On an elliptic-parabolic MEMS model with two free boundaries

References

• Esposito P, Ghoussoub N, Guo Y. Mathematical analysis of partial differential equations modeling electrostatic MEMS. Vol. 20, Courant lecture notes in mathematics. New York (NY): Courant Institute of Mathematical Sciences; 2010.
   Google Scholar
• Pelesko JA, Bernstein DH. Modeling MEMS and NEMS. Boca Raton (FL): Chapman & Hall/CRC; 2003.
   Google Scholar
• Pelesko JA, Triolo AA. Nonlocal problems in MEMS device control. In: Proceedings of modeling and simulation of microsystems 2000. San Diego, CA; 2000. p. 509–512.
   Google Scholar
• Pelesko JA, Triolo AA. Nonlocal problems in MEMS device control. J. Eng. Math. 2001;41:345–366.
   Web of Science ®Google Scholar
• Pelesko JA. Mathematical modeling of electrostatic MEMS with tailored dielectric properties. SIAM J. Appl. Math. 2002;62:888–908.
   Web of Science ®Google Scholar
• Abbaspour-Sani E, Afrang S. A low voltage MEMS structure for RF capacitive switches. Prog. Electromagn. Res. 2006;65:157–167.
   Web of Science ®Google Scholar
• Feng C, Tang Z, Yu J, Sun C. A MEMS device capable of measuring near-field thermal radiation between membranes. Sensors. 2013;13:1998–2010. doi:10.3390/s130201998.
   PubMed Web of Science ®Google Scholar
• Taylor GI. The coalescence of closely spaced drops when they are at different electric potentials. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 1968;306:423–434.
   Web of Science ®Google Scholar
• Ackerberg RC. On a nonlinear differential equation of electrohydrodynamics. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 1969;312:129–140.
   Web of Science ®Google Scholar
• Kohlmann M. A new model for electrostatic MEMS with two free boundaries. J. Math. Anal. Appl. 2013;408:513–524.
   Web of Science ®Google Scholar
• Kohlmann M. The abstract quasilinear Cauchy problem for a MEMS model with two free boundaries. Acta Appl. Math. Forthcoming. doi:10.1007/s10440-014-9962-4.
   Web of Science ®Google Scholar
• Flores G, Mercado G, Pelesko JA. Dynamics and touchdown in electrostatic MEMS. In: Proceedings of IDETC/CIE 2003, 19th ASME Biennal conference on mechanical vibration and noise; 2003; Chicago, IL. p. 1–8.
   Google Scholar
• Flores G, Mercado G, Pelesko JA, Smyth N. Analysis of the dynamics and touchdown in a model of electrostatic MEMS. SIAM J. Appl. Math. 2007;67:434–446.
   Web of Science ®Google Scholar
• Ghoussoub N, Guo Y. On the partial differential equations of electrostatic MEMS devices: stationary case. SIAM J. Math. Anal. 2007;38:1423–1449.
   Web of Science ®Google Scholar
• Ghoussoub N, Guo Y. On the partial differential equations of electrostatic MEMS devices. II. Dynamic case. NoDEA Nonlinear Differ. Equ. Appl. 2008;15:115–145.
   Web of Science ®Google Scholar
• Guo J-S, Hu B, Wang C-J. A nonlocal quenching problem arising in a micro-electro mechanical system. Quart. Appl. Math. 2009;67:725–734.
   Web of Science ®Google Scholar
• Guo Y. Global solutions of singular parabolic equations arising from electrostatic MEMS. J. Differ. Equ. 2008;245:809–844.
   Web of Science ®Google Scholar
• Guo Y. On the partial differential equations of electrostatic MEMS devices. III. Refined touchdown behavior. J. Differ. Equ. 2008;244:2277–2309.
   Web of Science ®Google Scholar
• Hui KM. The existence and dynamic properties of a parabolic nonlocal MEMS equation. Nonlinear Anal. 2011;74:298–316.
   Web of Science ®Google Scholar
• Lin F, Yang Y. Nonlinear non-local elliptic equation modelling electrostatic actuation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 2007;463:1323–1337.
   Web of Science ®Google Scholar
• Cimatti G. A free boundary problem in the theory of electrically actuated microdevices. Appl. Math. Lett. 2007;20:1232–1236.
   Web of Science ®Google Scholar
• Laurençot P, Walker C. A stationary free boundary problem modeling electrostatic MEMS. Arch. Rational Mech. Anal. 2013;207:139–158.
   Web of Science ®Google Scholar
• Escher J, Laurençot P, Walker C. A parabolic free boundary problem modeling electrostatic MEMS. Arch. Ration. Mech. Anal. 2014;211:389–417.
   Web of Science ®Google Scholar
• Brubaker ND, Pelesko JA. Non-linear effects on canonical MEMS models. European J. Appl. Math. 2011;22:455–470.
   Web of Science ®Google Scholar
• Brubaker ND, Pelesko JA. Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity. Nonlinear Anal. 2012;75:5086–5102.
   Web of Science ®Google Scholar
• Escher J, Laurençot P, Walker C. Dynamics of a free boundary problem with curvature modeling electrostatic MEMS. Trans. Amer. Math. Soc. Forthcoming; arXiv:1302.6026v1.
   Web of Science ®Google Scholar
• Laurençot P, Walker C. Sign-preserving property for some fourth-order elliptic operators in one dimension and radial symmetry. J. Anal. Math. Forthcoming; arXiv:1303.2237.
   Google Scholar
• Lega J, Lindsay AE. Quenching solutions of a fourth order PDE with a singular nonlinearity modelling a MEMS capacitor. SIAM J. Appl. Math. 2012;72:935–958.
   Web of Science ®Google Scholar
• Lindsay AE, Lega J, Glasner KB. Regularized model of post-touchdown configurations in electrostatic MEMS: equilibrium analysis. Physica D. 2014;280–281:95–108.
   Web of Science ®Google Scholar
• Lindsay AE, Lega J, Sayas FJ. The quenching set of a MEMS capacitor in two-dimensional geometries. J. Nonlinear Sci. 2013;23:807–834.
   Web of Science ®Google Scholar
• Guo Y. Dynamical solutions of singular wave equations modeling electrostatic MEMS. SIAM J. Appl. Dyn. Syst. 2010;9:1135–1163.
   Web of Science ®Google Scholar
• Kavallaris NI, Lacey AA, Nikolopoulos CV, Tzanetis DE. A hyperbolic non-local problem modelling MEMS technology. Rocky Mountain J. Math. 2011;41:505–534.
   Web of Science ®Google Scholar
• Nečas J. Les méthodes directes en théorie des équations elliptiques [Direct methods in the theory of elliptic equations]. Paris: Masson et Cie Editeurs; 1967.
   Google Scholar
• Amann H. Multiplication in Sobolev and Besov spaces. In: Ambrosetti A, Marino A, editors. Nonlinear analysis, a tribute in honor of G. Prodi. Pisa: Scuola Norm. Sup.; 1991. p. 27–50.
   Google Scholar
• Lunardi A. Analytic semigroups and optimal regularity in parabolic problems. Vol. 16, Progress in nonlinear differential equations and their applications. Basel: Birkhäuser; 1995.
   Google Scholar
• Adams RA. Sobolev spaces. New York (NY): Academic Press; 1975.
   Google Scholar
• Bernstein DH, Guidotti P, Pelesko JA. Analytical and numerical analysis of electrostatically actuated MEMS devices. In: Proceedings of modeling and simulation of microsystems 2000. San Diego, CA; 2000. p. 489–492.
   Google Scholar