Touchdown solutions in general MEMS models

Abstract

We study general problems modeling electrostatic microelectromechanical systems devices (Pλ )φ(r,−u′(r))=λ∫0rf(s)g(u(s))ds,r∈(0,1),00\lambda \gt 0 is a parameter. We obtain results on the existence and regularity of a touchdown solution to (Pλ{P}_{\lambda }) and find upper and lower bounds on the respective pull-in voltage. In the particular case, when φ(r,v)=rα∣v∣βv\varphi \left(r,v)={r}^{\alpha }{| v| }^{\beta }v, i.e., when the associated differential equation involves the operator r−γ(rα∣u′∣βu′)′{r}^{-\gamma }\left({r}^{\alpha }{| u^{\prime} | }^{\beta }u^{\prime} )^{\prime} , we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.

Keywords

• touchdown solution
• singular solution
• pull-in voltage
• subsolution and supersolution
• asymptotic behavior
• 34b15
• 35a24
• 35b09
• 35b40
• 35j75