## 月曜解析セミナー： Faraday cage principle for capacity and dichotomy of capacity density

2016年 　 5月 23日 14時 45分 ～ 2016年 　 5月 23日 16時 15分

In analysis, we often encounter situations where a priori weak estimates yield a stronger estimate. In 1975, Choquet found such a phenomenon for Newtonian capacity $C_2$ in $\mathbb{R}^3$. Namely, if $E, F\subset\mathbb{R}^3$ satisfy
$C_2(E\cap Q)\ge \kappa C_2(F\cap Q)\quad\text{for all cubes Q}$
with some constant $0<\kappa<1$, then, in fact,
$C_2(E\cap \Omega)\ge C_2(F\cap \Omega)\quad\text{for all bounded open sets \Omega.}$
For its reminiscence to a physical experiment, he named this result \emph{des cages de Faraday grillag\'ees} or \emph{the grounded Faraday cage}. In this talk, we show the same phenomenon for general convolution capacity and apply it to a 0-1 law for the global capacity density.

See http://www.math.sci.hokudai.ac.jp/~aik/ma/MonAnae.pdf.