball of radius *R*, then the pull-in threshold for the equation (3.3) over *Ω* has the lower bound(4.14)

## 5. Solution under clamped boundary condition

In this section, we consider equation (2.10) under the homogeneous boundary condition(5.1)where *ν* denotes the outnormal direction on the surface of ∂*Ω*, which corresponds to the case where the capacitive actuator at the boundary is clamped, giving rise to zero vertical displacement and zero slope. Since our equation is fourth-order and non-local, we no longer have the maximum principle and we are unable to obtain a complete description of the solutions in terms of the parameters *α* and *λ* as we had in §§3 and 4. Instead, we are only able to get a small-parameter result as described as follows.

Let *X* be the function space obtained by taking the completion under the norm of *W*^{4,p}(*Ω*) for the set of smooth functions that satisfy the boundary condition (5.1). It is well known that when 4*p*>*N*, the injection is continuous. Use to denote the norm of and consider the ball in defined by(5.2)Introduce a map *M*:→*X*, *M*(*u*)=*v*, by the relation(5.3)It is clear that such a relation is well defined because equation (5.3) has a unique solution *v*∈*X* for any given *u*∈. For the equation *T*Δ*v*−*D*Δ^{2}*v*=*f* (*v*∈*X*), the standard elliptic estimates give us(5.4)where *C*_{0}>0 depends only on *T*, *D* and the domain *Ω*. Applying equation (5.4) to (5.3), we have(5.5)From equation (5.5) and the continuous embedding , we see that when *α* is sufficiently small and *λ* is sufficiently small or *L* is sufficiently large, *v*∈. In other words, *M* maps into itself.

Besides, for *u*_{1}, *u*_{2}∈, let *v*_{1}=*M*(*u*_{1}), *v*_{2}=*M*(*u*_{2}). Then we have the similar estimate(5.6)which implies that when *λ* is small or *L* is large, the map *M*:→ is a contraction. Consequently, *M* has a unique fixed point *u* in that can be obtained in the *n*→∞ limit of the sequence defined by the iterative scheme *u*_{R}=*M*(*u*_{n−1}) (*n*=1, 2, …) starting from any *u*_{0}∈.

In the second-order equation (*D*=0) case, when equation (5.1) is replaced by equation (3.2), we use the function *W*^{2,p}(*Ω*) for *X* where 2*p*>*N*. We see that all the above constructions may be carried over straightforwardly.

## 6. Conclusions

In this paper, we have studied the nonlinear non-local elliptic differential equation(6.1)which models a basic technological procedure in the area of MEMS and NEMS called electrostatic actuation jointly governed physically by elastic deflection and electrostatics, under the influence of circuit series capacitance characterized by *Χ*>0 so that when the circuit series capacitance is negligible, *Χ*=0, the equation reduces to its local limit,(6.2)which further reduces to(6.3)in the zero capacitor-plate thickness limit (*D*=0) and also models the equilibrium state of two neighbouring charged liquid droplets suspended over two rings. For these equations, we have considered the problems of existence and construction of solutions and established the following results.

Under the pinned boundary condition (3.1), for any

*α*≥0, there is a critical satisfying when*α*is small, when*α*≥*λ*_{1}(*T*+*Dλ*_{1})*L*, and , where*λ*_{1}>0 is the first eigenvalue of the Laplace subject to the Dirichlet boundary condition, so that for any , the equation (6.2) has a classical maximal solution*u*_{α,λ}satisfying −*L*<*u*_{α,λ}<0 in*Ω*and such a solution can be constructed by the uniformly defined monotone iterative scheme (3.11) all starting from the zero function. Furthermore, for , there holds*u*_{α,λ′}>*u*_{α,λ″}and as ,*u*_{α,λ}tends in to a weak solution, say*U*, of equation (6.2) at the critical which represents the limiting elastic deflection of the electrostatic actuation plate at the pull-in voltage. In the one-dimensional situation,*U*remains above −*L*and the solution is still a classical solution, which indicates that the capacitor does not collapse even at the pull-in voltage. In the situation when*Ω*is a ball (or disc) in^{N}, say*Ω*=*B*_{R}, the solution*u*_{α,λ}of equation (6.2) for*λ*below is radially symmetric about the centre of*B*_{R}and the weak limit*U*of*u*_{α,λ}as can utmost attain the value −*L*at the centre of*B*_{R}. In other words, for a radially symmetric actuator, the pull-in collapse can only happen at the centre. Besides, in general, if*Ω*is contained in a ball of radius*R*>0, then for*α*satisfying equation (4.12), the critical has the lower estimate given in equation (4.14). As*α*,*λ*→0,*u*_{α,λ}→0 in any standard function space topology over*Ω*, which simply says that elastic deflection of the capacitor plate vanishes whenever electricity is switched off.Under the Dirichlet boundary condition (3.2), similar conclusions hold for equation (6.3) with the modification that (Pelesko & Bernstein 2003) and the weak convergence in the space is replaced by the space . If

*Ω*may be contained in a ball of radius*R*and*α*=0, has the lower estimate given in equation (4.6).Under the pinned boundary condition (3.1), the full non-local equation (6.1) has a classical solution

*u*_{Χ}satisfying −*L*<*u*_{Χ}<0 in*Ω*provided that*λ*,*Χ*, |*Ω*|,*L*satisfy the condition(6.4)where is the critical constant for the solvability of equation (6.2) described in (i) above. In other words, the presence of circuit series capacitance effectively enhances the pull-in voltage. Moreover, if ,*u*_{Χ}approaches (as*Χ*→0) a classical solution of equation (6.2) weakly in*W*^{4,2}(*Ω*) (or in*W*^{2,2}(*Ω*) when*D*=0 and the boundary condition is equation (3.2)).Under the clamped boundary condition (5.1), the full non-local equation (6.1) for any given

*Χ*≥0 has a unique small-amplitude solution when*α*>0 is small and*λ*>0 is small or*L*>0 is large and such a solution can be constructed via the contractive iteration scheme(6.5)and the convergence is achieved in*W*^{4,p}(*Ω*) for*p*>*N*/4 (or*W*^{2,p}(*Ω*) for*p*>*N*/2 if*D*=0 and equation (5.1) is replaced by equation (3.2)). Furthermore, as*α*,*λ*→0, the solution tends to zero in the same corresponding function space, which again means the absence of electricity makes the elastic deflection of the actuator plate vanish as in the pinned boundary situation. Moreover, the approximation scheme (6.5) is also valid for solution subject to the pinned boundary condition (3.1) when*α*is small and*λ*is small or*L*is large.

## Acknowledgments

Fanghua Lin was supported in part by NSF grant DMS-0201443. Yisong Yang was supported in part by NSF grant DMS-0406446.

## Footnotes

- Received September 27, 2006.
- Accepted January 4, 2007.

- © 2007 The Royal Society