DOI: 10.1515/anona-2023-0102 ISSN:
Touchdown solutions in general MEMS models
Rodrigo Clemente, João Marcos do Ó, Esteban da Silva, Evelina Shamarova Abstract
We study general problems modeling electrostatic microelectromechanical systems devices
(Pλ
)
φ
(
r
,
−
u
′
(
r
)
)
=
λ
∫
0
r
f
(
s
)
g
(
u
(
s
)
)
d
s
,
r
∈
(
0
,
1
)
,
0
<
u
(
r
)
<
1
,
r
∈
(
0
,
1
)
,
u
(
1
)
=
0
,
\left\{\begin{array}{ll}\varphi (r,-u^{\prime} \left(r))=\lambda \underset{0}{\overset{r}{\displaystyle \int }}\frac{f\left(s)}{g\left(u\left(s))}{\rm{d}}s,\hspace{1.0em}& r\in \left(0,1),\\ 0\lt u\left(r)\lt 1,\hspace{1.0em}& r\in \left(0,1),\\ u\left(1)=0,\hspace{1.0em}\end{array}\right.
where
φ
\varphi
,
g
g
, and
f
f
are some functions on
[
0
,
1
]
\left[0,1]
and
λ
>
0
\lambda \gt 0
is a parameter. We obtain results on the existence and regularity of a touchdown solution to (
P
λ
{P}_{\lambda }
) and find upper and lower bounds on the respective pull-in voltage. In the particular case, when
φ
(
r
,
v
)
=
r
α
∣
v
∣
β
v
\varphi \left(r,v)={r}^{\alpha }{| v| }^{\beta }v
, i.e., when the associated differential equation involves the operator
r
−
γ
(
r
α
∣
u
′
∣
β
u
′
)
′
{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.