CHAPTER 1

Introduction

Condensed information storage in miniature computers, infinitesimal machin-ery for space exploration, microscopic surgical tools, and modern telecommuni-cations are only a few of the vast number of applications that lie at the roots ofmicrosystem technology. Richard Feynman may not have been the only one toanticipate the need to develop this important area of modern technology, but he isdefinitely one of the pioneers in describing possible ways of locomotion for suchmicrodevices, by advocating for techniques and ideas based on fundamental prin-ciples of physics, chemistry, and biology, and ranging from electrostatic actuationto quantum computation at the atomic electron levels [68, 69].

Microelectromechanical systems (MEMS) and nanoelectromechanical systems(NEMS) are now a well-established sector of contemporary technology. A com-prehensive overview of their rapidly developing field can be found in the relativelyrecent monograph of Pelesko and Bernstein [140], which is a good source of infor-mation about the various areas of applications of MEMS. It also contains justifi-cations and derivations of the fundamental partial differential equations that modelsuch devices. More complicated models have also been proposed by E. M. Abdel-Rahman, M. I. Younis, and A. H. Nayfeh (see, for example, [1, 137, 162]).

It is the mathematical analysis of some of these equations that concerns us inthis monograph. We shall therefore settle in this introduction on a quick derivationof one of the most basic PDEs modeling electrostatic MEMS devices, and thenproceed towards stating some of the problems that interest applied analysts andengineers, while giving some numerical and heuristic evidence for various conjec-tures that will be addressed rigorously throughout this book.

1.1. Electrostatic Actuations and Nonlinear PDEs

A key component of some MEMS systems is the simple idealized electrostaticdevice shown in Figure 1.1. The upper part of this device consists of a thin and de-formable microplate that is held fixed along its boundary @�, where� is a boundeddomain in R2. The dielectric elastic microplate lies above a parallel rigid groundedplate, say at level z D �d , and has an upper surface that is coated with a negligi-bly thin metallic conducting film. When a voltage V is applied to the conductingfilm, it deflects towards the bottom plate, and if the applied voltage V is increasedbeyond a certain critical value V �, then the mechanical forces defined by the de-formable plate can no longer resist the opposing electrostatic force, thereby leadingit to touch the grounded plate. The steady state of the system is then lost, and wehave a snap-through at a finite time creating the so-called pull-in instability.

1

2 1. INTRODUCTION

FIGURE 1.1. The simple electrostatic MEMS device.

A mathematical model of this physical phenomenon when the plate is one di-mensional, i.e., a beam of length L, already leads to the following interesting par-tial differential equation for its dimensionless dynamic deflection:

@2u

@t2C a

@u

@tD .˛1kuxk

22 C T /

@2u

@x2� B

@4u

@x4�˛2f .t/

2

.d C u/2;

u.0; t/ D u.L; t/ D 0;@u

@x.0; t/ D

@u

@x.L; t/ D 0;

which was already derived by several authors. All the constants above—a, ˛1, T ,B , and ˛2—are assumed to be nonnegative. See for example, the survey paperof Marquès-Castello-Shkel [63], Pelesko-Bernstein [140], or [20, 73, 152] and thereferences therein.

The case of a capacitively actuated, two-dimensional microplate is much morecomplicated as described by Nayfeh et al. in [137], especially if one accounts formoderately large deflections, since one needs to use the dynamic analogue of thevon Kármán equations. We shall therefore only consider the case where the defor-mations are small, so that each material point moves vertically over its referenceposition, and so that the material response is essentially linear. In this case thedynamic deflection u D u.x; t/ of the membrane on a bounded domain� in R2 isassumed to satisfy the following evolution equation:

�A@2u

@t2C a

@u

@tD T�u � B�2u �

C 2V 2

.d C u/2for x 2 �; t > 0;(1.1a)

� 1 < u.x; t/ � 0 for x 2 �; t > 0;(1.1b)

u.x; t/ D B@u

@�.x; t/ D 0 for x 2 @�; t > 0;(1.1c)

u.x; 0/ D 0 and ut .x; 0/ � 0 for x 2 �;(1.1d)

where �2 WD ��.�� � / denotes the biharmonic operator, � denotes the outwardpointing unit normal to @�, and

1.1. ELECTROSTATIC ACTUATIONS AND NONLINEAR PDES 3

� � is the mass density per unit volume of the membrane and A is its thick-ness,� a is a damping intensity,� T is the tension constant in the stretching component of the energy,� B accounts for the bending energy and is given by B D 2A3Y

3.1��2/where

Y is the Young modulus and � is the Poisson ratio,� the uniformly distributed charges over the membrane and the bottom plate

are subjected to a capacitive influence of capacitance C and a fixed elec-tric voltage V ,� the initial condition in (1.1d) assumes that the membrane is initially un-

deflected and the voltage is suddenly applied to the upper surface of themembrane at time t D 0,� the boundary condition (1.1c) reflects the fact that the membrane is held

fixed along its boundary @� on the level z D 0.

Even then we have neglected several factors in the system. Indeed, while we in-clude an external viscous damping (linked to air friction), we have disregardedinternal structural damping generated by the molecular interaction in the materialdue to deformations; see [63]. We have also assumed the tension to be constant byconsidering only a small aspect ratio d

L� 1.

There are several issues that must be considered in the actual design of MEMSdevices. Typically one of the primary goals is to achieve the maximum possiblestable deflection before touchdown occurs, which is referred to as pull-in distance(cf. [99, 139]). Another consideration is to increase the stable operating rangeof the device by improving the pull-in voltage V � subject to the constraint thatthe range of the applied voltage is limited by the available power supply. Suchimprovements in the stable operating range are important for the design of certainMEMS devices, such as microresonators. One way of achieving larger values ofV � while simultaneously increasing the pull-in distance consists of introducing aspatially varying dielectric permittivity "2.x/ of the membrane. We shall see inSection 1.3 that the equation becomes of the form

(1.2) �A@2u

@t2C a

@u

@tD T�u � B�2u �

�"0

"2.x/.d C u/2for x 2 �; t > 0;

where "0 is the permittivity of free space, and � is a positive constant that is pro-portional to the square of the supply voltage.

In other situations, cf. [140, 141, 142], the capacitance C of the actuator maydepend on the deformation variable u according to the relation

(1.3) C D �

Z�

1

d C u.x/dx:

As a result, the voltage drop V at the actuator can no longer be kept at the constantsupply voltage Vs , but is instead given by the series circuit formula V D Vs

.1CC=Cf /

where Cf is the circuit series capacitance. Therefore, in view of (1.3), V depends

4 1. INTRODUCTION

on the deformation variable u according to

V DVs

.1C �R�

1dCu.x/

dx/;

where � D �Cf

reflects the influence of the circuit series capacitance, and weeventually arrive at the nonlocal equation

(1.4) �A@2u

@t2C a

@u

@tD T�u � B�2u �

�

.d C u/2.1C �R�

1dCu.x/

dx/2;

where � D C 2V 2s > 0 is a constant that is proportional to the square of the supplyvoltage Vs .

1.2. Derivation of the Model for Homogeneous Systems

A typical MEMS device is subject to electrostatic, elastic, and dynamic forces.We first deal with the case of a membrane with constant permittivity profile.

1.2.1. Analysis of the Simplified Electrostatic Problem. The Coulomb lawstates that the electrostatic force F between two charges q1 and q2 placed at adistance r apart is given in normalized units by

F Dq1q2

r2:

If the two charges are uniformly distributed over two parallel plates subject to acapacitive influence of capacitance C and a fixed electric voltage V , then we canwrite q1 and q2 as q1 D q D CV D �q2 and F becomes F D �C 2V 2=r2. Ifnow one of the plates stretches a small vertical distance dr , then the work done isF dr , which decreases the electric potential stored in the capacitor. As a conse-quence, the electric potentialW can be expressed as

(1.5) W D �C 2V 2

r:

In the more general situation, when the electric charges are not uniformly dis-tributed as a result of a varying distance r D 1Cu.x/, where d D 1 is the distancebetween the two plates in the absence of plate deformation, and u.x/ is the platedeformation variable, the electric potential (1.5) becomes

(1.6) EW .u/ D �

Z�

C 2V 2

1C u.x/dx;

where � is the domain in R2 limiting the membrane.

1.2. DERIVATION OF THE MODEL FOR HOMOGENEOUS SYSTEMS 5

1.2.2. Analysis of Elastic Forces. On the other hand, in the presence of aplate deformation characterized by u ¤ 0, the elastic energy has two components:the stretching energy sector given by

(1.7) ES .u/ D

Z�

T

2jruj2 dx;

where T > 0 is the tension constant, and the bending energy given by

(1.8) EB.u/ D

Z�

B

2.�u/2 dx;

where B D 2A3Y3.1��2/

, in which A is the plate thickness, Y is the Young modulus,and � is the Poisson ratio.

Indeed, the stretching energy in the elastic membrane is proportional to thechanges in the area of the membrane from its unstretched configuration. Since weassume the membrane is held fixed at its boundary, we may then write it as

(1.9) stretching energy WD TZ�

q1C jruj2 dx;

where the proportionality constant T is simply the tension in the membrane. Ourexpression for ES is then nothing but the linearization of the stretching energy. Onthe other hand, the bending energy is assumed to be proportional to the linearizedcurvature of the membrane, hence the expression for EB , where the constant B isthe flexural rigidity of the membrane.

Consequently, the total energy E D ES CEB C EW may be represented as

(1.10) E.u/ D

Z�

�T

2jruj2 C

B

2j�uj2 �

C 2V 2

1C u

�dx;

so that its Euler-Lagrange equation is

(1.11) T�u � B�2u �C 2V 2

.1C u/2D 0 in �:

1.2.3. Analysis of the Dynamic Problem. For the dynamic deflection u.x; t/of the membrane, we use Newton’s second law to write

(1.12) �A@2u

@t2DX

forces;

where � is the mass density per unit volume of the membrane andA is its thickness.As to the forces, we combine the superposition of the elastic and electrostatic forcesassembled in (1.11) with a damping force Fd that is linearly proportional to thevelocity, that is,

(1.13) Fd D �a@u

@t:

6 1. INTRODUCTION

FIGURE 1.2. The key component of an electrostatic MEMS.

We finally obtain the equation

(1.14) �A@2u

@t2D �a

@u

@tC T�u � B�2u �

C 2V 2

.1C u/2:

1.3. MEMS Models with Variable Permittivity Profiles

We now allow the elastic membrane to exhibit a spatial variation reflecting avarying dielectric permittivity. For that, we first need to formulate the equationsgoverning the electrostatic field within the device components and around it.

1.3.1. Tailored Permittivity Profiles. For simplicity, we shall consider a two-dimensional case where both the membrane and the ground plate have width andlength equal to L, and, in the undeformed state, are separated by a gap of length d ;see Figure 1.2. We assume that the ground plate, located now at z0 D 0, is a perfectconductor. The elastic membrane is assumed to be of uniform thickness A D 2 .The deflection of the membrane is specified by the deflection of its center plane,located at z0 D bu.x0; y0/. Hence the top surface is located at z0 D bu.x0; y0/C ,while the bottom of the membrane is located at z0 Dbu.x0; y0/ � .

We assume that the elastic membrane is held at potential V , while the fixedground plate is held at zero potential. The electrostatic potential 1 in the wholeregion D0 between the elastic plate, the ground plate, and the lateral region sur-rounding the device must satisfy the Laplace equation on D0, with boundary con-ditions 1 D 0 on the ground plate. In other words,

�1 D 0 on D0;(1.15)

1.x0; y0; 0/ D 0 in �0:(1.16)

Denote by "2.x0; y0/ the varying permittivity of the membrane; the potential 2inside the membrane must then satisfy

r � ."2r2/ D 0 forbu.x0; y0/ � � z0 �bu.x0; y0/C ;(1.17)

2.x0; y0;buC / D V in �0:(1.18)

1.3. MEMS MODELS WITH VARIABLE PERMITTIVITY PROFILES 7

At the interface between the gap and the membrane, the potential and the displace-ment fields must be continuous; hence we impose

(1.19) 1.x0; y0;bu.x0; y0/ � / D 2.x0; y0;bu.x0; y0/ � /;

and

(1.20) "2.x0; y0/r2 � n D "0r1 � n at z0 Dbu.x0; y0/ � ;

where "0 is the permittivity of free space and n is an outward unit normal.Now, introduce dimensionless variables

(1.21) 1 D1

V; 2 D

2

V; u D

bud; x D

x0

L; y D

y0

L; z D

z0

d;

to see that the electrostatic problem reduces to

@2 1

@z2C "2

�@2 1

@x2C@2 1

@y2

�D 0; 0 � z � u �

d;(1.22a)

"2.x; y/@2 2

@z2C "2

�@

@x

�"2.x; y/

@ 2

@x

�C

@

@y

�"2.x; y/

@ 2

@y

��D 0;(1.22b)

u �

d� z � uC

d;

1 D 0 on the ground plate z D 0;(1.23)

2 D 1 on the upper membrane surface z D uC

d;(1.24)

together with the continuity property of the potential and the displacement fieldsacross z D u � �

d, that is,

1

�x; y; u.x; y/ �

d

�D 2

�x; y; u.x; y/ �

d

�;(1.25)

"2."2.x; y/r? 2 � "0r? 1/r?u D "2.x; y/@ 2

@z� "0

@ 1

@z(1.26)

on the set z D u.x; y/ �

d:

Here , given by 1, 2, is the dimensionless potential scaled with respect to theapplied voltage V , and " � d=L is the so-called aspect ratio of the device. Weare using the notation�? to distinguish the differentiation in the domain variables.x; y/ 2 � from the differentiation in the space variables .x; y; z/ in � �R.

In general, one has little hope of finding an exact solution from (1.22a)–(1.26). However, we can simplify the system by examining a restricted parameterregime. In particular, we consider the small-aspect ratio limit " � d

L� 1. Phys-

ically, this means that the lateral dimensions of the device in Figure 1.2 are largercompared to the size of the gap between the undeflected membrane and groundplate. In this case, equation (1.22) gives @2 1=@z2 D 0 and @2 2=@z2 D 0. It

8 1. INTRODUCTION

follows that 1 and 2 are linear in z. Combined with the continuity of acrossz D w � �

d, it means that there exists a function 0.x; y/ such that

(1.27) D

( 1 D 0

zu��

; 0 � z � u � ;

2 D 1C.1� 0/2�

.z � .uC //; u � � z � uC :

Now to ensure that the displacement field is continuous across z D u� , we needfrom (1.26) that

"0@

@z

ˇ̌̌̌�

D "2.x; y/@

@z

ˇ̌̌̌C

on z D u.x; y/ �

d;

where the plus or minus signs indicate that @ @z

is to be evaluated on the upper orlower side of the bottom surface z D u � of the membrane, respectively. Thiscondition determines 0 in (1.27)

(1.28) 0.x; y/ D

�1C

2

u.x; y/ �

�"0

"2.x; y/

���1:

From (1.27) and (1.28), we observe that the electric field in the z-direction insidethe membrane is independent of z and is given by

(1.29)@

@zD

"0

"2.u � /

�1C

2

u �

"0

"2

��1

"0

"2ufor � 1:

In other words, in the small-aspect ratio limit "� 1 and for a very thin membrane � 1 and a ground plate located at z D �1, we have that the electric field in thez-direction at any point of the membrane is given by

(1.30)@

@zD

"0

"2.x; y/.1C u/:

1.3.2. MEMS Models with a Varying Permittivity Profile. In cases wherethe membrane has a varying permittivity profile, we have in view of (1.14) thefollowing system of equations for the membrane’s deflectionbu and the potential ,which couples the solution of the elastic problem to the solution of the electrostaticproblem

(1.31) �A@2bu@t 02C a

@bu@t 0� T�?buC B�2?bu D �"22 jr.x0; y0;bu/j2:

We are again using the notation �? to distinguish the Laplacian in the domainvariables .x0; y0/ 2 �0 from the Laplacian in the space variables .x0; y0; z0/ in�0 � R. The term on the right-hand side of (1.31) denotes the force on the elasticmembrane induced by the electric field with potential .x0; y0; z0/, where "2 is thepermittivity of the membrane. We have assumed that such a force is proportionalto the norm squared of the gradient of the potential. A derivation of such sourceterm may be found in [114]. Note that is coupled to the elastic problem via theboundary condition, which depends on the deformation of the membrane.

We now apply again dimensionless analysis to equation (1.31). We scale theelectrostatic potential with the applied voltage V , time with a damping timescale

1.3. MEMS MODELS WITH VARIABLE PERMITTIVITY PROFILES 9

of the system, the x0 (resp., y0) variable with a characteristic length and width ofthe device equal to L, and z0 andbu with the size of the gap d between the groundplate and the undeflected elastic membrane. So we define

(1.32) u Dbud; D

V; x D

x0

L; y D

y0

L; z D

z0

d; t D

T t 0

aL2;

and substitute these into equation (1.31) to find

(1.33) �2@2u

@t2C@u

@t��?uC ı�

2?u D ��

"2

"0

�"2jr? j

2 C

�@

@z

�2�in �;

where � is the dimensionless domain of the elastic membrane. Here the parame-ter � is defined as

(1.34) � D

p�TA

aL;

while ı measures the relative importance of tension and rigidity, defined by

(1.35) ı DB

TL2:

The parameter " is the aspect ratio of the system

(1.36) " Dd

L;

and the parameter � is a ratio of the reference electrostatic force to the referenceelastic force, defined by

(1.37) � DV 2L2"0

2Td3:

Therefore, in the small-aspect ratio limit " � 1 we can use expression (1.30) forthe field, and the governing equation (1.33) then reduces to

(1.38) �2@2u

@t2C@u

@t��?uC ı�

2?u D �

�"0

"2.x; y/.1C u/2in �:

We now call f .x; y/ D "0"2.x;y/

the permittivity profile of the device and � theapplied voltage, and we end up with

(1.39) �2@2u

@t2C@u

@t��?uC ı�

2?u D �

�f .x; y/

.1C u/2in � � .0;C1/:

Since the membrane is undeflected at the initial time, we must have u.x; y; 0/ D 0,and since the boundary of the membrane is held fixed, we also have u.x; y; t/ D 0on the boundary of � at any time t > 0.

10 1. INTRODUCTION

1.3.3. The Mathematical Equations under Study. The stationary state ofequation (1.39) will be considered in Part III of this monograph, where we assumethe deflection is upwards towards a ground plate at z D 1. We therefore con-sider the following fourth-order equation describing the deflection u D u.x/ of themembrane on a bounded domain � with a clamped boundary condition

(C�;f )

8̂<̂:�2u D �f.x/

.1�u/2in �;

0 � u < 1 in �;

u D @u@�D 0 on @�;

with � being the outward pointing unit normal to @�, and B and T are positiveconstants. We shall also consider the case of a pinned boundary, that is,

(P�;f )

8̂<̂:B�2u � T�u D �f.x/

.1�u/2in �;

0 � u < 1 in �;

u D �u D 0 on @�:

In the first part of the monograph, we shall actually consider the stationaryequation in the case where the membrane is assumed to be perfectly elastic and hasno rigidity, which means that ı D 0 in view of (1.35). In other words, we shallfocus on the familiar second-order nonlinear eigenvalue problem:

(S�;f )

8̂<̂:��u D �f.x/

.1�u/2in �;

0 � u < 1 in �;

u D 0 on @�:

The dynamic deflection u D u.x; t/ will be considered in Part II, but not beforewe make a huge simplification to the model by ignoring rotational inertia, whichwould have given the equation a hyperbolic character. In other words, we assumethat the membrane thickness A D 2 is negligible, which gives � � 1 in viewof (1.34). We therefore end up with the following parabolic equation on a timeinterval .0; T /:

(D�;f )

8̂<̂:@u@t��u D �f.x/

.1�u/2; .x; t/ 2 � � .0; T /;

u.x; t/ D 0; .x; t/ 2 @� � .0; T /;

u.x; 0/ D 0; x 2 �:

We note that this type of nonlinearities (with negative exponents) have also ap-peared in other models, especially those concerned with the dynamics of thin filmsof viscous fluids; see, for example, Bertozzi, Laugesen, Pugh [21, 22, 122, 123]and the references therein. Earlier work dealt with the determination of the equi-librium state of two neighboring charged liquid drops suspended over two circularrings that is governed by the equation

(1.40) �v D ˛ Cˇ

v2in �;

where ˛; ˇ > 0 are constants reflecting fluid-mechanical and electrostatic proper-ties of the drops, respectively, so that ˛ D 0 corresponds to the situation where

1.4. BIFURCATION DIAGRAMS AND NUMERICAL EVIDENCE 11

uncharged drops are flat (see Taylor [157] and also [33, 113, 120, 136]). The equa-tions

(1.41) �v D1

vp0 < p � 1;

have also been studied in the literature. The case where p D 1 is related to thestudy of singular minimal hypersurfaces with symmetry; see Meadows [131], Si-mon [153], and the references therein.

We shall concentrate in this monograph on the case when p D 2, but most ofthe results can be extended to more general nonlinearities of the form F.x; u/ D�f.x/.1�u/p

, where p > 0. We encourage the interested reader to work out these exten-sions.

Particular attention will be given to the much less studied case of a varyingpermittivity profile f . Some of the results are straightforward extensions of thecase where f is a constant already considered in [26, 135] such as the case whenf is assumed to be bounded away from zero. More interesting situations deal withsomewhere vanishing profiles, the easiest of which are power-law permittivities ofthe form f .x/ D jxj˛ . We shall see that they can sometimes dramatically changethe picture. We often try to cover both situations at once by considering profiles ofthe form

(1.42) f .x/ D� kYiD1

jx � pi j˛i�g.x/; g.x/ � C > 0 in �:

1.4. Bifurcation Diagrams and Numerical Evidence

In order to get an initial idea of what issues need to be addressed, we start byconsidering the equation

(S�)

8̂̂<̂:̂��u D �

.1�u/2in �;

0 � u < 1 in �;

u D 0 on @�;

where � is the unit ball B1.0/ RN .N � 1/. In this case, .S�/ possesses onlyradially symmetric solutions [85], in which case the equation reduces to

(1.43)

(�urr �

N�1rur D

�.1�u/2

; 0 < r � 1;

u0.0/ D 0; u.1/ D 0:

Here r D jxj and 0 < u D u.r/ < 1 for 0 < r < 1. This problem was studiedin depth by Joseph-Lundgren [116]. The ODE analysis of .S�/ points to a setof solutions that varies considerably with the dimension. It is illustrated in thebifurcation diagram Figure 1.3, which describes the graph

(1.44) G D f.�; u.0// 2 .0;C1/ � Œ0; 1/ W u is a solution of .S�/g:

Note that in this case, the maximum of a solution u necessarily occurs at the origin[79]. This diagram suggests the following structure for the solution set in more

12 1. INTRODUCTION

≡

≤ ≤

FIGURE 1.3. Plots of u.0/ versus � for profile f .x/ � 1 defined in theunit ball B1.0/ RN with different ranges of N .

general domains� and permittivity profiles f , and a substantial part of this mono-graph investigates the extent to which these properties of the global set of solutionscan be established mathematically in more general situations. Here are some of theissues that we address.

I. Estimate the pull-in voltage ��.�; f /, which is the parameter �� > 0

such that .S�;f / has at least one solution if � < ��, while it has noneif � > ��. What is the effect of the geometry of the domain and thepermittivity profile on the value of ��.�; f /?

II. In dimension 1, there exist exactly two branches of solutions for 0 < � <��, and one solution for � D ��. The bifurcation diagram disappears(goes back to .0; 1/) when it returns to � D 0.

III. For dimensions 2 � N � 7, there exists a curve .�.t/; u.t//t�0 in thesolution set

(1.45) V D f.�; u/ 2 .0;C1/ � C 1. N�/ W u is a solution of .S�;f /g

starting from .0; 0/ at t D 0 and going to “infinity”: ku.t/k1 ! 1 ast !C1, with infinitely many bifurcation or turning points in V .

IV. In dimension N � 2 and for any profile f , there exists a unique solutionfor small voltages �.

V. There is at most one solution at ��—the so-called extremal solution u�—and in dimensions 2 � N � 7, there exist exactly two solutions for � ina small left neighborhood of ��.

1.4. BIFURCATION DIAGRAMS AND NUMERICAL EVIDENCE 13

VI. Estimate the pull-in distance, which is the maximum of the extremal so-lution at �� (i.e., ku�k1) in terms of the geometry of the domain and thepermittivity profile f .

1.4.1. Bifurcation Diagrams for Power-Law Profiles. By carrying out theanalysis of Joseph and Lundgren for radially symmetric solutions, but in the caseof a power-law permittivity profile f .x/ D jxj˛ , that is for

(S�;˛)

8̂<̂:��u D �jxj˛

.1�u/2in �;

0 � u < 1 in �;

u D 0 on @�;

where � is the unit ball B1.0/ RN , one can already see a much richer situationinduced by the nonconstant profile. We shall do that, so as to present analytical andnumerical evidence for various conjectures relating to these profiles, some of whichwill be further discussed and proved for general domains in the next chapters. Inthis case, .S�;˛/ for radially symmetric solutions reduces to

(1.46)

(�urr �

N�1rur D

�r˛

.1�u/2; 0 < r � 1;

u0.0/ D 0 D u.1/:

Equivalently, we can consider the initial value problem

(1.47)

(U 00 C N�1

rU 0 D r˛

U 2; r > 0;

U 0.0/ D 0; U.0/ D 1:

Observe that U > 1 in .0;C1/. For any � > 0, we can define a solution u� .r/ of(1.46) as

u� .r/ D 1 � �13 ��

2C˛3 U.�r/:

The parameter � and the maximum value u� .0/ of u� depend on � in the followingway:

(1.48)

(u� .0/ D 1 �

1U.�/

;

� D �2C˛

U 3.�/;

where the second relation guarantees the boundary condition u� .1/ D 0.One can numerically integrate the initial value problem (1.47) and use the re-

sults to compute the complete bifurcation diagram for (1.46). We show such acomputation of u.0/ versus � defined in (1.48) for the slab domain .N D 1/ in Fig-ure 1.4. In this case, one observes from the numerical results that whenN D 1 and0 � ˛ � 1, there exist exactly two solutions for .S�;˛/ whenever � 2 .0; ��/. Onthe other hand, the situation becomes more complex for ˛ > 1 as u.0/! 1. Thisleads to the question of determining the asymptotic behavior of U.r/ as r ! 1.Towards this end, we proceed as follows:

Setting v.s/ D r�.2C˛/=3U.r/ > 0, r D es , we have that

(1.49) v00 C

�N �

2

3C2˛

3

�v0 C

2C ˛

3

�N �

4

3C˛

3

�v D v�2:

14 1. INTRODUCTION

FIGURE 1.4. Plots of u.0/ versus � for profile f .x/ D jxj˛ .˛ � 0/

defined in the slab domain (N D 1). The numerical experiments point toa constant ˛1 > 1 (analytically given in (1.50)) such that the bifurcationdiagrams are greatly different for different ranges of ˛: 0 � ˛ � 1,1 < ˛ � ˛1, and ˛ > ˛1.

We can already id entify from this equation the following regimes:

Case 1. Assume that N D 1 and 0 � ˛ � 1.In this case, there is no positive equilibrium point for (1.49), which means

that the bifurcation diagram disappears at � D 0. Then one infers that there existexactly two solutions for � 2 .0; ��/ and just one for � D ��.

Case 2. N and ˛ satisfy either one of the following conditions: N D 1 and˛ > 1, or N � 2.

There exists then a positive equilibrium point ve of (1.49),

ve D3

s9

.2C ˛/.3N C ˛ � 4/> 0:

Linearizing around this equilibrium point by writing

v D ve C ıe��; 0 < ı � 1;

we obtain that

�2 C3N C 2˛ � 2

3� C

.2C ˛/.3N C ˛ � 4/

3D 0:

Such an equation admits the following solutions:

�˙ D �3N C 2˛ � 2

6˙

p�

6;

with� D �8˛2 � .24N � 16/˛ C .9N 2 � 84N C 100/:

1.4. BIFURCATION DIAGRAMS AND NUMERICAL EVIDENCE 15

We note that �˙ < 0 whenever � � 0. Now define

(1.50) ˛1 D �1

2C3

4

p6; ˛N D

3N � 14 � 4p6

4C 2p6

.N � 8/:

Next, we discuss the ranges of N and ˛ such that � � 0 or � < 0.

Case 2A. N and ˛ satisfy one of the following two conditions:

N D 1 with 1 < ˛ � ˛1;(1.51a)

N � 8 with 0 � ˛ � ˛N :(1.51b)

In this case, we have4 � 0 and

v.s/

�9

.2C ˛/.3N C ˛ � 4/

� 13

C ı1e� 3NC2˛�2�

p�

6s C � � � as s !C1:

Further, we conclude that

U.r/ r2C˛3

�9

.2C ˛/.3N C ˛ � 4/

� 13

C ı1r�N�2

2Cp�6 C � � � as r !C1:

In both cases, the branch monotonically approaches the value 1 as � ! C1

(u� .0/ " 1 as � !C1). Moreover, since � D �2C˛=U 3.�/, we have

(1.52) � " �� D.2C ˛/.3N C ˛ � 4/

9as � !1;

which is an important critical threshold for the voltage. In the case (1.51a) illus-trated by Figure 1.4, we have �� < ��, and the number of solutions increases butremains finite as � approaches ��. On the other hand, in the case of (1.51b) illus-trated by Figure 1.5(b), we have �� D ��, and there seems to be only one branchof solutions.

Case 2B. N and ˛ satisfy one of the following conditions:

N D 1 with ˛ > ˛1;(1.53a)

2 � N � 7 with ˛ � 0;(1.53b)

N � 8 with ˛ > ˛N :(1.53c)

In this case, we have � < 0 and

v.s/

�9

.2C ˛/.3N C ˛ � 4/

�13

C ı1e� 3NC2˛�2

6s cos

�p��

6s C C2

�C � � � as s !C1:

We also have for r !C1,

U.r/ r2C˛3

�9

.2C ˛/.3N C ˛ � 4/

� 13

C ı1r�N�2

2 cos

�p��

6ln r C C2

�C � � � ;

(1.54)

and from the fact that � D �2C˛=U 3.�/ we get again that

� �� D.2C ˛/.3N C ˛ � 4/

9as � !1:

16 1. INTRODUCTION

FIGURE 1.5. Top: Plots of u.0/ versus � for 2 � N � 7, where u.0/oscillates around the value �� defined in (1.52) and u� is regular. Bot-tom: Plots of u.0/ versus � for N � 8: when 0 � ˛ � ˛N , there existsa unique solution for .S/� with � 2 .0; ��/ and u� is singular; when˛ > ˛N , u.0/ oscillates around the value �� defined in (1.52) and u� isregular.

Note the oscillatory behavior of U.r/ in (1.54) for large r , which means that � isexpected to oscillate around the value �� D

.2C˛/.3NC˛�4/9

and u� .0/ " 1 as

1.4. BIFURCATION DIAGRAMS AND NUMERICAL EVIDENCE 17

� !1. The diagrams above point to the existence of a sequence f�i g satisfying

�0 D 0; �2k % �� as k !1;

�1 D ��; �2k�1 & �� as k !1;

such that exactly 2k C 1 solutions for .S�;˛/ exist when � 2 .�2k; �2kC2/, whilethere are exactly 2k solutions when � 2 .�2kC1; �2k�1/. Furthermore, .S�;˛/has infinitely many solutions at � D ��. The three cases (1.53a), (1.53b), and(1.53c) considered here for N and ˛ are illustrated by the diagrams in Figure 1.4,Figure 1.5(a), and Figure 1.5(b), respectively.

We now infer from the above that the bifurcation diagrams show four possibleregimes—at least if the domain is a ball:

VII. There is exactly one branch of solutions for 0 < � < ��. This regime oc-

curs when N � 8 and 0 � ˛ � ˛N D3N�14�4

p6

4C2p6

. The above observa-

tions actually show that in this range, the first branch of solutions “disap-pears” at ��, which happens to be equal to ��.˛;N / D

.2C˛/.3NC˛�4/9

.VIII. There exists an infinite number of branches of solutions. This regime

occurs when� N D 1 and ˛ � ˛1 D �12 C

34

p6;

� 2 � N � 7 and ˛ � 0;� N � 8 and ˛ > ˛N .

In this case, ��.˛;N / < �� and the multiplicity becomes arbitrarily largeas � approaches—from either side—��.˛;N /, at which there is a touch-down solution u (i.e., kuk1 D 1).

IX. There exists a finite number of branches of solutions. In this case, we haveagain that ��.˛;N / < ��, but now the branch approaches the value 1monotonically, and the number of solutions increases but remains finiteas � approaches ��.˛;N /. This regime occurs when N D 1 and 1 <˛ � ˛1.

X. There exist exactly two branches of solutions for 0 < � < �� and onesolution for � D ��. The bifurcation diagram disappears when it returnsto � D 0. This regime occurs when N D 1 and 0 � ˛ � 1.

1.4.2. Asymptotic Analysis of Dynamic Quenching Profiles. Consider nowthe dynamic problem .D�;f / on a domain �. Numerical evidence shows thatfor small voltages �, a solution exists for all time and eventually converges to thecorresponding minimal solution of the stationary equation. On the other hand, forlarge voltages �, the solution must quench (i.e., touch the ground plate) at a finitetime T�.�; f /. This naturally leads to the following queries:

XI. Is the pull-in voltage �� the exact cutoff between a regime of globallyconvergent solutions, and the one where we have quenching in finitetime? Moreover, what is the mode of deflection when � D ��, and canwe have quenching in infinite time?

XII. Estimate the quenching time T�.�; f / in terms of the geometry of thedomain, the permittivity profile f , and the excess voltage � � ��.

18 1. INTRODUCTION

Besides the qualitative behavior of the solutions for various �, and the esti-mates on the quenching time T�.�; f /, one is also interested in describing thequenching profile of the deflection, including the information on quenching ratesas well as quenching locations and how they are affected by the permittivity. Withthis in mind, we now use elementary asymptotic analysis to identify the issues andthe facts that need to be established. We start with the simplest case.

Quenching Profile When f.x/ � 1. We also assume that quenching occursat x D 0 and t D T . In the absence of diffusion, the time-dependent behaviorof solutions for .D�;1/ is given by ut D �.1 � u/�2. Integrating this differentialequation and setting u.T / D 1, we get .1 � u/3 D �3�.t � T /. This solutionmotivates the introduction of a new variable v.x; t/ defined by

(1.55) v D1

3�.1 � u/3:

A simple calculation shows that .D�;1/ transforms to the following problem for v:

vt D �v �2

3vjrvj2 � 1; x 2 �; t > 0I(1.56a)

v.x; k/ D1

3�on .@� � .0;C1// [ .� � f0g/:(1.56b)

Notice that u D 1 maps to v D 0. We will find a formal power series for a radiallysymmetric solution to (1.56a) near v D 0, in the form

(1.57) v.x; t/ D v0.t/Cr2

2Šv2.t/C

r4

4Šv4.t/C � � � ;

where r D jxj. We then substitute (1.57) into (1.56a) and collect coefficients in r .In this way, we obtain the following coupled ordinary differential equations for v0and v2:

(1.58) v00 D �1CNv2; v02 D �4

3v0v22 C

.N C 2/

3v4:

We are interested in the solution to this system for which v0.T / D 0, with v00 < 0and v2 > 0 for T � t > 0 with T � t � 1. System (1.58) has a closure problemin that v2 depends on v4. However, we will assume that v4 � v22=v0 near thesingularity. With this assumption, (1.58) reduces to

(1.59) v00 D �1CNv2; v02 D �4

3v0v22 :

We now solve system (1.59) asymptotically as t ! T �. We first assume thatNv2 � 1 near t D T . This leads to v0 T � t , and the following differentialequation for v2:

(1.60) v02 �4

3.T � t/v22 as t ! T �:

1.4. BIFURCATION DIAGRAMS AND NUMERICAL EVIDENCE 19

By integrating (1.60), we obtain that

(1.61) v2 �3

4Œlog.T � t/�C

B0

Œlog.T � t/�2C � � � as t ! T �

for some unknown constant B0. From (1.61), we observe that the consistencycondition thatNv2 � 1 as t ! T � is indeed satisfied. Substituting (1.61) into theequation (1.59) for v0, we obtain for t ! T � that

(1.62) v00 D �1CN

��

3

4Œlog.T � t/�C

B0

Œlog.T � t/�2C � � �

�:

Using the method of dominant balance, we look for a solution to (1.62) as t ! T �

in the form

(1.63) v0 .T � t/C .T � t/

�C0

Œlog.T � t/�C

C1

Œlog.T � t/�2C � � �

�for some C0 and C1 to be found. A simple calculation yields that

v0 .T � t/C�3N.T � t/

4jlog.T � t/jC�N.B0 �

34/.T � t/

jlog.T � t/j2

C � � � as t ! T �:

(1.64)

The local form for v near quenching is v v0Cr2v2=2. Using the leading term inv2 from (1.61) and the first two terms in v0 from (1.64), we obtain the local form

(1.65) v .T � t/

�1 �

3N

4jlog.T � t/jC

3r2

8.T � t/jlog.T � t/jC � � �

�for r � 1 and t � T � 1. Finally, using the relation (1.55) between u and v, weconclude that

u 1� Œ3�.T � t /�13

�1�

3N

4jlog.T � t /jC

3r2

8.T � t /jlog.T � t /jC � � �

� 13

:(1.66)

An important question is then:

XIII. To what extent does (1.66) describe the profile of a quenching solution,and how valid is this description for general permittivities?

Before going further, we note that numerical experiments can be performedthanks to the following observation, which will also have a theoretical relevanceas well. Indeed, we observe that if we use the local behavior v .T � t/ C

3r2=Œ8jlog.T � t/j�, we get that

(1.67)jrvj2

v

�2

3jlog.T � t/j C

16.T � t/jlog.T � t/j2

9r2

��1:

Hence, the term jrvj2=v in (1.56a) is bounded for any r , even as t ! T �. Thisallows us to use a simple finite-difference scheme to compute numerical solutionsfor the transformed problem (1.56).

For the slab domain .�12; 12/, we define vmj for j D 1; : : : ; N C 2 to be the

discrete approximation to v.m�t;�12C.j �1/h/, where h D 1

NC1and�t are the

20 1. INTRODUCTION

−0.5 −0.25 0 0.25 0.50.50.7

0.76

0.82

0.88

0.94

1

x

1 −

u

FIGURE 1.6. Plots of 1�u.x; t/ versus x with � D 1 at different times.Here it is observed that the dynamic solution approaches a steady stateas the time t is increased.

spatial and temporal mesh sizes, respectively. A second-order accurate-in-space,and a first-order accurate-in-time discretization of (1.56) is

vmC1j D vmj C�t

�.vmjC1 � 2v

mj C v

mj�1/

h2� 1 �

.vmjC1 � vmj�1/

2

6vmj h2

�;

j D 2; : : : ; N C 1;

(1.68)

with vm1 D vmNC2 D .3�/�1 for m � 0. The initial condition is v0j D .3�/�1

for j D 1; : : : ; N C 2. The time step �t is chosen to satisfy �t < h2=4 for thestability of the discrete scheme. Using this argument, one can get useful numericalresults for the dynamic deflection u; see Figures 1.6, 1.7, and 1.8.

Quenching Profile for a Variable Permittivity. Suppose we are now dealingwith a spatially variable permittivity profile in a slab domain. Let u again be aquenching solution of .D�;f / at finite time T , and let x D x0 be a quenchingpoint of u. With the transformation

(1.69) v D1

3�.1 � u/3;

problem .D�;f / in the slab domain transforms to

vt D vxx �2

3vv2x � f .x/; �

1

2< x <

1

2; t > 0;(1.70a)

v D1

3�; x D ˙

1

2I v D

1

3�; t D 0;(1.70b)

where f .x/ is the permittivity profile.

1.4. BIFURCATION DIAGRAMS AND NUMERICAL EVIDENCE 21

−0.5 −0.25 0 0.25 0.50.55

0.7

0.85

1

x

1 −

u

FIGURE 1.7. Plots of 1 � u.x; t/ versus x with � D 1:3 at differenttimes. Here it is observed that the dynamic solution approaches a steadystate as the time t is increased.

−0.5 −0.25 0 0.25 0.5 0

0.2

0.4

0.6

0.8

1

1 −

u

x

FIGURE 1.8. Plots of 1�u.x; t/ versus x with � D 6 at different times.Here the quenching behavior is observed at finite time.

Look now for a quenching profile for (1.70) near x D x0 at time T in the form

v.x; t/ D v0.t/C.x � x0/

2

2Šv2.t/C

.x � x0/3

3Šv3.t/

C.x � x0/

4

4Šv4.t/C � � � :

(1.71)

22 1. INTRODUCTION

In order for v to be a quenching profile, it is clear that we must require

limt!T�

v0 D 0; v0 > 0 for t < T;v2 > 0 for t � T � 1:

(1.72)

We first discuss the case where f .x/ is analytic at x D x0 with f .x0/ > 0.Therefore, for x � x0 � 1, f itself has the convergent power series expansion

(1.73) f .x/ D f0 C f00.x � x0/C

f 000 .x � x0/2

2C � � � ;

where f0 WD f .x0/, f 00 D f 0.x0/, and f 000 WD f 00.x0/. Substituting (1.71) and(1.73) into (1.70), we equate the powers of x � x0 to obtain

v00 D �f0 C v2;(1.74a)

v02 D �4v223v0C v4 � f

000 ;(1.74b)

v3 D f00:(1.74c)

We now assume that v2 � 1 and v4 � 1 as t ! T �. This yields that v0 f0.T � t/, and

(1.75) v02 �4v22

3f0.T � t/� f 000 :

For t ! T �, we obtain from a simple dominant balance argument that

(1.76) v2 �3f0

4Œlog.T � t/�C � � � as t ! T �:

Substituting (1.76) into (1.74a) and integrating, we obtain that

(1.77) v0 f0.T � t/C�3f0.T � t/

4jlog.T � t/jC � � � as t ! T �:

Next, we substitute (1.76), (1.77), and (1.74c) into (1.71) to obtain the localquenching behavior

v f0.T � t/

�1 �

3

4jlog.T � t/jC

3.x � x0/2

8.T � t/jlog.T � t/j

Cf 00.x � x0/

3

6f0.T � t/C � � �

�;

(1.78)

for .x � x0/� 1 and t � T � 1. Finally, using the relation (1.69) between uand v, we conclude that

(1.79) u 1 � Œ3f .x0/�.T � t/�13

�

�1 �

3

4jlog.T � t/jC

3.x � x0/2

8.T � t/jlog.T � t/jCf 0.x0/.x � x0/

3

6f0.T � t/C � � �

�13

:

This formal analysis is obviously not satisfactory if the quenching point x0 hap-pens to be a zero of the profile f (i.e., if f .x0/ D 0), and leads to the followingimportant question:

1.4. BIFURCATION DIAGRAMS AND NUMERICAL EVIDENCE 23

XIV. Can finite-time quenching occur at the zero set of permittivity profile f ?

Again, formal asymptotic analysis points towards a negative answer to thisquestion by showing how to exclude the possibility that f .x0/ D 0. Indeed, as-sume f .x/ is analytic at x D x0, with f .x0/ D 0 and f 0.x0/ D 0, so thatf .x/ D a0.x � x0/

2 C O..x � x0/3/ as x ! x0 with a0 > 0. We then look

for a power series solution to (1.70) as in (1.71). In place of (1.74) for v3, we getv3 D 0, and

(1.80) v00 D v2; v02 D �4v223v0C v4 � 2a0:

Assuming that v4 � 1 as before, we can combine the equations in (1.80) to get

(1.81) v000 D �4.v00/

2

3v0� 2a0:

By solving (1.81) with v0.T / D 0, we obtain the exact solution

(1.82) v0 D �3a0

11.T � t/2 < 0; v2 D

6a0

11.T � t/:

Since the criteria (1.72) are not satisfied, the form (1.82) does not represent aquenching profile centered at x D x0. Therefore, the above asymptotical anal-ysis also shows that the point x D x0 satisfying f .x0/ D 0 is not a quenchingpoint of u.

1.4.3. Evidence and Conjectures for Fourth-Order MEMS Models. Thecase where the rigidity is not neglected leads to a substantial number of mathemat-ical challenges. The equation is then of the fourth order in space and one has todeal with the biharmonic operator, which is not as well understood mathematicallyas the Laplacian. For one, �2 does not satisfy the maximum principle on generaldomains when we are dealing with the physically relevant clamped boundary con-ditions. However, if the domain is a ball where�2 satisfies a comparison principleand minimal solutions are radially symmetric, numerical and analytical evidencepoints towards a strong analogy with the corresponding second-order model, pro-vided we increase all critical dimensions by 1. Specifically, the following pictureemerges for equation .C�;1/, at least when � is a ball:

There exists again a pull-in voltage �� > 0 such that .C�/ has at least onesolution if � < ��, while it has none if � > ��. However, the dependence of thebifurcation diagrams on the dimension changes as follows:

XV. In dimensions 1 and 2 there exist exactly two branches of solutions for0 < � < �� and one solution for � D ��. The second (unstable) solutionheads towards a quenching state when the bifurcation diagram returns to� D 0.

XVI. There exists at most one solution at �� in any dimension, while theuniqueness of solutions also occurs at small voltages � whenever N � 3.

XVII. In dimensions 3 � N � 8 there exist exactly two solutions for � in asmall left neighborhood of ��, as well as a curve .�.t/; u.t//t�0 in the

24 1. INTRODUCTION

solution set

(1.83) V D f.�; u/ 2 .0;C1/ � C 2. N�/ W u is a solution of .C�;1/g;

starting from .0; 0/ at t D 0 with ku.t/k1 ! 1 as t ! C1, whilehaving infinitely many turning points.

XVIII. Describe the situation in more general domains.

Queries I to XVIII and many more will be addressed throughout this mono-graph under various levels of generality.

1.5. Brief Outline

This text is divided into three major parts. The first one covers the simplestsituation, which deals with the stationary case when one ignores bending. Theresults here are the most satisfying from the mathematical point of view. Thestationary case for a nonelastic model is dealt with in Part III, where the results aresatisfactory only in the case of a radially symmetric domain. The dynamic model iscovered in Part II, but only in the parabolic case where we ignore rotational inertia,which would have given the equation a hyperbolic character.

Part I: Second-Order Equations with Singular Nonlinearities ModelingStationary MEMS. This part deals with the stationary deflection of an elasticmembrane satisfying the elliptic problem .S�;f /, where � > 0 represents the ap-plied voltage. The nonnegative function f .x/will describe the varying permittivityprofile of the elastic membrane, which will be allowed to vanish somewhere, butwill always be assumed to satisfy

(1.84) f 2 C ˇ . N�/ for some ˇ 2 .0; 1�; 0 � f � 1; and f 6� 0:

Chapter 2. We start by considering the pull-in voltage for .S�;f /, which isdefined as

(1.85) ��.�; f / D supf� > 0 j .S�;f / possesses at least one solutiong:

Our goal is to establish the basic properties of the pull-in voltage �� and to an-alyze its dependence on the size and the shape of the domain, as well as on thepermittivity profile.

Chapter 3. We study the branch � 7! u� of minimal solutions for .S�;f /,i.e., those that are below any other solution. We include a detailed analysis ofthe monotonicity, differentiability, and compactness properties of this branch withrespect to the voltage �. The stability and regularity properties of such minimalsolutions are also investigated, with a particular emphasis on the extremal solution,i.e., the one that corresponds to the pull-in voltage ��. The results confirm thecomplexity and richness of the problem and its dependence on the dimension ofthe ambient space and on the permittivity profile, as suggested by the bifurcationdiagrams (Figures 1.3–1.5). In particular, there is a critical dimension (N D 7)below which all minimal solutions—including the solution at the pull-in voltageu��—are regular, while u�� may “touch the plate” (i.e., u��.x/ D 1 at some pointx 2 �) for dimension N � 8.

1.5. BRIEF OUTLINE 25

On the other hand, a closer study of the minimal branch for powerlike profileson the unit ball uncovers a new and interesting phenomenon: certain permittivityprofiles can restore compactness and regularity to the problem at the pull-in volt-age. This analysis is done through a blowup procedure that we introduce in thischapter, but which we will explore more deeply in the following ones.

Chapter 4. One of the primary goals in the design of MEMS devices is tooptimize the pull-in distance over a certain allowable voltage range that is set by thepower supply. Here the pull-in distance refers to the maximum stable deflection ofthe elastic membrane before quenching occurs. It will be denoted by P.�; f;N /.Since the L1-norm of the minimal solutions u� for .S�;f / is strictly increasingin �, the pull-in distance is achieved exactly at � D ��, that is P.�; f;N / Dku�kL1.�/, where u� is the unique extremal steady state of .S�;f /. Our first goalin this chapter is to give upper and lower bounds for P.�; f;N / in terms of thepermittivity f , the dimension N , and the geometry of the domain. We also studythe effect of powerlike permittivity profiles f .x/ D jxj˛ and the dimension N onboth the pull-in voltage and the pull-in distance. In particular, we give rigorousproofs that explain various numerically observed phenomena, such as the curiousfact noted in [99], that on a two-dimensional disk, a change in the power law hasno effect on the pull-in distance, as well as the “rule of thumb” that the pull-indistance is about one-third the size of the zero voltage gap [140, p. 214].

Chapter 5. We continue the analysis of problem .S�;f / by considering the lin-earized operatorLu;� D ���2�f .x/=.1�u/3, at a solution u, and its eigenvaluesf k;�.u/I k D 1; 2; : : : g, the first one of which is simple and given by

(1.86) 1;�.u/ D inf

�hLu;�; iL2.�/ W 2 C

10 .�/;

Z�

2 dx D 1

�:

When the infimum is attained at a first, positive eigenfunction 1, the second eigen-value is then given by the formula

2;�.u/ D inf�hLu;�; iL2.�/ W 2 C

10 .�/;

Z�

2 dx D 1; andZ�

1 dx D 0

�:(1.87)

This construction can be iterated to obtain the kth eigenvalue k;�.u/ with theconvention that eigenvalues are repeated according to their multiplicities [23]. TheMorse indexm.u; �/ of a solution u is the largest k for which k;�.u/ is negative.

The compactness—in dimensions 1 � N � 7—of the minimal branch of semi-stable solutions, i.e., those with zero Morse index (or those verifying 1;�.u/ � 0),gives rise to a second unstable solution U� of .S�;f / for � in a small deletedleft neighborhood of ��. We give a “mountain pass” variational construction ofthis second solution for � close to �� whenever the minimal branch is compact.We then establish, in the right dimensions, the compactness of this first branch ofunstable solutions whose Morse index is equal to 1, i.e., those such that

(1.88) 1;�.u/ < 0 but 2;�.u/ � 0:

This eventually leads to the identification of a second bifurcation point. The maintool here for controlling the blowup behavior of a possible noncompact sequence

26 1. INTRODUCTION

of solutions is a linear instability result which states that there is no stable solutionfor the limiting problem (on RN ) if either 1 � N � 7 or if N � 8 and ˛ >

˛N . Roughly speaking, compactness is deduced from the fact that a blown-upsequence of solutions whose Morse indices do not exceed 1 cannot account forsuch instability at infinity.

Chapter 6. We investigate here the extent to which the properties of the globalset of solutions—as illustrated in the bifurcation diagram of Figure 1.3—can beestablished mathematically in more general situations. We start by extending theresults of the previous chapters and establishing the compactness of any set ofsolutions of .S�;f /, provided their Morse indices are uniformly bounded and notjust equal to 0 as in Chapter 3 or less than or equal to 1 as in Chapter 4. The resultwill again be true for dimensions 2 � N � 7 and for permittivity profilesf ofthe form (1.42). This will follow from a detailed blowup analysis of noncompactsequences of solutions, combined with a result stating that a solution u of finiteMorse index is either regular or “badly singular,” meaning that the set fx 2 � Wu.x/ D 1g has no isolated points. As a by-product, we show the existence of aquenching branch of solutions with an infinite number of turns (i.e., solutions witharbitrarily large Morse indices) as long as the extremal solution u�� is a classicalsolution, and in particular for dimensions 2 � N � 7. On the other hand, we showa uniqueness result for small voltage in the case of power-law profiles—withoutany restriction on the energy or on the Morse index—provided the domain � isstar-shaped and N � 3, or when it is strictly convex (or suitably symmetric) in thetwo-dimensional case.

Chapter 7. We consider in this chapter the particular case of a MEMS devicewith a power-law profile and whose domain satisfies various symmetry conditions.More information about the set of solutions can be given in this situation, which isstill complex—even in the radially symmetric case—as shown in the above bifur-cation diagrams.

Our main purpose here, however, is to introduce mathematical techniques thathave not been used so far. The first section includes an interesting and nonstandardone-dimensional Sobolev inequality. This will be followed by a powerful mono-tonicity formula that should have many applications. Section 3 deals with thecounterpart of the last chapter’s compactness result in the case of radially sym-metric solutions, where the role of the radial Morse indices is investigated. Thiswill allow for a more detailed description of the branches of radially symmetric so-lutions that closely reflects the numerical evidence. We finally pay special attentionto the two-dimensional case, which arguably is the most relevant for engineeringapplications.

Part II: Parabolic Equations with Singular Nonlinearities ModelingMEMS Dynamic Deflections. The second part of this monograph is devoted tothe study of dynamic deflection of .D�;f /. We note that the case when the per-mittivity profile f .x/ � 1 was studied in the 1980s (see [95, 96] and referencestherein). However, the case of a varying and somewhere vanishing profile turnedout to be a rich source of new and interesting mathematical phenomena.

1.5. BRIEF OUTLINE 27

Chapter 8. It is shown here that whenever � � ��.�; f /, the unique solutionu of .D�;f / exists for all time and must globally converge as t ! C1 to itscorresponding unique minimal steady state. On the other hand, if � > ��.�; f /

the unique solution u of .D�;f / must quench at finite time T�.�; f / in the sensethat u.x; t/ reaches 1 at time T�.�; f /.

We also study the set of finite-time quenching points, and we show amongother things that generically they cannot occur at zero points of the varying permit-tivity profile f .x/, a fact first observed numerically and conjectured in [99]. Wethen study the case of convex domains and also the particular case of radially sym-metric solutions on the ball, where often the only location of finite-time quenchingpoints is at the origin.

Chapter 9. We analyze and estimate the finite quenching time T�.�; f / of thedynamic solutions of .D�;f /. This often translates into useful information con-cerning the speed of the operation for many MEMS devices, such as RF switchesand microvalves. We start by giving comparison results for the quenching timeT�.�; f / in terms of the profile f and the voltage �. We then obtain various ana-lytic estimates for quenching times for large voltages �, while in the third sectionwe give analytic and numerical estimates on the quenching time for any � abovethe pull-in voltage ��. The last section deals with the particular but important casewhen the extremal steady state is regular, which occurs at least in low dimensions(1 � N � 7).

Chapter 10. The focus here is on the quenching profiles of the dynamic solu-tions of .D�;f /, when � exceeds the pull-in voltage ��. Since such solutions mustquench at a finite time T D T�.�; f /, we obtain—by adapting self-similaritymethods and center manifold analysis—estimates on the quenching rate and a rela-tively precise description of the solution near time T and around a given quenchingpoint a. Under a compactness assumption on the quenching set, we establish en-ergy quenching rates such as

(1.89) limt!T�

Z�

f .x/

.1 � u.x; t//�dx D C1

for any � > 32N . On the other hand, we show that in the radially symmetric case,

a solution quenching at 0 must do so in such a way that for any C > 0,

(1.90) limt!T�

1 � u.x; t/

.T � t/13

D .3�f .0//13

uniformly on jxj � CpT � t .

More precise quenching behavior is given in dimension N D 1. For example,it is shown that for sufficiently large voltage �, finite-time quenching necessarilyoccurs near the maximum points of the profile f , which refines considerably theresults obtained in Chapter 8 about finite-time quenching not occurring at the zeropoints of the permittivity profile f .

Part III: Fourth-Order Equations Modeling Nonelastic MEMS. The thirdpart of this monograph is devoted to the study of nonelastic MEMS models where

28 1. INTRODUCTION

bending effects are taken into consideration. Specifically, we shall consider equa-tion .C�;f / where we suppose a clamped boundary condition, as well as .P�;f /where a pinned boundary is assumed.

Chapter 11. We study here the fourth-order nonlinear boundary value problem�2u D �

.1�u/2on a ball B RN , again under Dirichlet boundary conditions

u D @u@�D 0 on @B ; there exists �� > 0 such that for � 2 .0; ��/, the equation

has a minimal (classical) solution u� that satisfies 0 < u� < 1. For � > �� thereare no solutions of any kind. In the extremal case � D ��; we prove the existenceof a weak solution that has finite energy and that is the unique solution even in avery weak sense. The main result asserts that the extremal solution u�� is regular.supB u�� < 1/ providedN � 8, while u�� is singular (supB u�� D 1) forN � 9,in which case 1 � C0jxj4=3 � u��.x/ � 1 � jxj4=3 on the unit ball, where

C0 WD

���

�

�13

and N� WD8.N � 2

3/.N � 8

3/

9:

Estimates on ��, as well as stability properties of minimal solutions, are also es-tablished.

Chapter 12. We consider here the fourth-order nonlinear equation B�2u �T�u D �

.1�u/2on a bounded smooth domain � RN , with the Navier boundary

conditions u D �u D 0 on @�, and where � > 0 is a parameter, and T > 0

and B > 0 are fixed constants. There again exists a well-defined pull-in voltage�� > 0 such that for � < ��, there is a branch of minimal solutions that are alsostable. The minimal solution is unique at � D �� and is regular in dimensionsN � 4. The novelty here is that, unlike the case when B D 0, there exists, at leaston a two-dimensional convex smooth domain, a second mountain pass solution forall � 2 .0; ��/, which shows that the two-dimensional bifurcation diagram of .P�)changes drastically when B > 0. The asymptotic behavior of the second solutionas �! 0 is also studied. The situation is again much clearer on a radially symmet-ric domain where the extremal solution u�� is regular .supB u�� < 1/ providedN � 8, while u�� is singular (supB u�� D 1) for N � 9 and T

Bis small enough.

Key ingredients for proving the singularity of the extremal solution in higher di-mensions (above N D 9) are the new Hardy-Rellich inequalities described in theAppendix.

Glossary of Notation. The following list of notation and abbreviations willbe used throughout this book. Let � be a smooth domain of Rn.

� C1.�;Rn/ (resp., C10 .�;Rn/) will denote the space of infinitely dif-

ferentiable functions (resp., the space of infinitely differentiable functionswith compact support) on �.� For 1 � p < C1, Lp.�/ will be the space of all integrable functionsu W �! R with norm

kukp D

�Z�

ju.x/jp dx

� 1p

:

1.5. BRIEF OUTLINE 29

For p D C1, L1.�/ will be the space of all measurable functionsu W �! R such that

kuk1 D ess sup limx2�ju.x/j < C1:

� For m 2 N and 1 � p � C1, W m;p.�/ will be the Banach space of(classes of) measurable functions u W � ! R such that D˛u 2 Lp.�/in the sense of distributions, for every multi-index ˛ with j˛j � m. Thespace W m;p.�/ will be equipped with the norm

kukWm;p.�/ DX

limj˛j�m

kD˛ukp:

Wm;p0 .�/will be the closure ofC10 .�;R/ inW m;p.�/, andW �m;q.�/

will denote the Banach space dual of W m;p0 .�/, where 1

pC 1

qD 1.

� W m;2.�/ (resp.,W m;20 .�/) will be denoted byHm.�/ (resp.,Hm

0 .�/).They will be Hilbert spaces once equipped with the scalar product

hu; vi D

Z�

u.x/v.x/dx:

The dual of H�m0 .�/ will be denoted by H�m.�/.