Shape optimization consists in the study of the minimization
\[\inf\{J(\Omega),\ \Omega \in S\}\]
where $S$ is a class of subsets of $\R^d$ and $J:S\to\R\cup\{\infty\}$ is a functional. There are many problems where one expects the Euclidean ball $B\subset\R^d$ to be the (unique, up to translation) minimizer of $J$ under volume constraint, as in the celebrated \textit{isoperimetric} problem which asserts that the ball is in fact the unique minimizer of the perimeter among sets of fixed volume. Under this assumption of minimality of the ball, one can wonder further about its stability: if $J(\Omega)\approx J(B)$, can we say that $\Omega$ is close to being a ball in some appropriate sense? In this talk we focus on a regularity-based approach for obtaining stability inequalities that has proven to be very efficient, which can be roughly described as: (i) proving stability for smooth perturbations of the ball and (ii) obtaining stability in general.
After setting up the context and illustrating the general strategy with some examples, we present some stability results that we have obtained for isoperimetric-type problems and Faber-Krahn-type problems for different classes $\Sad$, and put them in perspective by comparing the methods and difficulties.
This talk is based on several joint works with D. Bucur, J. Lamboley and M. Nahon.