arXiv subjects

Eleonora Cinti

Publications and source records attributed to Eleonora Cinti.

At least 19 recordsLinked to original sources

arXiv2026

Existence and non-existence results for a fractional Lane-Emden equation with nonlocal Neumann conditions

We consider the fractional Lane-Emden equation with a nonlocal Neumann condition in a half-space. We establish the existence of non-constant solutions for the critical problem in any dimension. On the other hand, we show that, in dimension $n=1$, the subcritical problem admits only the trivial solution. This result follows from a new Pohozaev-type identity, which we obtain by using suitable decay estimates for the solutions.

math.AP↗

arXiv2025

A strong quantitative form of the fractional isoperimetric inequality

We show a strong version of the fractional quantitative isoperimetric inequality, in which the isoperimetric deficit controls not only the Fraenkel asymmetry but also a sort of oscillation of the boundary. This generalizes the local result by Fusco and Julin in \cite{FJ}. The proof follows a regularization process as in \cite{FJ} but it is quite different in its spirit. Then, as a consequence of the quantitative inequality, we prove some stability estimates for a fractional Cheeger inequality.

math.AP↗

arXiv2025

On a fractional semilinear Neumann problem arising in Chemotaxis

We study a semilinear and nonlocal Neumann problem, which is the fractional analogue of the problem considered by Lin--Ni--Takagi in the '80s. The model under consideration arises in the description of stationary configurations of the Keller--Segel model for chemotaxis, when a nonlocal diffusion for the concentration of the chemical is considered. In particular, we extend to any fractional power $s\in (0,1)$ of the Laplacian (with homogeneous Neumann boundary conditions) the results obtained in [20] for $s=1/2$. We prove existence and some qualitative properties of non--constant solutions when the diffusion parameter $\varepsilon$ is small enough, and on the other hand, we show that for $\varepsilon$ large enough any solution must be necessarily constant.

math.AP↗

arXiv2024

On fractional Hardy-type inequalities in general open sets

We show that, when $sp>N$, the sharp Hardy constant $\mathfrak{h}_{s,p}$ of the punctured space $\mathbb R^N\setminus\{0\}$ in the Sobolev-Slobodecki\u{\i} space provides an optimal lower bound for the Hardy constant $\mathfrak{h}_{s,p}(\Omega)$ of an open $\Omega\subsetneq \mathbb R^N$. The proof exploits the characterization of Hardy's inequality in the fractional setting in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation and relies on the construction of suitable supersolutions by means of the distance function from the boundary of $\Omega$. Moreover, we compute the limit of $\mathfrak{h}_{s,p}$ as $s\nearrow 1$, as well as the limit when $p \nearrow \infty$. Finally, we apply our results to establish a lower bound for the non-local eigenvalue $\lambda_{s,p}(\Omega)$ in terms of $\mathfrak{h}_{s,p}$ when $sp>N$, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as $p\nearrow \infty$.

math.AP↗

arXiv2023

Existence and non-existence results for a semilinear fractional Neumann problem

We establish a priori $L^\infty$-estimates for non-negative solutions of a semilinear nonlocal Neumann problem. As a consequence of these estimates, we get non-existence of non-constant solutions under suitable assumptions on the diffusion coefficient and on the nonlinearity. Moreover, we prove an existence result for radial, radially non-decreasing solutions in the case of a possible supercritical nonlinearity, extending to the case $0<s\le 1/2$ the analysis started in [7].

math.AP↗

arXiv2023

Optimal regularity of isoperimetric sets with Hölder densities

We establish a regularity result for optimal sets of the isoperimetric problem with double density under mild ($α$-)Hölder regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach in any dimension the regularity class $C^{1,\fracα{2-α}}$. This class is indeed the optimal one for local minimizers of variational functionals with an integrand that depends $α$-Hölder continuous on the minimizer itself, and as such can (the boundary of) the isoperimetric set be locally written (with additional constraint).

math.AP↗

arXiv2022

A nonlocal supercritical Neumann problem

We establish existence of positive non-decreasing radial solutions for a nonlocal nonlinear Neumann problem both in the ball and in the annulus. The nonlinearity that we consider is rather general, allowing for supercritical growth (in the sense of Sobolev embedding). The consequent lack of compactness can be overcome, by working in the cone of non-negative and non-decreasing radial functions. Within this cone, we establish some a priori estimates which allow, via a truncation argument, to use variational methods for proving existence of solutions. As a side result, we prove a strong maximum principle for nonlocal Neumann problems, which is of independent interest.

math.AP↗

arXiv2021

Stable solutions to the fractional Allen-Cahn equation in the nonlocal perimeter regime

We study stable solutions to the fractional Allen-Cahn equation \linebreak $(-Δ)^{s/2} u = u-u^3$, $|u|<1$ in $\mathbb{R}^n$. For every $s\in (0,1)$ and dimension $n\geq 2$, we establish sharp energy estimates, density estimates, and the convergence of blow-downs to stable nonlocal $s$-minimal cones. As a consequence, we obtain a new classification result: if for some pair $(n,s)$, with $n\ge 3$, hyperplanes are the only stable nonlocal $s$-minimal cones in $\mathbb{R}^n\setminus\{0\}$, then every stable solution to the fractional Allen-Cahn equation in $\mathbb{R}^n$ is 1D, namely, its level sets are parallel hyperplanes. Combining this result with the classification of stable $s$-minimal cones in $\mathbb{R}^3\setminus\{0\}$ for $s\sim 1$ obtained by the authors in a recent paper, we give positive answers to the "stability conjecture" in $\mathbb{R}^3$ and to the "De Giorgi conjecture" in $\mathbb{R}^4$ for the fractional Allen-Cahn equation when the order $s\in (0,1)$ of the operator is sufficiently close to $1$.

math.AP↗

arXiv2021

A quantitative stability inequality for fractional capacities

The aim of this work is to show a non-sharp quantitative stability version of the fractional isocapacitary inequality. In particular, we provide a lower bound for the isocapacitary deficit in terms of the Fraenkel asymmetry. In addition, we provide the asymptotic behaviour of the $s$-fractional capacity when $s$ goes to $1$ and the stability of our estimate with respect to the parameter $s$.

math.AP↗

arXiv2020

Sharp quantitative stability for isoperimetric inequalities with homogeneous weights

We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequalities through the ABP method, we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromov's proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of Figalli-Maggi-Pratelli and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.

math.AP↗

arXiv2020

A quantitative stability estimate for the fractional Faber-Krahn inequality

We prove a quantitative version of the Faber-Krahn inequality for the first eigenvalue of the fractional Dirichlet-Laplacian of order s. This is done by using the so-called Caffarelli-Silvestre extension and adapting to the nonlocal setting a trick by Hansen and Nadirashvili. The relevant stability estimate comes with an explicit constant, which is stable as the fractional order of differentiability goes to 1.

math.AP↗

arXiv2018

Convex sets evolving by volume preserving fractional mean curvature flows

We consider the volume preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long time asymptotics approach round spheres. The proofs are based on apriori estimates on the inner and outer radii of the solutions.

math.AP↗

arXiv2018

On fractional Hardy inequalities in convex sets

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodecki\uı spaces of order $(s,p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every $1<p<\infty$ and $0<s<1$, with a constant which is stable as $s$ goes to $1$.

math.AP↗

arXiv2017

One-dimensional symmetry for the solutions of a three-dimensional water wave problem

We prove a one-dimensional symmetry result for a weighted Dirichlet-to-Neumann problem arising in a model for water waves in dimension 3. More precisely we prove that minimizers and bounded monotone solutions depend on only one Euclidean variable. The analogue of this result for the 2-dimensional case (and without weights) was established in an article by De La Llave and the third author. In this paper, a crucial ingredient in the proof is given by an energy estimate for minimizers obtained via a comparison argument.

math.AP↗

arXiv2017

Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s\sim 1$

We prove that half spaces are the only stable nonlocal $s$-minimal cones in $\mathbb{R}^3$, for $s\in(0,1)$ sufficiently close to $1$. This is the first classification result of stable $s$-minimal cones in dimension higher than two. Its proof can not rely on a compactness argument perturbing from $s=1$. In fact, our proof gives a quantifiable value for the required closeness of $s$ to $1$. We use the geometric formula for the second variation of the fractional $s$-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

math.AP↗

arXiv2016

$Γ$-convergence of variational functionals with boundary terms in Stein manifolds

Let $Ω$ be an open subset of a Stein manifold $Σ$ and let $M$ be its boundary. It is well known that $M$ inherits a natural contact structure. In this paper we consider a family of variational functionals $F_\varepsilon$ defined by the sum of two terms: a Dirichlet-type energy associated with a sub-Riemannian structure in $Ω$ and a potential term on the boundary $M$. We prove that the functionals $F_\varepsilon$ $Γ$-converge to the intrinsic perimeter in $M$ associated with its contact structure. Similar results have been obtained in the Euclidean space by Alberti, Bouchitté, Seppecher. We stress that already in the Euclidean setting the situation is not covered by the classical Modica-Mortola Theorem because of the presence of the boundary term. We recall also that Modica-Mortola type results (without a boundary term) have been proved in the Euclidean space for sub-Riemannian energies by Monti and Serra Cassano.

math.AP↗

arXiv2016

Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces

We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case. On the one hand, we establish universal $BV$-estimates in every dimension $n\ge 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has finite classical perimeter in $B_{1/2}$, with a universal bound. This nonlocal result is new even in the case of $s$-perimeters and its local counterpart (for classical stable minimal surfaces) was known only for simply connected two-dimensional surfaces immersed in $\mathbb R^3$. On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions $n=2,3$. More precisely, we show that a stable set in $B_R$, with $R$ large, is very close in measure to being a half space in $B_1$ ---with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane.

math.AP↗

arXiv2016

Neckpinch singularities in fractional mean curvature flows

In this paper we consider the evolution of sets by a fractional mean curvature flow. Our main result states that for any dimension $n > 2$, there exists an embedded surface in $\mathbb R^n$ evolving by fractional mean curvature flow, which developes a singularity before it can shrink to a point. When $n > 3$ this result generalizes the analogue result of Grayson for the classical mean curvature flow. Interestingly, when $n = 2$, our result provides instead a counterexample in the nonlocal framework to the well known Grayson Theorem, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point.

math.DG↗