Some singular equations modeling MEMS

Laurençot, Philippe; Walker, Christoph

doi:10.1090/bull/1563

Some singular equations modeling MEMS
HTML articles powered by AMS MathViewer

by Philippe Laurençot and Christoph Walker PDF

Bull. Amer. Math. Soc. 54 (2017), 437-479 Request permission

Abstract:

In the past fifteen years mathematical models for microelectromechanical systems (MEMS) have been the subject of several studies, in particular due to the interesting qualitative properties they feature. Still most research is devoted to an illustrative but simplified model, which is deduced from a more complex model when the aspect ratio of the device vanishes, the so-called vanishing (or small) aspect ratio model. The analysis of the aforementioned complex model involving a moving boundary has started only recently, and an outlook of the results obtained so far in this direction is provided in this survey.

References

Wolfgang Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, Evolutionary equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 2004, pp. 1–85. MR 2103696
M. Bao and H. Yang, Squeeze film air damping in MEMS, Sensors and Actuators A 136 (2007), 3–27.
R. C. Batra, M. Porfiri, and D. Spinello, Effects of van der Waals force and thermal stresses on pull-in instability of clamped rectangular microplates, Sensors 8 (2008), 1048–1069.
J. Regan Beckham and John A. Pelesko, Symmetry analysis of a canonical MEMS model, Methods Appl. Anal. 15 (2008), no. 3, 327–339. MR 2500850, DOI 10.4310/MAA.2008.v15.n3.a5
J. Regan Beckham and John A. Pelesko, An electrostatic-elastic membrane system with an external pressure, Math. Comput. Modelling 54 (2011), no. 11-12, 2686–2708. MR 2841813, DOI 10.1016/j.mcm.2011.06.051
D. H. Bernstein, P. Guidotti, and J. A. Pelesko, Analytical and numerical analysis of electrostatically actuated MEMS devices, Proceedings of Modeling and Simulation of Microsystems 2000, San Diego, CA, 2000, pp. 489–492.
T. Boggio, Sulle funzioni di Green d’ordine $m$, Rend. Circ. Mat. Palermo 20 (1905), 97–135.
N. D. Brubaker and A. E. Lindsay, The onset of multi-valued solutions of a prescribed mean curvature equation with singular non-linearity, European J. Appl. Math. 24 (2013), no. 5, 631–656. MR 3104284, DOI 10.1017/S0956792513000077
Nicholas D. Brubaker and John A. Pelesko, Non-linear effects on canonical MEMS models, European J. Appl. Math. 22 (2011), no. 5, 455–470. MR 2834014, DOI 10.1017/S0956792511000180
Nicholas D. Brubaker and John A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal. 75 (2012), no. 13, 5086–5102. MR 2927572, DOI 10.1016/j.na.2012.04.025
N. D. Brubaker, J. I. Siddique, E. Sabo, R. Deaton, and J. A. Pelesko, Refinements to the study of electrostatic deflections: theory and experiment, European J. Appl. Math. 24 (2013), no. 3, 343–370. MR 3053651, DOI 10.1017/S0956792512000435
B. Buffoni, E. N. Dancer, and J. F. Toland, The sub-harmonic bifurcation of Stokes waves, Arch. Ration. Mech. Anal. 152 (2000), no. 3, 241–271. MR 1764946, DOI 10.1007/s002050000087
Daniele Cassani, Luisa Fattorusso, and Antonio Tarsia, Global existence for nonlocal MEMS, Nonlinear Anal. 74 (2011), no. 16, 5722–5726. MR 2819313, DOI 10.1016/j.na.2011.05.060
D. Cassani, L. Fattorusso, and A. Tarsia, Nonlocal dynamic problems with singular nonlinearities and applications to MEMS, Analysis and topology in nonlinear differential equations, Progr. Nonlinear Differential Equations Appl., vol. 85, Birkhäuser/Springer, Cham, 2014, pp. 187–206. MR 3330730
Daniele Cassani, Barbara Kaltenbacher, and Alfredo Lorenzi, Direct and inverse problems related to MEMS, Inverse Problems 25 (2009), no. 10, 105002, 22. MR 2545971, DOI 10.1088/0266-5611/25/10/105002
Daniele Cassani, João Marcos do Ó, and Nassif Ghoussoub, On a fourth order elliptic problem with a singular nonlinearity, Adv. Nonlinear Stud. 9 (2009), no. 1, 177–197. MR 2473155, DOI 10.1515/ans-2009-0109
Yan-Hsiou Cheng, Kuo-Chih Hung, and Shin-Hwa Wang, Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS, Nonlinear Anal. 89 (2013), 284–298. MR 3073332, DOI 10.1016/j.na.2013.04.018
Giovanni Cimatti, A free boundary problem in the theory of electrically actuated microdevices, Appl. Math. Lett. 20 (2007), no. 12, 1232–1236. MR 2384253, DOI 10.1016/j.aml.2006.07.016
Christian Clason and Barbara Kaltenbacher, Optimal control of a singular PDE modeling transient MEMS with control or state constraints, J. Math. Anal. Appl. 410 (2014), no. 1, 455–468. MR 3109854, DOI 10.1016/j.jmaa.2013.08.058
Craig Cowan, Pierpaolo Esposito, and Nassif Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 1033–1050. MR 2644777, DOI 10.3934/dcds.2010.28.1033
Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub, and Amir Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal. 198 (2010), no. 3, 763–787. MR 2729319, DOI 10.1007/s00205-010-0367-x
Craig Cowan and Nassif Ghoussoub, Estimates on pull-in distances in microelectromechanical systems models and other nonlinear eigenvalue problems, SIAM J. Math. Anal. 42 (2010), no. 5, 1949–1966. MR 2684306, DOI 10.1137/090752857
Craig Cowan and Nassif Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 291–305. MR 3148117, DOI 10.1007/s00526-012-0582-4
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180. MR 341212, DOI 10.1007/BF00282325
Juan Dávila, Isabel Flores, and Ignacio Guerra, Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann. 348 (2010), no. 1, 143–193. MR 2657438, DOI 10.1007/s00208-009-0476-8
Juan Dávila, Kelei Wang, and Juncheng Wei, Qualitative analysis of rupture solutions for a MEMS problem, Ann. Inst. H. Poincaré C Anal. Non Linéaire 33 (2016), no. 1, 221–242. MR 3436432, DOI 10.1016/j.anihpc.2014.09.009
Juan Dávila and Juncheng Wei, Point ruptures for a MEMS equation with fringing field, Comm. Partial Differential Equations 37 (2012), no. 8, 1462–1493. MR 2957549, DOI 10.1080/03605302.2012.679990
Miroslav Engliš and Jaak Peetre, A Green’s function for the annulus, Ann. Mat. Pura Appl. (4) 171 (1996), 313–377. MR 1441874, DOI 10.1007/BF01759391
Joachim Escher, Philippe Laurençot, and Christoph Walker, Finite time singularity in a free boundary problem modeling MEMS, C. R. Math. Acad. Sci. Paris 351 (2013), no. 21-22, 807–812 (English, with English and French summaries). MR 3128966, DOI 10.1016/j.crma.2013.10.004
Joachim Escher, Philippe Laurençot, and Christoph Walker, A parabolic free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal. 211 (2014), no. 2, 389–417. MR 3149061, DOI 10.1007/s00205-013-0656-2
Joachim Escher, Philippe Laurençot, and Christoph Walker, Dynamics of a free boundary problem with curvature modeling electrostatic MEMS, Trans. Amer. Math. Soc. 367 (2015), no. 8, 5693–5719. MR 3347187, DOI 10.1090/S0002-9947-2014-06320-4
Joachim Escher and Christina Lienstromberg, A qualitative analysis of solutions to microelectromechanical systems with curvature and nonlinear permittivity profile, Comm. Partial Differential Equations 41 (2016), no. 1, 134–149. MR 3439465, DOI 10.1080/03605302.2015.1105259
Pierpaolo Esposito and Nassif Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl. Anal. 15 (2008), no. 3, 341–353. MR 2500851, DOI 10.4310/MAA.2008.v15.n3.a6
Pierpaolo Esposito, Nassif Ghoussoub, and Yujin Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math. 60 (2007), no. 12, 1731–1768. MR 2358647, DOI 10.1002/cpa.20189
Pierpaolo Esposito, Nassif Ghoussoub, and Yujin Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lecture Notes in Mathematics, vol. 20, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010. MR 2604963, DOI 10.1090/cln/020
A. Fargas Marquès, R. Costa Castelló, and A. M. Shkel, Modelling the electrostatic actuation of MEMS: state of the art 2005, Technical Report, Universitat Politècnica de Catalunya (2005).
H. O. Fattorini, Second order linear differential equations in Banach spaces, North-Holland Mathematics Studies, vol. 108, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 99. MR 797071
Peng Feng and Zhengfang Zhou, Multiplicity and symmetry breaking for positive radial solutions of semilinear elliptic equations modelling MEMS on annular domains, Electron. J. Differential Equations (2005), No. 146, 14. MR 2195542
R. P. Feynman, There’s plenty of room at the bottom, IEEE J. Microelectromech. Syst. 1 (1992), 60–66.
Marek Fila, Josephus Hulshof, and Pavol Quittner, The quenching problem on the $N$-dimensional ball, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 183–196. MR 1167839, DOI 10.1007/978-1-4612-0393-3_{1}4
G. Flores, G. Mercado, J. A. Pelesko, and N. Smyth, Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM J. Appl. Math. 67 (2006/07), no. 2, 434–446. MR 2285871, DOI 10.1137/060648866
Gilberto Flores, Dynamics of a damped wave equation arising from MEMS, SIAM J. Appl. Math. 74 (2014), no. 4, 1025–1035. MR 3233101, DOI 10.1137/130914759
Gilberto Flores and Noel F. Smyth, A numerical study of the pull-in instability in some free boundary models for MEMS, Appl. Math. Model. 40 (2016), no. 17-18, 7962–7970. MR 3529672, DOI 10.1016/j.apm.2016.04.022
Masafumi Fukunishi and Tatsuya Watanabe, Variational approach to MEMS model with fringing field, J. Math. Anal. Appl. 423 (2015), no. 2, 1262–1283. MR 3278198, DOI 10.1016/j.jmaa.2014.10.053
Filippo Gazzola, Hans-Christoph Grunau, and Guido Sweers, Polyharmonic boundary value problems, Lecture Notes in Mathematics, vol. 1991, Springer-Verlag, Berlin, 2010. Positivity preserving and nonlinear higher order elliptic equations in bounded domains. MR 2667016, DOI 10.1007/978-3-642-12245-3
Filippo Gazzola and Guido Sweers, On positivity for the biharmonic operator under Steklov boundary conditions, Arch. Ration. Mech. Anal. 188 (2008), no. 3, 399–427. MR 2393435, DOI 10.1007/s00205-007-0090-4
Nassif Ghoussoub and Yujin Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math. Anal. 38 (2006/07), no. 5, 1423–1449. MR 2286013, DOI 10.1137/050647803
Nassif Ghoussoub and Yujin Guo, Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS, Methods Appl. Anal. 15 (2008), no. 3, 361–376. MR 2500853, DOI 10.4310/MAA.2008.v15.n3.a8
Nassif Ghoussoub and Yujin Guo, On the partial differential equations of electrostatic MEMS devices. II. Dynamic case, NoDEA Nonlinear Differential Equations Appl. 15 (2008), no. 1-2, 115–145. MR 2408347, DOI 10.1007/s00030-007-6004-1
Hans-Christoph Grunau, Positivity, change of sign and buckling eigenvalues in a one-dimensional fourth order model problem, Adv. Differential Equations 7 (2002), no. 2, 177–196. MR 1869560
Hans-Christoph Grunau and Frédéric Robert, Positivity and almost positivity of biharmonic Green’s functions under Dirichlet boundary conditions, Arch. Ration. Mech. Anal. 195 (2010), no. 3, 865–898. MR 2591975, DOI 10.1007/s00205-009-0230-0
Hans-Christoph Grunau and Guido Sweers, Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions, Math. Nachr. 179 (1996), 89–102. MR 1389451, DOI 10.1002/mana.19961790106
J.-G. Guo and Y.-P. Zhao, Influence of van der Waals and Casimir forces on electrostatic torsional actuators, J. Microelectromech. Syst. 13 (2004), 1027–1035.
Jong-Shenq Guo, Bei Hu, and Chi-Jen Wang, A nonlocal quenching problem arising in a micro-electro mechanical system, Quart. Appl. Math. 67 (2009), no. 4, 725–734. MR 2588232, DOI 10.1090/S0033-569X-09-01159-5
Jong-Shenq Guo and Bo-Chih Huang, Hyperbolic quenching problem with damping in the micro-electro mechanical system device, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 2, 419–434. MR 3170192, DOI 10.3934/dcdsb.2014.19.419
Jong-Shenq Guo and Nikos I. Kavallaris, On a nonlocal parabolic problem arising in electrostatic MEMS control, Discrete Contin. Dyn. Syst. 32 (2012), no. 5, 1723–1746. MR 2871333, DOI 10.3934/dcds.2012.32.1723
Jong-Shenq Guo and Philippe Souplet, No touchdown at zero points of the permittivity profile for the MEMS problem, SIAM J. Math. Anal. 47 (2015), no. 1, 614–625. MR 3305368, DOI 10.1137/140970070
Yujin Guo, Global solutions of singular parabolic equations arising from electrostatic MEMS, J. Differential Equations 245 (2008), no. 3, 809–844. MR 2422528, DOI 10.1016/j.jde.2008.03.012
Yujin Guo, On the partial differential equations of electrostatic MEMS devices. III. Refined touchdown behavior, J. Differential Equations 244 (2008), no. 9, 2277–2309. MR 2413842, DOI 10.1016/j.jde.2008.02.005
Yujin Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS, SIAM J. Appl. Dyn. Syst. 9 (2010), no. 4, 1135–1163. MR 2736315, DOI 10.1137/09077117X
Yujin Guo, Zhenguo Pan, and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math. 66 (2005), no. 1, 309–338. MR 2179754, DOI 10.1137/040613391
Zongming Guo, Baishun Lai, and Dong Ye, Revisiting the biharmonic equation modelling electrostatic actuation in lower dimensions, Proc. Amer. Math. Soc. 142 (2014), no. 6, 2027–2034. MR 3182022, DOI 10.1090/S0002-9939-2014-11895-8
Zongming Guo and Zhongyuan Liu, Further study of a fourth-order elliptic equation with negative exponent, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 3, 537–549. MR 2805617, DOI 10.1017/S0308210509001061
Zongming Guo and Juncheng Wei, Entire solutions and global bifurcations for a biharmonic equation with singular non-linearity in $\Bbb R^3$, Adv. Differential Equations 13 (2008), no. 7-8, 753–780. MR 2479029
Zongming Guo and Juncheng Wei, Infinitely many turning points for an elliptic problem with a singular non-linearity, J. Lond. Math. Soc. (2) 78 (2008), no. 1, 21–35. MR 2427049, DOI 10.1112/jlms/jdm121
Zongming Guo and Juncheng Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal. 40 (2008/09), no. 5, 2034–2054. MR 2471911, DOI 10.1137/070703375
J. Hadamard, Sur certains cas intéressants du problème biharmonique, Œuvres de Jacques Hadamard. Tome II, Editions du Centre National de la Recherche Scientifique, Paris, 1968, pp. 1297–1299.
A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal. 100 (1988), no. 2, 191–206. MR 913963, DOI 10.1007/BF00282203
Antoine Henrot and Michel Pierre, Variation et optimisation de formes, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 48, Springer, Berlin, 2005 (French). Une analyse géométrique. [A geometric analysis]. MR 2512810, DOI 10.1007/3-540-37689-5
Kin Ming Hui, Global and touchdown behaviour of the generalized MEMS device equation, Adv. Math. Sci. Appl. 19 (2009), no. 2, 347–370. MR 2605723
Kin Ming Hui, The existence and dynamic properties of a parabolic nonlocal MEMS equation, Nonlinear Anal. 74 (2011), no. 1, 298–316. MR 2734998, DOI 10.1016/j.na.2010.08.045
Kin Ming Hui, Quenching behaviour of a nonlocal parabolic MEMS equation, Adv. Math. Sci. Appl. 21 (2011), no. 1, 173–185. MR 2883880
J. D. Jackson, Classical electrodynamics, John Wiley and Sons, New York, London, Sydney, 1962.
Reza N. Jazar, Nonlinear mathematical modeling of microbeam MEMS, Nonlinear approaches in engineering applications, Springer, New York, 2012, pp. 69–104. MR 3204615, DOI 10.1007/978-1-4614-1469-8_{3}
Reza N. Jazar, Nonlinear modeling of squeeze-film phenomena: in microbeam MEMS, Nonlinear approaches in engineering applications, Springer, New York, 2012, pp. 41–68. MR 3204614, DOI 10.1007/978-1-4614-1469-8_{2}
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241–269. MR 340701, DOI 10.1007/BF00250508
Stanley Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math. 16 (1963), 305–330. MR 160044, DOI 10.1002/cpa.3160160307
Nikos I. Kavallaris, Andrew A. Lacey, and Christos V. Nikolopoulos, On the quenching of a nonlocal parabolic problem arising in electrostatic MEMS control, Nonlinear Anal. 138 (2016), 189–206. MR 3485145, DOI 10.1016/j.na.2016.02.001
N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos, and D. E. Tzanetis, A hyperbolic non-local problem modelling MEMS technology, Rocky Mountain J. Math. 41 (2011), no. 2, 505–534. MR 2794451, DOI 10.1216/RMJ-2011-41-2-505
Nikos I. Kavallaris, Andrew A. Lacey, Christos V. Nikolopoulos, and Dimitrios E. Tzanetis, On the quenching behaviour of a semilinear wave equation modelling MEMS technology, Discrete Contin. Dyn. Syst. 35 (2015), no. 3, 1009–1037. MR 3277183, DOI 10.3934/dcds.2015.35.1009
Nikos I. Kavallaris, Tosiya Miyasita, and Takashi Suzuki, Touchdown and related problems in electrostatic MEMS device equation, NoDEA Nonlinear Differential Equations Appl. 15 (2008), no. 3, 363–385. MR 2458644, DOI 10.1007/s00030-008-7081-5
J. Kisyński, Sur les équations hyperboliques avec petit paramètre, Colloq. Math. 10 (1963), 331–343 (French). MR 156104, DOI 10.4064/cm-10-2-331-343
Martin Kohlmann, A new model for electrostatic MEMS with two free boundaries, J. Math. Anal. Appl. 408 (2013), no. 2, 513–524. MR 3085048, DOI 10.1016/j.jmaa.2013.06.012
Martin Kohlmann, The abstract quasilinear Cauchy problem for a MEMS model with two free boundaries, Acta Appl. Math. 138 (2015), 171–197. MR 3365586, DOI 10.1007/s10440-014-9962-4
Martin Kohlmann, On an elliptic-parabolic MEMS model with two free boundaries, Appl. Anal. 94 (2015), no. 10, 2175–2199. MR 3384040, DOI 10.1080/00036811.2015.1033410
M. C. Kropinski, A. E. Lindsay, and M. J. Ward, Asymptotic analysis of localized solutions to some linear and nonlinear biharmonic eigenvalue problems, Stud. Appl. Math. 126 (2011), no. 4, 347–408. MR 2827648, DOI 10.1111/j.1467-9590.2010.00507.x
A. A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math. 43 (1983), no. 6, 1350–1366. MR 722946, DOI 10.1137/0143090
Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
Baishun Lai, On the partial differential equations of electrostatic MEMS devices with effects of Casimir force, Ann. Henri Poincaré 16 (2015), no. 1, 239–253. MR 3296644, DOI 10.1007/s00023-014-0322-8
Philippe Laurençot and Christoph Walker, On a three-dimensional free boundary problem modeling electrostatic MEMS, Interfaces Free Bound. 18 (2016), no. 3, 393–411. MR 3563783, DOI 10.4171/IFB/368
Philippe Laurençot and Christoph Walker, A variational approach to a stationary free boundary problem modeling MEMS, ESAIM Control Optim. Calc. Var. 22 (2016), no. 2, 417–438. MR 3491777, DOI 10.1051/cocv/2015012
Ph. Laurençot and Ch. Walker, Vanishing aspect ratio limit for a fourth-order MEMS model, Ann. Mat. Pura Appl., DOI: 10.1007/s10231-016-0628-x.
Philippe Laurençot and Christoph Walker, A stationary free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal. 207 (2013), no. 1, 139–158. MR 3004769, DOI 10.1007/s00205-012-0559-7
Philippe Laurençot and Christoph Walker, A fourth-order model for MEMS with clamped boundary conditions, Proc. Lond. Math. Soc. (3) 109 (2014), no. 6, 1435–1464. MR 3293155, DOI 10.1112/plms/pdu037
Philippe Laurençot and Christoph Walker, A free boundary problem modeling electrostatic MEMS: I. Linear bending effects, Math. Ann. 360 (2014), no. 1-2, 307–349. MR 3263165, DOI 10.1007/s00208-014-1032-8
Philippe Laurençot and Christoph Walker, A free boundary problem modeling electrostatic MEMS: II. Nonlinear bending effects, Math. Models Methods Appl. Sci. 24 (2014), no. 13, 2549–2568. MR 3260278, DOI 10.1142/S0218202514500298
Philippe Laurençot and Christoph Walker, Sign-preserving property for some fourth-order elliptic operators in one dimension or in radial symmetry, J. Anal. Math. 127 (2015), 69–89. MR 3421988, DOI 10.1007/s11854-015-0024-2
Philippe Laurençot and Christoph Walker, The time singular limit for a fourth-order damped wave equation for MEMS, Elliptic and parabolic equations, Springer Proc. Math. Stat., vol. 119, Springer, Cham, 2015, pp. 233–246. MR 3375174, DOI 10.1007/978-3-319-12547-3_{1}0
Howard A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann. Mat. Pura Appl. (4) 155 (1989), 243–260. MR 1042837, DOI 10.1007/BF01765943
Howard A. Levine, Advances in quenching, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 319–346. MR 1167848
Jingyu Li and Chuangchuang Liang, Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS, Discrete Contin. Dyn. Syst. 36 (2016), no. 2, 833–849. MR 3392906, DOI 10.3934/dcds.2016.36.833
Chuangchuang Liang, Jingyu Li, and Kaijun Zhang, On a hyperbolic equation arising in electrostatic MEMS, J. Differential Equations 256 (2014), no. 2, 503–530. MR 3121704, DOI 10.1016/j.jde.2013.09.010
Chuangchuang Liang and Kaijun Zhang, Global solution of the initial boundary value problem to a hyperbolic nonlocal MEMS equation, Comput. Math. Appl. 67 (2014), no. 3, 549–554. MR 3149731, DOI 10.1016/j.camwa.2013.11.012
Chuangchuang Liang and Kaijun Zhang, Asymptotic stability and quenching behavior of a hyperbolic nonlocal MEMS equation, Commun. Math. Sci. 13 (2015), no. 2, 355–368. MR 3291373, DOI 10.4310/CMS.2015.v13.n2.a5
Christina Lienstromberg, A free boundary value problem modelling microelectromechanical systems with general permittivity, Nonlinear Anal. Real World Appl. 25 (2015), 190–218. MR 3351019, DOI 10.1016/j.nonrwa.2015.03.008
Christina Lienstromberg, On qualitative properties of solutions to microelectromechanical systems with general permittivity, Monatsh. Math. 179 (2016), no. 4, 581–602. MR 3474882, DOI 10.1007/s00605-015-0744-5
Fanghua Lin and Yisong Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2081, 1323–1337. MR 2313813, DOI 10.1098/rspa.2007.1816
A. E. Lindsay, Regularized model of post-touchdown configurations in electrostatic MEMS: bistability analysis, J. Engrg. Math. 99 (2016), 65–77. MR 3520093, DOI 10.1007/s10665-015-9820-z
A. E. Lindsay and J. Lega, Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor, SIAM J. Appl. Math. 72 (2012), no. 3, 935–958. MR 2968757, DOI 10.1137/110832550
A. E. Lindsay, J. Lega, and K. G. Glasner, Regularized model of post-touchdown configurations in electrostatic MEMS: Equilibrium analysis, Phys. D 280-281 (2014), 95–108.
A. E. Lindsay, J. Lega, and K. B. Glasner, Regularized model of post-touchdown configurations in electrostatic MEMS: interface dynamics, IMA J. Appl. Math. 80 (2015), no. 6, 1635–1663. MR 3477204, DOI 10.1093/imamat/hxv011
A. E. Lindsay, J. Lega, and F. J. Sayas, The quenching set of a MEMS capacitor in two-dimensional geometries, J. Nonlinear Sci. 23 (2013), no. 5, 807–834. MR 3101835, DOI 10.1007/s00332-013-9169-2
A. E. Lindsay and M. J. Ward, Asymptotics of some nonlinear eigenvalue problems for a MEMS capacitor. I. Fold point asymptotics, Methods Appl. Anal. 15 (2008), no. 3, 297–325. MR 2500849, DOI 10.4310/MAA.2008.v15.n3.a4
A. E. Lindsay and M. J. Ward, Asymptotics of some nonlinear eigenvalue problems modelling a MEMS capacitor. Part II: multiple solutions and singular asymptotics, European J. Appl. Math. 22 (2011), no. 2, 83–123. MR 2774778, DOI 10.1017/S0956792510000318
Zhe Liu and Xiaoliu Wang, On a parabolic equation in MEMS with fringing field, Arch. Math. (Basel) 98 (2012), no. 4, 373–381. MR 2914353, DOI 10.1007/s00013-012-0363-5
Xue Luo and Stephen S.-T. Yau, On the quenching behavior of the MEMS with fringing field, Quart. Appl. Math. 73 (2015), no. 4, 629–659. MR 3432276, DOI 10.1090/qam/1396
Tosiya Miyasita, Global existence of radial solutions of a hyperbolic MEMS equation with nonlocal term, Differ. Equ. Appl. 7 (2015), no. 2, 169–186. MR 3356272, DOI 10.7153/dea-07-10
Amir Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. Differential Equations 248 (2010), no. 3, 594–616. MR 2557908, DOI 10.1016/j.jde.2009.09.011
H. C. Nathanson, W. E. Newell, R. A. Wickstrom, and J. R. Davis, The resonant gate transistor, IEEE Trans. on Electron Devices 14 (1967), 117 – 133.
A. H. Nayfeh and M. I. Younis, A new approach to the modeling and simulation of flexible microstructures under the effect of squeeze-film damping, J. Micromech. Microeng. 14 (2004), 170–181.
Ali H. Nayfeh, Mohammad I. Younis, and Eihab M. Abdel-Rahman, Reduced-order models for MEMS applications, Nonlinear Dynam. 41 (2005), no. 1-3, 211–236. MR 2157181, DOI 10.1007/s11071-005-2809-9
Janpou Nee, Nonlinear problems arising in electrostatic actuation in MEMS, J. Math. Anal. Appl. 352 (2009), no. 2, 836–845. MR 2501929, DOI 10.1016/j.jmaa.2008.11.031
M. P. Owen, Asymptotic first eigenvalue estimates for the biharmonic operator on a rectangle, J. Differential Equations 136 (1997), no. 1, 166–190. MR 1443328, DOI 10.1006/jdeq.1996.3235
Hongjing Pan and Ruixiang Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to MEMS models, Nonlinear Anal. Real World Appl. 13 (2012), no. 5, 2432–2445. MR 2911927, DOI 10.1016/j.nonrwa.2012.02.012
Hongjing Pan and Ruixiang Xing, On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models, Discrete Contin. Dyn. Syst. 35 (2015), no. 8, 3627–3682. MR 3320140, DOI 10.3934/dcds.2015.35.3627
J. A. Pelesko, Electrostatic field approximations and implications for MEMS devices, ESA 2001 Proceedings (2001), 126–137.
John A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math. 62 (2001/02), no. 3, 888–908. MR 1897727, DOI 10.1137/S0036139900381079
John A. Pelesko and David H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1955412
J. A. Pelesko and X. Y. Chen, Electrostatic deflections of circular elastic membranes, J. Electrostatics 57 (2003), no. 1, 1–12.
John A. Pelesko and Tobin A. Driscoll, The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models, J. Engrg. Math. 53 (2005), no. 3-4, 239–252. MR 2230109, DOI 10.1007/s10665-005-9013-2
J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control, J. Engrg. Math. 41 (2001), no. 4, 345–366. MR 1872152, DOI 10.1023/A:1012292311304
Edoardo Sassone, Positivity for polyharmonic problems on domains close to a disk, Ann. Mat. Pura Appl. (4) 186 (2007), no. 3, 419–432. MR 2317647, DOI 10.1007/s10231-006-0012-3
Johann Schröder, Operator inequalities, Mathematics in Science and Engineering, vol. 147, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 578001
J. I. Siddique, R. Deaton, E. Sabo, and J. A. Pelesko, An experimental study of the theory of electrostatic deflection, journal of electrostatics, J. Electrostatics 69 (2011), no. 2, 1–6.
Piero Villaggio, Mathematical models for elastic structures, Cambridge University Press, Cambridge, 1997. MR 1486043, DOI 10.1017/CBO9780511529665
Qi Wang, Estimates for the quenching time of a MEMS equation with fringing field, J. Math. Anal. Appl. 405 (2013), no. 1, 135–147. MR 3053493, DOI 10.1016/j.jmaa.2013.03.054
Qi Wang, Dynamical solutions of singular parabolic equations modeling electrostatic MEMS, NoDEA Nonlinear Differential Equations Appl. 22 (2015), no. 4, 629–650. MR 3385615, DOI 10.1007/s00030-014-0298-6
Juncheng Wei and Dong Ye, On MEMS equation with fringing field, Proc. Amer. Math. Soc. 138 (2010), no. 5, 1693–1699. MR 2587454, DOI 10.1090/S0002-9939-09-10226-5
M. I. Younis, MEMS. Linear and nonlinear statics and dynamics, Springer, New York, Dordrecht, Heidelberg, London, 2011.
Ruifeng Zhang and Na Li, Mathematical analysis of nonlinear differential equation arising in MEMS, Bull. Korean Math. Soc. 49 (2012), no. 4, 705–714. MR 2978411, DOI 10.4134/BKMS.2012.49.4.705