Revisiting the biharmonic equation modelling electrostatic actuation in lower dimensions

Abstract

Let B ⊂ R N B \subset \mathbb {R}^N be the unit ball. We study the structure of solutions to the semilinear biharmonic problem \[ { Δ 2 u = λ ( 1 − u ) − p a m p ; in B , 0 > u > 1 a m p ; in B , u = ∂ ν = 0 ( resp.~ u = Δ u = 0 ) a m p ; on ∂ B , \begin {cases} \Delta ^2 u=\lambda (1-u)^{-p} & \text {in $B$},\\ 0>u>1 & \text {in $B$},\\ u=\partial _\nu =0\; (\text {resp.~$u = \Delta u = 0$}) & \text {on $\partial B$}, \end {cases} \] where p , λ > 0 p, \lambda >0 , which arises in the study of the deflection of charged plates in electrostatic actuators. We study in particular the structure of solutions for N = 2 N=2 or 3 3 and show the existence of mountain-pass solutions under suitable conditions on p p . Our results contribute to completing the picture of solutions in previous works. Moreover, we also analyze the asymptotic behavior of the constructed mountain-pass solutions as λ → 0 \lambda \to 0 .

A fourth-order model for MEMS with clamped boundary conditions

“…More precisely, for each λ ∈ (0, λ * ) there are at least two values 0 < s 1 < s * < s 2 with Λ(s j ) = λ for j = 1, 2 and U (s 2 ) U (s 1 ) in B 1 with U (s 2 ) = U (s 1 ). In [15], a similar result is shown by a different technique when T = 0, the second solution being constructed by a Mountain Pass Principle.…”

“…was enjoying the kind hospitality of the Institut für Angewandte Mathematik of the Leibniz Universität Hannover. We also thank Dong Ye for pointing out [15] to us.…”

On a three-dimensional free boundary problem modeling electrostatic MEMS

“…We note that this small gap approximation is a singular equation but no longer a free boundary problem. We refer to [5,7,13,17] and the references therein for more information on this case.…”

Some singular equations modeling MEMS

“…• Control and inverse problems for MEMS using the vanishing aspect ratio model (1.9) are investigated in [16,19]. • Finally, the extension to (1 + u) −p for p > 1 of the above mentioned results has been studied and shown to depend strongly upon both p and d, see [17,20,22,26,62,70,117,125,138,140] and the references therein.…”