On MEMS equation with fringing field
Juncheng Wei, Dong Ye · Proceedings of the American Mathematical Society · 2009
We consider the MEMS equation with fringing field \[ − Δ u = λ ( 1 + δ | ∇ u | 2 ) ( 1 − u ) − 2 in Ω , u = 0 on ∂ Ω , -\Delta u = \lambda (1 + \delta | abla u|^2)(1 - u)^{-2} \ \mbox {in} \ \Omega , \ u=0 \ \mbox {on} \ \partial \Omega , \] where λ , δ > 0 \lambda , \delta >0 and Ω ⊂ R n \Omega \subset \mathbb {R}^n is a smooth and bounded domain. We show that when the fringing field exists (i.e. δ > 0 \delta > 0 ), given any μ > 0 \mu > 0 , we have a uniform upper bound of classical solutions u u away from the rupture level 1 for all λ ≥ μ \lambda \geq \mu