arXiv subjects

Giorgio Stefani

Publications and source records attributed to Giorgio Stefani.

At least 19 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

On blow-ups of sets with finite fractional variation

Given $\alpha\in(0,1)$ and a set $E\subset\mathbb{R}^N$ with locally finite fractional $\alpha$-variation, we show that for $|D^\alpha\mathbf 1_E|$-a.e. $x$, every non-trivial tangent set of $E$ at $x$ with locally finite integer perimeter is a half-space oriented by the fractional inner unit normal of $E$ at $x$.

math.AP↗

arXiv2025

A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces

We give a partial negative answer to a question left open in a previous work by Brasco and the first and third-named authors concerning the sharp constant in the fractional Hardy inequality on convex sets. Our approach has a geometrical flavor and equivalently reformulates the sharp constant in the limit case $p=1$ as the Cheeger constant for the fractional perimeter and the Lebesgue measure with a suitable weight. As a by-product, we obtain new lower bounds on the sharp constant in the $1$-dimensional case, even for non-convex sets, some of which optimal in the case $p=1$.

math.AP↗

arXiv2025

On a weighted version of the BBM formula

We prove a weighted version of the Bourgain-Brezis-Mironescu (BBM) formula, both in the pointwise and $Γ$-convergence sense, together with a compactness criterion for energy-bounded sequences. The non-negative weights need only be $L^\infty$ convergent to a bounded and uniformly continuous limit. We apply the BBM formula to show a Poincaré-type inequality and the stability of the first eigenvalues relative to the energies. Finally, we discuss a non-local analogue of the weighted BBM formula.

math.AP↗

arXiv2025

On the $\Gamma$-limit of weighted fractional energies

Given $p\in[1,\infty)$ and a bounded open set $\Omega\subset\mathbb R^d$ with Lipschitz boundary, we study the $\Gamma$-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 $u\in L^p(\Omega)$, where $\tilde{u}=u$ on $\Omega$ and $\tilde{u}=0$ on $\mathbb R^d\setminus\Omega$. 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 $(1-s)[u]_{s,p,f_s}$ $\Gamma$-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

Properties of Lipschitz smoothing heat semigroups

We prove several functional and geometric inequalities only assuming the linearity and a quantitative $\mathrm{L}^\infty$-to-Lipschitz smoothing of the heat semigroup in metric-measure spaces. Our results comprise a Buser inequality, a lower bound on the size of the nodal set of a Laplacian eigenfunction, and different estimates involving the Wasserstein distance. The approach works in large variety settings, including Riemannian manifolds with a variable Kato-type lower bound on the Ricci curvature tensor, $\mathsf{RCD}(K,\infty)$ spaces, and some sub-Riemannian structures, such as Carnot groups, the Grushin plane and the $\mathbb{SU}(2)$ group.

math.FA↗

arXiv2025

Sharp conditions for the BBM formula and asymptotics of heat content-type energies

Given $p\in[1,\infty)$, we provide sufficient and necessary conditions on the non-negative measurable kernels $(\rho_t)_{t\in(0,1)}$ ensuring convergence of the associated Bourgain-Brezis-Mironescu (BBM) energies $(\mathscr{F}_{t,p})_{t\in(0,1)}$ to a variant of the $p$-Dirichlet energy on $\mathbb R^N$ as $t\to0^+$ both in the pointwise and in the $\Gamma$-sense. We also devise sufficient conditions on $(\rho_t)_{t\in(0,1)}$ yielding local compactness in $L^p(\mathbb R^N)$ of sequences with bounded BBM energy. Moreover, we give sufficient conditions on $(\rho_t)_{t\in(0,1)}$ implying pointwise and $\Gamma$-convergence and compactness of $(\mathscr{F}_{t,p})_{t\in(0,1)}$ when the limit $p$-energy is of non-local type. Finally, we apply our results to provide asymptotic formulas in the pointwise and $\Gamma$-sense for heat content-type energies both in the local and non-local settings.

math.AP↗

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

Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula

Given $α\in(0,1]$ and $p\in[1,+\infty]$, we define the space $\mathscr{DM}^{α,p}(\mathbb R^n)$ of $L^p$ vector fields whose $α$-divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the $α$-divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss-Green formula. The sharpness of our results is discussed via some explicit examples.

math.FA↗

arXiv2024

Topological singularities arising from fractional-gradient energies

We prove that, on a planar regular domain, suitably scaled functionals of Ginzburg-Landau type, given by the sum of quadratic fractional Sobolev seminorms and a penalization term vanishing on the unitary sphere, $Γ$-converge to vortex-type energies with respect to the flat convergence of Jacobians. The compactness and the $Γ$-$\liminf$ follow by comparison with standard Ginzburg-Landau functionals depending on Riesz potentials. The $Γ$-$\limsup$, instead, is achieved via a direct argument by joining a finite number of vortex-like functions suitably truncated around the singularity.

math.AP↗

arXiv2024

On sets with finite distributional fractional perimeter

We continue the study of the fine properties of sets having locally finite distributional fractional perimeter. We refine the characterization of their blow-ups and prove a Leibniz rule for the intersection of sets with locally finite distributional fractional perimeter with sets with finite fractional perimeter. As a byproduct, we provide a description of non-local boundaries associated with the distributional fractional perimeter.

math.FA↗

arXiv2023

Lipschitz approximation of almost $\mathbb G$-perimeter minimizing boundaries in plentiful groups

We prove that the boundary of an almost minimizer of the intrinsic perimeter in a plentiful group can be approximated by intrinsic Lipschitz graphs. Plentiful groups are Carnot groups of step~$2$ whose center of the Lie algebra is generated by any co-dimension one horizontal subspace. For example, $H$-type groups not isomorphic to the first Heisenberg group are plentiful. Our results provide the first extension of the regularity theory of intrinsic minimal surfaces beyond the family of Heisenberg groups.

math.DG↗

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

On the monotonicity of non-local perimeter of convex bodies

Under mild assumptions on the kernel $K\ge0$, the non-local $K$-perimeter $P_K$ satisfies the monotonicity property on nested convex bodies, i.e., if $A\subset B\subset\mathbb{R}^n$ are two convex bodies, then $P_K(A)\le P_K(B)$. In this note, we prove quantitative lower bounds on the difference of the $K$-perimeters of $A$ and $B$ in terms of their Hausdorff distance, provided that $K$ satisfies suitable symmetry properties.

math.MG↗

arXiv2023

On the Steiner property for planar minimizing clusters. The isotropic case

We consider the isoperimetric problem for clusters in the plane with a double density, that is, perimeter and volume depend on two weights. In this paper we consider the isotropic case, in the parallel paper "On the Steiner property for planar minimizing clusters. The anisotropic case", the anisotropic case is studied. Here we prove that, in a wide generality, minimal clusters enjoy the "Steiner property", which means that the boundaries are made by ${\rm C}^{1,γ}$ regular arcs, meeting in finitely many triple points with the $120^\circ$ property.

math.AP↗

arXiv2023

On the Steiner property for planar minimizing clusters. The anisotropic case

In this paper we discuss the Steiner property for minimal clusters in the plane with an anisotropic double density. This means that we consider the classical isoperimetric problem for clusters, but volume and perimeter are defined by using two densities. In particular, the perimeter density may also depend on the direction of the normal vector. The classical "Steiner property" for the Euclidean case (which corresponds to both densities being equal to $1$) says that minimal clusters are made by finitely many ${\rm C}^{1,γ}$ arcs, meeting in finitely many "triple points". We can show that this property holds under very weak assumptions on the densities. In the parallel paper "On the Steiner property for planar minimizing clusters. The isotropic case" we consider the isotropic case, i.e., when the perimeter density does not depend on the direction, which makes most of the construction much simpler. In particular, in the present case the three arcs at triple points do not necessarily meet with three angles of $120^\circ$, which is instead what happens in the isotropic case.

math.AP↗

arXiv2023

Existence and uniqueness theorems for some semi-linear equations on locally finite graphs

We study some semi-linear equations for the $(m,p)$-Laplacian operator on locally finite weighted graphs. We prove existence of weak solutions for all $m\in\mathbb{N}$ and $p\in(1,+\infty)$ via a variational method already known in the literature by exploiting the continuity properties of the energy functionals involved. When $m=1$, we also establish a uniqueness result in the spirit of the Brezis-Strauss Theorem. We finally provide some applications of our main results by dealing with some Yamabe-type and Kazdan-Warner-type equations on locally finite weighted graphs.

math.AP↗

arXiv2023

On the convex components of a set in $\mathbb{R}^n$

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in $\mathbb{R}^n$ for all $n\ge2$. Our result generalizes and improves the inequalities previously obtained in M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, "Convex components", in Communications in Contemporary Mathematics, Vol. 21, No. 06, 1850036 (2019) and in M. La Civita and F. Leonetti, "Convex components of a set and the measure of its boundary", Atti. Sem. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008-2009) 71-78.

math.MG↗