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Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS

Pierpaolo Esposito Università degli Studi Roma Tre, Rome, Italy
Nassif Ghoussoub University of British Columbia, Vancouver, BC, Canada
Yujin Guo University of Minnesota, Minneapolis, MN
A co-publication of the AMS and Courant Institute of Mathematical Sciences at New York University
Available Formats:
Softcover ISBN: 978-0-8218-4957-6
Product Code: CLN/20
List Price: $52.00 MAA Member Price:$46.80
AMS Member Price: $41.60 Electronic ISBN: 978-1-4704-1763-5 Product Code: CLN/20.E List Price:$49.00
MAA Member Price: $44.10 AMS Member Price:$39.20
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AMS Member Price: $62.40 Click above image for expanded view Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS Pierpaolo Esposito Università degli Studi Roma Tre, Rome, Italy Nassif Ghoussoub University of British Columbia, Vancouver, BC, Canada Yujin Guo University of Minnesota, Minneapolis, MN A co-publication of the AMS and Courant Institute of Mathematical Sciences at New York University Available Formats:  Softcover ISBN: 978-0-8218-4957-6 Product Code: CLN/20  List Price:$52.00 MAA Member Price: $46.80 AMS Member Price:$41.60
 Electronic ISBN: 978-1-4704-1763-5 Product Code: CLN/20.E
 List Price: $49.00 MAA Member Price:$44.10 AMS Member Price: $39.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$78.00 MAA Member Price: $70.20 AMS Member Price:$62.40
• Book Details

Courant Lecture Notes
Volume: 202010; 318 pp
MSC: 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 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.

Graduate students and research mathematicians interested in PDEs and applications.

• Chapters
• Chapter 1. Introduction
• Second-order equations modeling stationary MEMS
• Chapter 2. Estimates for the pull-in voltage
• Chapter 3. The branch of stable solutions
• Chapter 4. Estimates for the pull-in distance
• Chapter 5. The first branch of unstable solutions
• Chapter 6. Description of the global set of solutions
• Chapter 7. Power-law 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. Fourth-order equations modeling nonelastic MEMS
• Chapter 11. A fourth-order model with a clamped boundary on a ball
• Chapter 12. A fourth-order model with a pinned boundary on convex domains
• Appendix A. Hardy–Rellich inequalities

• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 202010; 318 pp
MSC: 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 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.

Graduate students and research mathematicians interested in PDEs and applications.

• Chapters
• Chapter 1. Introduction
• Second-order equations modeling stationary MEMS
• Chapter 2. Estimates for the pull-in voltage
• Chapter 3. The branch of stable solutions
• Chapter 4. Estimates for the pull-in distance
• Chapter 5. The first branch of unstable solutions
• Chapter 6. Description of the global set of solutions
• Chapter 7. Power-law 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. Fourth-order equations modeling nonelastic MEMS
• Chapter 11. A fourth-order model with a clamped boundary on a ball
• Chapter 12. A fourth-order model with a pinned boundary on convex domains
• Appendix A. Hardy–Rellich inequalities
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
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