Pavel Exner, Martin Fraas, Evans M. Harrell II
On the critical exponent in an isoperimetric inequality for chords
(209K, pdf)
ABSTRACT. The problem of maximizing the $L^p$ norms of
chords connecting points on a closed curve separated by arclength
$u$ arises in electrostatic and quantum--mechanical problems. It
is known that among all closed curves of fixed length, the unique
maximizing shape is the circle for $1 \le p \le 2$, but this is
not the case for sufficiently large values of $p$. Here we
determine the critical value $p_c(u)$ of $p$ above which the
circle is not a local maximizer finding, in particular, that
$p_c(\frac12 L)=\frac52$. This corrects a claim made in
\cite{EHL}.