Dynamics of a free boundary problem with curvature modeling electrostatic MEMS

Escher, Joachim; Laurençot, Philippe; Walker, Christoph

doi:10.1090/S0002-9947-2014-06320-4

Dynamics of a free boundary problem with curvature modeling electrostatic MEMS
HTML articles powered by AMS MathViewer

by Joachim Escher, Philippe Laurençot and Christoph Walker PDF

Trans. Amer. Math. Soc. 367 (2015), 5693-5719 Request permission

Abstract:

The dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference across the device. More precisely, the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane. The gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation. The main feature of the model considered herein is that, unlike most previous research, the small deformation assumption is dropped, and curvature for the deformation of the membrane is taken into account which leads to a quasilinear parabolic equation. The free boundary problem is shown to be well-posed, locally in time for arbitrary voltage values and globally in time for small voltage values. Furthermore, existence of asymptotically stable steady-state configurations is proved in case of small voltage values as well as non-existence of steady-state solutions if the applied voltage difference is large. Finally, convergence of solutions of the free boundary problem to the solutions of the well-established small aspect ratio model is shown.

References

Herbert Amann, Quasilinear evolution equations and parabolic systems, Trans. Amer. Math. Soc. 293 (1986), no. 1, 191–227. MR 814920, DOI 10.1090/S0002-9947-1986-0814920-4
Herbert Amann, Multiplication in Sobolev and Besov spaces, Nonlinear analysis, Sc. Norm. Super. di Pisa Quaderni, Scuola Norm. Sup., Pisa, 1991, pp. 27–50. MR 1205371
Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992) Teubner-Texte Math., vol. 133, Teubner, Stuttgart, 1993, pp. 9–126. MR 1242579, DOI 10.1007/978-3-663-11336-2_{1}
Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. MR 1345385, DOI 10.1007/978-3-0348-9221-6
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
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
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, 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
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
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
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
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
Pierre Grisvard, Équations différentielles abstraites, Ann. Sci. École Norm. Sup. (4) 2 (1969), 311–395 (French). MR 270209
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
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
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 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
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
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
David Jerison and Carlos E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), no. 1, 161–219. MR 1331981, DOI 10.1006/jfan.1995.1067
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
V. Leus and D. Elata. On the dynamic response of electrostatic MEMS switches. IEEE J. Microelectromechanical Systems 17 (2008), 236–243.
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
L. Lorenzi, A. Lunardi, G. Metafune, and D. Pallara. Analytic semigroups and reaction-diffusion problems. Internet Seminar 2004–2005. (Available online at: http://www.math.unipr.it/~lunardi/LectureNotes/I-Sem2005.pdf )
Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. MR 1329547, DOI 10.1007/978-3-0348-9234-6
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
R. Seeley, Interpolation in $L^{p}$ with boundary conditions, Studia Math. 44 (1972), 47–60. MR 315432, DOI 10.4064/sm-44-1-47-60
Hans Triebel, Interpolation theory, function spaces, differential operators, 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995. MR 1328645