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Antonio Greco

Publications and source records attributed to Antonio Greco.

9 recordsLinked to original sources
arXiv2026

Event Detection in Videos: A Framework for the Development of New Methods

Event detection tasks in videos, the most important aspect of video surveillance, aim to detect events either at the pixel-level, frame-level, or clip-level. Plenty of methods intended for event detection in different environments, for various applications, and within different acquisition techniques were introduced. Naturally, the attempts were made as well to classify these algorithms in terms of detection of performance or in terms of real-time abilities. Nevertheless, the lack of a large-scale dataset as well as rigorous performance evaluation methods have biased such comparisons as well as the development of the methods. Given the diversity of existing approaches, we believe it is essential for researchers to position their work within such a rich landscape. Thus, we propose a rigorous framework for developing new methods in event detection for videos. Specifically, this framework is based on three main pillars: datasets, performance evaluation, and scenarios for deploying methods.

cs.CV↗
arXiv2026

Overdetermined problems for the rotationally invariant Poisson equation in model manifolds

We present rigidity results for overdetermined problems associated to the rotationally invariant Poisson equation $-\Delta_{g_\mathcal{M}} u = f(r)$ in a model manifold $\mathcal{M} = [0,S) \times_h \mathbb S^{N-1}$ with warping function $h$. The variable $r$ ranges in the interval $[0,S)$, whose endpoint $S$ is positive and possibly infinite. The first part of the paper deals with the problem \[ \begin{array}{ll} -\Delta_{g_\mathcal{M}} {u}=f(r) &\mbox{in $\Omega$}, u=\varphi(r) &\mbox{on $\partial \Omega$}, \frac{\partial u}{\partial \nu} = \kappa(r) &\mbox{on $\partial \Omega$}, \end{array} \] where $\Omega \subset \mathcal{M}$ is a bounded domain containing the point $O \in \mathcal{M}$ corresponding to $r = 0$, $\nu$ is the exterior unit normal vector on $\partial \Omega$, and $f$, $\varphi$, $\kappa$ are three prescribed functions. In the second part of the paper, we consider a similar overdetermined problem for the exterior Bernoulli problem in a domain $\Omega \setminus \overline B_{R_0}(O)$, where $B_{R_0}(O)$ denotes the geodesic ball centered at $O$ with radius $R_0$, within the class of functions that vanish on $\partial B_{R_0}(O)$. In both cases, we give conditions on $f$, $\varphi$ and $\kappa$ implying that the solution $u$ is radial and $\Omega$ is a geodesic ball centered at $O$. Our results apply in particular to the three space forms $\mathbb{R}^N$, $\mathbb{H}^N$ and $\mathbb{S}^N$.

math.AP↗
arXiv2022

Non-autonomous overdetermined problems for the normalized p-Laplacian

We present existence and nonexistence results on the solution of an overdetermined problem for the normalized p-Laplacian in a bounded open set, with p ranging from 1 to infinity. More precisely we consider a non-constant Neumann condition at the boundary. The definitions and statements needed to understand the main results are recalled in detail.

math.AP↗
arXiv2022

Symmetry and monotonicity results for solutions of semilinear PDEs in sector-like domains

In this manuscript we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates $(r, θ, z)$, we investigate the shape of solutions whose derivative in $θ$ vanishes at the boundary. We prove that any solution with Morse index less than two must be either independent of $θ$ or strictly monotone with respect to $θ$. In the special case of a planar domain, the result holds in a circular sector as well as in an annular, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as an extension to unbounded domains. The proof is based on a rotating-plane argument: a convenient manifold is introduced in order to avoid overlapping the domain with its reflected image in the case when its opening is larger than $π$.

math.AP↗
arXiv2020

An overdetermined problem associated to the Finsler Laplacian

We prove a rigidity result for the anisotropic Laplacian. More precisely, the domain of the problem is bounded by an unknown surface supporting a Dirichlet condition together with a Neumann-type condition which is not translation-invariant. Using a comparison argument, we show that the domain is in fact a Wulff shape. We also consider the more general case when the unknown surface is required to have its boundary on a given conical surface: in such a case, the domain of the problem is bounded by the unknown surface and by a portion of the given conical surface, which supports a homogeneous Neumann condition. We prove that the unknown surface lies on the boundary of a Wulff shape.

math.AP↗
arXiv2020

On the solvability of a two-dimensional Ventcel problem with variable coefficients

This paper deals with the following mixed boundary value problem \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} -Δu = f &\mbox{in $Ω$,} \\ u = φ&\mbox{on $Γ_{\! D}$,} \\ u_ν- a_2 \, Δ_{τ\,} u + a_0 \, u = g &\mbox{on $Γ_{\! ν}$,} \end{cases} \end{equation} where $Ω$ is some bounded domain of $\mathbb{R}^2$ with $\partial Ω=Γ_{\!D}\cup Γ_{\! ν}$, $ν$ indicating the normal unit vector to $Γ_{\! ν}$ and $Δ_τ$ the Laplace--Beltrami operator along~$Γ_{\! ν}$. Additionally, $f(\bf x)$, $φ(\bf x)$, $a_2(\bf x)$, $a_0(\bf x)$ and $g(\bf x)$ are convenient functions defined on $Ω$, $Γ_{\!D}$ and $Γ_{\! ν}$, and ${\bf x} = (x,y)$ denotes a two-dimensional array. Under suitable assumptions on the data, we first give the definition of a weak solution $u$ to the problem and then we prove that it is uniquely solvable. Further, we consider a particular case of \eqref{ProblemAbstract} arising in real-world applications: we discuss the resulting model and provide an explicit solution.

math.AP↗
arXiv2017

Existence and convexity of solutions of the fractional heat equation

We prove that the initial-value problem for the fractional heat equation admits a solution provided that the (possibly unbounded) initial datum has a conveniently moderate growth at infinity. Under the same growth condition we also prove that the solution is unique. Our result does not require any sign assumption, thus complementing the Widder's type theorem of Barrios et al. (Arch. Rational Mech. Anal. 213 (2014) 629-650) for positive solutions. Finally, we show that the fractional heat flow preserves convexity of the initial datum. Incidentally, several properties of stationary convex solutions are established.

math.AP↗
arXiv2008

Radial Balanced metrics on the unit disk

Let $Φ$ be a strictly plurisubharmonic and radial function on the unit disk ${\cal D}\subset {\complex}$ and let $g$ be the \K metric associated to the \K form $ω=\frac{i}{2}\partial\bar\partialΦ$. We prove that if $g$ is $g_{eucl}$-balanced of height 3 (where $g_{eucl}$ is the standard Euclidean metric on ${\complex}={\real}^2$), and the function $h(x)=e^{-Φ(z)}$, $x=|z|^2$, extends to an entire analytic function on ${\real}$, then $g$ equals the hyperbolic metric. The proof of our result is based on a interesting characterization of the function $f(x)=1-x$.

math.DG↗