 05300 Pavel Exner, Evans M. Harrell, and Michael Loss
 Inequalities for means of chords, with application to isoperimetric
problems
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Aug 31, 05

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Abstract. We consider a pair of isoperimetric problems
arising in physics. The first concerns a Schr\"odinger operator in
$L^2(\R^2)$ with an attractive interaction supported on a closed
curve $\Gamma$, formally given by $\Delta\alpha
\delta(x\Gamma)$; we ask which curve of a given length maximizes
the ground state energy. In the second problem we have a
loopshaped thread $\Gamma$ in $\R^3$, homogeneously charged but
not conducting, and we ask about the (renormalized)
potentialenergy minimizer. Both problems reduce to purely
geometric questions about inequalities for mean values of chords
of $\Gamma$. We prove an isoperimetric theorem for $p$means of
chords of curves when $p \leq 2$, which implies in particular that
the global extrema for the physical problems are always attained
when $\Gamma$ is a circle. The article finishes with a discussion
of the $p$means of chords when $p > 2$.
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