3 problems

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

Upper critical dimension conjecture. The upper critical dimension for the prudent walk is precisely five.