Article Contents
Article Contents

# On a free boundary model for three-dimensional MEMS with a hinged top plate II: Parabolic case

• A parabolic 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 fourth-order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The strength of the coupling is tuned by a parameter $\lambda$ which is proportional to the square of the applied voltage difference. It is proven that the model is locally well-posed in time and that, for $\lambda$ sufficiently small, solutions exist globally in time. In addition, touchdown of the top plate on the ground plate is shown to be the only possible finite time singularity.

Mathematics Subject Classification: 35K91, 35R35, 35M33, 35Q74, 35B44.

 Citation:

• Figure 1.  Cross section of the idealized MEMS device

•  [1] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), vol. 133 of Teubner-Texte Math. Teubner, Stuttgart, (1993), 9–126. doi: 10.1007/978-3-663-11336-2\_1. [2] H. Amann, Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9221-6. [3] J. Escher, Ph. Laurençot and Ch. Walker, A parabolic free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal., 211 (2014), 389-417.  doi: 10.1007/s00205-013-0656-2. [4] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985. [5] D. Guidetti, On elliptic problems in Besov spaces, Math. Nachr., 152 (1991), 247-275.  doi: 10.1002/mana.19911520120. [6] D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460.  doi: 10.1007/BF02571401. [7] D. Guidetti, On elliptic systems in $L^1$, Osaka J. Math., 30 (1993), 397-429. [8] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1995. [9] Ph. Laurençot and Ch. Walker, A free boundary problem modeling electrostatic MEMS: I. Linear bending effects, Math. Ann., 360 (2014), 307-349.  doi: 10.1007/s00208-014-1032-8. [10] Ph. Laurençot and Ch. Walker, A free boundary problem modeling electrostatic MEMS: II. Nonlinear bending, Math. Models Methods Appl. Sci., 24 (2014), 2549-2568.  doi: 10.1142/S0218202514500298. [11] Ph. Laurençot and Ch. Walker, The time singular limit for a fourth-order damped wave equation for MEMS, Springer Proc. Math. Stat., 119 (2015), 233-246.  doi: 10.1007/978-3-319-12547-3\_10. [12] Ph. Laurençot and Ch. Walker, On a three-dimensional free boundary problem modeling electrostatic MEMS, Interfaces Free Bound., 18 (2016), 393-411.  doi: 10.4171/IFB/368. [13] Ph. Laurençot and Ch. Walker, A variational approach to a stationary free boundary problem modeling MEMS, ESAIM Control Optim. Calc. Var., 22 (2016), 417-438.  doi: 10.1051/cocv/2015012. [14] Ph. Laurençot and Ch. Walker, Shape derivative of the Dirichlet energy for a transmission problem, Arch. Ration. Mech. Anal., 237 (2020), 447-496.  doi: 10.1007/s00205-020-01512-8. [15] Ph. Laurençot and Ch. Walker, Touchdown is the only finite time singularity in a three-dimensional MEMS model, Ann. Math. Blaise Pascal, 27 (2020), 65-81. [16] A. F. Marques, R. C. Castelló and A. M. Shkel, Modelling the electrostatic actuation of MEMS: state of the art 2005, Technical Report, Universitat Politècnica de Catalunya, (2005). [17] K. Nik, On a free boundary model for three-dimensional MEMS with a hinged top plate I: Stationary sase, preprint, arXiv: 2103.06772. [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. [19] J. A. Pelesko and  D. H. Bernstein,  Semigroups of Linear Operators and Applications to Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, FL, 2003. [20] G. Sweers and K. Vassi, Positivity for a hinged convex plate with stress, SIAM J. Math. Anal., 50 (2018), 1163-1174.  doi: 10.1137/17M1138790. [21] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2$^nd$ edition, Johann Ambrosius Barth, Heidelberg, 1995. [22] M. I. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York, 2011.
Open Access Under a Creative Commons license

Figures(1)