We show that the capacity of a class of plane condensers is comparable
to the capacity of corresponding "dyadic condensers". As an application,
we show that for plane condensers in that class the capacity blows up as
the distance between the plates shrinks, but there can be no asymptotic
estimate of the blow-up.
Capacity of shrinking condensers in the plane
Let be an open region in the complex plane and let E and K be disjoint
subsets of , with F closed and K compact. The capacity of the condenser
(F, K) in is
Cap(F, K) = inf u 2
L2() : u 1 on K, u 0 on F .
The sets F and K are the plates of the condenser. The infimum is taken over
functions u which are C1
in and continuous on F K. The capacity of
a condenser, a notion arising in electrostatics, became part of mainstream Po-