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Francesco Chiacchio

Publications and source records attributed to Francesco Chiacchio.

At least 19 recordsLinked to original sources
arXiv2026

A sharp Gaussian harmonic-mean inequality for Neumann eigenvalues of the Ornstein-Uhlenbeck operator

Let $N\geq2$ and let $\Omega\subset\R^N$ be a connected Lipschitz domain, possibly unbounded, symmetric with respect to the origin, and such that $0<\gammaN(\Omega)<1$. We assume that the Gaussian Sobolev embedding $H^1(\Omega,\gammaN)\hookrightarrow L^2(\Omega,\gammaN)$ is compact; a sufficient condition is the existence of a bounded Gaussian Sobolev extension operator from $\Omega$ to $\R^N$. Denote by \[ 0=\mu_0(\Omega)<\mu_1(\Omega)\leq\mu_2(\Omega)\leq\cdots \] the Neumann eigenvalues of the positive Ornstein--Uhlenbeck operator $-\Delta+x\cdot\nabla$ in $\Omega$. We prove the sharp reciprocal-sum inequality \[ \sum_{k=1}^{N}\frac{1}{\mu_k(\Omega)} \geq \frac{N}{\mu_1(B_R)}, \] where $B_R$ is the Euclidean ball centred at the origin and satisfying $\gammaN(B_R)=\gammaN(\Omega)$. Equality holds if and only if $\Omega=B_R$. The proof combines a coupled $N$-dimensional Ritz argument with a Gaussian raywise rearrangement. The angular imbalance is encoded by a symmetric trace-free matrix, whose contribution is controlled by a finite-dimensional convexity inequality.

math.SP↗
arXiv2026

Kohler-Jobin inequality for $p$-Laplace operator

A sharp lower bound for the first Dirichlet eigenvalue of the $p$-laplacian is derived for sets with prescribed $p$-torsional rigidity. The result provides an extension of the classical spectral inequality due to Kohler-Jobin. The proof is based on a careful analysis of the generalized $p$-torsional rigidity and on a sharp mass comparison result.

math.AP↗
arXiv2026

Kohler-Jobin inequality for $p$-Laplace operator in the Gauss space

A sharp lower bound for the first Dirichlet eigenvalue of the $p$-laplacian in Gaussian space is derived for sets with prescribed generalized torsional rigidity. The result provides an extension of the classical spectral inequality due to Kohler-Jobin. The proof is based on a careful analysis of the generalized torsional rigidity and on a sharp mass comparison result. Furthermore, a Payne-Rayner type inequality is established.

math.AP↗
arXiv2025

Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights

We study the following class of Steklov eigenvalue problems: \[ \nabla \cdot \bigl( w \nabla u \bigr) = 0 \quad \text{in } \Omega, \qquad \frac{\partial u}{\partial \nu} = \gamma v u \quad \text{on } \partial \Omega, \] where $w$ and $v$ are prescribed positive radial functions, $\Omega$ is a Lipschitz domain in $\mathbb{R}^N$ with $N \geq 2$ and $\nu$ denotes its outward unit normal. Extending classical results in the unweighted case due to Weinstock, the first author, and others, we establish isoperimetric inequalities for low-order eigenvalues under suitable symmetry assumptions on the domain. In the first part, we consider the case $w(x) = |x|^{\alpha}$ and $v(x) = |x|^{\beta-\alpha}$, where the parameters $\alpha, \beta \in \mathbb{R}$ satisfy appropriate constraints. Our analysis relies on an explicit computation of the spectrum in the radial case, variational principles, and a family of weighted isoperimetric inequalities with ``double density''. In the second part, we address the case $v \equiv 1$ and $w(x) = W(|x|)$, where $W$ is a non-decreasing, log-convex function. In this setting, the proof relies, among other tools, on a new weighted isoperimetric inequality, which may be of independent interest.

math.AP↗
arXiv2025

Shape of extremal functions for weighted Sobolev-type inequalities

We study the shape of solutions to some variational problems in Sobolev spaces with weights that are powers of |x|. In particular, we detect situations when the extremal functions lack symmetry properties such as radial symmetry and antisymmetry. We also prove an isoperimetric inequality for the first non-zero eigenvalue of a weighted Neumann problem.

math.OC↗
arXiv2025

On the reverse isoperimetric inequality in Gauss space

In this paper we investigate the reverse isoperimetric inequality with respect to the Gaussian measure for convex sets in $\mathbb{R}^{2}$. While the isoperimetric problem for the Gaussian measure is well understood, many relevant aspects of the reverse problem have not yet been investigated. In particular, to the best of our knowledge, there seem to be no results on the shape that the isoperimetric set should take. Here, through a local perturbation analysis, we show that smooth perimeter-maximizing sets have locally flat boundaries. Additionally, we derive sharper perimeter bounds than those previously known, particularly for specific classes of convex sets such as the convex sets symmetric with respect to the axes. Finally, for quadrilaterals with vertices on the coordinate axes, we prove that the set maximizing the perimeter "degenerates" into the x-axis, traversed twice.

math.AP↗
arXiv2024

Eigenvalue Estimates for $p$-Laplace Problems on Domains Expressed in Fermi Coordinates

We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if $γ$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric with respect to the $y$-axis, let $D\subset \mathbb{R}^2$ denote the domain of points that lie on one side of $γ$ and within a prescribed distance $δ(s)$ from $γ(s)$ (here $s$ denotes the arc length parameter for $γ$). Write $μ_1^{odd}(D)$ for the lowest nonzero eigenvalue of the Neumann $p$-Laplacian with an eigenfunction that is odd with respect to the $y$-axis. For all $p>1$, we provide a lower bound on $μ_1^{odd}(D)$ when the distance function $δ$ and the signed curvature $k$ of $γ$ satisfy certain geometric constraints. In the linear case ($p=2$), we establish sufficient conditions to guarantee $μ_1^{odd}(D)=μ_1(D)$. We finally study the asymptotics of $μ_1(D)$ as the distance function tends to zero. We show that in the limit, the eigenvalues converge to the lowest nonzero eigenvalue of a weighted one-dimensional Neumann $p$-Laplace problem.

math.AP↗
arXiv2019

Some weighted isoperimetric problems on $\mathbb{R}^N _+ $ with stable half balls have no solutions

We show the counter-intuitive fact that some weighted isoperimetric problems on the half-space $ \mathbb{R}^N _+ $, for which half-balls centered at the origin are stable, have no solutions. A particular case is the measure $dμ= x_N ^{α} \, dx$, with $α\in (-1,0)$. Some results on stability and nonexistence for weighted isoperimetric problems on $\mathbb{R}^N $ are also obtained.

math.AP↗
arXiv2019

Some isoperimetric inequalities with respect to monomial weights

We solve a class of isoperimetric problems on $\mathbb{R}^2_+ :=\left\{ (x,y)\in \mathbb{R} ^2 : y>0 \right\}$ with respect to monomial weights. Let $α$ and $β$ be real numbers such that $0\le α<β+1$, $β\le 2 α$. We show that, among all smooth sets $Ω$ in $\mathbb{R} ^2_+$ with fixed weighted measure $\iint_{Ω} y^β dxdy$, the weighted perimeter $\int_{\partial Ω} y^α\, ds$ achieves its minimum for a smooth set which is symmetric w.r.t. to the $y$--axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a lower bound for the first eigenvalue of a class of nonlinear problems.

math.AP↗
arXiv2018

The isoperimetric problem for a class of non-radial weights and applications

We study a class of isoperimetric problems on $\mathbb{R}^{N}_{+} $ where the densities of the weighted volume and weighted perimeter are given by two different non-radial functions of the type $|x|^k x_N^α$. Our results imply some sharp functional inequalities, like for instance, Caffarelli-Kohn-Nirenberg type inequalities.

math.AP↗
arXiv2018

On weighted isoperimetric inequalities with non-radial densities

We consider a class of isoperimetric problems on $\mathbb{R}^{N}_{+} $ where the volume and the area element carry two different weights of the type $|x|^lx_N^α$. We solve them in a special case while a more detailed study is contained in \cite{ABCMP2}. Our results imply a weighted Polya-Szëgo principle and a priori estimates for weak solutions to a class of boundary value problems for degenerate elliptic equations

math.AP↗
arXiv2017

New Pólya-Szegö-type inequalities and an alternative approach to comparison results for PDE's

We prove some Pólya-Szegö type inequalities which involve couples of functions and their rearrangements. Our inequalities reduce to the classical Pólya-Szegö principle when the two functions coincide. As an application, we give a different proof of a comparison result for solutions to Dirichlet boundary value problems for Laplacian equations proved by A. Alvino, G. Trombetti, J. I. Diaz and P. L. Lions.

math.AP↗