On the critical exponent in an isoperimetric inequality for chords

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 ≤ p ≤ 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 ( If Γ(s) describes a planar curve of parametrized by arclength s and L is its total lengthdescribes the L p -mean of the Euclidean length of the chords connecting points separated by arclength u. A reasonable geometric question is to determine the shape that maximizes this quantity for any given value of p. Some phys-1

A spectral isoperimetric inequality for cones

“…Finding the curve which minimizes this functional is a particular problem in the theory of knots. The literature on knots and their energies is quite extensive; see [1,18,19,31], the monograph [32] and the references therein. For our problem it is proven that circles are unique minimizers under reasonable assumptions on f .…”

On geometric perturbations of critical Schrödinger operators with a surface interaction

“…It claims that if the coupling is constant along the loop the ground state eigenvalue is maximized in the class of all loops of a fixed length by a circle. The result has interested connections to both the classical electrodynamics [1,3] and a class of isoperimetric inequalities of a purely geometric nature [3,4,5].…”

Resonance free regions and non-Hermitian spectral optimization for Schrödinger point interactions

“…, N (see [1,2,3,6] and Section 2 for basic definitions). The question of optimization of the principal eigenvalue of self-adjoint Schrödinger Hamiltonians with δ-type or point interactions attracted recently considerable attention especially in a quantum mechanics context [14,17,16,18,36]. This line of research was motivated by the isoperimetric problem posed in [14].…”