A new isoperimetric inequality for the elasticae
A new isoperimetric inequality for the elasticae
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一种新的弹性等周不等式
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
A. Henrot
中科院分区：
文献类型：
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作者：
D. Bucur;A. Henrot
对于一条光滑曲线$gamma$，我们定义它的弹性能为$E（gamma）= frac 12 int_{gamma} k^2（s）ds$其中$k（s）$是曲率。本文的主要目的是证明在$mathbb{R}^2 $中所有光滑的、单连通的、有界的、具有规定面积的开集中，圆盘具有弹性能最小的边界.换句话说，对于任何有界单连通域$Omega$，下列等周不等式成立：$E^2（partial Omega）A（Omega）geq pi ^3$。分析依赖于最小化的弹性能量的液滴包围一个规定的区域，我们也给出了一个解析的答案。
For a smooth curve $gamma$, we define its elastic energy as $E(gamma)= frac 12 int_{gamma} k^2 (s) ds$ where $k(s)$ is the curvature. The main purpose of the paper is to prove that among all smooth, simply connected, bounded open sets of prescribed area in $mathbb{R}^2$, the disc has the boundary with the least elastic energy. In other words, for any bounded simply connected domain $Omega$, the following isoperimetric inequality holds: $E^2(partial Omega)A(Omega)geq pi ^3$. The analysis relies on the minimization of the elastic energy of drops enclosing a prescribed area, for which we give as well an analytic answer.