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Rossano Sannipoli

Publications and source records attributed to Rossano Sannipoli.

16 recordsLinked to original sources
arXiv2026

The Makai inequality in higher dimensions: qualitative and quantitative aspects

In this paper, given a convex, bounded, open set $Ω\subset \mathbb{R}^n$ we prove a sharp inequality involving the Laplacian torsional rigidity and both the perimeter and the measure of the domain. Our result generalizes to arbitrary dimensions the inequality established by Makai in the plane which, as conjectured in arXiv:2007.02549. Furthermore, we establish quantitative estimates that provide key insights into the geometric structure and the thickness of the underlying optimizing sequences.

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arXiv2026

Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition

In this paper, we deal with functionals involving the torsion and the first eigenvalue of the Laplacian with Robin boundary conditions (to which we refer as Robin Torsion and Robin Eigenvalue), with other geometrical quantities, in the class of convex sets. Firstly, we prove an upper bound for the Robin Torsion in terms of the $L^1$ and $L^2$ norms of the distance function from the boundary, which allows us to prove a generalization of the Makai inequality involving the Robin Torsion, the Lebeasgue measure, and the inradius of a convex set. Subsequently, we prove quantitative estimates for the Robin Makai functional and for the Robin Pólya functionals, which link the Lebesgue measure and the perimeter with the Robin Torsion and the Robin Eigenvalue respectively. In particular, we prove that the optimal values of all these shape functionals are achieved by slab domains.

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arXiv2026

Sharp bounds and geometric properties of the first non trivial Steklov Neumann Eigenvalue

In this article, we study the mixed Steklov--Neumann eigenvalue problem on doubly connected domains. First, we show that among all doubly connected domains in $\mathbb{R}^n$ of the form $B_{R_2}\setminus \overline{B_{R_1}}$, where $B_{R_1}$ and $B_{R_2}$ are open balls of fixed radii satisfying $\overline{B_{R_1}} \subset B_{R_2}$, the first non-zero Steklov--Neumann eigenvalue attains its maximal value when the balls are concentric. Next, we establish bounds for the first non-zero Steklov--Neumann eigenvalue on a doubly connected star-shaped domain contained in a hypersurface equipped with a revolution-type metric. We also derive the asymptotic behavior of the first non-zero Steklov--Neumann eigenvalue on a bounded domain with a spherical hole in $\mathbb{R}^n$ as the radius of the hole approaches zero. Finally, we study the number of nodal domains of the eigenfunction corresponding to the first non zero Steklov--Neumann eigenvalue on a bounded domain in $\mathbb{R}^n$ having a spherical hole.

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arXiv2026

On the optimal sets in Pólya and Makai type inequalities

In this paper, we examine some shape functionals, introduced by Pólya and Makai, involving the torsional rigidity and the first Dirichlet-Laplacian eigenvalue for bounded, open and convex sets of $\mathbb{R}^n$. We establish new quantitative bounds, which give us key properties and information on the behavior of the optimizing sequences. In particular, we consider two kinds of reminder terms that provide information about the structure of these minimizing sequences, such as information about the thickness.

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arXiv2025

Estimates for the first and second Steklov-Dirichlet eigenvalues

In this paper, we deal with the Steklov-Dirichlet eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $Ω_r = Ω_0 \setminus \overline{B}_r$, where $Ω_0 \subset \mathbb{R}^n$, $n \geq 2$, is an open, bounded set with a Lipschitz boundary, and $B_r$ is the ball centered at the origin with radius $r > 0$, such that $\overline{B}_r \subset Ω_0$. In the first part of the paper, we focus on the first Steklov-Dirichlet eigenvalue $σ_1(Ω_r)$ and prove that the sequence of corresponding normalized eigenfunctions converges to a particular constant as $r \to 0^+$. This will allow us to prove an isoperimetric inequality for $ σ_1(Ω_r)$ when $r$ is small enough, under a measure constraint. The second part is focused on the second Steklov-Dirichlet eigenvalue $σ_2(Ω_r)$. We prove that it converges to the first non-trivial Steklov eigenvalue $\overlineσ_1(Ω_0)$ of the non-perforated domain $Ω_0$. This result, together with the Brock and Weinstock inequalities, respectively, allows us to prove two isoperimetric inequalities for small holes.

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arXiv2025

Some isoperimetric inequalities involving the boundary momentum

The aim of this paper is twofold. In the first part we focus on a functional involving a weighted curvature integral and the quermassintegrals. We prove upper and lower bounds for this functional in the class of convex sets, which provide a stronger form of the classical Aleksandrov-Fenchel inequality involving the $(n-1)$ and $(n-2)$-quermassintegrals, and consequently a stronger form of the classical isoperimetric inequality in the planar case. Moreover, quantitative estimates are proved. In the second part we deal with a shape optimization problem for a functional involving the boundary momentum. It is known that in dimension two the ball is a maximizer among simply connected sets when the perimeter and centroid is fixed. We show that the result still holds in the class of undecomposable sets. In higher dimensions the same result does not hold and we consider a new scaling invariant functional that might be a good candidate to generalize the planar case. For this functional we prove that the ball is a stable maximizer in the class of nearly spherical sets in any dimension.

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arXiv2025

On a 3D Stokes eigenvalue problem under Navier slip-with-friction boundary conditions and applications to Navier-Stokes equations

In this paper we consider, by means of a precise spectral analysis, the 3D Navier-Stokes equations endowed with Navier slip-with-friction boundary conditions. We study the problem in a very simple geometric situation as the region between two parallel planes, with periodicity along the two planes. This setting, which is often used in the theory of boundary layers, requires some special treatment for what concerns the functional setting and allows us to characterize in a rather explicit manner eigenvalues and eigenfunctions of the associated Stokes problem. These, will be then used in order to identify infinite dimensional classes of data leading to global strong solutions for the corresponding evolution Navier-Stokes equations.

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arXiv2024

Energy conservation for 3D Euler and Navier-Stokes equations in a bounded domain. Applications to Beltrami flows

In this paper we consider the incompressible 3D Euler and Navier-Stokes equations in a smooth bounded domain. First, we study the 3D Euler equations endowed with slip boundary conditions and we prove the same criteria for energy conservation involving the gradient, already known for the Navier-Stokes equations. Subsequently, we utilise this finding, which is based on a proper approximation of the velocity (and doesn't require estimates or additional assumptions on the pressure), to explore energy conservation for Beltrami flows. Finally, we explore Beltrami solutions to the Navier-Stokes equations and demonstrate that conditions leading to energy conservation are significantly distinct from those implying regularity. This remains true even when making use of the bootstrap regularity improvement, stemming from the solution being a Beltrami vector field.

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arXiv2024

Velocity-vorticity geometric constraints for the energy conservation of 3D ideal incompressible fluids

In this paper we consider the 3D Euler equations and we first prove a criterion for energy conservation for weak solutions with velocity satisfying additional assumptions in fractional Sobolev spaces with respect to the space variables, balanced by proper integrability with respect to time. Next, we apply the criterion to study the energy conservation of solution of the Beltrami type, carefully applying properties of products in (fractional and possibly negative) Sobolev spaces and employing a suitable bootstrap argument.

math.AP↗
arXiv2024

On a weighted anisotropic eigenvalue problem

In this paper we deal with a weighted eigenvalue problem for the anisotropic $(p,q)$-Laplacian with Dirichlet boundary conditions. We study the main properties of the first eigenvalue and prove a reverse Hölder type inequality for the corresponding eigenfunctions.

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arXiv2023

On a Steklov-Robin eigenvalue problem

In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $Ω=Ω_0 \setminus \overline{B}_{r}$, where $B_{r}$ is the ball centered at the origin with radius $r>0$ and $Ω_0\subset\mathbb{R}^n$, $n\geq 2$, is an open, bounded set with Lipschitz boundary, such that $\overline{B}_{r}\subset Ω_0$. We impose a Steklov condition on the outer boundary and a Robin condition involving a positive $L^{\infty}$-function $β(x)$ on the inner boundary. Then, we study the first eigenvalue $σ_β(Ω)$ and its main properties. In particular, we investigate the behaviour of $σ_β(Ω)$ when we let vary the $L^1$-norm of $β$ and the radius of the inner ball. Furthermore, we study the asymptotic behaviour of the corresponding eigenfunctions when $β$ is a positive parameter that goes to infinity.

math.AP↗
arXiv2022

Sharp and quantitative estimates for the $p-$Torsion of convex sets

Let $Ω\subset\mathbb{R}^n$, $n\geq 2$, be a bounded, open and convex set and let $f$ be a positive and non-increasing function depending only on the distance from the boundary of $Ω$. We consider the $p-$torsional rigidity associated to $Ω$ for the Poisson problem with Dirichlet boundary conditions, denoted by $T_{f,p}(Ω)$. Firstly, we prove a Pólya type lower bound for $T_{f,p}(Ω)$ in any dimension; then, we consider the planar case and we provide two quantitative estimates in the case $f\equiv 1 $.

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arXiv2022

An isoperimetric inequality for the first Steklov-Dirichlet Laplacian eigenvalue of convex sets with a spherical hole

In this paper we prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint. More precisely, if $Ω=Ω_0 \setminus \bar{B}_{R_1}$, where $B_{R_1}$ is the ball centered at the origin with radius $R_1>0$ and $Ω_0\subset\mathbb{R}^n$, $n\geq 2$, is an open bounded and convex set such that $B_{R_1}\Subset Ω_0$, then the first Steklov-Dirichlet eigenvalue $σ_1(Ω)$ has a maximum when $R_1$ and the measure of $Ω$ are fixed. Moreover, if $Ω_0$ is contained in a suitable ball, we prove that the spherical shell is the maximum.

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arXiv2021

Comparison results for the solutions to the anisotropic Laplacian with Robin boundary conditions

In this paper we consider PDE's problems involving the anisotropic Laplacian operator, with Robin boundary conditions. By means of Talenti techniques, widely used in the last decades, we prove a comparison result between the solutions of the above-mentioned problems and the solutions of the symmetrized ones. As a consequence of these results, a Bossel-Daners type inequality can be shown in dimension 2.

math.AP↗
arXiv2021

Some properties of the torsion function with Robin boundary conditions

In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the $L^{\infty}$ and $L^p$ norms, for $p\ge 1$, of the torsion function, seen as a functional on a bounded simply connected open set $Ω\subset \mathbb{R}^n$, and prove that the balls are critical shapes for these functionals, when the volume of $Ω$ is preserved.

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