Laurençot, Philippe; Walker, Christoph Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. (English) Zbl 1458.35241 Discrete Contin. Dyn. Syst., Ser. S 14, No. 2, 677-694 (2021). Summary: 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. Cited in 1 Document MSC: 35K86 Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators 74H20 Existence of solutions of dynamical problems in solid mechanics 35Q74 PDEs in connection with mechanics of deformable solids 74M25 Micromechanics of solids 35M86 Unilateral problems for nonlinear PDEs of mixed type and variational inequalities with nonlinear partial differential operators of mixed type 35K25 Higher-order parabolic equations Keywords:microelectromechanical system; gradient flow; transmission problem; obstacle problem; fourth-order equation PDFBibTeX XMLCite \textit{P. Laurençot} and \textit{C. Walker}, Discrete Contin. Dyn. Syst., Ser. S 14, No. 2, 677--694 (2021; Zbl 1458.35241) Full Text: DOI arXiv References: [1] H. Amann; P. Quittner, Semilinear parabolic equations involving measures and low regularity data, Trans. Amer. Math. Soc., 356, 1045-1119 (2004) · Zbl 1072.35094 [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. · Zbl 1145.35001 [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, 1-168 (1972) · Zbl 0237.35001 [6] L. A. Caffarelli; A. Friedman, The obstacle problem for the biharmonic operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 6, 151-184 (1979) · Zbl 0405.31007 [7] F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, EDP Sciences, Les Ulis, 2012. · Zbl 1239.46001 [8] J. Frehse, On the regularity of the solution of the biharmonic variational inequality, Manuscripta Math., 9, 91-103 (1973) · Zbl 0252.35031 [9] A. Henrot and M. Pierre, Variation et Optimisation de Formes. Mathématiques and Applications, Vol. 48, Springer, Berlin, 2005. · Zbl 1098.49001 [10] A. Henrot and M. Pierre, Shape Variation and Optimization. EMS Tracts in Mathematics, Vol. 28, European Mathematical Society (EMS), Zürich, 2018. · Zbl 1392.49001 [11] Ph. Laurençot; Ch. Walker, Some singular equations modeling MEMS, Bull. Amer. Math. Soc. (N.S.), 54, 437-479 (2017) · Zbl 1456.35193 [12] Ph. Laurençot; Ch. Walker, Shape derivative of the Dirichlet energy for a transmission problem, Arch. Rational Mech. Anal., 237, 447-496 (2020) · Zbl 1440.35045 [13] A. E. Lindsay; J. Lega; K. G. Glasner, Regularized model of post-touchdown configurations in electrostatic MEMS: Equilibrium analysis, Phys. D, 280-281, 95-108 (2014) · Zbl 1349.74127 [14] A. E. Lindsay; J. Lega; K. G. Glasner, Regularized model of post-touchdown configurations in electrostatic MEMS: Interface dynamics, IMA J. Appl. Math., 80, 1635-1663 (2015) · Zbl 1334.35341 [15] M. Novaga; S. Okabe, Regularity of the obstacle problem for the parabolic biharmonic equation, Math. Ann., 363, 1147-1186 (2015) · Zbl 1327.35145 [16] J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62, 888-908 (2001/02) · Zbl 1007.78005 [17] J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. [18] C. Pozzolini; A. Léger, A stability result concerning the obstacle problem for a plate, J. Math. Pures Appl., 90, 505-519 (2008) · Zbl 1153.49019 [19] B. Schild, On the coincidence set in biharmonic variational inequalities with thin obstacles, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13, 559-616 (1986) · Zbl 0704.35055 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.