11 problems

Let $(u_\rho,\psi_{u_\rho})$ denote the stationary solutions constructed in Theorem 3, where the construction uses a sequence of parameters tending to infinity, and let $\lambda$ d…

Let $(S)_\lambda$ be the stationary electrostatic MEMS problem on a domain $\Omega\subset\mathbb{R}^N$, with critical voltage $\lambda^*$ and positive permittivity profile $f$. Pos…

Let $\Omega$ be the domain, let $f$ be the permittivity profile on $\Omega$, let $\lambda^*$ be the pull-in voltage, and let $\phi_\Omega$ be the first eigenfunction of the relevan…

Let $(S)_{\lambda}$ denote the stationary electrostatic MEMS problem on a domain $\Omega\subset\mathbb{R}^N$, with critical voltage $\lambda^*$ and permittivity profile $f$. Nonexi…

Assume $(B,T)$, $(u_{0},u_{1})$, $d$, and $\Omega$ are given, and let $u$ be the unique maximal solution to the hyperbolic problem. Critical-threshold conjecture. There exist criti…

Let $\Omega\subset\mathbb{R}^N$, with $1\leq N\leq 3$, be a bounded domain with smooth boundary, and let $u_0\in H^2\cap H_0^1(\Omega)$. Let $\lambda^{*}$ be the critical parameter…

Let $\Omega\subset\mathbb{R}^N$, with $1\leq N\leq 3$, be a bounded domain with smooth boundary, and let $u_0\in H^2\cap H_0^1(\Omega)$. Consider equation (1.1), its solution…

Consider the free-boundary model with $\gamma=0$ and $\beta>0$, for which the paper studies stationary solutions and their dependence on the voltage parameter $\lambda$. The vanish…

Consider the typical MEMS problem with spatially dependent coefficient $|x|^m$, whose solution $u$ has quenching set $\mathcal{Q}$; a point is a quenching point if it is approached…

Critical-threshold conjecture. Above a certain threshold value of $clambda$, solutions cease to exist globally in time and there are no steady states, whereas below this critical v…

Let $T$ denote the finite quenching time of the solution of the MEMS equation with fringing parameter $\delta>0$ and voltage parameter $\lambda>\lambda_\delta^*$, where…