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February  2021, 14(2): 677-694. doi: 10.3934/dcdss.2020360

Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties

1. 

Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, F–31062 Toulouse Cedex 9, France

2. 

Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, D–30167 Hannover, Germany

* Corresponding author

Dedicated to Michel Pierre on the occasion of his 70th birthday

Received  October 2019 Published  February 2021 Early access  May 2020

Fund Project: Partially supported by the CNRS Projet International de Coopération Scientifique PICS07710

The existence of weak solutions to the obstacle problem for a nonlocal semilinear fourth-order parabolic equation is shown, using its underlying gradient flow structure. The model governs the dynamics of a microelectromechanical system with heterogeneous dielectric properties.

Citation: Philippe Laurençot, Christoph Walker. Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 677-694. doi: 10.3934/dcdss.2020360
References:
[1]

H. Amann and P. Quittner, Semilinear parabolic equations involving measures and low regularity data, Trans. Amer. Math. Soc., 356 (2004), 1045-1119.  doi: 10.1090/S0002-9947-03-03440-8.

[2]

V. R. Ambati, A. Asheim, J. B. van den Berg, et al., Some studies on the deformation of the membrane in an RF MEMS switch, In Proceedings of the 63rd European Study Group Mathematics with Industry, Centrum voor Wiskunde en Informatica Syllabus, Netherlands, 2008, 65–84. /http://eprints.ewi.utwente.nl/14950

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[4]

D. H. Bernstein and P. Guidotti, Modeling and analysis of hysteresis phenomena in electrostatic zipper actuators, In Proceedings of Modeling and Simulation of Microsystems 2001, Hilton Head Island, SC, 2001,306–309.

[5]

H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl., 51 (1972), 1-168. 

[6]

L. A. Caffarelli and A. Friedman, The obstacle problem for the biharmonic operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 6 (1979), 151-184. 

[7]

F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, EDP Sciences, Les Ulis, 2012. doi: 10.1007/978-1-4471-2807-6.

[8]

J. Frehse, On the regularity of the solution of the biharmonic variational inequality, Manuscripta Math., 9 (1973), 91-103.  doi: 10.1007/BF01320669.

[9]

A. Henrot and M. Pierre, Variation et Optimisation de Formes. Mathématiques and Applications, Vol. 48, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.

[10]

A. Henrot and M. Pierre, Shape Variation and Optimization. EMS Tracts in Mathematics, Vol. 28, European Mathematical Society (EMS), Zürich, 2018. doi: 10.4171/178.

[11]

Ph. Laurençot and Ch. Walker, Some singular equations modeling MEMS, Bull. Amer. Math. Soc. (N.S.), 54 (2017), 437-479.  doi: 10.1090/bull/1563.

[12]

Ph. Laurençot and Ch. Walker, Shape derivative of the Dirichlet energy for a transmission problem, Arch. Rational Mech. Anal., 237 (2020), 447-496.  doi: 10.1007/s00205-020-01512-8.

[13]

A. E. LindsayJ. Lega and K. G. Glasner, Regularized model of post-touchdown configurations in electrostatic MEMS: Equilibrium analysis, Phys. D, 280-281 (2014), 95-108. 

[14]

A. E. LindsayJ. Lega and K. G. Glasner, Regularized model of post-touchdown configurations in electrostatic MEMS: Interface dynamics, IMA J. Appl. Math., 80 (2015), 1635-1663.  doi: 10.1093/imamat/hxv011.

[15]

M. Novaga and S. Okabe, Regularity of the obstacle problem for the parabolic biharmonic equation, Math. Ann., 363 (2015), 1147-1186.  doi: 10.1007/s00208-015-1200-5.

[16]

J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2001/02), 888-908.  doi: 10.1137/S0036139900381079.

[17]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003.

[18]

C. Pozzolini and A. Léger, A stability result concerning the obstacle problem for a plate, J. Math. Pures Appl., 90 (2008), 505-519.  doi: 10.1016/j.matpur.2008.07.005.

[19]

B. Schild, On the coincidence set in biharmonic variational inequalities with thin obstacles, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 559-616. 

show all references

References:
[1]

H. Amann and P. Quittner, Semilinear parabolic equations involving measures and low regularity data, Trans. Amer. Math. Soc., 356 (2004), 1045-1119.  doi: 10.1090/S0002-9947-03-03440-8.

[2]

V. R. Ambati, A. Asheim, J. B. van den Berg, et al., Some studies on the deformation of the membrane in an RF MEMS switch, In Proceedings of the 63rd European Study Group Mathematics with Industry, Centrum voor Wiskunde en Informatica Syllabus, Netherlands, 2008, 65–84. /http://eprints.ewi.utwente.nl/14950

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[4]

D. H. Bernstein and P. Guidotti, Modeling and analysis of hysteresis phenomena in electrostatic zipper actuators, In Proceedings of Modeling and Simulation of Microsystems 2001, Hilton Head Island, SC, 2001,306–309.

[5]

H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl., 51 (1972), 1-168. 

[6]

L. A. Caffarelli and A. Friedman, The obstacle problem for the biharmonic operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 6 (1979), 151-184. 

[7]

F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, EDP Sciences, Les Ulis, 2012. doi: 10.1007/978-1-4471-2807-6.

[8]

J. Frehse, On the regularity of the solution of the biharmonic variational inequality, Manuscripta Math., 9 (1973), 91-103.  doi: 10.1007/BF01320669.

[9]

A. Henrot and M. Pierre, Variation et Optimisation de Formes. Mathématiques and Applications, Vol. 48, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.

[10]

A. Henrot and M. Pierre, Shape Variation and Optimization. EMS Tracts in Mathematics, Vol. 28, European Mathematical Society (EMS), Zürich, 2018. doi: 10.4171/178.

[11]

Ph. Laurençot and Ch. Walker, Some singular equations modeling MEMS, Bull. Amer. Math. Soc. (N.S.), 54 (2017), 437-479.  doi: 10.1090/bull/1563.

[12]

Ph. Laurençot and Ch. Walker, Shape derivative of the Dirichlet energy for a transmission problem, Arch. Rational Mech. Anal., 237 (2020), 447-496.  doi: 10.1007/s00205-020-01512-8.

[13]

A. E. LindsayJ. Lega and K. G. Glasner, Regularized model of post-touchdown configurations in electrostatic MEMS: Equilibrium analysis, Phys. D, 280-281 (2014), 95-108. 

[14]

A. E. LindsayJ. Lega and K. G. Glasner, Regularized model of post-touchdown configurations in electrostatic MEMS: Interface dynamics, IMA J. Appl. Math., 80 (2015), 1635-1663.  doi: 10.1093/imamat/hxv011.

[15]

M. Novaga and S. Okabe, Regularity of the obstacle problem for the parabolic biharmonic equation, Math. Ann., 363 (2015), 1147-1186.  doi: 10.1007/s00208-015-1200-5.

[16]

J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2001/02), 888-908.  doi: 10.1137/S0036139900381079.

[17]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003.

[18]

C. Pozzolini and A. Léger, A stability result concerning the obstacle problem for a plate, J. Math. Pures Appl., 90 (2008), 505-519.  doi: 10.1016/j.matpur.2008.07.005.

[19]

B. Schild, On the coincidence set in biharmonic variational inequalities with thin obstacles, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 559-616. 

Figure 1.  Geometry of $ \Omega(u) $ for a state $ u = v $ with empty coincidence set (green)
Figure 2.  Geometry of $ \Omega(u) $ for a state $ u = w $ with non-empty coincidence set (blue)
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