On MEMS equation with fringing field
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- by Juncheng Wei and Dong Ye
- Proc. Amer. Math. Soc. 138 (2010), 1693-1699
- DOI: https://doi.org/10.1090/S0002-9939-09-10226-5
- Published electronically: December 30, 2009
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Abstract:
We consider the MEMS equation with fringing field \[ -\Delta u = \lambda (1 + \delta |\nabla u|^2)(1 - u)^{-2} \ \mbox {in} \ \Omega , \ u=0 \ \mbox {on} \ \partial \Omega ,\] where $\lambda , \delta >0$ and $\Omega \subset \mathbb {R}^n$ is a smooth and bounded domain. We show that when the fringing field exists (i.e. $\delta > 0$), given any $\mu > 0$, we have a uniform upper bound of classical solutions $u$ away from the rupture level 1 for all $\lambda \geq \mu$. Moreover, there exists $\overline \lambda _{\delta }^{*}>0$ such that there are at least two solutions when $\lambda \in (0, \overline \lambda _{\delta }^{*})$; a unique solution exists when $\lambda = \overline \lambda _{\delta }^{*}$; and there is no solution when $\lambda >\overline \lambda _{\delta }^{*}$. This represents a dramatic change of behavior with respect to the zero fringing field case (i.e., $\delta =0$) and confirms the simulations in a paper by Pelesko and Driscoll as well as a paper by Lindsay and Ward.References
- Haïm Brezis, Thierry Cazenave, Yvan Martel, and Arthur Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations 1 (1996), no. 1, 73–90. MR 1357955
- H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), no. 6, 601–614. MR 509489, DOI 10.1080/03605307708820041
- Pierpaolo Esposito and Nassif Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl. Anal. 15 (2008), no. 3, 341–353. MR 2500851, DOI 10.4310/MAA.2008.v15.n3.a6
- Pierpaolo Esposito, Nassif Ghoussoub, and Yujin Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math. 60 (2007), no. 12, 1731–1768. MR 2358647, DOI 10.1002/cpa.20189
- Flores G., Mercado G.A. and Pelesko J.A., Dynamics and touchdown in electrostatic MEMS, Proceedings of ASME DETC’03, 1-8, IEEE Computer Soc. (2003).
- Nassif Ghoussoub and Yujin Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math. Anal. 38 (2006/07), no. 5, 1423–1449. MR 2286013, DOI 10.1137/050647803
- Nassif Ghoussoub and Yujin Guo, Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS, Methods Appl. Anal. 15 (2008), no. 3, 361–376. MR 2500853, DOI 10.4310/MAA.2008.v15.n3.a8
- Yujin Guo, Zhenguo Pan, and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math. 66 (2005), no. 1, 309–338. MR 2179754, DOI 10.1137/040613391
- Zongming Guo and Juncheng Wei, Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity, Commun. Pure Appl. Anal. 7 (2008), no. 4, 765–786. MR 2393396, DOI 10.3934/cpaa.2008.7.765
- Zongming Guo and Juncheng Wei, Infinitely many turning points for an elliptic problem with a singular non-linearity, J. Lond. Math. Soc. (2) 78 (2008), no. 1, 21–35. MR 2427049, DOI 10.1112/jlms/jdm121
- A. E. Lindsay and M. J. Ward, Asymptotics of some nonlinear eigenvalue problems for a MEMS capacitor. I. Fold point asymptotics, Methods Appl. Anal. 15 (2008), no. 3, 297–325. MR 2500849, DOI 10.4310/MAA.2008.v15.n3.a4
- Julián López-Gómez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations 127 (1996), no. 1, 263–294. MR 1387266, DOI 10.1006/jdeq.1996.0070
- John A. Pelesko and David H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1955412
- John A. Pelesko and Tobin A. Driscoll, The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models, J. Engrg. Math. 53 (2005), no. 3-4, 239–252. MR 2230109, DOI 10.1007/s10665-005-9013-2
- Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI 10.1016/0022-1236(71)90030-9
- Renate Schaaf, Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry, Adv. Differential Equations 5 (2000), no. 10-12, 1201–1220. MR 1785673
Bibliographic Information
- Juncheng Wei
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
- MR Author ID: 339847
- ORCID: 0000-0001-5262-477X
- Email: wei@math.cuhk.edu.hk
- Dong Ye
- Affiliation: LMAM, UMR 7122, Université de Metz, 57045 Metz, France
- Email: dong.ye@univ-metz.fr
- Received by editor(s): August 13, 2009
- Published electronically: December 30, 2009
- Additional Notes: The research of the first author is supported by the General Research Fund from the Research Grant Council of Hong Kong
The second author is supported by the French ANR project ANR-08-BLAN-0335-01 - Communicated by: Matthew J. Gursky
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 1693-1699
- MSC (2010): Primary 35B45; Secondary 35J15
- DOI: https://doi.org/10.1090/S0002-9939-09-10226-5
- MathSciNet review: 2587454