Proc. R. Soc. A (2007) 463, 1323–1337 doi:10.1098/rspa.2007.1816 Published online 27 February 2007

Nonlinear non-local elliptic equation modelling electrostatic actuation B Y F ANGHUA L IN 1

AND

Y ISONG Y ANG 2, *

1

Courant Institute of Mathematical Sciences, New York University, New York, NY 10021, USA 2 Department of Mathematics, Polytechnic University, Brooklyn, NY 11201, USA

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. Keywords: electrostatic actuation; singularities; nonlinear methods

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). * Author for correspondence ([email protected]). Received 27 September 2006 Accepted 4 January 2007

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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 smallamplitude 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 q1 and q2 placed at a distance r apart is given in normalized units by FZq1q2/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 q1 and q2 as q1ZqZCVZKq2 and F satisfies FZKC 2V 2/r 2. Imagine one of the plates stretches a small vertical distance dr. Then the work done is F dr which decreases the electric potential stored in the capacitor. As a consequence, the electric potential W can be expressed as C 2V 2 W ZK : ð2:1Þ r In the general situation, when the electric charges are not uniformly distributed as a result of varying distance rZLCu(x), where LO0 is the distance between the two plates in the absence of plate deformation and u(x) is the plate deformation Proc. R. Soc. A (2007)

Equation modelling electrostatic actuation

variable, we need to replace equation (2.1) by ð aV 2 W ZK dx; U L C uðxÞ

1325

ð2:2Þ

where U is a bounded domain in R2 and aO0 is a constant related to permittivity. On the other hand, in the presence of elastic deformation characterized by us0, it is well known that the elastic energy collects contributions from two sectors, i.e. the stretching energy sector given by ð T PZ ð2:3Þ jVuj2 dx; U 2 where TO0 is the tension constant and the bending energy sector given by ð D jDuj2 dx; ð2:4Þ QZ U 2 where DZ2h3Y/3(1Kn2) in which h is the plate thickness, Y is the Young modulus and n is the Poisson ratio. Consequently, the total energy EZPCQCW may be represented as  ð
T D aV 2 2 2 EðuÞ Z jVuj C jDuj K dx; ð2:5Þ 2 L Cu U 2 so that its Euler–Lagrange equation is aV 2 TDuKDD2 u Z ; x 2U: ð2:6Þ ðL C uÞ2 In the zero plate thickness limit DZ0, 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) b Dv Z a C 2 ; x 2U; ð2:7Þ v where a, bO0 are constants reflecting fluid-mechanical and electrostatic properties of the drops, respectively, so that aZ0 corresponds to the situation where uncharged drops are flat. Note also that the equation DvZvKp (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) ð 1 C Zg dx: ð2:8Þ L C uðxÞ U Besides, as a result of the circuit series capacitance Cf 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 Vs, but is instead given by the series circuit formula VZVs/(1CC/Cf). Therefore, in view of equation (2.8), V depends on the deformation variable u according to V Ð s 1 VZ ; ð2:9Þ 1 C c U LCuðxÞ dx Proc. R. Soc. A (2007)

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where cZ1/Cf counts the influence of the circuit series capacitance and we arrive at the non-local equation (Pelesko & Bernstein 2003) l TDuKDD2 u Z ð2:10Þ  2 ; x 2U; Ð 2 1 ðL C uÞ 1 C c U LCuðxÞ dx where lO0 is a constant which is proportional to the supply voltage squared, Vs2 . 3. Solution under pinned boundary condition We first consider equation (2.10) subject to the pinned boundary condition u Z 0; Du Z 0; x 2vU; ð3:1Þ 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 DZ0, the boundary condition is simply the Dirichlet boundary condition u Z 0 on vU: ð3:2Þ 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 Du(x)O0 for x2U. We begin by considering the situation that the influence of the circuit series capacitance is negligible, cZ0. Thus, equation (2.10) becomes l TDuKDD2 u Z a C ; x 2U; ð3:3Þ ðL C uÞ2 where we add a constant aR0 in order to accommodate the equation (2.7) when DZ0. Of course, a classical solution of equation (3.3) also satisfies LCu(x)O0, and other properties u!0 and DuO0 in U still hold. We shall see that it is not hard to construct a classical solution of equation (3.3). We first show that when lO0 is sufficiently large, equation (3.3) has no classical solution. The physical reason is that as the supply voltage Vs becomes sufficiently large, the charged plates undergo a deflection so that they collapse onto each other, rZLCuZ0, and the system fails to operate. Such a situation is called a ‘pull-in’ situation and the level of Vs when pull-in takes place is called the pull-in voltage and the corresponding l value is called the pull-in value, lc. We use l1 to denote the first eigenvalue of the operator KD subject to equation (3.2) and U1 an associated eigenfunction satisfying U1O0 in U. Then TDU1KDD2U1ZKl1(TCDl1)U1. If u is a classical solution of equation (3.3), we add l1(TCDl1)u to both sides of equation (3.3), multiply the resulting relation by U1 and integrate it over U. We obtain ð 0 Z ðTDU1 KDD2 U1 C l1 ðT C Dl1 ÞU1 Þu dx U

ð 

 l Z U1 dx: a C l1 ðT C Dl1 Þu C ð3:4Þ ðL C uÞ2 U Consider the function f(u)Zl1(TCDl1)uCl/(LCu)2 for KL!u%0. Then f(0)Zl/LO0 and f(u)/N as u/KLC. We see that we can find a condition Proc. R. Soc. A (2007)

Equation modelling electrostatic actuation

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so that min ff ðuÞgR 0;

ð3:5Þ

KL!u%0

which implies that equation (3.4) cannot hold and equation (3.3) has no solution. Indeed, the only critical point of f(u) is  1=3 2l u0 Z KL: ð3:6Þ l1 ðT C Dl1 Þ If u 0R0, then f 0 (u)%0 for all KL!u%0 and f(u)Rf(0)O0 (KL!u%0). Therefore, the only situation to be checked is when u 0!0. We have 2=3

ð3:7Þ f ðu 0 Þ Z l1=3 l1 ðT C Dl1 Þ2=3 ð21=3 C 2 K2=3 ÞK l1 ðT C Dl1 ÞL: Requiring f(u 0)R0 in equation (3.7), we arrive at the non-existence condition 4 lR l1 ðT C Dl1 ÞL3 ; c aR 0: ð3:8Þ 27 Similarly, we have another non-existence condition expressed in terms of a, ð3:9Þ aR l1 ðT C Dl1 ÞL; c lR 0: To obtain a solution of equation (3.3), we assume that it has a subsolution u
which satisfies KL!u
%0 in U and the inequalities l ; x 2U; TDu
KDD2 u
R a C ðL C u
Þ2 u
% 0;

Du
R 0;

x 2vU:

ð3:10Þ

With such a function u
, we can use the maximum principle to show straightforwardly that the iterative scheme TDun KDD2 un Z a C un Z 0;

l ðL C unK1 Þ2

Dun Z 0

in U;

on vU;

u 0 Z 0;

ð3:11Þ

gives rise to a monotone sequence {u n} satisfying KL! u
%/% un %/% u 1 % u 0 Z 0:

ð3:12Þ

In particular, the limit uZlim 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 a and l are small enough, equation (3.3) has a subsolution. To see this, we use l1 and U1 to denote the first eigenpair of KD
so that U1Z0 on vU1 and U1O0 in over a large bounded domain U1 containing U U1. Let 3O0 be a small number so that 3U1!L/2. Then (TDKDD2)(K3U1)Z
when l1(TCDl1)3U1, which can be made greater than aCl/(LK3U1)2 over U a and l are sufficiently small. Of course, K3U1!0 and D(K3U1)Z3l1U1O0 on vU. Therefore, K3U1 is a subsolution of equation (3.3) satisfying KL!3U1!0. As a consequence, a solution of equation (3.3) exists, which can be constructed by the iterative scheme equation (3.11). Proc. R. Soc. A (2007)

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For given aR0, we define La Z flO 0j ð3:3Þ has a classical solution at lg: ð3:13Þ We show that La is either empty or a nontrivial bounded interval. In fact, if La is not empty and l02L, we can easily prove that (0, l0]3La. Indeed, let ul0 be a solution of equation (3.3) at lZl0. Then, for any 0!l!l0, we see that u 0 is a subsolution of equation (3.3) at l. Therefore it follows that l2La and the claim is verified. As a by-product, we have seen in the above that if l 0 , l 00 2La and l 0 !l 00 , then the corresponding maximal solutions u l 0 and u l 00 satisfy ul0 O ul00

in U:

ð3:14Þ

Physically, this is a natural relation because higher supply voltage results in greater elastic deformation or deflection. /, then If Las0 lac h supfl 2La gO 0; ð3:15Þ which determines the pull-in voltage. We now study the limit of u l as l/ laK c . For simplicity, we first consider the situation when DZ0 and see that, after we rescale the tension constant T to unity to save some notation, u l satisfies l Dul Z a C : ð3:16Þ ðL C ul Þ2 The monotonicity condition (3.14) implies that there is a well-defined function U so that U ðxÞ Z limaK ul ðxÞ; KL% U ðxÞ! 0; x 2U: ð3:17Þ l/lc

In order to find the meaning of U, we again use (l1, U1) to denote a first eigenpair of KD under the boundary condition (3.2) for which U1 satisfies 0!U1%1. Multiplying equation (3.16) by U1 and integrating, we have  ð ð
lU1 aU1 C ð3:18Þ dx Z l1 jul U1 jdx % l1 LjUj: ðL C ul Þ2 U U Since U1 only vanishes at vU, equation (3.18) gives us the uniform bound ð l l1 LjUj h C ðKÞ; ð3:19Þ dx % 2 inffU1 ðxÞjx 2Kg K ðL C ul Þ for any compact subset K of U. For any subdomain U033U (i.e. U0 3U), insert another subdomain U1 satisfying U033U133U. Let h be a cut-off function satisfying hZ1 on U0, supp(h)3U1 and 0%h%1. Multiplying equation (3.16) by h2u l and integrating, we have  ð ð ð
lh2 ul 2 2 2 dx ZK ah ul C h jVu j dx K 2hul Vh$Vul dx: ð3:20Þ l ðL C ul Þ2 U U U Therefore, in view of the Schwartz inequality, we have ð ð ð Ll 1 2 2 2 dx R h jVul j dx K2L jVhj2 dx: 2 2 U1 U1 ðL C ul Þ U1 Proc. R. Soc. A (2007)

ð3:21Þ

Equation modelling electrostatic actuation

In particular

ð

ð l 2 jVul j % 2L dx C 4L jVhj2 dx: 2 U0 U1 ðL C ul Þ U1 Combining equations (3.19) and (3.22), we arrive at the uniform bound 2

ð

kVul kL2 ðU0 Þ % C1 ðU0 Þ;

c U0 33 U:

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ð3:22Þ

ð3:23Þ

Consequently, we obtain that, for any U033U, there holds lim ul Z U

l/laK c

weakly in W 1;2 ðU0 Þ:

ð3:24Þ

As a corollary, this shows that U is a distributional solution of the equation (3.16) at the critical (pull-in) lac , lac DU Z a C in U; ð3:25Þ ðL C U Þ2 a so that U 2W 1;2 loc ðUÞ and KL%U!0 in U. In other words, lc is attainable (i.e. a a L Z ð0; lc ) for distributional solutions in any dimensions. Note that the Hausdorff dimension of the zero set S of a W 1,p(U) solution vZLCU of the critical equation (3.25) has been studied in Jiang & Lin (2004) where S is defined by
  ð  1  S Z x 0 2U lim jvðxÞjdx Z 0 : ð3:26Þ r/0 jBr ðx 0 Þj Br ðx 0 Þ In the special case, when we consider a one-dimensional situation, the distributional solution of equation (3.25) is continuous and S is the zero set of LCU in the classical sense. Using the result of Jiang & Lin (2004), we must have /. In other words, UOKL everywhere and we obtain a classical solution at lac SZ0 as well. Therefore, in dimension one, we have a complete description of the set La, La Z ð0; lac : ð3:27Þ N Another special case is when U is a disc or ball in R . Assume UZBR (a ball of radius R centred at the origin). Using the symmetry theorem of Gidas et al. (1979), we know that the solution u l 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 nondecreasing along any radial direction. Using the upper bound equation (3.19), we see that U can only attain the value KL at the origin. In other words, when the domain U 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 pullin voltage at all. In order to derive similar results for the equation (3.3), we set vZLCu. Hence 0!v!L and the maximum principle implies that TvKDDv!TL in U3RN. Now define LT jxj2 $ : ð3:28Þ w ZvC D 2N Then w satisfies the inequality DDwO TvO 0: Proc. R. Soc. A (2007)

ð3:29Þ

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In other words, w is a bounded positive subharmonic function in U whose gradient necessarily has L2-local bounds. In fact, multiplying equation (3.29) by h2w where h is a smooth cut-off function used earlier and integrating by parts, we have ð ð h2 jVwj2 dx !K2 hwVh$Vw dx: ð3:30Þ U

U

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 h and the uniform bound of w over U. Hence the claim follows. Therefore, we have ð h2 jVuj2 dx % C ðhÞ: ð3:31Þ U

On the other hand, as a consequence of equation (3.4), we also note that by replacing equation (3.19), we have ð l l ðT C Dl1 ÞLjUj h C ðKÞ; ð3:32Þ dx % 1 2 inffU1 ðxÞjx 2Kg K ðL C uÞ for any compact subset K of U. 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 lac . For this purpose, we multiply equation (3.3) by h4u and integrate it twice by parts to get ð ðh4 jDuj2 C 8h3 Vh$VuDu C 12h2 ujVhj2 Du C 4h3 uDhDuÞdx U ð ð ð T l h4 u a 4 2 3 ZK ðh jVuj C 4h uVh$VuÞdx K dx K h4 u dx: ð3:33Þ 2 D U D U ðL C uÞ D U Applying equations (3.31) and (3.32), and the Schwartz inequality in equation (3.33), we obtain the uniform interior bound ð h4 jDuj2 dx % C1 ðhÞ: ð3:34Þ U

As before, we use u l to denote the classical solution obtained at lO0 which is below the pull-in threshold value lac . Then equations (3.31) and (3.34), and standard elliptic theory (Gilbarg & Trudinger 1977) imply that, for any subdomain U033U, we have the uniform bound kul kW 2;2 ðU0 Þ % C ðU0 Þ: In particular, there is a function U

2W 2;2 loc ðUÞ

lim ul Z U

l/laK c

ð3:35Þ

such that

weakly in W 2;2 loc ðUÞ:

ð3:36Þ

Consequently, U is a weak solution of the equation (3.3) at the critical parameter lac , lac TDU KDD2 U Z a C in U; ð3:37Þ ðL C U Þ2 and in the sense of weak solutions, the critical value lac is again attainable. Besides, when NZ1 (a one-dimensional device again), the result of Jiang & Lin (2004) implies that the zero set of the function vZLCU is empty. Hence, in this situation, Proc. R. Soc. A (2007)

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Equation modelling electrostatic actuation

the solution of equation (3.37) is a classical solution satisfying KL!U!0 in U as in the second-order (DZ0) 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 lac O 0 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 l ! lac : ð1 C cjUj=LÞ2

ð3:38Þ

In fact, we can consider the closed convex set S 3L2 ðUÞ defined by S Z fw 2L2 ðUÞju 0 % w% 0

ð3:39Þ

a:e: on Ug;

where u 0 is the (maximum) classical solution of the equation TDuKDD2 u Z a C

l ; ðL C uÞ ð1 C cjUj=LÞ2

ð3:40Þ

2

whose existence is guaranteed by the condition (3.38). For any w2S, since KL!u 0!w%0, we see that u 0 satisfies TDu 0 KDD2 u 0 R a C

2



l

ðL C u 0 Þ 1 C c

Ð

1 U LCw

dx

2 :

ð3:41Þ

Therefore u 0 is a subsolution of the equation TDuKDD2 u Z a C

l  2 ; Ð 1 ðL C uÞ 1 C c U LCw dx 2

ð3:42Þ

subject to the boundary condition (3.1) (or (3.2) if DZ0). As a consequence of our earlier discussion, equation (3.42) has a unique maximal solution, say u, satisfying u 0%u!0 in U. In particular, we have thus defined a map M:S/S with M(w)Zu. Since the functions in M(S) are uniformly bounded away from KL, we can show by L2-estimates that M(S) is a bounded subset of W 4,2(U) (or W 2,2(U) if DZ0). Hence M is a completely continuous map. As a result, M has a fixed point in S, which is of course a classical solution of equation (2.10). Let the solution of equation (2.10) obtained above be denoted by u c (which may not be unique). Suppose now 0! l! lac . We can take u 0 in equation (3.39) to be the classical solution of equation (3.40) with setting cZ0. Then our discussion shows that the family {u c}cO0 is bounded in W 4,2(U) (or W 2,2(U) if DZ0). Therefore, in the sense of subsequence, we see that lim uc Z a solution of ð3:3Þ weakly in W 4;2 ðUÞ ðor W 2;2 ðUÞ if D Z 0Þ:

c/0

ð3:43Þ

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 cZ0. The solution obtained for the equation (2.10) or (3.3) is denoted by u a,l. We have seen that when a, l are sufficiently small, u a,l are uniformly bounded away by the classical solutions of equation (3.3) corresponding to the larger values of a and l. As a consequence, when we take a, l/0, we see immediately through the Proc. R. Soc. A (2007)

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well-known elliptic estimates that u a,l/0 in any standard function space topology over U, which says correctly that when the applied voltage is switched off, the elastic deflection vanishes in the special case where aZ0.

4. Symmetric solutions and applications Using the method of moving planes as in Gidas et al. (1979), we can show that when UZBR (the ball in RN of radius RO0 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 lac . Let UZBR in equation (3.3). We first consider the easier situation when DZ0 and try a solution of the form u ZKL C Ar k ; AO 0; ð4:1Þ Kk which where rZjxj. The boundary condition uZ0 at rZR implies AZLR automatically gives us the desired condition KL!u!0 for r!R. Assuming aZ0 and inserting equation (4.1), we have l ð4:2Þ TAr kK2 ðk½k K1 C k½N K1Þ Z 2 2k : Ar Thus kZ2/3, NR2, and the parameter l is determined by     2 4 2 4 3 K2 3 NK A T Z N K L R T: l Z lR h ð4:3Þ 3 3 3 3 To see the meaning of this solution given in equation (4.1), we set uZu RhL(K1C [r/R]2/3). We have jVu Rj2Z4LRK4/3r K2/3/9. Therefore, u R2W 1,p(BR) for any p!3N. Besides, jvi vju RjZO(r K4/3) for r small, which implies vi vju R2Lp(BR) for p!3N/4. Note that the classical solutions we obtained below the pull-in threshold in §3 are all smooth. So u R with l given in equation (4.3) is a new (non-classical) solution at the limit DZ0 and aZ0. For 0!l!lR, consider the function l t=ðL C uR Þ2 : ð4:4Þ f ðtÞ Z R l=ðL C tuR Þ2 Since limt/1f(t)ZlR/lO1, we see that f(t)O1 uniformly in BR when t is close to 1. Therefore, when t is close to 1, we have lR t l TDðtuR Þ Z O : ð4:5Þ 2 ðL C uR Þ ðL C tuR Þ2 In other words, tuR is a subsolution of the equation (3.3) when DZ0 and aZ0. Therefore, there is a classical solution u satisfying tuR!u!0 in BR which implies lR % l0c for UZBR. Proc. R. Soc. A (2007)

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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 lac ðUÞ to denote the dependence of the pull-in threshold on the domain U, we have the monotonicity relation lac ðU1 ÞR lac ðU2 Þ when U13U2. Consequently, if U can be contained in a ball of radius R, then the critical value l0c of the equation (3.3) with DZ0 and aZ0 over U satisfies the lower estimate   2 4 3 K2 0 N K L R T; N R 2: lc R lR Z ð4:6Þ 3 3 Similarly, we can also get an exact radial solution of equation (2.10) when DZ0 using the same ansatz equation (4.1). Of course, we still have kZ2/3 and NR2. However, the parameter l is now replaced by    2 2 4 cRN sN K2 l Z lR h N K LR T L C ; N R 2; ð4:7Þ 3 3 ðN K2=3Þ where sN is the surface area of the unit sphere in RN. We now turn to another limiting case of equation (3.3) when TZ0 and aZ0. With UZBR and u given in equation (4.1), we have kZ4/3, NR3 and 8 ð4:8Þ l Z lR h DL3 R K4 ð9½N K1½N K3 C 6½N K1K5Þ: 81 Note that unfortunately the most interesting situation in which NZ2 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 l0c as before. If U is contained in a ball of radius R, then the critical value l0c for the equation (3.3) with TZ0 and aZ0 has the lower bound given in equation (4.8). Likewise, for the non-local equation (2.10) with TZ0, we have  2 8 cRN sN K4 l Z lR h DLR ð9½N K1½N K3 C 6½N K1K5Þ L C : ð4:9Þ 81 ðN K4=3Þ We must point out that the exact radial solution just obtained for the equations (3.3) with TZ0 and aZ0; and (2.10) with TZ0, i.e.  r 4=3 uR ZKL C L ; ð4:10Þ R does not satisfy the full Navier boundary condition (4.1). Instead, we have   4 2 K2=3 K4=3 DuR Z LR NK r O 0; ð4:11Þ 3 3 everywhere, which suggests that we can use it as a subsolution for the full equation (3.3) even when aO0. Indeed, equation (4.11) implies that, if   4 2 a K1=2 LR ð4:12Þ NK O ; 3 3 T then u R satisfies the inequality lR TDuR KDD2 uR O a C in BR ; ð4:13Þ ðL C uR Þ2 Proc. R. Soc. A (2007)

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where u RZ0 and Du RO0 on vBR and lR is defined by equation (4.8). Hence, tuR is a classical subsolution of equation (3.3) (equation (3.10)) for 0!l!lR and t (0!t!1) close to 1 and, consequently, the existence of a classical solution of equation (3.3) follows. Hence lac ðBR ÞR lR . In particular, if U is an arbitrary domain which is contained in a ball of radius R, then the pull-in threshold lac for the equation (3.3) over U has the lower bound 8 lac R lR Z DL3 R K4 ð9½N K1½N K3 C 6½N K1K5Þ; N R 3: ð4:14Þ 81 5. Solution under clamped boundary condition In this section, we consider equation (2.10) under the homogeneous boundary condition vu u Z 0; Z 0; x 2vU; ð5:1Þ vn where n denotes the outnormal direction on the surface of vU, 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 a and l 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(U) for the set of smooth functions that satisfy the boundary condition
is continuous. (5.1). It is well known that when 4pON, the injection X / CðUÞ
and consider the ball B in C ðUÞ
defined by Use k$k to denote the norm of C ðUÞ
kuk! L=2g: B Z fu 2CðUÞj ð5:2Þ Introduce a map M:B/X, M(u)Zv, by the relation l TDvKDD2 v Z a C  ð 2 ðL C uÞ 1 C c

ð5:3Þ 2 : 1 dx U L Cu It is clear that such a relation is well defined because equation (5.3) has a unique solution v2X for any given u2B. For the equation TDvKDD2vZf (v2X ), the standard elliptic estimates give us kvkX % C0 kf kLp ðUÞ ; ð5:4Þ where C0O0 depends only on T, D and the domain U. Applying equation (5.4) to (5.3), we have   4l ð5:5Þ kvkX % C0 a C 4 jUj: L
we see that when From equation (5.5) and the continuous embedding X / C ðUÞ, a is sufficiently small and l is sufficiently small or L is sufficiently large, v2B. In other words, M maps B into itself. Besides, for u 1, u 22B, let v 1ZM(u 1), v 2ZM(u 2). Then we have the similar estimate   16l 2 kv1 K v2 kX % C0 5 1 C 2 cjUj jUj ku 1 K u 2 k; ð5:6Þ L L Proc. R. Soc. A (2007)

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which implies that when l is small or L is large, the map M:B/B is a contraction. Consequently, M has a unique fixed point u in B that can be obtained in the n/N limit of the sequence defined by the iterative scheme u RZM(u nK1) (nZ1, 2, .) starting from any u 02B. In the second-order equation (DZ0) case, when equation (5.1) is replaced by equation (3.2), we use the function W 2,p(U) for X where 2pON. 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 l TDuKDD2 u Z a C ð6:1Þ  2 ; x 2U; Ð 1 ðL C uÞ2 1 C c U LCuðxÞ dx 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 cO0 so that when the circuit series capacitance is negligible, cZ0, the equation reduces to its local limit, l TDuKDD2 u Z a C ; x 2U; ð6:2Þ ðL C uÞ2 which further reduces to l ; x 2U; ð6:3Þ TDu Z a C ðL C uÞ2 in the zero capacitor-plate thickness limit (DZ0) 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. (i) Under the pinned boundary condition (3.1), for any aR0, there is a critical lac R 0 satisfying lac O 0 when a is small, lac Z 0 when aRl1(TCDl1)L, and lac % 4l1 ðT C Dl1 ÞL3 =27, where l1O0 is the first eigenvalue of the Laplace subject to the Dirichlet boundary condition, so that for any l 2ð0; lac Þ, the equation (6.2) has a classical maximal solution u a,l satisfying KL!u a,l!0 in U and such a solution can be constructed by the uniformly defined monotone iterative scheme (3.11) all starting from the zero function. Furthermore, for 0! l 0 ! l 00 ! lac , there holds u a,l 0 Ou a,l 00 and as l/ lac , u a,l tends in W 2;2 loc ðUÞ to a weak solution, say U, of equation (6.2) at the critical lac which represents the limiting elastic deflection of the electrostatic actuation plate at the pull-in voltage. In the one-dimensional situation, U remains above KL 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 U is a ball (or disc) in RN, say UZBR, the solution u a,l of equation (6.2) for l below lac is radially symmetric about the centre of BR and the weak limit U of u a,l as l/ lac can utmost attain the value KL at the centre of BR. In other words, for a radially symmetric Proc. R. Soc. A (2007)

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actuator, the pull-in collapse can only happen at the centre. Besides, in general, if U is contained in a ball of radius RO0, then for a satisfying equation (4.12), the critical lac has the lower estimate given in equation (4.14). As a, l/0, u a,l/0 in any standard function space topology over U, which simply says that elastic deflection of the capacitor plate vanishes whenever electricity is switched off. (ii) Under the Dirichlet boundary condition (3.2), similar conclusions hold for equation (6.3) with the modification that lac % 4l1 TL3 =27 (Pelesko & Bernstein 2003) and the weak convergence in the space W 2;2 loc ðUÞ is replaced by the space W 1;2 ðUÞ. If U may be contained in a ball of radius R and aZ0, l0c has loc the lower estimate given in equation (4.6). (iii) Under the pinned boundary condition (3.1), the full non-local equation (6.1) has a classical solution u c satisfying KL!u c!0 in U provided that l, c, jUj, L satisfy the condition l ! lac ; ð6:4Þ ð1 C cjUj=LÞ2 where lac 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 0! l! lac , u c approaches (as c/0) a classical solution of equation (6.2) weakly in W 4,2(U) (or in W 2,2(U) when DZ0 and the boundary condition is equation (3.2)). (iv) Under the clamped boundary condition (5.1), the full non-local equation (6.1) for any given cR0 has a unique small-amplitude solution when aO0 is small and lO0 is small or LO0 is large and such a solution can be constructed via the contractive iteration scheme l TDun KDD2 un Z a C  2 ; x 2U; Ð 1 ð6:5Þ dx ðL C unK1 Þ2 1 C c U LCunK 1 ðxÞ u 0 Z 0; and the convergence is achieved in W 4,p(U) for pON/4 (or W 2,p(U) for pON/2 if DZ0 and equation (5.1) is replaced by equation (3.2)). Furthermore, as a, l/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 a is small and l is small or L is large. Fanghua Lin was supported in part by NSF grant DMS-0201443. Yisong Yang was supported in part by NSF grant DMS-0406446.

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