Local stability for a class of Saint-Venant type inequalities

We establish a local stability result for a class of Saint-Venant type inequalities.

Given the solution $u$ of the Dirichlet torsion problem in a domain $\Omega$, we consider shape functionals $\mathcal{J}(\Omega)$ involving the integral of $j(u)$, where $j$ is convex and satisfies suitable structural assumptions.

By Talenti's comparison principle, balls maximize $\mathcal{J}$ among sets of prescribed measure.

We prove that this extremal property is stable in the class of nearly spherical sets: the deficit from the optimal value controls the square of the $H^{1/2}$-norm of the boundary perturbation.

The argument relies on shape derivative techniques, including the computation of the second variation and the introduction of an adjoint state.

As applications, the result covers several relevant examples, including the torsional rigidity, $L^p$-norms of the torsion function for $p\ge 2$, and Moser-Trudinger functional in dimension two.