Mathematical analysis of partial differential equations modeling electrostatic MEMS:
"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. Ev...
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Courant Inst. of Math. Sciences [u.a.]
2010
|
| Schriftenreihe: | Courant lecture notes in mathematics
20 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "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."--Publisher's description. |
| Beschreibung: | XIII, 318 S. graph. Darst. |
| ISBN: | 9780821849576 |
Internformat
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| 245 | 1 | 0 | |a Mathematical analysis of partial differential equations modeling electrostatic MEMS |c Pierpaolo Esposito ; Nassif Ghoussoub ; Yujin Guo |
| 264 | 1 | |a New York, NY |b Courant Inst. of Math. Sciences [u.a.] |c 2010 | |
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| 520 | 3 | |a "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."--Publisher's description. | |
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Datensatz im Suchindex
| _version_ | 1804140851657441280 |
|---|---|
| adam_text | Contents
Preface
xi
Chapter
1.
Introduction
1
1.1.
Electrostatic Actuations and Nonlinear PDEs
1
1.2.
Derivation of the Model for Homogeneous Systems
4
1.3.
MEMS Models with Variable Permittivity Profiles
6
1.4.
Bifurcation Diagrams and Numerical Evidence
11
1.5.
Brief Outline
24
Part
1.
Second-Order Equations Modeling Stationary MEMS
31
Chapter
2.
Estimates for the Pull-In Voltage
33
2.1.
Existence of the Pull-In Voltage
33
2.2.
Lower Estimates for the Pull-in Voltage
39
2.3.
Upper Bounds for the Pull-in Voltage
43
2.4.
Numerics for the Pull-In Voltage
46
Further Comments
50
Chapter
3.
The Branch of Stable Solutions
51
3.1.
Spectral Properties of Minimal Solutions
51
3.2.
Energy Estimates and Regularity of Solutions
55
3.3.
Linear Instability and Compactness
62
3.4.
Effect of an Advection on the Minimal Branch
69
Further Comments
74
Chapter
4.
Estimates for the Pull-In Distance
77
4.1.
Lower Estimates on the Pull-In Distance in General Domains
77
4.2.
Upper Estimate for the Pull-In Distance in General Domains
80
4.3.
Upper Bounds for the Pull-In Distance in the Radial Case
82
4.4.
Effect of Power-Law Profiles on Pull-In Distances
85
4.5.
Asymptotic Behavior of Stable Solutions near the Pull-In Voltage
90
Further Comments
92
Chapter
5.
The First Branch of Unstable Solutions
93
5.1.
Existence of
Nonminimal
Solutions
94
5.2.
Blowup Analysis for Noncompact Sequences of Solutions
98
5.3.
Compactness along the First Branch of Unstable Solutions
103
5.4.
Second Bifurcation Point
110
Further Comments
112
viii CONTENTS
Chapter
6.
Description of the Global Set of Solutions
115
6.1.
Compactness along the Unstable Branches
116
6.2.
Quenching Branch of Solutions in General Domains
125
6.3.
Uniqueness of Solutions for Small Voltage in Star-Shaped Domains
129
6.4.
One-Dimensional Problem
137
Further Comments
139
Chapter
7.
Power-Law Profiles on Symmetric Domains
141
7.1.
A One-Dimensional Sobolev Inequality
141
7.2.
Monotonicity
Formula and Applications
145
7.3.
Compactness of Higher Branches of Radial Solutions
152
7.4.
Two-Dimensional MEMS on Symmetric Domains
162
Further Comments
172
Part
2.
Parabolic Equations Modeling MEMS Dynamic Deflections
175
Chapter
8.
Different Modes of Dynamic Deflection
177
8.1.
Global Convergence versus Quenching
178
8.2.
Quenching Points and the Zero Set of the Profile
187
8.3.
The Quenching Set on Convex Domains
192
Further Comments
198
Chapter
9.
Estimates on Quenching Times
199
9.1.
Comparison Results for Quenching Times
199
9.2.
General Asymptotic Estimates for Quenching Time
201
9.3.
Upper Estimates for Quenching Times for all A
>
λ*
203
9.4.
Quenching Time Estimates in Low Dimension
210
Further Comments
215
Chapter
10.
Refined Profile of Solutions at Quenching Time
217
10.1.
Integral and Gradient Estimates for Quenching Solutions
217
10.2.
Refined Quenching Profile
221
10.3.
Refined Quenching Profiles in Dimension
N =
Ì
229
10.4.
Refined Quenching Profiles in the Radially Symmetric Case
233
10.5.
More on the Location of Quenching Points
240
Further Comments
242
Part
3.
Fourth-Order Equations Modeling Nonelastic MEMS
243
Chapter
11.
A Fourth-Order Model with a Clamped Boundary
on a Ball
245
11.1.
Boggio s Principle
245
11.2.
Pull-in Voltage
249
11.3.
Stability of the Minimal Branch of Solutions
255
1
1.4.
Regularity of the Extremal Solution for
1 < N < 8 260
11.5.
The Extremal Solution Is Singular for
N > 9 263
Further Comments
268
CONTENTS ix
Chapter
12.
A Fourth-Order Model with a Pinned Boundary
on Convex Domains
269
12.1.
The Minimal Solutions up to the Pull-In Voltage
269
12.2.
Stability of Minimal Solutions
274
12.3.
Regularity of the Extremal Solution on General Domain
f
or ./V
< 4 279
12.4.
Uniform Energy Bounds for Solutions in Convex Domains
280
12.5.
The Solution Set on Convex Domains in
Ж2
283
12.6.
Regularity of the Extremal Solution on Balls for
N < 8 289
12.7.
Singularity of the Extremal Solution on Balls for
N > 9 291
Further Comments
296
Appendix A. Hardy-Rellich Inequalities
299
A.
1.
Improved Hardy-Rellich Inequalities in
Н£(В)
299
A.2. Improved Hardy-Rellich Inequalities in H2(B)
Π
Я,,1 (В)
302
Bibliography
309
Index
317
|
| any_adam_object | 1 |
| author | Esposito, Pierpaolo Ghoussoub, Nassif 1953- Guo, Yujin |
| author_GND | (DE-588)141633743 (DE-588)137025246 (DE-588)141633824 |
| author_facet | Esposito, Pierpaolo Ghoussoub, Nassif 1953- Guo, Yujin |
| author_role | aut aut aut |
| author_sort | Esposito, Pierpaolo |
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| dewey-ones | 621 - Applied physics |
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| dewey-tens | 620 - Engineering and allied operations |
| discipline | Mathematik |
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| id | DE-604.BV035875186 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:06:33Z |
| institution | BVB |
| isbn | 9780821849576 |
| language | English |
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| physical | XIII, 318 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Courant Inst. of Math. Sciences [u.a.] |
| record_format | marc |
| series | Courant lecture notes in mathematics |
| series2 | Courant lecture notes in mathematics |
| spelling | Esposito, Pierpaolo Verfasser (DE-588)141633743 aut Mathematical analysis of partial differential equations modeling electrostatic MEMS Pierpaolo Esposito ; Nassif Ghoussoub ; Yujin Guo New York, NY Courant Inst. of Math. Sciences [u.a.] 2010 XIII, 318 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Courant lecture notes in mathematics 20 "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."--Publisher's description. Mathematisches Modell Mathematical analysis Microelectromechanical systems Mathematical models MEMS (DE-588)4824724-8 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s MEMS (DE-588)4824724-8 s DE-604 Ghoussoub, Nassif 1953- Verfasser (DE-588)137025246 aut Guo, Yujin Verfasser (DE-588)141633824 aut Erscheint auch als Online-Ausgabe 978-1-4704-1763-5 Courant lecture notes in mathematics 20 (DE-604)BV012714106 20 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018732878&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Esposito, Pierpaolo Ghoussoub, Nassif 1953- Guo, Yujin Mathematical analysis of partial differential equations modeling electrostatic MEMS Courant lecture notes in mathematics Mathematisches Modell Mathematical analysis Microelectromechanical systems Mathematical models MEMS (DE-588)4824724-8 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4824724-8 (DE-588)4044779-0 |
| title | Mathematical analysis of partial differential equations modeling electrostatic MEMS |
| title_auth | Mathematical analysis of partial differential equations modeling electrostatic MEMS |
| title_exact_search | Mathematical analysis of partial differential equations modeling electrostatic MEMS |
| title_full | Mathematical analysis of partial differential equations modeling electrostatic MEMS Pierpaolo Esposito ; Nassif Ghoussoub ; Yujin Guo |
| title_fullStr | Mathematical analysis of partial differential equations modeling electrostatic MEMS Pierpaolo Esposito ; Nassif Ghoussoub ; Yujin Guo |
| title_full_unstemmed | Mathematical analysis of partial differential equations modeling electrostatic MEMS Pierpaolo Esposito ; Nassif Ghoussoub ; Yujin Guo |
| title_short | Mathematical analysis of partial differential equations modeling electrostatic MEMS |
| title_sort | mathematical analysis of partial differential equations modeling electrostatic mems |
| topic | Mathematisches Modell Mathematical analysis Microelectromechanical systems Mathematical models MEMS (DE-588)4824724-8 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Mathematisches Modell Mathematical analysis Microelectromechanical systems Mathematical models MEMS Partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018732878&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV012714106 |
| work_keys_str_mv | AT espositopierpaolo mathematicalanalysisofpartialdifferentialequationsmodelingelectrostaticmems AT ghoussoubnassif mathematicalanalysisofpartialdifferentialequationsmodelingelectrostaticmems AT guoyujin mathematicalanalysisofpartialdifferentialequationsmodelingelectrostaticmems |