Positivity of the Shape Hessian and Instability of some Equilibrium Shapes

Research paper by Antoine Henrot, Michel Pierre, Mounir Rihani

Indexed on: 01 Apr '04Published on: 01 Apr '04Published in: Mediterranean Journal of Mathematics


We study the positivity of the second shape derivative around an equilibrium for a 2-dimensional functional involving the perimeter of the shape and its the Dirichlet energy under volume constraint. We prove that, generally, convex equilibria lead to strictly positive second derivatives. We also exhibit some examples where strict positivity of the second order derivative holds at an equilibrium while existence of a minimum does not.