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.
Publié le : 2004-07-05
Classification:
Shape Optimization,
Domain Variations,
Shape Hessian,
Stability,
Energy Functionals,
Dirichlet Problem,
49Q10, 49K40, 49Q12, 35J25,
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
@article{hal-00141917,
author = {Henrot, Antoine and Pierre, Michel and Rihani, Mounir},
title = {Positivity of the shape Hessian and instability of some equilibrium shapes},
journal = {HAL},
volume = {2004},
number = {0},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00141917}
}
Henrot, Antoine; Pierre, Michel; Rihani, Mounir. Positivity of the shape Hessian and instability of some equilibrium shapes. HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00141917/