Abstract
Existence of stationary solutions to a nonlocal fourth-order elliptic obstacle problem arising from the modelling of microelectromechanical systems with heterogeneous dielectric properties is shown. The underlying variational structure of the model is exploited to construct these solutions as minimizers of a suitably regularized energy, which allows us to weaken considerably the assumptions on the model used in a previous article.
Highlights
Idealized electrostatically actuated microelectromechanical systems (MEMS) are made up of an elastic conducting plate which is clamped on its boundary and suspended above a rigid conducting ground plate
Their dynamics results from the competition between mechanical and electrostatic forces in which the elastic plate is deformed by a Coulomb
When the electrostatic forces dominate the mechanical ones, the elastic plate comes into contact with the ground plate, thereby generating a short circuit and leading to the occurrence of a touchdown singularity in the related mathematical models, see [5, 6, 9, 12, 18, 19] and the references therein
Summary
Idealized electrostatically actuated microelectromechanical systems (MEMS) are made up of an elastic conducting plate which is clamped on its boundary and suspended above a rigid conducting ground plate. While ∂ S0 (u) accounts for the non-penetrability of the insulating layer, the fourth- and second-order terms in (1.1) represent forces due to plate bending and plate stretching, respectively These forces are balanced by the electrostatic force g(u) acting on the elastic plate, which is derived in [13] and involves the electrostatic potential u in the device. Since we no longer impose assumption (1.10) in Theorem 1.3, the boundedness from below of the functional E is a priori unclear, due to the negative contribution from the electrostatic energy Ee. We shall work with regularized coercive functionals instead (see (2.1) below) and use comparison principle arguments to derive a priori bounds on minimizers of the regularized functionals, see Sect. If the coincidence set C(u) ⊂ D of a solution u ∈ S0 to (1.1) is non-empty, it is an interval
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