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Electronic ISBN:  9781470417635 
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Book DetailsCourant Lecture NotesVolume: 20; 2010; 318 ppMSC: Primary 35; 74; 58;Nassif Ghoussoub is the winner of the 2010 CMS David Borwein Award
Micro and nanoelectromechanical systems (MEMS and NEMS), which combine electronics with miniaturesize 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 fourthorder 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 welldeveloped 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.ReadershipGraduate students and research mathematicians interested in PDEs and applications.

Table of Contents

Chapters

Chapter 1. Introduction

Secondorder equations modeling stationary MEMS

Chapter 2. Estimates for the pullin voltage

Chapter 3. The branch of stable solutions

Chapter 4. Estimates for the pullin distance

Chapter 5. The first branch of unstable solutions

Chapter 6. Description of the global set of solutions

Chapter 7. Powerlaw profiles on symmetric domains

Part 2. Parabolic equations modeling MEMS dynamic deflections

Chapter 8. Different modes of dynamic deflection

Chapter 9. Estimates on quenching times

Chapter 10. Refined profile of solutions at quenching time

Part 3. Fourthorder equations modeling nonelastic MEMS

Chapter 11. A fourthorder model with a clamped boundary on a ball

Chapter 12. A fourthorder model with a pinned boundary on convex domains

Appendix A. Hardy–Rellich inequalities


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Micro and nanoelectromechanical systems (MEMS and NEMS), which combine electronics with miniaturesize 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 fourthorder 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 welldeveloped 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.
Graduate students and research mathematicians interested in PDEs and applications.

Chapters

Chapter 1. Introduction

Secondorder equations modeling stationary MEMS

Chapter 2. Estimates for the pullin voltage

Chapter 3. The branch of stable solutions

Chapter 4. Estimates for the pullin distance

Chapter 5. The first branch of unstable solutions

Chapter 6. Description of the global set of solutions

Chapter 7. Powerlaw profiles on symmetric domains

Part 2. Parabolic equations modeling MEMS dynamic deflections

Chapter 8. Different modes of dynamic deflection

Chapter 9. Estimates on quenching times

Chapter 10. Refined profile of solutions at quenching time

Part 3. Fourthorder equations modeling nonelastic MEMS

Chapter 11. A fourthorder model with a clamped boundary on a ball

Chapter 12. A fourthorder model with a pinned boundary on convex domains

Appendix A. Hardy–Rellich inequalities