arXiv subjects

Esteban da Silva

Publications and source records attributed to Esteban da Silva.

2 recordsLinked to original sources

arXiv2022

Touchdown solutions in general MEMS models

We study general equations modeling electrostatic MEMS devices \begin{equation} \begin{cases} \label{P} φ\big(r,- u'(r)\big)=λ\int_0^r\frac{f(s)}{g(u(s))}\,\mathrm{d}s, & r\in(0,1), \\ 0 < u(r) < 1, & r\in(0,1), \\ u(1) = 0, \tag{$P_λ$} \end{cases} \end{equation} 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 \eqref{P} 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.

math.AP↗

arXiv2022

Singular solutions to $k$-Hessian equations with fast-growing nonlinearities

We study a class of elliptic problems, involving a $k$-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An essential ingredient of this study is analyzing the number of intersection points between the singular and regular solutions for rescaled problems. In the particular case of the exponential nonlinearity, we obtain the convergence of regular solutions to the singular and analyze the intersection number depending on the parameter $k$ and the dimension $d$.

math.AP↗