A MEMS model with an insulating layer is considered and its reinforced limit is derived by means of a Gamma convergence approach when the thickness of the layer tends to zero. The limiting model inherits the dielectric properties of the insulating layer.

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A model for a MEMS device, consisting of a fixed bottom plate and an elastic plate, is studied. It was derived in a previous work as a reinforced limit when the thickness of the insulating layer covering the bottom plate tends to zero. This asymptotic model inherits the dielectric properties of the insulating layer. It involves the electrostatic potential in the device and the deformation of the elastic plate defining the geometry of the device. The electrostatic potential is given by an elliptic equation with mixed boundary conditions in the possibly non-Lipschitz region between the two plates. The deformation of the elastic plate is supposed to be a critical point of an energy functional which, in turn, depends on the electrostatic potential due to the force exerted by the latter on the elastic plate. The energy functional is shown to have a minimizer giving the geometry of the device. Moreover, the corresponding Euler–Lagrange equation is computed and the maximal regularity of the electrostatic potential is established.

A stationary free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a nonlocal fourth-order equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the free domain between the two plates. The strength of the coupling is tuned by a parameter \lambda which is proportional to the square of the applied voltage difference. Existence of a stable stationary solution is established for small values of \lambda . Nonexistence of stationary solutions is obtained when \lambda is large enough.

Starting with the periodic waves earlier constructed for the gravity Whitham equation, we parameterise the solution curves through relative wave height, and use a limiting argument to obtain a full family of solitary waves. The resulting branch starts from the zero solution, traverses unique points in the wave speed–wave height space, and reaches a singular highest wave at φ ( 0 ) = μ 2 \varphi (0) = \frac {\mu }{2} . The construction is based on uniform estimates improved from earlier work on periodic waves for the same equation, together with limiting arguments and a Galilean transform to exclude vanishing waves and waves levelling off at negative surface depth. In fact, the periodic waves can be proved to converge locally uniformly to a wave with negative tails, which is then transformed to the desired branch of solutions. The paper also contains some proof concerning uniqueness and continuity for signed solutions (improved touching lemma).