 | Esposito P. Mathematical analysis of partial differential equations modeling electrostatic MEMS / P.Esposito, N.Ghoussoub, Y.Guo. - New York: Courant Institute of Mathematical Sciences; Providence: American Mathematical Society, 2010. - xiii, 318 p.: ill. - (Courant lecture notes in mathematics; 20). - Bibliogr.: p.309-315. - Ind.: p.317-318. - ISBN 978-0-8218-4957-6
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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
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 λ > 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 = 1 ............ 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
11.4 Regularity of the Extremal Solution for 1 ≤ N ≤ 8 ........ 260
11.5 The Extremal Solution Is Singular for N ≥ 9 .............. 263
Further Comments .............................................. 268
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
for N ≤ 4 ................................................ 279
12.4 Uniform Energy Bounds for Solutions in Convex Domains .... 280
12.5 The Solution Set on Convex Domains in  2 ................. 283
12.6 Regularity of the Extremal Solution on Balls for
TV ≤ 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.l. Improved Hardy-Rellich Inequalities in H02(В) ............. 299
A.2. Improved Hardy-Rellich Inequalities in H2(B) ∩ H01(B) ..... 302
Bibliography .................................................. 309
Index ......................................................... 317
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