Positivity of the shape Hessian and instability of some equilibrium shapes

Antoine Henrot (1) , Michel Pierre (2) , Mounir Rihani (3)

1 CORIDA - Robust control of infinite dimensional systems and applications
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 Département de Mathématiques [Casablanca]

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.

Mots clés: Shape Optimization; Domain Variations; Shape Hessian; Stability; Energy Functionals; Dirichlet Problem

Domaines: Equations aux dérivées partielles [math.AP]

Antoine Henrot, Michel Pierre, Mounir Rihani. Positivity of the shape Hessian and instability of some equilibrium shapes. Mediterranean Journal of Mathematics, 2004, 1 (2), pp.195-214. ⟨10.1007/s00009-004-0011-9⟩. ⟨hal-00141917⟩