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Alessandro Iacopetti

Publications and source records attributed to Alessandro Iacopetti.

At least 19 recordsLinked to original sources
arXiv2026

Entire spacelike radial graphs with prescribed mean curvature in the Lorentz--Minkowski space

In this paper we address the existence and uniqueness of entire spacelike hypersurfaces in the Lorentz--Minkowski space $\mathbb{L}^{m+1}$ with prescribed mean curvature that are star-shaped with respect to a point and asymptotic to a light cone. We also establish a Willmore-type inequality and prove a non-existence result for spacelike radial graphs asymptotic to the light cone whose mean curvature belongs to $L^p$ for $1 \leq p\leq m$, in particular in the case of compactly supported mean curvature.

math.AP↗
arXiv2024

A shape optimization problem in cylinders and related overdetermined problems

In this paper, we study a shape optimization problem for the torsional energy associated with a domain contained in an infinite cylinder, under a volume constraint. We prove that a minimizer exists for all fixed volumes and show some of its geometric and topological properties. As this issue is closely related to the question of characterizing domains in cylinders that admit solutions to an overdetermined problem, our minimization result allows us to deduce interesting consequences in that direction. In particular, we find that, for some cylinders and some volumes, the ``trivial" domain given by a bounded cylinder is not the only domain where the overdetermined problem has a solution. Moreover, it is not even a minimizer, which indicates that solutions with flat level sets are not always the best candidates for optimizing the torsional energy.

math.AP↗
arXiv2023

A sharp gradient estimate and $W^{2,q}$ regularity for the prescribed mean curvature equation in the Lorentz-Minkowski space

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely \begin{equation*} -\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= ρ\quad \hbox{in }\mathbb{R}^N, \end{equation*} where $N\geq 3$. We first prove a new gradient estimate for classical solutions with smooth data $ρ$. As a consequence we obtain that the unique weak solution of the equation satisfying a homogeneous boundary condition at infinity is locally of class $W^{2,q}$ and strictly spacelike in $\mathbb{R}^N$, provided that $ρ\in L^q(\mathbb{R}^N) \cap L^m(\mathbb{R}^N)$ with $q>N$ and $m\in[1,\frac{2N}{N+2}]$.

math.AP↗
arXiv2023

Energy stability for a class of semilinear elliptic problems

In this paper, we consider semilinear elliptic problems in a bounded domain $Ω$ contained in a given unbounded Lipschitz domain $\mathcal C \subset \mathbb R^N$. Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain $Ω$ inside $\mathcal C$. Once a rigorous variational approach to this question is set, we focus on the cases when $\mathcal C$ is a cone or a cylinder and we consider spherical sectors and radial solutions or bounded cylinders and special one-dimensional solutions, respectively. In these cases, we show both stability and instability results, which have connections with related overdetermined problems.

math.AP↗
arXiv2022

Overdetermined problems and relative Cheeger sets in unbounded domains

In this paper we study a partially overdetermined mixed boundary value problem for domains $Ω$ contained in an unbounded set $\mathcal C$. We introduce the notion of Cheeger set relative to $\mathcal C$ and show that if a domain $Ω\subset \mathcal C$ admits a solution of the overdetermined problem, then it coincides with its relative Cheeger set. We also study the related problem of characterizing constant mean curvature surfaces $Γ$ inside $\mathcal C$. In the case when $\mathcal C$ is a cylinder we obtain further results whenever the relative boundary of $Ω$ or the surface $Γ$ is a graph on the base of the cylinder.

math.AP↗
arXiv2021

Existence of nonradial domains for overdetermined and isoperimetric problems in nonconvex cones

In this work we address the question of the existence of nonradial domains inside a nonconvex cone for which a mixed boundary overdetermined problem admits a solution. Our approach is variational, and consists in proving the existence of nonradial minimizers, under a volume constraint, of the associated torsional energy functional. In particular we give a condition on the domain $D$ on the sphere spanning the cone which ensures that the spherical sector is not a minimizer. Similar results are obtained for the relative isoperimetric problem in nonconvex cones.

math.AP↗
arXiv2020

On the non-existence of compact surfaces of genus one with prescribed, almost constant mean curvature, close to the singular limit

In Euclidean 3-space endowed with a Cartesian reference system we consider a class of surfaces, called Delaunay tori, constructed by bending segments of Delaunay cylinders with neck-size $a$ and $n$ lobes along circumferences centered at the origin. Such surfaces are complete and compact, have genus one and almost constant, say 1, mean curvature, when $n$ is large. Considering a class of mappings $H\colon\mathbb{R}^{3}\to\mathbb{R}$ such that $H(X)\to 1$ as $|X|\to\infty$ with some decay of inverse-power type, we show that for $n$ large and $|a|$ small, in a suitable neighborhood of any Delaunay torus with $n$ lobes and neck-size $a$ there is no parametric surface constructed as normal graph over the Delaunay torus and whose mean curvature equals $H$ at every point.

math.AP↗
arXiv2019

New concentration phenomena for a class of radial fully nonlinear equations

We study radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values, showing that new concentration phenomena occur. Finally we define a suitable weighted energy for these solutions and compute its limit value.

math.AP↗
arXiv2019

Sign-changing bubble-tower solutions to fractional semilinear elliptic problems

We study the asymptotic and qualitative properties of least energy radial sign-changing solutions to fractional semilinear elliptic problems of the form \[ \begin{cases} (-Δ)^s u = |u|^{2^*_s-2-\varepsilon}u &\text{in } B_R, \\ u = 0 &\text{in }\mathbb{R}^n \setminus B_R, \end{cases} \] where $s \in (0,1)$, $(-Δ)^s$ is the s-Laplacian, $B_R$ is a ball of $\mathbb{R}^n$, $2^*_s := \frac{2n}{n-2s}$ is the critical Sobolev exponent and $\varepsilon>0$ is a small parameter. We prove that such solutions have the limit profile of a "tower of bubbles", as $ \varepsilon \to 0^+$, i.e. the positive and negative parts concentrate at the same point with different concentration speeds. Moreover, we provide information about the nodal set of these solutions.

math.AP↗
arXiv2018

On the structure of the nodal set and asymptotics of least energy sign-changing radial solutions of the fractional Brezis-Nirenberg problem

In this paper we study the asymptotic and qualitative properties of least energy radial sign-changing solutions of the fractional Brezis--Nirenberg problem ruled by the s-laplacian, in a ball of $\mathbb{R}^n$, when $s \in (0,1)$ and $n > 6s$. As usual, $λ$ is the (positive) parameter in the linear part in $u$, and we consider $λ$ close to zero. We prove that if such solutions vanish at the center of the ball then they vanish everywhere, we establish a bound on the number of sign-changes and, when $s$ is close to $1$, for a suitable value of the parameter $λ$ such solutions change sign exactly once. Moreover, for any $s \in (0,1)$ and $λ$ sufficiently small we prove that the number of connected components of the complement of the nodal set corresponds to the number of sign-changes plus one. In addition, for any $s \in (\frac{1}{2},1)$, we prove that least energy nodal solutions which change sign exactly once have the limit profile of a "tower of bubbles", as $λ\to 0^+$, i.e. the positive and negative parts concentrate at the same point (which is the center of the ball) with different concentration speeds.

math.AP↗
arXiv2018

On the regularity of the minimizer of the electrostatic Born-Infeld energy

We consider the electrostatic Born-Infeld energy \begin{equation*} \int_{\mathbb{R}^N}\left(1-{\sqrt{1-|\nabla u|^2}}\right)\, dx -\int_{\mathbb{R}^N}ρu\, dx, \end{equation*} where $ρ\in L^{m}(\mathbb{R}^N)$ is an assigned charge density, $m \in [1,2_*]$, $2_*:=\frac{2N}{N+2}$, $N\geq 3$. We prove that if $ρ\in L^q(\mathbb{R}^N) $ for $q>2N$, the unique minimizer $u_ρ$ is of class $W_{loc}^{2,2}(\mathbb{R}^N)$. Moreover, if the norm of $ρ$ is sufficiently small, the minimizer is a weak solution of the associated PDE \begin{equation}\label{eq:BI-abs} \tag{$\mathcal{BI}$} -\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= ρ\quad\hbox{in }\mathbb{R}^N, \end{equation} with the boundary condition $\lim_{|x|\to\infty}u(x)=0$ and it is of class $C^{1,α}_{loc}(\mathbb{R}^N)$, for some $α\in (0,1)$.

math.AP↗
arXiv2016

Existence of isovolumetric extremals for capillarity functionals

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in R3 as the sum of the area integral and a non homogeneous term of suitable form. Here we consider the case of a class of non homogenous terms vanishing at infinity for which the corresponding capillarity functional has no volume-constrained S2-type minimal surface. Using variational techniques, we prove existence of extremals characterized as saddle-type critical points.

math.DG↗
arXiv2015

Existence of stable H-surfaces in cones and their representation as radial graphs

In this paper we study the Plateau problem for disk-type surfaces contained in conic regions of $\mathbb{R}^{3}$ and with prescribed mean curvature $H$. Assuming a suitable growth condition on $H$, we prove existence of a least energy $H$-surface $X$ spanning an arbitrary Jordan curve $Γ$ taken in the cone. Then we address the problem of describing such surface $X$ as radial graph when the Jordan curve $Γ$ admits a radial representation. Assuming a suitable monotonicity condition on the mapping $λ\mapstoλH(λp)$ and some strong convexity-type condition on the radial projection of the Jordan curve $Γ$, we show that the $H$-surface $X$ can be represented as a radial graph.

math.AP↗
arXiv2015

Sign-changing blowing-up solutions for the Brezis--Nirenberg problem in dimensions four and five

We consider the Brezis-Nirenberg problem: $$-Δu =λu + |u|^{p-1}u\qquad \mbox{in}\,\, Ω,\quad u=0\,\, \mbox{on}\,\,\ \partialΩ,$$ where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$, $p=\frac{N+2}{N-2}$ and $λ>0$. In this paper we prove that, if $Ω$ is symmetric and $N=4,5$, there exists a sign-changing solution whose positive part concentrates and blows-up at the center of symmetry of the domain, while the negative part vanishes, as $λ\rightarrow λ_1$, where $λ_1=λ_1(Ω)$ denotes the first eigenvalue of $-Δ$ on $Ω$, with zero Dirichlet boundary condition.

math.AP↗
arXiv2015

A nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions

We consider the Brezis-Nirenberg problem: \begin{equation*} \begin{cases} -Δu = λu + |u|^{2^* -2}u & \hbox{in}\ Ω\\ u=0 & \hbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$, $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent and $λ>0$ a positive parameter. The main result of the paper shows that if $N=4,5,6$ and $λ$ is close to zero there are no sign-changing solutions of the form $$u_λ=PU_{δ_1,ξ}-PU_{δ_2,ξ}+w_λ, $$ where $PU_{δ_i}$ is the projection on $H_0^1(Ω)$ of the regular positive solution of the critical problem in $\mathbb{R}^N$, centered at a point $ξ\in Ω$ and $w_λ$ is a remainder term. Some additional results on norm estimates of $w_λ$ and about the concentrations speeds of tower of bubbles in higher dimensions are also presented.

math.AP↗