P I E R P A O L O E S P O S I T O

N A S S I F G H O U S S O U B

Y U J I N G U O

Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS

American Mathematical SocietyCourant Institute of Mathematical Sciences

C O U R A N T 20LECTURE

NOTES

Mathematical Analysis of Partial Differential

Equations Modeling Electrostatic MEMS

Courant Lecture Notes in Mathematics

Executive EditorJalal Shatah

Managing EditorPaul D. Monsour

Assistant Editor Reeva Goldsmith

Copy Editor Will Klump

Pierpaolo EspositoUniversità degli Studi Roma Tre

Nassif Ghoussoub University of British Columbia

Yujin Guo University of Minnesota

20 Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS

Courant Institute of Mathematical SciencesNew York University New York, New York

American Mathematical SocietyProvidence, Rhode Island

http://dx.doi.org/10.1090/cln/020

2000 Mathematics Subject Classification. Primary 35J60, 35B45, 35B35, 35B40, 35P30,74K15, 74F15, 35J20, 58E07, 74M05.

For additional information and updates on this book, visitwww.ams.org/bookpages/cln-20

Library of Congress Cataloging-in-Publication Data

Mathematical analysis of partial differential equations modeling electrostatic MEMS / PierpaoloEsposito, Nassif Ghoussoub, Yujin Guo.

p. cm. — (Courant lecture notes ; 20)Includes bibliographical references and index.ISBN 978-0-8218-4957-6 (alk. paper)1. Microelectromechanical systems—Mathematical models. 2. Mathematical analysis.

I. Ghoussoub, Nassif. II. Guo, Yujin.

TK7875.M385 2010621—dc22

2009045518

Copying and reprinting. Individual readers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Requests for suchpermission should be addressed to the Acquisitions Department, American Mathematical Society,201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made bye-mail to reprint-permission@ams.org.

c© 2010 by the authors. All rights reserved.Printed in the United States of America.

©∞ The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10

Dedication

To Valerio Esposito

To Michelle Ghoussoub

To Chenghui Guo

Contents

Preface xi

Chapter 1. Introduction 11.1. Electrostatic Actuations and Nonlinear PDEs 11.2. Derivation of the Model for Homogeneous Systems 41.3. MEMS Models with Variable Permittivity Profiles 61.4. Bifurcation Diagrams and Numerical Evidence 111.5. Brief Outline 24

Part 1. Second-Order Equations Modeling Stationary MEMS 31

Chapter 2. Estimates for the Pull-In Voltage 332.1. Existence of the Pull-In Voltage 332.2. Lower Estimates for the Pull-In Voltage 392.3. Upper Bounds for the Pull-In Voltage 432.4. Numerics for the Pull-In Voltage 46Further Comments 50

Chapter 3. The Branch of Stable Solutions 513.1. Spectral Properties of Minimal Solutions 513.2. Energy Estimates and Regularity of Solutions 553.3. Linear Instability and Compactness 623.4. Effect of an Advection on the Minimal Branch 69Further Comments 74

Chapter 4. Estimates for the Pull-In Distance 774.1. Lower Estimates on the Pull-In Distance in General Domains 774.2. Upper Estimate for the Pull-In Distance in General Domains 804.3. Upper Bounds for the Pull-In Distance in the Radial Case 824.4. Effect of Power-Law Profiles on Pull-In Distances 854.5. Asymptotic Behavior of Stable Solutions near the Pull-In Voltage 90Further Comments 92

Chapter 5. The First Branch of Unstable Solutions 935.1. Existence of Nonminimal Solutions 945.2. Blowup Analysis for Noncompact Sequences of Solutions 985.3. Compactness along the First Branch of Unstable Solutions 1035.4. Second Bifurcation Point 110Further Comments 112

vii

viii CONTENTS

Chapter 6. Description of the Global Set of Solutions 1156.1. Compactness along the Unstable Branches 1166.2. Quenching Branch of Solutions in General Domains 1256.3. Uniqueness of Solutions for Small Voltage in Star-Shaped Domains 1296.4. One-Dimensional Problem 137Further Comments 139

Chapter 7. Power-Law Profiles on Symmetric Domains 1417.1. A One-Dimensional Sobolev Inequality 1417.2. Monotonicity Formula and Applications 1457.3. Compactness of Higher Branches of Radial Solutions 1527.4. Two-Dimensional MEMS on Symmetric Domains 162Further Comments 172

Part 2. Parabolic Equations Modeling MEMS Dynamic Deflections 175

Chapter 8. Different Modes of Dynamic Deflection 1778.1. Global Convergence versus Quenching 1788.2. Quenching Points and the Zero Set of the Profile 1878.3. The Quenching Set on Convex Domains 192Further Comments 198

Chapter 9. Estimates on Quenching Times 1999.1. Comparison Results for Quenching Times 1999.2. General Asymptotic Estimates for Quenching Time 2019.3. Upper Estimates for Quenching Times for all � > �� 2039.4. Quenching Time Estimates in Low Dimension 210Further Comments 215

Chapter 10. Refined Profile of Solutions at Quenching Time 21710.1. Integral and Gradient Estimates for Quenching Solutions 21710.2. Refined Quenching Profile 22110.3. Refined Quenching Profiles in Dimension N D 1 22910.4. Refined Quenching Profiles in the Radially Symmetric Case 23310.5. More on the Location of Quenching Points 240Further Comments 242

Part 3. Fourth-Order Equations Modeling Nonelastic MEMS 243

Chapter 11. A Fourth-Order Model with a Clamped Boundaryon a Ball 245

11.1. Boggio’s Principle 24511.2. Pull-In Voltage 24911.3. Stability of the Minimal Branch of Solutions 25511.4. Regularity of the Extremal Solution for 1 � N � 8 26011.5. The Extremal Solution Is Singular for N � 9 263Further Comments 268

CONTENTS ix

Chapter 12. A Fourth-Order Model with a Pinned Boundaryon Convex Domains 269

12.1. The Minimal Solutions up to the Pull-In Voltage 26912.2. Stability of Minimal Solutions 27412.3. Regularity of the Extremal Solution on General Domain for N � 4 27912.4. Uniform Energy Bounds for Solutions in Convex Domains 28012.5. The Solution Set on Convex Domains in R2 28312.6. Regularity of the Extremal Solution on Balls for N � 8 28912.7. Singularity of the Extremal Solution on Balls for N � 9 291Further Comments 296

Appendix A. Hardy-Rellich Inequalities 299A.1. Improved Hardy-Rellich Inequalities in H 2

0 .B/ 299A.2. Improved Hardy-Rellich Inequalities in H 2.B/\H 1

0 .B/ 302

Bibliography 309

Index 317

Preface

Microelectromechanical systems (MEMS) and nanoelectromechanical systems(NEMS), which combine electronics with miniature-size mechanical devices, areessential components of the modern technology that is currently driving telecom-munications, commercial systems, biomedical engineering, and space exploration.These are only a few of the vast number of applications that lie at the roots of mi-crosystem technology. Over the years, and in order to provide accurate, controlled,and stable locomotion for such microdevices, researchers have proposed a varietyof modes, based upon thermal, biological, or electrostatic forces. It is the math-ematical model describing the method of “electrostatic actuation” that we shalladdress in this monograph. The process is based on an electrostatically controlledtunable capacitor that is widely used in microresonators, optical microswitches,chemical sensors, micromirrors, accelerometers for airbag development of auto-mobiles, micropumps for inkjet printer heads, microvalves, shuffle motors, micro-and nanotweezers, among many other devices.

There now exist many variations in electrostatic actuation technology. Theyare all, however, based on a simple physical principle relating

� the elastic deformation which—by elementary plate theory—depends onthe Laplacian of the deformation variable (to account for stretching), andon its bi-Laplacian (for bending),� the electrostatic force which—by the classical Coulomb law—is propor-

tional to the inverse square of the distance between the two charged plates,itself a function of the deformation variable.

Unfortunately, models for electrostatically actuated microplates that accountfor moderately large deflections and which do not assume that each material pointmoves vertically over its reference position are quite complicated and not yet amen-able to rigorous mathematical analysis. In this book we deal with much simplifiedmodels that still lead to very interesting second- and fourth-order nonlinear ellipticequations (in the stationary case) and to nonlinear parabolic equations (in the dy-namic case). The nonlinearity is of an inverse square type, which until recently hasnot received much attention as a mathematical problem. It was therefore rewardingto see, besides the above practical considerations, that the model is actually a veryrich source of interesting mathematical phenomena. Numerics and formal asymp-totic analysis give lots of information and point to many conjectures, but even inthe simplest idealized versions of electrostatic MEMS, one essentially needs thefull available arsenal of modern nonlinear analysis and PDE techniques “to do” therequired mathematics. Indeed, while nonlinear eigenvalue problems—where the

xi

xii PREFACE

simplified MEMS models seem to fit—are a well-developed field of PDEs, the typeof nonlinearity that appears here helps to shed a new light on the class of singularsupercritical problems and their specific challenges. Furthermore, these fourth-order models for MEMS are also amplifying the need for a better and deeper un-derstanding of equations involving the biharmonic Laplacian, which remain quiteelusive in spite of recent advances. The dynamic case presents its own challenges,which have only been tackled in the parabolic setting so far, while its second-orderwavelike counterpart is still completely open to mathematical inquiry.

It is therefore our objective to present in this text a rigorous mathematicalanalysis for various phenomena related to some of the simplest proposed mod-els, many of which were observed either numerically or via ODE methods in theradially symmetric case. Our goal is to try to contribute to the practical needs ofengineers and manufacturers, while satisfying at the same time the intellectual cu-riosity and the quest for rigor of research mathematicians. A case in point are theestimates on “pull-in voltages” and “pull-in distances” that depend on the size andgeometry of the domain and on the permittivity profile of the membrane, whichhave obvious practical considerations. On the other hand, pull-in voltage estimatesthat also depend on the dimension of the ambient space may only be of interestto mathematicians whenever one goes beyond two-dimensional space. A similardependence occurs for the refined properties of steady states—such as regularity,stability, uniqueness, multiplicity, energy estimates, and comparison results. Thesame complexity carries to the dynamic case where issues related to the “quench-ing profile”—in finite or infinite time—or to global convergence towards a stablesteady state, present many interesting mathematical challenges.

From the pedagogical point of view, this monograph is definitely meant forthose already familiar with the modern theory of partial differential equations. Itmay, however, offer an unusual opportunity as an advanced graduate text: a moti-vational introduction to the most recent methods of nonlinear analysis and PDEsthrough the analysis of a set of equations that have enormous practical signifi-cance. Indeed, as mentioned above, the analysis of this most simple idealizedversion of electrostatic MEMS seems to require the “kitchen sink” of modern toolsin PDEs: the notions of weak, sub- and supersolutions, bifurcation diagrams andtheir connection to Morse theory, energy estimates via Sobolev spaces and Moser’siteration, compactness via blowup phenomena and nonlinear Liouville theorems,uniqueness via monotonicity formulae and Pohozaev identities, as well as profileanalysis via maximum principles and moving plane methods. None of these re-quired tools is detailed here (an impossible task), but our hope is that their efficacyis displayed enough so that this book can serve as a motivational reference forlearning and practicing these powerful tools of mathematical analysis.

Most of the results in this volume have been obtained in close collaborationbetween Nassif Ghoussoub, his graduate students Yujin Guo, Craig Cowan, andAmir Moradifam, and his former postdoctoral fellow Pierpaolo Esposito. We areindebted to Xavier Cabré, Daniele Castorina, Daniele Cassani, Joao Marcos Do O,

PREFACE xiii

Changfeng Gui, Zongming Guo, Stephen Gustafson, Abbas Moameni, Peter Po-lacik, Berardino Sciunzi, and Tai-Peng Tsai for their tremendous help in numerousdiscussions on the subject over the last five years.

We are particularly thankful to Michael J. Ward, who introduced us to thisimportant model and for sharing with us his computations and simulations, to LouisNirenberg who led us (as usual) in the right direction by pointing out the pioneeringwork of Joseph and Lundgren, and to Juncheng Wei for his extreme generosityin sharing with us his ideas and his files. We are also grateful to Jalal Shatah,Paul Monsour, and the Courant Institute for the support and for facilitating thepublication of this volume.

Last but not least, we are enormously grateful to our respective spouses Clau-dia, Louise, and Pengxia for their constant support, but also to our children Valerio,Mireille, Michelle, Joseph, and Chenghui for being so forgiving during the devel-opment of this project.

P. EspositoN. Ghoussoub

Y. GuoJune 2009

Bibliography

[1] Abdel-Rahman, E. M., Younis, M. I., and Nayfeh, A. H. Characterization of the mechanicalbehavior of an electrically actuated microbeam. J. Micromech. Microeng. 12: 759–766, 2002.

[2] Adimurthi, and Sekar, A. Role of the fundamental solution in Hardy-Sobolev-type inequali-ties. Proc. Roy. Soc. Edinburgh Sect. A 136(6): 1111–1130, 2006.

[3] Agmon, S., Douglis, A., and Nirenberg, L. Estimates near the boundary for solutions of ellip-tic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl.Math. 12: 623–727, 1959.

[4] Ai, J., Chou, K.-S., and Wei, J. Self-similar solutions for the anisotropic affine curve shorten-ing problem. Calc. Var. Partial Differential Equations 13(3): 311–337, 2001.

[5] Alama, S., and Tarantello, G. Elliptic problems with nonlinearities indefinite in sign. J. Funct.Anal. 141(1): 159–215, 1996.

[6] . On the solvability of a semilinear elliptic equation via an associated eigenvalue prob-lem. Math. Z. 221(3): 467–493, 1996.

[7] Ambrosetti, A., Brezis, H., and Cerami, G. Combined effects of concave and convex nonlin-earities in some elliptic problems. J. Funct. Anal. 122(2): 519–543, 1994.

[8] Ambrosetti, A., and Rabinowitz, P. H. Dual variational methods in critical point theory andapplications. J. Functional Analysis 14: 349–381, 1973.

[9] Arioli, G., Gazzola, F., Grunau, H.-C., and Mitidieri, E. A semilinear fourth order ellipticproblem with exponential nonlinearity. SIAM J. Math. Anal. 36(4): 1226–1258, 2005 (elec-tronic).

[10] Ascher, U., Christiansen, J., and Russell, R. D. A collocation solver for mixed order systemsof boundary value problems. Math. Comp. 33(146): 659–679, 1979.

[11] Bahri, A., and Lions, P.-L. Solutions of superlinear elliptic equations and their Morse indices.Comm. Pure Appl. Math. 45(9): 1205–1215, 1992.

[12] Bandle, C. Isoperimetric inequalities and applications. Monographs and Studies in Mathe-matics, 7. Pitman, Boston–London, 1980.

[13] Bank, R. E. PLTMG: a software package for solving elliptic partial differential equations.Users’ guide 8.0. Software, Environments, and Tools, 5. Society for Industrial and AppliedMathematics (SIAM), Philadelphia, Pa., 1998.

[14] Baras, P., and Cohen, L. Complete blow-up after Tmax for the solution of a semilinear heatequation. J. Funct. Anal. 71(1): 142–174, 1987.

[15] Bebernes, J., and Eberly, D. Mathematical problems from combustion theory. Applied Mathe-matical Sciences, 83. Springer, New York, 1989.

[16] Bellout, H. A criterion for blow-up of solutions to semilinear heat equations. SIAM J. Math.Anal. 18(3): 722–727, 1987.

[17] Berchio, E., Gazzola, F., and Mitidieri, E. Positivity preserving property for a class of bihar-monic elliptic problems. J. Differential Equations 229(1): 1–23, 2006.

[18] Berchio, E., Gazzola, F., and Weth, T. Radial symmetry of positive solutions to nonlinearpolyharmonic Dirichlet problems. J. Reine Angew. Math. 620: 165–183, 2008.

[19] Berestycki, H., Kiselev, A., Novikov, A., and Ryzhik, L. The explosion problem in a flow.J. Anal. Math, to appear. Available online at: http://www.math.wisc.edu/~kiselev/combustion.html.

309

310 BIBLIOGRAPHY

[20] Bernstein, D., Guidott, P., and Pelesko, J. A. Mathematical analysis of an electrostatically ac-tuated MEMS device. Technical Proceedings of the 2000 International Conference on Model-ing and Simulation of Microsystems, 489–492. Nano Science and Tehcnology Institute, Cam-bridge, Ma., 2000. Available online at: http://www.math.uci.edu/~gpatrick/source/papers/BGP00.pdf or http://www.nsti.org/procs/MSM2000/11/W41.03.

[21] Bertozzi, A. L., and Pugh, M. C. Long-wave instabilities and saturation in thin film equations.Comm. Pure Appl. Math. 51(6): 625–661, 1998.

[22] . Finite-time blow-up of solutions of some long-wave unstable thin film equations.Indiana Univ. Math. J. 49(4): 1323–1366, 2000.

[23] Birman, M. S., and Solomjak, M. Z. Spectral theory of selfadjoint operators in Hilbert space.Mathematics and Its Applications (Soviet Series). Reidel, Dordrecht, 1987.

[24] Boggio, T. Sulle funzioni di Green d’ordine m. Rend. Circ. Mat. Palermo 20: 97–135, 1905.[25] Brau, T. A decomposition method with respect to dual cones and its application to

higher order Sobolev spaces. Preprint, 2006. Available online at: http://www-ian.math.uni-magdeburg.de/home/grunau/papers/BraeuStudEnglisch.pdf.

[26] Brauner, C.-M., and Nicolaenko, B. Sur une classe de problèmes elliptiques non linéaires.C. R. Acad. Sci. Paris Sér. A-B 286(21): A1007–A1010, 1978.

[27] . On nonlinear eigenvalue problems which extend into free boundaries problems. Bi-furcation and nonlinear eigenvalue problems (Proc., Session, Univ. Paris XIII, Villetaneuse,1978), 61–100. Lecture Notes in Mathematics, 782. Springer, Berlin–New York, 1980.

[28] Brezis, H., Cazenave, T., Martel, Y., and Ramiandrisoa, A. Blow up for ut � �u D g.u/

revisited. Adv. Differential Equations 1(1): 73–90, 1996.[29] Brezis, H., and Merle, F. Uniform estimates and blow-up behavior for solutions of ��u D

V.x/eu in two dimensions. Comm. Partial Differential Equations 16(8-9): 1223–1253, 1991.[30] Brezis, H., and Nirenberg, L. H1 versus C 1 local minimizers. C. R. Acad. Sci. Paris Sér. I

Math. 317(5): 465–472, 1993.[31] Brezis, H., and Vázquez, J. L. Blow-up solutions of some nonlinear elliptic problems. Rev.

Mat. Univ. Complut. Madrid 10(2): 443–469, 1997.[32] Buffoni, B., Dancer, E. N., and Toland, J. F. The sub-harmonic bifurcation of Stokes waves.

Arch. Ration. Mech. Anal. 152(3): 241–271, 2000.[33] Burelbach, J. P., Bankoff, S. G., and Davis, S. H. Nonlinear stability of evaporating/condensing

liquid films. J. Fluid Mech. 195: 463–494, 1988.[34] Cabré, X. Extremal solutions and instantaneous complete blow-up for elliptic and parabolic

problems. Perspectives in nonlinear partial differential equations, 159–174. ContemporaryMathematics, 446. American Mathematical Society, Providence, R.I., 2007.

[35] Cabré, X., and Capella, A. On the stability of radial solutions of semilinear elliptic equationsin all of Rn. C. R. Math. Acad. Sci. Paris 338(10): 769–774, 2004.

[36] . Regularity of radial minimizers and extremal solutions of semilinear elliptic equa-tions. J. Funct. Anal. 238(2): 709–733, 2006.

[37] Cabré, X., and Martel, Y. Weak eigenfunctions for the linearization of extremal elliptic prob-lems. J. Funct. Anal. 156(1): 30–56, 1998.

[38] Caffarelli, L. A., Gidas, B., and Spruck, J. Asymptotic symmetry and local behavior of semi-linear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42(3): 271–297, 1989.

[39] Cassani, D., do Ó, J. M., Ghoussoub, N. On a fourth order elliptic problem with a singularnonlinearity. Adv. Nonlinear Stud. 9(1): 177–197, 2009.

[40] Castorina, D., Esposito, P., and Sciunzi, B. p-MEMS equation on a ball. Methods Appl. Anal.15(3): 277–284, 2008.

[41] . Degenerate elliptic equations with singular nonlinearities. Calc. Var. Partial Differ-ential Equations 34(3): 279–306, 2009.

[42] . Low dimensional instability for semilinear and quasilinear problems in RN . Comm.Pure Appl. Anal., to appear.

BIBLIOGRAPHY 311

[43] Chang, K.-C. Infinite-dimensional Morse theory and multiple solution problems. Progress inNonlinear Differential Equations and Their Applications, 6. Birkäuser, Boston, 1993.

[44] Chen, W. X., and Li, C. Classification of solutions of some nonlinear elliptic equations. DukeMath. J. 63(3): 615–623, 1991.

[45] Cortazar, C., Elgueta, M., and Rossi, J. D. The blow-up problem for a semilinear parabolicequation with a potential. J. Math. Anal. Appl. 335(1): 418–427, 2007.

[46] Cowan, C. Hardy inequalities for general elliptic operators with improvements. Comm. PureAppl. Anal., in press.

[47] Cowan, C., Esposito, P., Ghoussoub, N., and Moradifam, A. The critical dimension for a 4thorder problem with singular nonlinearity. arXiv: 0904.2414, 2009. Arch. Ration. Mech. Anal.,in press.

[48] Cowan, C., and Ghoussoub, N. Regularity of the extremal solution in a MEMS model withadvection. Methods Appl. Anal. 15(3): 355–360, 2008.

[49] . Estimates on pull-in distances in MEMS models and other nonlinear eigenvalueproblems. arXiv :0903.4464, 2009. Submitted.

[50] Crandall, M. G., and Rabinowitz, P. H. Bifurcation, perturbation of simple eigenvalues andlinearized stability. Arch. Rational Mech. Anal. 52: 161–180, 1973.

[51] . Some continuation and variational methods for positive solutions of nonlinear ellipticeigenvalue problems. Arch. Rational Mech. Anal. 58(3): 207–218, 1975.

[52] Dancer, E. N. Infinitely many turning points for some supercritical problems. Ann. Mat. PuraAppl (4). 178: 225–233, 2000.

[53] . Finite Morse index solutions of exponential problems. Ann. Inst. H. Poincaré Anal.Non Linéaire 25(1) 173–179, 2008.

[54] Dávila, J., Dupaigne, L., Guerra, I., and Montenegro, M. Stable solutions for the bilaplacianwith exponential nonlinearity. SIAM J. Math. Anal. 39(2): 565–592, 2007 (electronic).

[55] Diaz, J. I., Morel, J.-M., and Oswald, L. An elliptic equation with singular nonlinearity. Comm.Partial Differential Equations 12(12): 1333–1344, 1987.

[56] Dold, J. W., Galaktionov, V. A., Lacey, A. A., and Vázquez, J. L. Rate of approach to a singu-lar steady state in quasilinear reaction-diffusion equations. Ann. Scuola Norm. Sup. Pisa Cl.Sci. (4) 26(4): 663–687, 1998.

[57] Druet, O., Hebey, E., and Robert, F. Blow-up theory for elliptic PDEs in Riemannian geometry.Mathematical Notes, 45. Princeton University Press, Princeton, N.J., 2004.

[58] Esposito, P. Compactness of a nonlinear eigenvalue problem with a singular nonlinearity.Commun. Contemp. Math. 10(1): 17–45, 2008.

[59] . Linear instability of entire solutions for a class of non-autonomous elliptic equations.Proc. Roy. Soc. Edinburgh, Sect. A 138(5): 1005–1018, 2008.

[60] Esposito, P., and Ghoussoub, N. Uniqueness of solutions for an elliptic equation modelingMEMS. Methods Appl. Anal. 15(3): 341–354, 2008.

[61] Esposito, P., Ghoussoub, N., and Guo, Y. Compactness along the branch of semistable andunstable solutions for an elliptic problem with a singular nonlinearity. Comm. Pure Appl.Math. 60(12): 1731–1768, 2007.

[62] Evans, L. C. Partial differential equations. Graduate Studies in Mathematics, 19. AmericanMathematical Society, Providence, R.I., 1998.

[63] Fargas Marquès, A., Costa Castelló, R., and Shkel, A. M. Modelling the electrostatic ac-tuation of MEMS: state of the art 2005. Universitat Politècnica De Catalunya, TechnicalReport IOC-DT-P-2005-18, 2005. Available online at: http://bibliotecnica.upc.es/reports/ioc/IOC-DT-P-2005-18.pdf.

[64] Farina, A. On the classification of solutions of the Lane-Emden equation on unbounded do-mains of RN . J. Math. Pures Appl. (9) 87(5): 537–561, 2007.

[65] . Stable solutions of ��u D eu on RN . C. R. Math. Acad. Sci. Paris 345(2): 63–66,2007.

312 BIBLIOGRAPHY

[66] Feng, P., and Zhou, Z. Multiplicity and symmetry breaking for positive radial solutions ofsemilinear elliptic equations modelling MEMS on annular domains. Electron. J. DifferentialEquations 2005(146): 1–14, 2005 (electronic).

[67] Ferrero, A., and Warnault, G. On solutions of second and fourth order elliptic equations withpower-type nonlinearities. Nonlinear Anal. Theory, Methods Appl. 70(8): 2889–2902, 2009.

[68] Feynman, R. P. There’s plenty of room at the bottom. IEEE J. Microelectromech. Syst. 1(1):60–66, 1992.

[69] . Infinitesimal machinery. IEEE J. Microelectromech. Syst. 2(1): 4–14, 1993.[70] Fila, M., and Hulshof, J. A note on the quenching rate. Proc. Amer. Math. Soc. 112(2): 473–

477, 1991.[71] Fila, M., Hulshof, J., and Quittner, P. The quenching problem on the N -dimensional ball.

Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), 183–196.Progress in Nonlinear Differential Equations and their Applications, 7. Birkhäuser, Boston,1992.

[72] Filippas, S., and Kohn, R. V. Refined asymptotics for the blowup of ut � �u D up . Comm.Pure Appl. Math. 45(7): 821–869, 1992.

[73] Flores, G., Mercado, G., Pelesko, J. A., and Smyth, N. Analysis of the dynamics and touch-down in a model of electrostatic MEMS. SIAM J. Appl. Math. 67(2): 434–446, 2006/07 (elec-tronic).

[74] Friedman, A., Friedman, J., and McLeod, B. Concavity of solutions of nonlinear ordinarydifferential equations. J. Math. Anal. Appl. 131(2): 486–500, 1988.

[75] Friedman, A., and McLeod, B. Blow-up of positive solutions of semilinear heat equations.Indiana Univ. Math. J. 34(2): 425–447, 1985.

[76] Fujita, H. On the nonlinear equations�uC eu D 0 and @v=@t D �vC ev . Bull. Amer. Math.Soc. 75: 132-135, 1969.

[77] Galaktionov, V. A., and King, J. R. Stabilization to a singular steady state for the Frank-Kamenetskii equation in the critical dimension. Proc. Roy. Soc. Edinburgh Sect. A 135(4):777–787, 2005.

[78] Garcia Azorero, J. P., and Alonso, I. P. Hardy inequalities and some critical elliptic and para-bolic problems. J. Differential Equations 144(2): 441–476, 1998.

[79] Gazzola, F., and Malchiodi, A. Some remarks on the equation ��u D �.1C u/p for varying�; p and varying domains. Comm. Partial Differential Equations 27(3-4): 809–845, 2002.

[80] Ghoussoub, N. Duality and perturbation methods in critical point theory. Cambridge Tractsin Mathematics, 107. Cambridge University Press, Cambridge, 1993.

[81] Ghoussoub, N., and Guo, Y. On the partial differential equations of electrostatic MEMS de-vices: stationary case. SIAM J. Math. Anal. 38(5): 1423–1449, 2006/07 (electronic).

[82] . Estimates for the quenching time of a parabolic equation modeling electrostaticMEMS. Methods Appl. Anal. 15(3): 361–376, 2008.

[83] . On the partial differential equations of electrostatic MEMS devices. II. Dynamiccase. NoDEA Nonlinear Differential Equations Appl. 15(1-2): 115–145, 2008.

[84] Ghoussoub, N., and Moradifam, A. Bessel pairs and optimal Hardy and Hardy-Rellich in-equalities. Math. Ann., submitted, 2008. arXiv: 0709.1954, 2007.

[85] Gidas, B., Li, W. M., and Nirenberg, L. Symmetry and related properties via the maximumprinciple. Comm. Math. Phys. 68(3): 209–243, 1979.

[86] Gidas, B., and Spruck, J. Global and local behavior of positive solutions of nonlinear ellipticequations. Comm. Pure Appl. Math. 34(4): 525–598, 1981.

[87] Giga, Y. On elliptic equations related to self-similar solutions for nonlinear heat equations.Hiroshima Math. J. 16(3): 539–552, 1986.

[88] Giga, Y., and Kohn, R. V. Asymptotically self-similar blow-up of semilinear heat equations.Comm. Pure Appl. Math. 38(3): 297–319, 1985.

[89] . Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36(1): 1–40, 1987.

BIBLIOGRAPHY 313

[90] . Nondegeneracy of blowup for semilinear heat equations. Comm. Pure Appl. Math.42(6): 845–884, 1989.

[91] Gilbarg, D., and Trudinger, N. S. Elliptic partial differential equations of second order. Reprintof the 1998 ed. Classics in Mathematics. Springer, Berlin, 2001.

[92] Grunau, H.-C., and Sweers, G. Positivity properties of elliptic boundary value problems ofhigher order. Nonlinear Anal. 30(8): 5251–5258, 1997.

[93] Gui, C., and Lin, F.-H. Regularity of an elliptic problem with a singular nonlinearity. Proc.Roy. Soc. Edinburgh Sect. A 123(6): 1021–1029, 1993.

[94] Guo, H., Guo, Z., and Li, K. Positive solutions of a semilinear elliptic equation with singularnonlinearity. J. Math. Anal. Appl. 323(1): 344–359, 2006.

[95] Guo, J.-S. On the quenching behavior of the solution of a semilinear parabolic equation.J. Math. Anal. Appl. 151(1): 58–79, 1990.

[96] . On the semilinear elliptic equation�w� 12y �rwC�w�w�ˇ D 0 in Rn. Chinese

J. Math. 19(4): 355–377, 1991.[97] Guo, Y. Global solutions of singular parabolic equations arising from electrostatic MEMS.

J. Differential Equations 245(3): 809–844, 2008.[98] . On the partial differential equations of electrostatic MEMS devices. III. Refined

touchdown behavior. J. Differential Equations 244(9): 2277–2309, 2008.[99] Guo, Y., Pan, Z., and Ward, M. J. Touchdown and pull-in voltage behavior of a MEMS device

with varying dielectric properties. SIAM J. Appl. Math. 66(1): 309–338, 2005.[100] Guo, Z., and Webb, J. R. L. Large and small solutions of a class of quasilinear elliptic eigen-

value problems. J. Differential Equations 180(1): 1–50, 2002.[101] Guo, Z., and Wei, J. Hausdorff dimension of ruptures for solutions of a semilinear elliptic

equation with singular nonlinearity. Manuscripta Math. 120(2): 193–209, 2006.[102] . On the Cauchy problem for a reaction-diffusion equation with a singular nonlinear-

ity. J. Differential Equations 240(2): 279–323, 2007.[103] . Symmetry of non-negative solutions of a semilinear elliptic equation with singular

nonlinearity. Proc. Royal. Soc. Edinburgh. Sect. A 137(5): 963–994, 2007.[104] . Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic

problem with a singular nonlinearity. Comm. Pure Appl. Anal. 7(4): 765–786, 2008.[105] . Entire solutions and global bifurcations for a biharmonic equation with singular non-

linearity in R3. Adv. Differential Equations 13(7-8): 753–780, 2008.[106] . Infinitely many turning points for an elliptic problem with a singular non-linearity.

J. Lond. Math. Soc. (2) 78(1): 21–35, 2008.[107] . On solutions with point ruptures for a semilinear elliptic problem with singularity.

Methods Appl. Anal. 15(3): 377–390, 2008.[108] . On a fourth order nonlinear elliptic equation with negative exponent. SIAM J. Math.

Anal. 40(5): 2034–2054, 2008/09.[109] Han, Q., and Lin, F. Elliptic partial differential equations. Courant Lecture Notes in Math-

ematics, 1. New York University, Courant Institute of Mathematical Sciences, New York;American Mathematical Society, Providence, R.I., 1997.

[110] Han, Z.-C. Asymptotic approach to singular solutions for nonlinear elliptic equations involv-ing critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linèaire 8(2): 159–174, 1991.

[111] Hartman, P. Ordinary differential equations. Wiley, New York–London–Sydney, 1964.[112] Hui, K. M. Global and touchdown behaviour of the generalized MEMS device equation.

Preprint, 2008. arXiv: 0808.0110v1.[113] Hwang, C.-C., Lin, C.-K., and Uen, W.-Y. A nonlinear three-dimensional rupture theory of

thin liquid films. J. Colloid Interface Sci. 190(1): 250–252, 1997.[114] Jackson, J. D. Classical electrodynamics. 2nd ed. Wiley, New York–London–Sydney, 1975.[115] Jiang, H., and Lin, F. Zero set of Sobolev functions with negative power of integrability. Chi-

nese Ann. Math. Ser. B 25(1): 65–72, 2004.[116] Joseph, D. D., and Lundgren, T. S. Quasilinear Dirichlet problems driven by positive sources.

Arch. Rational Mech. Anal. 49: 241–269, 1972/73.

314 BIBLIOGRAPHY

[117] Kavallaris, N. I., Miyasita, T., and Suzuki, T. Touchdown and related problems in electro-static MEMS device equation. NoDEA Nonlinear Differential Equations Appl. 15(3): 363–385, 2008.

[118] Keener, J. P., and Keller, H. B. Positive solutions of convex nonlinear eigenvalue problems.J. Differential Equations 16: 103–125, 1974.

[119] Keller, J. B., and Lowengrub, J. S. Asymptotic and numerical results for blowing-up solu-tions to semilinear heat equations. Singularities in fluids, plasmas and optics (Heraklion,1992), 111–129. NATO Advanced Science Institutes Series C: Mathematical and PhysicalSciences, 404. Kluwer, Dordrecht, 1993.

[120] Lacey, A. A. Mathematical analysis of thermal runaway for spatially inhomogeneous reac-tions. SIAM J. Appl. Math. 43(6): 1350–1366, 1986.

[121] Ladyženskaja, O. A., Solonnikov, V. A., and Ural0ceva, N. N. Linear and quasilinear equa-tions of parabolic type. Translations of Mathematical Monographs, 23. American Mathemati-cal Society, Providence, R.I., 1967.

[122] Laugesen, R. S., and Pugh, M. C. Linear stability of steady states for thin film and Cahn-Hilliard type equations. Arch. Ration. Mech. Anal. 154(1): 3–51, 2000.

[123] . Properties of steady states for thin film equations. European J. Appl. Math. 11(3):293–351, 2000.

[124] . Energy levels of steady states for thin-film-type equations. J. Differential Equations182(2): 377–415, 2002.

[125] Levine, H. A. Quenching, nonquenching, and beyond quenching for solution of some para-bolic equations. Ann. Mat. Pura Appl. (4) 155: 243–260, 1989.

[126] Lin, F., and Yang, Y. Nonlinear non-local elliptic equation modelling electrostatic actuation.Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463(2081), 1323–1337, 2007.

[127] Lindsay, A., and Ward, M. J. Asymptotics of some nonlinear eigenvalue problems for a MEMScapacitor: Part I: Fold point asymptotics. Methods Appl. Anal. 15(3): 297–326, 2008.

[128] Luckhaus, S. Existence and regularity of weak solutions to the Dirichlet problem for semilin-ear elliptic systems of higher order. J. Reine Angew. Math. 306: 192–207, 1979.

[129] Ma, L., and Wei, J. C. Properties of positive solutions to an elliptic equation with negativeexponent. J. Funct. Anal. 254(4): 1058–1087, 2008.

[130] Martel, Y. Uniqueness of weak extremal solutions of nonlinear elliptic problems. Houston J.Math. 23(1): 161–168, 1997.

[131] Meadows, A. M. Stable and singular solutions of the equation �u D 1=u. Indiana Univ.Math. J. 53(6): 1681–1703, 2004.

[132] Mignot, F. and Puel, J.-P. Sur une classe de problèmes non linéaires avec non linéairité posi-tive, croissante, convexe. Comm. Partial Differential Equations 5(8): 791–836, 1980.

[133] Moradifam, A. On the critical dimension of a fourth order elliptic problem with negativeexponent. J. Differential Equations, in press. arXiv: 0905.1940, 2009.

[134] . The singular extremal solutions of the bilaplacian with exponential nonlinearity.Proc. Amer. Math. Soc., forthcoming. arXiv: 0905.1937, 2009.

[135] Moreau, J.-J. Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuelle-ment polaires. C. R. Acad. Sci. Paris 255: 238–240, 1962.

[136] Nathanson, H. C., Newell, W. E., Wickstrom, R. A. and Davis Jr., J. R. The resonant gatetransistor. IEEE Trans. Electron. Devices 14(3): 117–133, 1967.

[137] Nayfeh, A. H., Younis, M. I., and Abdel-Rahman, E. M. Reduced-order models for MEMSapplications. Nonlinear Dynam. 41(1-3): 211–236, 2005.

[138] Nedev, G. Regularity of the extremal solution of semilinear elliptic equations. C. R. Acad. Sci.Paris Sér. I Math. 330(11): 997–1002, 2000.

[139] Pelesko, J. A. Mathematical modeling of electrostatic MEMS with tailored dielectric proper-ties. SIAM J. Appl. Math. 62(3): 888–908, 2001/02, (electronic).

[140] Pelesko, J. A., and Bernstein, D. H. Modeling MEMS and NEMS. Chapman and Hall/CRC,Boca Raton, Fla., 2003.

BIBLIOGRAPHY 315

[141] Pelesko, J. A., Bernstein, D. H., and McCuan, J. Symmetry and symmetry breaking in elec-trostatically actuated MEMS. Nanotech 2003, vol. 1, 304–307. Nano Science and TechnologyInstitute, Cambridge, Mass., 22003.

[142] Pelesko, J. A., and Triolo, A. A. Nonlocal problems in MEMS device control. J. Engrg. Math.41(4): 345–366, 2001.

[143] Peral, I., and Vázquez, J. L. On the stability or instability of the singular solution of the semi-linear heat equation with exponential reaction term. Arch. Rational Mech. Anal. 129(3): 201–224, 1995.

[144] Prajapat, J., and Tarantello, G. On a class of elliptic problems in R2: symmetry and uniquenessresults. Proc. Roy. Soc. Edinburgh Sect. A 131(4): 967–985, 2001.

[145] Pucci, P., and Serrin, J. A general variational identity. Indiana Univ. Math. J. 35(3): 681–703,1986.

[146] Rabinowitz, P. H. Minimax methods in critical point theory with applications to differentialequations. CBMS Regional Conference Series in Mathematics, 65. American MathematicalSociety, Providence, R.I., 1986.

[147] Rébaï, Y. Solutions of semilinear elliptic equations with one isolated singularity. DifferentialIntegral Equations 12(4): 563–581, 1999.

[148] Richtmyer, R. D. Principles of advanced mathematical physics. Vol. 1. Texts and Monographsin Physics. Springer, New York–Heidelberg, 1978.

[149] Sacks, J., and Uhlenbeck, K. The existence of minimal immersions of 2-spheres. Ann. ofMath. (2) 113(1): 1–24, 1981.

[150] Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S. P., and Mikhailov, A. P. Blow-up in quasi-linear parabolic equations. de Gruyter Expositions in Mathematics, 19. de Gruyter, Berlin,1995.

[151] Schaaf, R. Uniqueness for semilinear elliptic problems: supercritical growth and domain ge-ometry. Adv. Differential Equations 5(10-12): 1201–1220, 2000.

[152] Seeger, J. I., and Crary, S. B. Stabilization of electrostatically actuated mechanical devices.Transducers ’97. International Conference on Solid State Sensors and Actuators (Chicago,1997), vol. 2, 1133–1136. IEEE, Los Alamitos, Calif., 1998.

[153] Simon, L. Asymptotics for a class of nonlinear evolution equations, with applications to geo-metric problems. Ann. of Math. (2) 118(3): 525–571, 1983.

[154] Soranzo, R. A priori estimates and existence of positive solutions of a superlinear polyhar-monic equation. Dynam. Systems Appl. 3(4): 465–487, 1994.

[155] Struwe, M. Variational methods. Applications to nonlinear partial differential equations andHamiltonian systems. Springer, Berlin, 1990.

[156] Sweers, G. No Gidas-Ni-Nirenberg type result for semilinear biharmonic problems. Math.Nachr. 246/247: 202–206, 2002.

[157] Taylor, G. The coalescence of closely spaced drops when they are at different electric poten-tials. Proc. R. Soc. Lond. A 306(1487): 423–434, 1968.

[158] Troy, W. C. Symmetry properties in systems of semilinear elliptic equations. J. DifferentialEquations 42(3): 400–413, 1981.

[159] Wang, Z., and Ruan, L. On a class of semilinear elliptic problems with singular nonlinearities.Appl. Math. Comput. 193(1): 89–105, 2007.

[160] Warnault, G. Regularity of the extremal solution for a biharmonic problem with general non-linearity. Comm. Pure Appl. Anal. 8(5): 1709–1723, 2009.

[161] Ye, D., and Zhou, F. Boundedness of the extremal solution for semilinear elliptic problems.Commun. Contemp. Math. 4(3): 547–558, 2002.

[162] Younis, M. I., Abdel-Rahman, E. M., and Nayfeh, A. H. Static and dynamic behavior of anelectrically excited resonant microbeam. 43rd AIAA/AME/ASCE/AHS/ASC Structures, Struc-tural Dynamics, and Materials Conference, Denver, CO, AIAA, 2002.

[163] Zheng, G. New results on the formation of singularities for parabolic problems. Ph.D. thesis,The Chinese University of Hong Kong, 2005.

Index

Admissible pair, 271Ambient space, 24, 51Applied voltage, 24Aspect ratio, 3, 7, 9Asymptotic behavior, 13, 28, 90, 235, 269

Backward heat equation, 182Badly singular, 26Bending energy, 3, 5Bessel function, 47Bifurcation

diagram, 11–13, 17, 24, 296point, 25, 93, 110, 139, 161theory, 92

Biharmonic operator, 2, 253, 256, 275Blowup analysis, 98, 113, 115Boggio’s principle, 246, 257

Calculus of variations, 270Capacitance, 3, 4Cauchy-Schwarz inequality, 205Characteristic function, 247Charged liquid drops, 10Chemical catalyst kinetics, 173Clamped boundary, 10, 23, 28, 245Classical iterative scheme, 251, 273COLSYS, 47, 49Comparison principle, 209Concavity method, 204Conducting film, 1Coulomb law, 4Crank-Nicholson scheme, 87, 187, 196Critical

dimension, 23, 24, 51, 74, 75, 92point, 95, 236threshold, 15

Damping intensity, 3Dielectric permittivity, 6, 240Dimensionless analysis, 8Dirichlet boundary, 28

Eigenfunction expansion, 179Electrostatic

actuation, 1force, 1, 4, 5MEMS, 1potential, 8

Elliptic regularity theory, 37, 68, 251, 273Energy quenching rate, 27Equilibrium

point, 14state, 10

Euler equation, 160Euler-Lagrange equation, 5Evolution equation, 2Exponential nonlinearity, 75External viscous damping, 3Extremal function, 54Extremal solution, 13, 24

Fatou’s lemma, 37Formal analysis, 22Fourth-order equation, 10Fredholm operator, 128Free space, 3, 7

Gelfand problem, 74, 92Global convergence, 177, 181Governing equation, 9Green’s function, 38, 39, 246Green’s theorem, 207

Hardy inequality, 70Harnack inequality, 68, 189Heat equation, 193Heat semigroup, 183Hodge decomposition, 70Hopf’s lemma, 101, 103, 247

Implicit function theorem, 34, 62, 112, 272,279

Improved Hardy-Rellich inequality, 299

317

318 INDEX

Initialcondition, 3, 20data, 178, 198time, 9

Initial value problem, 13, 201, 236Isoperimetric inequality, 39, 200

Kernel, 53Krein-Rutman theory, 70

Laplace-Beltrami operator, 303Laplacian, 8, 35Laplacian resolvent, 127Lebesgue’s theorem, 144Leray-Schauder degree, 127Lyapunov functional, 180Lyapunov-Schmidt reduction, 128

Maple, 82, 84Mass density, 3, 5MEMS, 1Method of dominant balance, 19, 22Microplate, 1, 2Microresonator, 3Microsystem, 1Microvalve, 27, 199Minimal solution, 17, 23, 24, 51Moreau decomposition, 248, 256, 277Morse index, 25, 115, 126, 127Moser-Trudinger inequality, 57Moser-type iteration scheme, 63Mountain pass theorem, 95, 97, 284Moving plane method, 192, 198, 281

Navier boundary, 28, 269Negative exponent, 10NEMS, 1Newton’s second law, 5Newtonian potential, 81Non-self-adjoint eigenvalue problem, 75Nonlocal equation, 4Nonoscillatory criterion, 239Numerical quadrature, 47

One-dimensional Sobolev inequality, 141

Parabolic equation, 10, 26, 177, 198, 204Pinned boundary, 10, 28, 269, 296Pohozaev identity, 45, 130Poisson ratio, 3Poisson ratio, 5Power series solution, 23Power-law profiles, 48Power-law permittivity, 11, 43

Principal eigenpair, 70, 71Pull-in

distance, 3, 13, 25instability, 1voltage, 12, 23, 33, 51, 78, 252

Quenchingfinite time, 218set, 191, 192, 194, 198

Quenching point, 177Quenching time, 177

Recursive scheme, 38, 85Refined quenching profile, 221, 229, 233RF switches, 27, 199Rotational inertia, 24Rotational inertial, 10

Saddle-node point, 47, 50Schauder’s estimate, 220Schwarz symmetrization, 40, 200Semistable, 52Singular minimal hypersurfaces, 11, 172Slab domain, 47, 187Small-aspect ratio limit, 8, 9Snap-through, 1Sobolev inequality, 26, 172Stretching energy, 5Sub/supersolutions, 33Suitably symmetric domain, 135Super level set, 40Superstable, 90

Temporal mesh size, 20Tension constant, 3, 5Truncation, 246Turning points, 12

Weak iterative scheme, 250, 272Weierstrass’s approximation theorem, 247

Young modulus, 3, 5Young’s inequality, 278

Titles in This Series

20 Pierpaolo Esposito, Nassif Ghoussoub, and Yujin Guo, Mathematical analysis ofpartial differential equations modeling electrostatic MEMS, 2010

19 Stephen Childress, An introduction to theoretical fluid mechanics, 2009

18 Percy Deift and Dimitri Gioev, Random matrix theory: Invariant ensembles anduniversality, 2009

17 Ping Zhang, Wigner measure and semiclassical limits of nonlinear Schrodinger equations,

2008

16 S. R. S. Varadhan, Stochastic processes, 2007

15 Emil Artin, Algebra with Galois theory, 2007

14 Peter D. Lax, Hyperbolic partial differential equations, 2006

13 Oliver Buhler, A brief introduction to classical, statistical, and quantum mechanics, 2006

12 Jurgen Moser and Eduard J. Zehnder, Notes on dynamical systems, 2005

11 V. S. Varadarajan, Supersymmetry for mathematicians: An introduction, 2004

10 Thierry Cazenave, Semilinear Schrodinger equations, 2003

9 Andrew Majda, Introduction to PDEs and waves for the atmosphere and ocean, 2003

8 Fedor Bogomolov and Tihomir Petrov, Algebraic curves and one-dimensional fields,2003

7 S. R. S. Varadhan, Probability theory, 2001

6 Louis Nirenberg, Topics in nonlinear functional analysis, 2001

5 Emmanuel Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities,2000

3 Percy Deift, Orthogonal polynomials and random matrices: A Riemann-Hilbertapproach, 2000

2 Jalal Shatah and Michael Struwe, Geometric wave equations, 2000

1 Qing Han and Fanghua Lin, Elliptic partial differential equations, 2000

Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS

PIERPAOLO ESPOSITO, NASSIF GHOUSSOUB, YUJIN GUO

CLN/20

New York UniversityAMS on the Webwww.ams.org

For additional informationand updates on this book, visit

www.ams.org/bookpages/cln-20

Micro- and nanoelectromechanical systems (MEMS and NEMS), which combine electronics with miniature-size mechanical devices, are essen-tial components of modern technology. It is the mathematical model describing “electrostatically actuated” MEMS that is addressed in this monograph. Even the simplified models that the authors deal with still lead to very interesting second- and fourth-order nonlinear elliptic equations (in the stationary case) and to nonlinear parabolic equations (in the dynamic case). While nonlinear eigenvalue problems—where the stationary MEMS models fit—are a well-developed field of PDEs, the type of inverse square nonlinearity that appears here helps shed a new light on the class of singular supercritical problems and their specific challenges.

Besides the practical considerations, the model is a rich source of inter-esting mathematical phenomena. Numerics, formal asymptotic analysis, and ODE methods give lots of information and point to many conjectures. However, even in the simplest idealized versions of electrostatic MEMS, one essentially needs the full available arsenal of modern PDE techniques to do the required rigorous mathematical analysis, which is the main objective of this volume. This monograph could therefore be used as an advanced graduate text for a motivational introduction to many recent methods of nonlinear analysis and PDEs through the analysis of a set of equations that have enormous practical significance.