Inequalities for means of chords, with application to isoperimetric problems

Abstract

We consider a pair of isoperimetric problems arising in physics. The first concerns a Schrödinger operator in $L^2({\mathbb{R}}^2)$ 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 ${\mathbb{R}}^3$, 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 $p\le 2$, 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 $p>2$.

LaTeX2e, 11 pages

References in corpus (3)

• Geometrically induced spectrum in curved leaky wires
• An isoperimetric problem for leaky loops and related mean-chord inequalities
• An isoperimetric problem for point interactions

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• Spectral theory for Schrödinger operators with $\delta$-interactions supported on curves in ${\mathbb{R}}^3$
• Optimization of the lowest eigenvalue of a soft quantum ring
• On the critical exponent in an isoperimetric inequality for chords
• On the maximization of a class of functionals on convex regions, and the characterization of the farthest convex set
• An optimization problem for finite point interaction families
• Distributions of points on non-extensible closed curves in ${\mathbb{R}}^3$ realizing maximum energies