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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

Philippe Laurençot 1,, and  Christoph Walker 2,

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  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.

Keywords: MEMS, gradient flow, transmission problem, obstacle problem, fourth-order equation.
Mathematics Subject Classification: Primary: 35K86, 74H20, 35Q74; Secondary: 35M86, 35K25.
Citation: Philippe Laurençot, Christoph Walker. Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. Discrete & 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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

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

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

A. E. Lindsay, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

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

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

A. E. Lindsay, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.   Google Scholar

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