05-300 Pavel Exner, Evans M. Harrell, and Michael Loss
Inequalities for means of chords, with application to isoperimetric problems (29K, LaTeX) 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 loop-shaped thread $\Gamma$ in $\R^3$, homogeneously charged but not conducting, and we ask about the (renormalized) potential-energy 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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