漏环的一个等周问题及相关的平均弦不等式
An isoperimetric problem for leaky loops and related mean-chord inequalities
摘要：
我们考虑一类具有吸引相互作用的$L^2(\R^2)$中的哈密顿算子，它由固定长度的$L$的分段$C^2$光滑环$Γ$支持，形式上由$-Δ-αδ(x-Γ)$和$α>0$给出.结果表明，该算子的基态是局部最大化的圆形$Γ$。我们还推测，这一属性持有全球性的问题是一个有趣的家庭有关的几何不等式的平均值的弦$Γ$。
英文摘要：
We consider a class of Hamiltonians in $L^2(\R^2)$ with attractive interaction supported by piecewise $C^2$ smooth loops $Γ$ of a fixed length $L$, formally given by $-Δ-αδ(x-Γ)$ with $α>0$. It is shown that the ground state of this operator is locally maximized by a circular $Γ$. We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of $Γ$.
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