Elasticae and inradius

详细信息    查看全文

• 作者：Antoine Henrot ; Othmane Mounjid
• 关键词：Elastic energy ; Convex geometry ; Inradius ; Shape optimization
• 刊名：Archiv der Mathematik
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：108
• 期：2
• 页码：181-196
• 全文大小：650KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1420-8938
• 卷排序：108

文摘

The elastic energy of a convex curve bounding a planar convex body \(\Omega \) is defined by \(E(\Omega )=\frac{1}{2}\,\int _{\partial \Omega } k^2(s)\,ds\) where k(s) is the curvature of the boundary. In this paper we are interested in the minimization problem of \(E(\Omega )\) with a constraint on the inradius of \(\Omega \). In contrast to all the other minimization problems involving this elastic energy (with a perimeter, area, diameter, or circumradius constraints) for which the solution is always the disk, we prove here that the solution of this minimization problem is not the disk and we completely characterize it in terms of elementary functions.