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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On MEMS equation with fringing field


Authors: Juncheng Wei and Dong Ye
Journal: Proc. Amer. Math. Soc. 138 (2010), 1693-1699
MSC (2010): Primary 35B45; Secondary 35J15
Published electronically: December 30, 2009
MathSciNet review: 2587454
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Abstract: We consider the MEMS equation with fringing field

$\displaystyle -\Delta u = \lambda(1 + \delta\vert\nabla u\vert^2)(1 - u)^{-2}$   in$\displaystyle \Omega, u=0$   on$\displaystyle \partial \Omega,$

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.


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Additional Information

Juncheng Wei
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email: wei@math.cuhk.edu.hk

Dong Ye
Affiliation: LMAM, UMR 7122, Université de Metz, 57045 Metz, France
Email: dong.ye@univ-metz.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10226-5
PII: S 0002-9939(09)10226-5
Keywords: MEMS, rupture, fringing field, bifurcation
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.