Estimates on Pull-in Distances in MEMS Models and other Nonlinear Eigenvalue Problems

Details Estimates on Pull-in Distances in MEMS Models and other Nonlinear Eigenvalue Problems Estimates on Pull-in Distances in MEMS Models and other Nonlinear Eigenvalue Problems

2009-03-25

arxiv.org/abs/0903.4464v1

Mathematics

Analysis of PDEs

17 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.birs.ca/~nassif/

Scientific paper

Motivated by certain mathematical models for Micro-Electro-Mechanical Systems (MEMS), we give upper and lower $L^\infty$ estimates for the minimal solutions of nonlinear eigenvalue problems of the form $-\Delta u = \lambda f(x) F(u)$ on a smooth bounded domain $ \Omega$ in $\IR^N$. We are mainly interested in the {\it pull-in distance}, that is the $L^\infty-$norm of the extremal solution $u^*$ and how it depends on the geometry of the domain, the dimension of the space, and the so-called {\it permittivity profile} $f$. In particular, our results provide mathematical proofs for various observed phenomena, as well as rigorous derivations for several estimates obtained numerically by Pelesko \cite{P}, Guo-Pan-Ward \cite{GPW} and others in the case of the MEMS non-linearity $F(u)=\frac{1}{(1-u)^2}$ and for power-law permittivity profiles $f(x)=|x|^\alpha$.

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