Elastic energy of a convex body

Abstract

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 setwhere 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: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.

The elastica problem under area constraint

“…where r and R are the inradius and outer radius of Ω γ . We emphasize that in view of the classical isoperimetric inequality L ≥ √ 4πA our new inequality (1) is stronger than (3).…”

“…One of the first studies on these was done by Radon [23], When dealing with closed elasticae several different constraints can be considered. In the planar case, for instance, together with fixed length an additional area constraint has been considered among others by [1,3,28,29,30]. Another possibility is to restrict the set of curves to those which lie inside a given domain, see for instance [4].…”

“…In spite of all the literature on the subject, it has been noted in [3] that there exists a very simple question pertaining to elasticae that is apparently still unsolved.…”

“…However, a special case of Theorem 1.1 has recently been pointed out in [3] as being a consequence of a famous result of Gage. Proposition 1.2.…”

A new isoperimetric inequality for the elasticae

“…This ODE is issued from the optimality conditions on a free branch of a minimizer for our problem, see Theorem 2.6. We also refer the reader to reference [2] for related analysis. Clearly, C ≥ − 3 4 2 1 3 ≈ −0.944, otherwise the right hand side is negative.…”