no. 21-22
Partial differential equations
Finite time singularity in a free boundary problem modeling MEMS
[Singularité en temps fini pour un modèle de microsystème électromécanique à frontière libre]

Présenté par : the Editorial Board

Escher, Joachim1 ; Laurençot, Philippe2 ; Walker, Christoph1
1 Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, 30167 Hannover, Germany
2 Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, 31062 Toulouse cedex 9, France
Comptes Rendus. Mathématique, Tome 351 (2013) no. 21-22, pp. 807-812.

Lʼapparition dʼune singularité en temps fini est établie pour un problème à frontière libre décrivant lʼévolution spatio-temporelle dʼun microsystème électromécanique lorsque la tension appliquée est suffisamment élevée. Le modèle inclut un terme de réaction singulier et un terme non linéaire de courbure, prenant en compte les grandes déformations.

The occurrence of a finite time singularity is shown for a free boundary problem modeling microelectromechanical systems (MEMS) when the applied voltage exceeds some value. The model involves a singular nonlocal reaction term and a nonlinear curvature term taking into account large deformations.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.10.004
Escher, Joachim 1 ; Laurençot, Philippe 2 ; Walker, Christoph 1

1 Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, 30167 Hannover, Germany
2 Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, 31062 Toulouse cedex 9, France 
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Escher, Joachim; Laurençot, Philippe; Walker, Christoph. Finite time singularity in a free boundary problem modeling MEMS. Comptes Rendus. Mathématique, Tome 351 (2013) no. 21-22, pp. 807-812. doi : 10.1016/j.crma.2013.10.004. http://www.numdam.org/articles/10.1016/j.crma.2013.10.004/

[1] Brubaker, N.D.; Pelesko, J.A. Non-linear effects on canonical MEMS models, Eur. J. Appl. Math., Volume 22 (2011), pp. 455-470

[2] Brubaker, N.D.; Pelesko, J.A. Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal., Volume 75 (2012), pp. 5086-5112

[3] Cimatti, G. A free boundary problem in the theory of electrically actuated microdevices, Appl. Math. Lett., Volume 20 (2007), pp. 1232-1236

[4] Escher, J.; Laurençot, Ph.; Walker, C. A parabolic free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal. (2013) (in press)

[5] Escher, J.; Laurençot, Ph.; Walker, C. Dynamics of a free boundary problem with curvature modeling electrostatic MEMS (submitted for publication) | arXiv

[6] Esposito, P.; Ghoussoub, N.; Guo, Y. Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Courant Lect. Notes Math., vol. 20, Courant Institute of Mathematical Sciences, New York, 2010

[7] Flores, G.; Mercado, G.; Pelesko, J.A.; Smyth, N. Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM J. Appl. Math., Volume 67 (2007), pp. 434-446

[8] Guo, Y.; Pan, Z.; Ward, M.J. Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math., Volume 66 (2005), pp. 309-338

[9] Laurençot, P.; Walker, C. A stationary free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal., Volume 207 (2013), pp. 139-158

[10] Pelesko, J.A.; Bernstein, D.H. Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003

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