弦平均的不等式及其在等周问题中的应用
Inequalities for means of chords, with application to isoperimetric problems
摘要：
我们考虑物理学中的一对等周问题。第一个问题涉及$L^2(\mathbb{R}^2)$中的薛定谔算子，其中吸引相互作用支持在一条闭合曲线$Γ$上，形式上由$-Δ-αδ(x-Γ)$给出;我们问给定长度的哪条曲线使基态能量最大化。在第二个问题中，我们在$\mathbb{R}^3$中有一个环形的线$Γ$，均匀带电但不导电，我们询问（重整化）势能最小化。这两个问题减少到纯粹的几何问题不等式的平均值的和弦$Γ$。本文证明了当$p \leq 2$为圆时，$p$ -曲线弦的平均值的等周定理，特别是当$Γ$为圆时，这意味着物理问题的全局极值总是能得到.文章最后讨论了$p$ --$p > 2$时的和弦手段。
英文摘要：
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 $Γ$, formally given by $-Δ-αδ(x-Γ)$; we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread $Γ$ 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 $Γ$. 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 $Γ$ is a circle. The article finishes with a discussion of the $p$--means of chords when $p > 2$.
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