TY - JOUR

T1 - A survey on second-order free boundary value problems modelling MEMs with general permittivity profile

AU - Escher, Joachim

AU - Lienstromberg, Christina

PY - 2017/7

Y1 - 2017/7

N2 - In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semior quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the inuence of different classes of permittivity profiles. The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account - the direction of the membrane's deection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.

AB - In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semior quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the inuence of different classes of permittivity profiles. The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account - the direction of the membrane's deection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.

KW - Finite-time singularity

KW - Free boundary value problem

KW - General permittivity profile

KW - MEMS

UR - http://www.scopus.com/inward/record.url?scp=85018456400&partnerID=8YFLogxK

U2 - 10.3934/dcdss.2017038

DO - 10.3934/dcdss.2017038

M3 - Article

AN - SCOPUS:85018456400

VL - 10

SP - 745

EP - 771

JO - Discrete and Continuous Dynamical Systems - Series S

JF - Discrete and Continuous Dynamical Systems - Series S

SN - 1937-1632

IS - 4

ER -