On MEMS equation with fringing field

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.