A variational approach to a stationary free boundary problem modeling MEMS∗
Institut de Mathématiques de Toulouse, UMR 5219, Université de
Toulouse, CNRS, 31062
Toulouse cedex 9,
2 Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, 30167 Hannover, Germany.
Revised: 20 January 2015
A variational approach is employed to find stationary solutions to a free boundary problem modeling an idealized electrostatically actuated MEMS device made of an elastic plate coated with a thin dielectric film and suspended above a rigid ground plate. The model couples a non-local fourth-order equation for the elastic plate deflection to the harmonic electrostatic potential in the free domain between the elastic and the ground plate. The corresponding energy is non-coercive reflecting an inherent singularity related to a possible touchdown of the elastic plate. Stationary solutions are constructed using a constrained minimization problem. A by-product is the existence of at least two stationary solutions for some values of the applied voltage.
Mathematics Subject Classification: 35J35 / 35R35 / 35Q74
Key words: MEMS / free boundary / stationary solutions / multiplicity / constrained minimization
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