Touchdown is the Only Finite Time Singularity in a Three-Dimensional MEMS Model
[La désactivation est la seule singularité en temps fini possible dans un modèle de MEMS tridimensionnel]
Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 1, pp. 65-81.

Nous montrons que la désactivation est la seule singularité en temps fini pouvant se produire dans un problème à frontière libre décrivant un microsystème électromécanique tridimensionnel. La démonstration repose sur la structure variationnelle du modèle et utilise les propriétés régularisantes du semi-groupe engendré dans L 1 par le bi-Laplacien avec conditions aux bords encastrées.

Touchdown is shown to be the only possible finite time singularity that may take place in a free boundary problem modeling a three-dimensional microelectromechanical system. The proof relies on the energy structure of the problem and uses smoothing effects of the semigroup generated in L 1 by the bi-Laplacian with clamped boundary conditions.

Publié le :
DOI : 10.5802/ambp.391
Classification : 35K91, 35R35, 35M33, 35Q74, 35B44
Keywords: Microelectromechanical system, quenching, free boundary problem, bi-Laplacian
Mot clés : Microsystème électromécanique, désactivation, problème à frontière libre, bi-Laplacien
Philippe Laurençot 1 ; Christoph Walker 2

1 Institut de Mathématiques de Toulouse, UMR 5219 Université de Toulouse, CNRS 31062 Toulouse Cedex 9, France
2 Leibniz Universität Hannover Institut für Angewandte Mathematik Welfengarten 1 30167 Hannover, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AMBP_2020__27_1_65_0,
     author = {Philippe Lauren\c{c}ot and Christoph Walker},
     title = {Touchdown is the {Only} {Finite} {Time} {Singularity} in a {Three-Dimensional} {MEMS} {Model}},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {65--81},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {27},
     number = {1},
     year = {2020},
     doi = {10.5802/ambp.391},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.391/}
}
TY  - JOUR
AU  - Philippe Laurençot
AU  - Christoph Walker
TI  - Touchdown is the Only Finite Time Singularity in a Three-Dimensional MEMS Model
JO  - Annales mathématiques Blaise Pascal
PY  - 2020
SP  - 65
EP  - 81
VL  - 27
IS  - 1
PB  - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.391/
DO  - 10.5802/ambp.391
LA  - en
ID  - AMBP_2020__27_1_65_0
ER  - 
%0 Journal Article
%A Philippe Laurençot
%A Christoph Walker
%T Touchdown is the Only Finite Time Singularity in a Three-Dimensional MEMS Model
%J Annales mathématiques Blaise Pascal
%D 2020
%P 65-81
%V 27
%N 1
%I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.391/
%R 10.5802/ambp.391
%G en
%F AMBP_2020__27_1_65_0
Philippe Laurençot; Christoph Walker. Touchdown is the Only Finite Time Singularity in a Three-Dimensional MEMS Model. Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 1, pp. 65-81. doi : 10.5802/ambp.391. https://ambp.centre-mersenne.org/articles/10.5802/ambp.391/

[1] Herbert Amann Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992) (Teubner-Texte zur Mathematik), Volume 133, Teubner, 1993, pp. 9-126 | DOI | MR

[2] Herbert Amann Linear and quasilinear parabolic problems. Vol. I. Abstract linear theory, Monographs in Mathematics, 89, Birkhäuser, 1995, xxxvi+335 pages | DOI | MR

[3] David H. Bernstein; Patrick Guidotti; John 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

[4] Joachim Escher; Philippe Laurençot; Christoph Walker Finite time singularity in a free boundary problem modeling MEMS, C. R. Math. Acad. Sci. Paris, Volume 351 (2013) no. 21-22, pp. 807-812 | DOI | MR

[5] Pierpaolo Esposito; Nassif Ghoussoub; Yujin Guo Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lecture Notes in Mathematics, 20, Courant Institute of Mathematical Sciences; American Mathematical Society, 2010, xiv+318 pages | MR | Zbl

[6] A. Fargas Marquès; R. Costa Castelló; A. M. Shkel Modelling the electrostatic actuation of MEMS: state of the art 2005 (2005) (Technical Report, Universitat Politècnica de Catalunya)

[7] G. Flores; G. Mercado; John A. Pelesko; N. Smyth Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM J. Appl. Math., Volume 67 (2007) no. 2, pp. 434-446 | DOI | MR | Zbl

[8] Davide Guidetti On elliptic problems in Besov spaces, Math. Nachr., Volume 152 (1991), pp. 247-275 | DOI | MR

[9] Davide Guidetti On interpolation with boundary conditions, Math. Z., Volume 207 (1991) no. 3, pp. 439-460 | DOI | MR

[10] Yujin Guo; Zhenguo Pan; Michael J. Ward Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math., Volume 66 (2005) no. 1, pp. 309-338 | DOI | MR

[11] Philippe Laurençot; Christoph Walker A free boundary problem modeling electrostatic MEMS: I. Linear bending effects, Math. Ann., Volume 360 (2014) no. 1-2, pp. 307-349 | DOI | MR

[12] Philippe Laurençot; Christoph Walker On a three-dimensional free boundary problem modeling electrostatic MEMS, Interfaces Free Bound., Volume 18 (2016) no. 3, pp. 393-411 | DOI | MR

[13] Philippe Laurençot; Christoph Walker A variational approach to a stationary free boundary problem modeling MEMS, ESAIM, Control Optim. Calc. Var., Volume 22 (2016) no. 2, pp. 417-438 | DOI | MR

[14] Philippe Laurençot; Christoph Walker Some singular equations modeling MEMS, Bull. Am. Math. Soc., Volume 54 (2017) no. 3, pp. 437-479 | DOI | MR

[15] Philippe Laurençot; Christoph Walker Heterogeneous dielectric properties in models for microelectromechanical systems, SIAM J. Appl. Math., Volume 78 (2018) no. 1, pp. 504-530 | DOI | MR

[16] Jacques-Louis Lions; Enrico Magenes Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, 17, Dunod, 1968, xx+372 pages | MR

[17] Jindřich Nečas Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, 2012, xvi+372 pages (Translated from the 1967 French original by Gerard Tronel and Alois Kufner, Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader) | DOI | MR

[18] John A. Pelesko Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., Volume 62 (2001/02) no. 3, pp. 888-908 | DOI | MR

[19] John A. Pelesko; David H. Bernstein Modeling MEMS and NEMS, Chapman & Hall/CRC, 2003, xxiv+357 pages | MR | Zbl

Cité par Sources :