Estimates on Pull-In Distances in Microelectromechanical Systems Models and Other Nonlinear Eigenvalue Problems

Source abstract

Motivated by certain mathematical models for microelectromechanical 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 $\mathbb{R}^N$. We are mainly interested in the 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 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 [SIAM J. Appl. Math., 62 (2002), pp. 888–908], Guo, Pan, and Ward [SIAM J. Appl. Math., 66 (2005), pp. 309–338], and others in the case of the MEMS nonlinearity $F(u)=\frac{1}{(1-u)^2}$ and for power-law permittivity profiles $f(x)=|x|^\alpha$.