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Simone Cito

Publications and source records attributed to Simone Cito.

13 recordsLinked to original sources
arXiv2026

On the approximation of finite perimeter sets

We prove that if $\Omega\subseteq\mathbb{R}^N$ is a set with finite perimeter with $\mathscr{H}^{N-1}(\partial \Omega\setminus\partial^* \Omega)=0$, then any set of finite perimeter $E\subseteq\mathbb{R}^N$ can be approximated by a polyhedral or smooth bounded set $F$ in such a way that both the total perimeter of $E$ and the perimeter of $E$ inside $\Omega$ are approximated by those of $F$, and the boundary of $F$ has negligible intersection with the boundary of $\Omega$. In addition, we address the approximation for perimeter and volume with densities, and we present counterexamples illustrating the sharpness of our assumptions. Our constructions rely on a technical result that replaces $E$ with a set $F$ which agrees with $E$ and has the same boundary inside $\Omega$, while sharing no common boundary with $\Omega$, and does so without substantially altering the perimeter or the volume of the original set.

math.FA↗
arXiv2025

Relations between principal eigenvalue and torsional rigidity with Robin boundary conditions

We consider the torsional rigidity and the principal eigenvalue related to the Laplace operator with Dirichlet and Robin boundary conditions. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in the class of Lipschitz domains. The threshold exponent for the Robin case is explicitly recovered and shown to be strictly smaller than in the Dirichlet one.

math.AP↗
arXiv2025

A quantitative Talenti-type comparison result with Robin boundary conditions

The purpose of this paper is to establish a quantitative version of the Talenti comparison principle for solutions to the Poisson equation with Robin boundary conditions. This quantitative enhancement is proved in terms of the asymmetry of domain. The key role is played by a careful analysis of the propagation of asymmetry for the level sets of the solutions of a PDE. As a byproduct, we obtain an alternative proof of the quantitative Saint-Venant inequality for the Robin torsion and, in the planar case, of the quantitative Faber-Krahn inequality for the first Robin eigenvalue. In addition, we complete the framework of the rigidity result of the Talenti inequalities with Robin boundary conditions.

math.AP↗
arXiv2025

On the optimization of the Robin eigenvalues in some classes of polygons

Given the eigenvalue problem for the Laplacian with Robin boundary conditions, (with $\beta\in\R\setminus\{0\}$ the Robin parameter), we consider shape minimization problems if $\beta>0$ and shape maximization problems if $\beta<0$. Both problems are settled in a suitable class of generalized polygons with an upper bound on the number of sides, under either perimeter or volume constraint.

math.AP↗
arXiv2025

Optimality and stability of the radial shapes for the Sobolev trace constant

In this work we establish the optimality and the stability of the ball for the Sobolev trace operator $W^{1,p}(\Omega)\hookrightarrow L^q(\partial\Omega)$ among convex sets of prescribed perimeter for any $1< p <+\infty$ and $1\le q\le p$. More precisely, we prove that the trace constant $\sigma_{p,q}$ is maximal for the ball and the deficit is estimated from below by the Hausdorff asymmetry. With similar arguments, we prove the optimality and the stability of the spherical shell for the Sobolev exterior trace operator $W^{1,p}(\Omega_0\setminus\overline{\Theta})\hookrightarrow L^q(\partial\Omega_0)$ among open sets obtained removing from a convex set $\Omega_0$ a suitably smooth open hole $\Theta\subset\subset\Omega_0$, with $\Omega_0\setminus\overline{\Theta}$ satisfying a volume and an outer perimeter constraint.

math.AP↗
arXiv2023

Minimization of the $k$ -th eigenvalue of the Robin-Laplacian with perimeter constraint

In this paper we address the problem of the minimization of the $k$-th Robin eigenvalue $\lambda_{k,\beta}$ with parameter $\beta>0$ among bounded open Lipschitz sets with prescribed perimeter. The perimeter constraint allows us to naturally generalize the problem to a setting involving more general admissible geometries made up of sets of finite perimeter with inner cracks, along with a suitable generalization of the Robin-Laplacian operator with properties which look very similar to those of the classical setting. Within this extended framework we establish existence of minimizers, and prove that the associated eigenvalue coincides with the infimum of those achieved by regular domains.

math.AP↗
arXiv2023

Stability of the Gaussian Faber-Krahn inequality

We prove a quantitative version of the Gaussian Faber-Krahn type inequality proved by Betta, Chiacchio and Ferone for the first Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator, estimating the deficit in terms of the Gaussian Fraenkel asymmetry. As expected, the multiplicative constant only depends on the prescribed Gaussian measure.

math.AP↗
arXiv2023

Minimization of the buckling load of a clamped plate with perimeter constraint

We look for minimizers of the buckling load problem with perimeter constraint in any dimension. In dimension 2, we show that the minimizing plates are convex; in higher dimension, by passing through a weaker formulation of the problem, we show that any optimal set is open and connected. For higher eigenvalues, we prove that minimizers exist among convex sets with prescribed perimeter.

math.AP↗
arXiv2023

A stability result for the first Robin-Neumann eigenvalue: A double perturbation approach

Let $\Omega=\Omega_0\setminus \overline{\Theta}\subset \mathbb{R}^n$, $n\geq 2$, where $\Omega_0$ and $\Theta$ are two open, bounded and convex sets such that $\overline{\Theta}\subset \Omega_0$ and let $\beta<0$ be a given parameter. We consider the eigenvalue problem for the Laplace operator associated to $\Omega$, with Robin boundary condition on $\partial \Omega_0$ and Neumann boundary condition on $\partial \Theta$. In [Paoli-Piscitelli-Trani, ESAIM-COCV '20] it is proved that the spherical shell is the only maximizer for the first Robin-Neumann eigenvalue in the class of domains $\Omega$ with fixed outer perimeter and volume. We establish a quantitative version of the afore-mentioned isoperimetric inequality; the main novelty consists in the introduction of a new type of hybrid asymmetry, that turns out to be the suitable one to treat the different conditions on the outer and internal boundary. Up to our knowledge, in this context, this is the first stability result in which both the outer and the inner boundary are perturbed.

math.AP↗
arXiv2022

Local Regularity of very weak $s$-harmonic functions via fractional difference quotients

The aim of this paper is to give a new proof that any very weak $s$-harmonic function $u$ in the unit ball $B$ is smooth. As a first step, we improve the local summability properties of $u$. Then, we exploit a suitable version of the difference quotient method tailored to get rid of the singularity of the integral kernel and gain Sobolev regularity and local linear estimates of the $H^{s}_{\rm loc}$ norm of $u$. Finally, by applying more standard methods, such as elliptic regularity and Schauder estimates, we reach real analyticity of $u$. Up to the authors' knowledge, the difference quotient techniques are new.

math.AP↗
arXiv2021

Asymptotics of the $s$-fractional Gaussian perimeter as $s\to 0^+$

We study the asymptotic behaviour of the renormalised $s$-fractional Gaussian perimeter of a set $E$ inside a domain $\Omega$ as $s\to 0^+$. Contrary to the Euclidean case, as the Gaussian measure is finite, the shape of the set at infinity does not matter, but, surprisingly, the limit set function is never additive.

math.AP↗
arXiv2021

Gamma-convergence of fractional Gaussian perimeter

We prove the $\Gamma$-convergence of the renormalised fractional Gaussian $s$-perimeter to the Gaussian perimeter as $s\to 1^-$. Our definition of fractional perimeter comes from that of the fractional powers of Ornstein-Uhlenbeck operator given via Bochner subordination formula. As a typical feature of the Gaussian setting, the constant appearing in front of the $\Gamma$-limit does not depend on the dimension.

math.AP↗