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 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 by the bi-Laplacian with clamped boundary conditions.
CC-BY 4.0
@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] 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] Linear and quasilinear parabolic problems. Vol. I. Abstract linear theory, Monographs in Mathematics, 89, Birkhäuser, 1995, xxxvi+335 pages | DOI | MR
[3] 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] 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] 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] Modelling the electrostatic actuation of MEMS: state of the art 2005 (2005) (Technical Report, Universitat Politècnica de Catalunya)
[7] 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] On elliptic problems in Besov spaces, Math. Nachr., Volume 152 (1991), pp. 247-275 | DOI | MR
[9] On interpolation with boundary conditions, Math. Z., Volume 207 (1991) no. 3, pp. 439-460 | DOI | MR
[10] 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] 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] On a three-dimensional free boundary problem modeling electrostatic MEMS, Interfaces Free Bound., Volume 18 (2016) no. 3, pp. 393-411 | DOI | MR
[13] 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] Some singular equations modeling MEMS, Bull. Am. Math. Soc., Volume 54 (2017) no. 3, pp. 437-479 | DOI | MR
[15] Heterogeneous dielectric properties in models for microelectromechanical systems, SIAM J. Appl. Math., Volume 78 (2018) no. 1, pp. 504-530 | DOI | MR
[16] Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, 17, Dunod, 1968, xx+372 pages | MR
[17] 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] 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] Modeling MEMS and NEMS, Chapman & Hall/CRC, 2003, xxiv+357 pages | MR | Zbl
Cité par Sources :