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Giorgio Saracco

Publications and source records attributed to Giorgio Saracco.

At least 19 recordsLinked to original sources
arXiv2026

Convex sets with vanishing relative width and the reverse Cheeger inequality

We study the reverse Cheeger inequality, which bounds from above the ratio of the first Dirichlet Laplacian eigenvalue to the square of the Cheeger constant. We first extend this inequality from convex sets to a broader class. We then analyze maximizing sequences among convex bodies. To quantify domain collapse, we introduce the notion of principal widths. For three-dimensional convex bodies, we prove that if the ratio of the first principal width to the second principal width vanishes along a sequence of convex bodies, then such sequence is maximizing. Finally, in arbitrary dimensions, we prove that the same property holds for the class of rhomboid-like sets.

math.SP↗
arXiv2026

Spectral inequalities for weighted $p$-Laplacians via Talenti symmetrization

We consider the weighted $p$-Laplacian associated with a measure $μ$ that is absolutely continuous with respect to the Lebesgue measure on an open connected subset $X\subset\mathbb{R}^N$. We prove that Talenti's weighted Pólya--Szegő inequality -- originally established for Lipschitz functions on $X$ -- extends to Sobolev functions with zero boundary trace on arbitrary Borel subsets $Ω\subset X$. This yields Faber--Krahn-type inequalities for the first $(p,q)$-eigenvalue of the weighted Dirichlet $p$-Laplacian. We present several examples fitting this abstract framework, including classical Euclidean and Gaussian cases alongside new results for homogeneous weights in convex cones, anisotropic Gaussians, and log-concave Gaussian perturbations.

math.AP↗
arXiv2025

On the $Γ$-limit of weighted fractional energies

Given $p\in[1,\infty)$ and a bounded open set $Ω\subset\mathbb R^d$ with Lipschitz boundary, we study the $Γ$-convergence of the weighted fractional seminorm \[ [u]_{s,p,f}^p = \int_{\mathbb R^d} \int_{\mathbb R^d} \frac{|\tilde{u}(x)- \tilde{u}(y)|^p}{\|x-y\|^{d+sp}}\,f(x)\,f(y)\,\mathrm{d} x\,\mathrm{d} y \] as $s\to1^-$ for $u\in L^p(Ω)$, where $\tilde{u}=u$ on $Ω$ and $\tilde{u}=0$ on $\mathbb R^d\setminusΩ$. Assuming that $(f_s)_{s\in(0,1)}\subset L^\infty(\mathbb R^d;[0,\infty))$ and $f\in\mathrm{Lip}_b(\mathbb R^d;(0,\infty))$ are such that $f_s\to f$ in $L^\infty(\mathbb R^d)$ as $s\to1^-$, we show that $(1-s)[u]_{s,p,f_s}$ $Γ$-converges to the Dirichlet $p$-energy weighted by $f^2$. In the case $p=2$, we also prove the convergence of the corresponding gradient flows.

math.AP↗
arXiv2025

The Cheeger problem in abstract measure spaces

We consider non-negative $σ$-finite measure spaces coupled with a proper functional $P$ that plays the role of a perimeter. We introduce the Cheeger problem in this framework and extend many classical results on the Cheeger constant and on Cheeger sets to this setting, requiring minimal assumptions on the pair measure space-perimeter. Throughout the paper, the measure space will never be asked to be metric, at most topological, and this requires the introduction of a suitable notion of Sobolev spaces, induced by the coarea formula with the given perimeter.

math.MG↗
arXiv2025

A reverse isoperimetric inequality for the Cheeger constant under width constraint

Henrot and Lucardesi, in Commun. Contemp. Math. (2024), conjectured that among planar convex sets with prescribed minimal width, the equilateral triangle uniquely maximizes the Cheeger constant. In this short note, we confirm this conjecture. Moreover, we establish a stability result for the inequality in terms of the Hausdorff distance.

math.OC↗
arXiv2024

On the $N$-Cheeger problem for component-wise increasing norms

We study Cheeger and $p$-eigenvalue partition problems depending on a given evaluation function $Φ$ for $p\in[1,\infty)$. We prove existence and regularity of minima, relations among the problems, convergence, and stability with respect to $p$ and to $Φ$.

math.FA↗
arXiv2024

Cylindrical estimates for the Cheeger constant and applications

We prove a lower bound for the Cheeger constant of a cylinder $Ω\times (0,L)$, where $Ω$ is an open and bounded set. As a consequence, we obtain existence of minimizers for the shape functional defined as the ratio between the first Dirichlet eigenvalue of the $p$-Laplacian and the $p$-th power of the Cheeger constant, within the class of bounded convex sets in any $\mathbb{R}^N$. This positively solves open conjectures raised by Parini (J. Convex Anal. (2017)) and by Briani-Buttazzo-Prinari (Ann. Mat. Pura Appl. (2023)).

math.AP↗
arXiv2024

Geometric criteria for the existence of capillary surfaces in tubes

We review some geometric criteria and prove a refined version, that yield existence of capillary surfaces in tubes $Ω\times \mathbb{R}$ in a gravity free environment, in the case of physical interest, that is, for bounded, open, and simply connected $Ω\subset \mathbb{R}^2$. These criteria rely on suitable weak one-sided bounds on the curvature of the boundary of the cross-section $Ω$.

math.AP↗
arXiv2023

On the monotonicity of weighted perimeters of convex bodies

We prove that, among weighted isotropic perimeters, only constant multiples of the Euclidean perimeter satisfy the monotonicity property on nested convex bodies. Although the analogous result fails for general weighted anisotropic perimeters, a similar characterization holds for radially-weighted anisotropic densities.

math.MG↗
arXiv2023

Optimization of the anisotropic Cheeger constant with respect to the anisotropy

Given an open, bounded set $Ω$ in $\mathbb{R}^N$, we consider the minimization of the anisotropic Cheeger constant $h_K(Ω)$ with respect to the anisotropy $K$, under a volume constraint on the associated unit ball. In the planar case, under the assumption that $K$ is a convex, centrally symmetric body, we prove the existence of a minimizer. Moreover, if $Ω$ is a ball, we show that the optimal anisotropy $K$ is not a ball and that, among all regular polygons, the square provides the minimal value.

math.OC↗
arXiv2023

Isoperimetric sets and $p$-Cheeger sets are in bijection

Given an open, bounded, planar set $Ω$, we consider its $p$-Cheeger sets and its isoperimetric sets. We study the set-valued map $\mathfrak{V}:[\frac12,+\infty)\rightarrow\mathcal{P}((0,|Ω|])$ associating to each $p$ the set of volumes of $p$-Cheeger sets. We show that whenever $Ω$ satisfies some geometric structural assumptions (convex sets are encompassed), the map is injective, and continuous in terms of $Γ$-convergence. Moreover, when restricted to $(\frac 12, 1)$ such a map is univalued and is in bijection with its image. As a consequence of our analysis we derive some fine boundary regularity result.

math.AP↗
arXiv2022

The isoperimetric problem in $2$d domains without necks

We give a complete characterization of all isoperimetric sets contained in a domain of the Euclidean plane, that is bounded by a Jordan curve and satisfies a no-neck property. Further, we prove that the isoperimetric profile of such domain is convex above the volume of the largest ball contained in it, and that its square is globally convex.

math.AP↗
arXiv2020

A sufficient criterion to determine planar self-Cheeger sets

We show a sufficient criterion to determine if a planar set $Ω$ is a minimizer of the prescribed curvature functional among all of its subsets. As a special case, we derive a sufficient criterion to determine if $Ω$ is a self-Cheeger set, i.e. if it minimizes the ratio $P(E)/|E|$ among all of its subsets. Specifically, if a Jordan domain $Ω$ possesses the interior disk property of radius $|Ω|/P(Ω)$, then it is a self-Cheeger set; if it possesses the strict interior disk property then it is a minimal Cheeger set, i.e. the unique minimizer. As a side effect we provide a way to build self-Cheeger sets.

math.AP↗
arXiv2020

Minimizers of the prescribed curvature functional in a Jordan domain with no necks

We provide a geometric characterization of the minimal and maximal minimizer of the prescribed curvature functional $P(E)-κ|E|$ among subsets of a Jordan domain $Ω$ with no necks of radius $κ^{-1}$, for values of $κ$ greater than or equal to the Cheeger constant of $Ω$. As an application, we describe all minimizers of the isoperimetric profile for volumes greater than the volume of the minimal Cheeger set, relative to a Jordan domain $Ω$ which has no necks of radius $r$, for all $r$. Finally, we show that for such sets and volumes the isoperimetric profile is convex.

math.AP↗
arXiv2020

Rigidity and trace properties of divergence-measure vector fields

We consider a $φ$-rigidity property for divergence-free vector fields in the Euclidean $n$-space, where $φ(t)$ is a non-negative convex function vanishing only at $t=0$. We show that this property is always satisfied in dimension $n=2$, while in higher dimension it requires some further restriction on $φ$. In particular, we exhibit counterexamples to \textit{quadratic rigidity} (i.e., when $φ(t) = ct^2$) in dimension $n\ge 4$. The validity of the quadratic rigidity, which we prove in dimension $n=2$, implies the existence of the trace of a divergence-measure vector field $ξ$ on a $\mathcal{H}^{1}$-rectifiable set $S$, as soon as its weak normal trace $[ξ\cdot ν_S]$ is maximal on $S$. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.

math.AP↗
arXiv2019

A discrete districting plan

The outcome of elections is strongly dependent on the districting choices, making thus possible (and frequent) the gerrymandering phenomenon, i.e.\ politicians suitably changing the shape of electoral districts in order to win the forthcoming elections. While so far the problem has been treated using continuous analysis tools, it has been recently pointed out that a more reality-adherent model would use the discrete geometry of graphs or networks. Here we propose a parameter-dependent discrete model for choosing an "optimal" districting plan. We analyze several properties of the model and lay foundations for further analysis on the subject.

math.OC↗