arXiv subjects

Francesco Salerno

Publications and source records attributed to Francesco Salerno.

6 recordsLinked to original sources

arXiv2026

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.

math.AP↗

arXiv2026

Remarks on Brunn-Minkowski-type inequalities related to the Ornstein-Uhlenbeck operator

We investigate Brunn-Minkowski-type inequalities for the torsional rigidity $T_γ$ and the first eigenvalue $λ_γ$ associated with the Ornstein-Uhlenbeck operator. Counterexamples are provided showing that neither concavity nor convexity properties hold for $T_γ$ on general bounded convex sets. We also demonstrate that log-concavity and log-convexity properties fail in this setting. In the case of centrally symmetric sets, we answer a question raised by Cordero-Erausquin and Eskenazis by showing that $T_γ^{1/(n+2)}$ is neither convex nor concave. On the positive side, we prove that $T_γ^{1/3}$ is convex with respect to Minkowski addition when restricted to Euclidean balls centered at the origin. For $λ_γ$, we answer negatively a question posed by Colesanti, Francini, Livshyts, and Salani by showing that the inequality $λ_γ(Ω_t)^{-1/2} \geq (1-t)λ_γ(Ω_0)^{-1/2} + tλ_γ(Ω_1)^{-1/2}$ does not hold, even for centrally symmetric sets.

math.AP↗

arXiv2026

Sharp lower bound for the Monge-Ampère torsion on convex sets

The \emph{Monge-Ampère} torsion deficit of an open, bounded convex set $Ω\subset\R^n$ of class $C^2$ is the normalized gap between the value of the torsion functional evaluated on $Ω$ and its value on the ball with the same $(n-1)$-quermassintegral as $Ω$. Using the technique of the \emph{shape derivative}, we prove that the ratio between this deficit and to a geometric deficit arising from the \emph{Alexandrov-Fenchel inequality}, for any given family of open, bounded convex sets of $\R^n$ ($n\geq2$) of class $C^2$, smoothly converging to a ball, is bounded from below by a dimensional constant. We also show that this ratio is always bounded from above by a constant.

math.AP↗

arXiv2026

Talenti comparison results for solutions to $p$-Laplace equation on multiply connected domains

In the last years comparison results of Talenti type for Elliptic Problems have been widely investigated. In this paper we obtain a comparison result for the $p$-Laplace operator in multiply connected domains with Robin boundary condition on the exterior boundary and non-homogeneous Dirichlet boundary conditions on the interior one, generalizing the results obtained in \cite{ANT, AGM} to this type of domains. This will be a generalization to Robin boundary condition of the results obtained in \cite{B, B2}, with an improvement of the $L^2$ comparison in the case $p=2$. As a consequence, we obtain a Bossel-Daners and Saint-Venant type inequalities for multiply connected domains.

math.AP↗

arXiv2025

Some shape functionals for the $k$-Hessian equation

For a non-empty, bounded, open, and convex set of class $C^2$, we consider the Torsional Rigidity associated to the $k$-Hessian operator. We first prove Pólya type lower bound for the $k$-Torsional Rigidity in any dimension; then, in order to investigate optimal sets in the Pólya type inequality, we provide two quantitative estimates.

math.AP↗

arXiv2025

A quantitative result for the $k$-Hessian equation

In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a Pólya-Szeg\H o type inequality holds. We refine this symmetrization to obtain a quantitative improvement of the Pólya-Szeg\H o inequality for the $k$-Hessian integral, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso \cite{tso} for solutions to the $k$-Hessian equation. As an application of the first result, we prove a quantitative version of the Faber-Krahn and Saint-Venant inequalities for these equations.

math.AP↗