Touchdown solutions in general MEMS models

2022/10/30 by Rodrigo Clemente, João Marcos Ó, Clemente, Rodrigo +5
Computer Science · Physics and Astronomy · #34B15 #35A24 #35B09 #35B40 #35J75 #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.2210.16911

openalex publication_date 2022/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study general equations modeling electrostatic MEMS devices \begincases φ(r,- u'(r))=λ∫₀^r(f(s))/(g(u(s))) ds, amp; r∈(0,1),
0 lt; u(r) lt; 1, amp; r∈(0,1),
u(1) = 0, \endcases where φ, g, f are some functions on [0,1] and λ>0 is a parameter. We obtain results on the existence and regularity of a touchdown solution to \eqrefP and find upper and lower bounds on the respective pull-in voltage. In the particular case, when φ(r,v) = r^α|v|^βv, i.e., when the associated differential equation involves the operator r^-γ(r^α|u'|^βu')', we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.

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