**Courant Lecture Notes**

Volume: 20;
2010;
318 pp;
Softcover

MSC: Primary 35; 74; 58;

Print ISBN: 978-0-8218-4957-6

Product Code: CLN/20

List Price: $49.00

Individual Member Price: $39.20

**Electronic ISBN: 978-1-4704-1763-5
Product Code: CLN/20.E**

List Price: $49.00

Individual Member Price: $39.20

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#### Supplemental Materials

# Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS

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*Pierpaolo Esposito; Nassif Ghoussoub; Yujin Guo*

A co-publication of the AMS and the Courant Institute of Mathematical Sciences at New York University

Nassif Ghoussoub is the winner of the 2010 CMS David Borwein
Award

Micro- and nanoelectromechanical systems (MEMS and NEMS), which
combine electronics with miniature-size mechanical devices, are
essential 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
interesting 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.

Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

#### Table of Contents

# Table of Contents

## Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS

- Cover Cover11
- Title page iii4
- Dedication v6
- Contents vii8
- Preface xi12
- Introduction 116
- Part 1. Second-order equations modeling stationary MEMS 3146
- Estimates for the pull-in voltage 3348
- The branch of stable solutions 5166
- Estimates for the pull-in distance 7792
- The first branch of unstable solutions 93108
- Description of the global set of solutions 115130
- Power-law profiles on symmetric domains 141156
- Part 2. Parabolic equations modeling MEMS dynamic deflections 175190
- Different modes of dynamic deflection 177192
- Estimates on quenching times 199214
- Refined profile of solutions at quenching time 217232
- Part 3. Fourth-order equations modeling nonelastic MEMS 243258
- A fourth-order model with a clamped boundary on a ball 245260
- A fourth-order model with a pinned boundary on convex domains 269284
- Appendix A. Hardy–Rellich inequalities 299314
- Bibliography 309324
- Index 317332
- Back Cover Back Cover1338

#### Readership

Graduate students and research mathematicians interested in PDEs and applications.