# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021110
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## On a free boundary model for three-dimensional mems with a hinged top plate II: Parabolic case

 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A–1090 Vienna, Austria

Received  January 2021 Revised  May 2021 Early access June 2021

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.

Citation: Katerina Nik. On a free boundary model for three-dimensional mems with a hinged top plate II: Parabolic case. Communications on Pure &amp; Applied Analysis, doi: 10.3934/cpaa.2021110
##### References:
 [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.  Google Scholar [2] H. Amann, Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar [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.  Google Scholar [4] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.  Google Scholar [5] D. Guidetti, On elliptic problems in Besov spaces, Math. Nachr., 152 (1991), 247-275.  doi: 10.1002/mana.19911520120.  Google Scholar [6] D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460.  doi: 10.1007/BF02571401.  Google Scholar [7] D. Guidetti, On elliptic systems in $L^1$, Osaka J. Math., 30 (1993), 397-429.   Google Scholar [8] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1995.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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). Google Scholar [17] K. Nik, On a free boundary model for three-dimensional MEMS with a hinged top plate I: Stationary sase, preprint, arXiv: 2103.06772. Google Scholar [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.  Google Scholar [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.   Google Scholar [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.  Google Scholar [21] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2$^nd$ edition, Johann Ambrosius Barth, Heidelberg, 1995. Google Scholar [22] M. I. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York, 2011. Google Scholar

show all references

##### References:
 [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.  Google Scholar [2] H. Amann, Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar [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.  Google Scholar [4] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.  Google Scholar [5] D. Guidetti, On elliptic problems in Besov spaces, Math. Nachr., 152 (1991), 247-275.  doi: 10.1002/mana.19911520120.  Google Scholar [6] D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460.  doi: 10.1007/BF02571401.  Google Scholar [7] D. Guidetti, On elliptic systems in $L^1$, Osaka J. Math., 30 (1993), 397-429.   Google Scholar [8] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1995.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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). Google Scholar [17] K. Nik, On a free boundary model for three-dimensional MEMS with a hinged top plate I: Stationary sase, preprint, arXiv: 2103.06772. Google Scholar [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.  Google Scholar [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.   Google Scholar [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.  Google Scholar [21] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2$^nd$ edition, Johann Ambrosius Barth, Heidelberg, 1995. Google Scholar [22] M. I. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York, 2011. Google Scholar
Cross section of the idealized MEMS device
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