Stationary soap film bridge formed by a small electrostatic force
Author:
Lina Sophie Schmitz
Journal:
Quart. Appl. Math.
MSC (2020):
Primary 35R35, 35B35, 34L10, 47J07, 35Q99
DOI:
https://doi.org/10.1090/qam/1707
Published electronically:
January 21, 2025
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We consider two models, a free boundary problem and a simplification thereof, which describe a soap film bridge subjected to an electrostatic force. For both models, we construct stationary solutions if the force is small, analyse their stability, and examine how their shape is influenced by small changes in the strength of the force.
References
- H. Amann, Maximum principles and principal eigenvalues, Ten mathematical essays on approximation in analysis and topology, Elsevier B. V., Amsterdam, 2005, pp. 1–60. MR 2162975, DOI 10.1016/B978-044451861-3/50001-X
- Herbert Amann, Gewöhnliche Differentialgleichungen, de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter & Co., Berlin, 1983 (German). MR 713040
- Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. MR 1345385, DOI 10.1007/978-3-0348-9221-6
- Paul Binding and Hans Volkmer, Eigencurves for two-parameter Sturm-Liouville equations, SIAM Rev. 38 (1996), no. 1, 27–48. MR 1379040, DOI 10.1137/1038002
- Boris Buffoni and John Toland, Analytic theory of global bifurcation, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003. An introduction. MR 1956130, DOI 10.1515/9781400884339
- Joachim Escher, Philippe Laurençot, and Christoph Walker, A parabolic free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal. 211 (2014), no. 2, 389–417. MR 3149061, DOI 10.1007/s00205-013-0656-2
- Joachim Escher, Philippe Laurençot, and Christoph Walker, Dynamics of a free boundary problem with curvature modeling electrostatic MEMS, Trans. Amer. Math. Soc. 367 (2015), no. 8, 5693–5719. MR 3347187, DOI 10.1090/S0002-9947-2014-06320-4
- Joachim Escher and Christina Lienstromberg, A survey on second-order free boundary value problems modelling MEMS with general permittivity profile, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 4, 745–771. MR 3640536, DOI 10.3934/dcdss.2017038
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
- Jürgen Jost and Xianqing Li-Jost, Calculus of variations, Cambridge Studies in Advanced Mathematics, vol. 64, Cambridge University Press, Cambridge, 1998. MR 1674720
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
- Philippe Laurençot and Christoph Walker, A free boundary problem modeling electrostatic MEMS: I. Linear bending effects, Math. Ann. 360 (2014), no. 1-2, 307–349. MR 3263165, DOI 10.1007/s00208-014-1032-8
- Philippe Laurençot and Christoph Walker, Some singular equations modeling MEMS, Bull. Amer. Math. Soc. (N.S.) 54 (2017), no. 3, 437–479. MR 3662914, DOI 10.1090/bull/1563
- L. Lorenzi, A. Lunardi, G. Metafune, and D. Pallara, Analytic semigroups and reaction-diffusion problems, Internet Seminar, 2004-2005, available online at: https://www.math.tecnico.ulisboa.pt/~czaja/ISEM/08internetseminar200405.pdf.
- Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. [2013 reprint of the 1995 original] [MR1329547]. MR 3012216
- Derek E. Moulton, Mathematical modeling of field driven mean curvature surfaces, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–University of Delaware. MR 2712151
- D. E. Moulton and J. A. Pelesko, Catenoid in an electric field, SIAM J. Appl. Math. 70 (2009), no. 1, 212–230. MR 2505086, DOI 10.1137/070698579
- Katerina Nik, On a free boundary model for three-dimensional MEMS with a hinged top plate: stationary case, Port. Math. 78 (2021), no. 2, 211–232. MR 4301807, DOI 10.4171/pm/2067
- L. S. Schmitz, Analysis of a soap film catenoid driven by an electrostatic force, Ph.D. thesis, Gottfried Wilhelm Leibniz Universität, Hannover, 2024.
- L. S. Schmitz, Qualitative properties of a soap film bridge driven by an electrostatic force: the case of balanced forces, preprint, 2024.
- L. S. Schmitz, Dynamical behaviour of a soap film bridge driven by an electrostatic force, preprint 2024.
- Junping Shi, A radially symmetric anti-maximum principle and applications to fishery management models, Electron. J. Differential Equations (2004), No. 27, 13. MR 2036211
- Wolfgang Walter, Gewöhnliche Differentialgleichungen, 5th ed., Springer-Lehrbuch. [Springer Textbook], Springer-Verlag, Berlin, 1993 (German, with German summary). Eine Einführung. [An introduction]. MR 1231977, DOI 10.1007/978-3-642-97467-0
References
- H. Amann, Maximum principles and principal eigenvalues, Ten mathematical essays on approximation in analysis and topology, Elsevier B. V., Amsterdam, 2005, pp. 1–60. MR 2162975, DOI 10.1016/B978-044451861-3/50001-X
- Herbert Amann, Gewöhnliche Differentialgleichungen, de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter & Co., Berlin, 1983 (German). MR 713040
- Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Abstract linear theory, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. MR 1345385, DOI 10.1007/978-3-0348-9221-6
- Paul Binding and Hans Volkmer, Eigencurves for two-parameter Sturm-Liouville equations, SIAM Rev. 38 (1996), no. 1, 27–48. MR 1379040, DOI 10.1137/1038002
- Boris Buffoni and John Toland, Analytic theory of global bifurcation: An introduction, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003. MR 1956130, DOI 10.1515/9781400884339
- Joachim Escher, Philippe Laurençot, and Christoph Walker, A parabolic free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal. 211 (2014), no. 2, 389–417. MR 3149061, DOI 10.1007/s00205-013-0656-2
- Joachim Escher, Philippe Laurençot, and Christoph Walker, Dynamics of a free boundary problem with curvature modeling electrostatic MEMS, Trans. Amer. Math. Soc. 367 (2015), no. 8, 5693–5719. MR 3347187, DOI 10.1090/S0002-9947-2014-06320-4
- Joachim Escher and Christina Lienstromberg, A survey on second-order free boundary value problems modelling MEMS with general permittivity profile, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 4, 745–771. MR 3640536, DOI 10.3934/dcdss.2017038
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
- Jürgen Jost and Xianqing Li-Jost, Calculus of variations, Cambridge Studies in Advanced Mathematics, vol. 64, Cambridge University Press, Cambridge, 1998. MR 1674720
- Tosio Kato, Perturbation theory for linear operators, reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. MR 1335452
- Philippe Laurençot and Christoph Walker, A free boundary problem modeling electrostatic MEMS: I. Linear bending effects, Math. Ann. 360 (2014), no. 1-2, 307–349. MR 3263165, DOI 10.1007/s00208-014-1032-8
- P. Laurençot and C. Walker, Some singular equations modeling MEMS, Bull. Amer. Math. Soc. (N.S.) 54 (2017), no. 3, 437–479. MR 3662914
- L. Lorenzi, A. Lunardi, G. Metafune, and D. Pallara, Analytic semigroups and reaction-diffusion problems, Internet Seminar, 2004-2005, available online at: https://www.math.tecnico.ulisboa.pt/~czaja/ISEM/08internetseminar200405.pdf.
- Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, 2013 reprint of the 1995 original [MR1329547], Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. MR 3012216
- Derek E. Moulton, Mathematical modeling of field driven mean curvature surfaces, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–University of Delaware. MR 2712151
- D. E. Moulton and J. A. Pelesko, Catenoid in an electric field, SIAM J. Appl. Math. 70 (2009), no. 1, 212–230. MR 2505086, DOI 10.1137/070698579
- Katerina Nik, On a free boundary model for three-dimensional MEMS with a hinged top plate: stationary case, Port. Math. 78 (2021), no. 2, 211–232. MR 4301807, DOI 10.4171/pm/2067
- L. S. Schmitz, Analysis of a soap film catenoid driven by an electrostatic force, Ph.D. thesis, Gottfried Wilhelm Leibniz Universität, Hannover, 2024.
- L. S. Schmitz, Qualitative properties of a soap film bridge driven by an electrostatic force: the case of balanced forces, preprint, 2024.
- L. S. Schmitz, Dynamical behaviour of a soap film bridge driven by an electrostatic force, preprint 2024.
- Junping Shi, A radially symmetric anti-maximum principle and applications to fishery management models, Electron. J. Differential Equations (2004), No. 27, 13. MR 2036211
- Wolfgang Walter, Gewöhnliche Differentialgleichungen, Eine Einführung [An introduction], 5, Springer-Lehrbuch. [Springer Textbook], Springer-Verlag, Berlin, 1993 (German, with German summary). MR 1231977, DOI 10.1007/978-3-642-97467-0
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2020):
35R35,
35B35,
34L10,
47J07,
35Q99
Retrieve articles in all journals
with MSC (2020):
35R35,
35B35,
34L10,
47J07,
35Q99
Additional Information
Lina Sophie Schmitz
Affiliation:
Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany
ORCID:
0000-0002-8008-5811
Email:
schmitz@ifam.uni-hannover.de
Keywords:
Free boundary problem,
qualitative properties,
stability,
surface tension,
electrostatics
Received by editor(s):
October 14, 2024
Received by editor(s) in revised form:
December 23, 2024
Published electronically:
January 21, 2025
Article copyright:
© Copyright 2025
Brown University
