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Elastic energy of a convex body
Author(s) -
Bianchini Chiara,
Henrot Antoine,
Takahashi Takéo
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400256
Subject(s) - perimeter , mathematics , regular polygon , planar , plane (geometry) , energy (signal processing) , diagram , elastic energy , convex set , convex body , mathematical analysis , geometry , combinatorics , convex hull , convex optimization , physics , statistics , computer graphics (images) , quantum mechanics , computer science
In this paper a Blaschke‐Santaló diagram involving the area, the perimeter and the elastic energy of planar convex bodies is considered. More precisely we give a description of set E : = ( x , y ) ∈ R 2 , x = 4 π A ( Ω ) P ( Ω ) 2, y = E ( Ω ) P ( Ω ) 2 π 2, Ω convex , where A is the area, P is the perimeter and E is the elastic energy, that is a Willmore type energy in the plane. In order to do this, we investigate the following shape optimization problem:min Ω ∈ C{ E ( Ω ) + μ A ( Ω ) } , where C is the class of convex bodies with fixed perimeter and μ ⩾ 0 is a parameter. Existence, regularity and geometric properties of solutions to this minimum problem are shown.