Abstract
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.
Highlights
We consider a model for a three-dimensional microelectromechanical system (MEMS) including two components, a rigid ground plate of shape D ⊂ R2 and an elastic plate of the same shape which is suspended above the rigid one and clamped on its boundary, see Figure 1
When applying a sufficiently large voltage difference, a well-known phenomenon that might occur is that the two plates come into contact; that is, the elastic plate touches down on the rigid plate
Since the pioneering works [3, 7, 10, 18], their mathematical analysis has been the subject of numerous papers
Summary
We consider a model for a three-dimensional microelectromechanical system (MEMS) including two components, a rigid ground plate of shape D ⊂ R2 and an elastic plate of the same shape (at rest) which is suspended above the rigid one and clamped on its boundary, see Figure 1. Both plates being conducting, holding them at different voltages generates a Coulomb force across the device. Its local in time well-posedness can be shown in a suitable functional setting, as we recall below, and the aim of this note is to improve the criterion for global existence derived in [12]
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