Monday Analysis Seminar: Faraday cage principle for capacity and dichotomy of capacity density

Date
2016-5-14 14:45 - 2016-5-23 16:15
Place
Faculty of Science Buliding #3 Room 210
Speaker/Organizer
Hiroaki Aikawa (Hokkaido University)
 
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.