Energy minimizers for an asymptotic MEMS model with heterogeneous dielectric properties

Abstract

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.

Stationary states to a free boundary transmission problem for an electrostatically actuated plate

“…The proof of Theorem 1.2 now follows from Theorem 2.1 as in [2]. Indeed, Theorem 2.1 guarantees that any minimizer of the total energy E on S0 satisfies the Euler-Lagrange equation (1.7).…”

“…Indeed, Theorem 2.1 guarantees that any minimizer of the total energy E on S0 satisfies the Euler-Lagrange equation (1.7). In case that a > 0, the total energy E is coercive and thus the existence of a minimizer of E in S0 can be shown as in [2,Section 7]. In the more complex case a = 0, the total energy E need not be coercive.…”

“…It is worth already pointing out here that the derivation of the explicit formula (1.8) of g(u) does not require the explicit computation of the derivative of the electrostatic potential ψ u with respect to u. Once the formula (1.8) is established, the existence of minimizers of E in S0 follows along the lines of [2] as described above.…”

“…In fact, we are not able to prove directly that E is bounded below and thus have to proceed differently. In this case, the coercivity of the functional can be enforced by adding a penalty term which vanishes when u is bounded, an idea already used in [2]. The minimizers of the penalized energy functional on S0 then satisfy the Euler-Lagrange equation (1.7) with an additional term.…”