  We are interested in the question of stability in the field of shape
optimization, with focus on the strategy using second order shape derivative.
More precisely, we identify structural hypotheses on the hessian of the
considered shape function, so that critical stable domains (i.e. such that the
first order derivative vanishes and the second order one is positive) are local
minima for smooth perturbations; as we are in an infinite dimensional
framework, and that in most applications there is a norm-discrepancy
phenomenon, this type of result require a lot of work. We show that these
hypotheses are satisfied by classical functionals, involving the perimeter, the
Dirichlet energy or the first Laplace-Dirichlet eigenvalue. We also explain how
we can easily deal with constraints and/or invariance of the functionals. As an
application, we retrieve or improve previous results from the existing
literature, and provide new local stability results. We finally test the
sharpness of our results by showing that the local minimality is in general not
valid for non-smooth perturbations.
