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 set $$\mathcal{E}:=\left\{(x,y)\in \R^2, x=\frac{4πA(Ω)}{P(Ω)^2},y=\frac{E(Ω)P(Ω)}{2π^2},\,Ω\mbox{convex} \right\},$$ 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: $$\underset{\Omega \in \mathcal{C}}{\min }\{E(\Omega )+\mu A(\Omega )\},$$ where $\mathcal{C}$ is the class of convex bodies with fixed perimeter and $\mu \ge 0$ is a parameter. Existence, regularity and geometric properties of solutions to this minimum problem are shown.

Equipe Equations aux derivees partielles