arXiv subjects

Stefano Buccheri

Publications and source records attributed to Stefano Buccheri.

8 recordsLinked to original sources

arXiv2026

Well-posedness results for superlinear Fokker-Planck equations

In this manuscript we deal with a class of nonlinear Fokker-Planck equations with the following structure \[ \partial_t u - ÷\big(M\nabla u+ E h(u)\big)=0, \] with $M$ a bounded elliptic matrix, $E$ a vector field in a suitable Lebesgue space, and $h(u)$ featuring a superlinear growth for $u$ large. We provide existence results of $C([0,T),L^1)$ distributional solutions to initial-boundary value problems related to the equation above together with some qualitative properties of solutions.

math.AP↗

arXiv2025

Recovering functions via doubly homogeneous nonlocal gradients

We investigate a class of nonlocal gradients featuring distinct homogeneities at zero and infinity. We establish a representation formula for such doubly homogeneous operators and derive associated Sobolev-type inequalities. We also propose open questions linked to our results, suggesting directions for future research inspired by the work of Haim Brezis.

math.FA↗

arXiv2024

A metric counterpart of the Gu-Yung formula

In this note we consider a generalisation to the metric setting of the recent work [Gu-Yung, JFA 281 (2021), 109075]. In particular, we show that under relatively weak conditions on a metric measure space $(X,d,ν)$, it holds true that \[ \bigg[ \frac{u(x)-u(y)}{d(x,y)^{\frac{s}{p}}} \bigg]_{L^p_w(X \times X, ν\otimes ν)} \approx \| u \|_{L^p(X,ν)}, \] where $s$ is a generalised dimension associated to $X$ and $[\cdot]_{L^p_w}$ is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal.

math.FA↗

arXiv2023

The best approximation of a given function in $L^2$-norm by Lipschitz functions with gradient constraint

The starting point of this paper is the study of the asymptotic behavior, as $p\to\infty$, of the following minimization problem $$ \min\left\{\frac1{p}\int|\nabla v|^{p}+\frac12\int(v-f)^2 \,, \quad \ v\in W^{1,p} (Ω)\right\}. $$ We show that the limit problem provides the best approximation, in the $L^2$-norm, of the datum $f$ among all Lipschitz functions with Lipschitz constant less or equal than one. Moreover such approximation verifies a suitable PDE in the viscosity sense. After the analysis of the model problem above, we consider the asymptotic behavior of a related family of nonvariational equations and, finally, we also deal with some functionals involving the $(N-1)$-Hausdorff measure of the jump set of the function.

math.AP↗

arXiv2022

Viscosity solutions for nonlocal equations with space-dependent operators

We consider a class of elliptic and parabolic problems, featuring a specific nonlocal operator of fractional-laplacian type, where integration is taken on variable domains. Both elliptic and parabolic problems are proved to be uniquely solvable in the viscosity sense. Moreover, some spectral properties of the elliptic operator are investigated, proving existence and simplicity of the first eigenvalue. Eventually, parabolic solutions are proven to converge to the corresponding limiting elliptic solution in the long-time limit.

math.AP↗

arXiv2021

An Agmon-Allegretto-Piepenbrink principle for Schroedinger operators

We prove that each Borel function $V : Ω\to [-\infty, +\infty]$ defined on an open subset $Ω\subset \mathbb{R}^{N}$ induces a decomposition $Ω= S \cup \bigcup_{i} D_{i}$ such that every function in $W^{1,2}_{0}(Ω) \cap L^{2}(Ω; V^{+} dx)$ is zero almost everywhere on $S$ and existence of nonnegative supersolutions of $-Δ+ V$ on each component $D_{i}$ yields nonnegativity of the associated quadratic form $\int_{D_{i}} (|\nabla ξ|^2+Vξ^2)$.

math.AP↗

arXiv2021

A singular Schrödinger-Maxwell system

In this paper we are concerned with existence of positive solutions for a Schrödinger-Maxwell system with singular or strongly-singular terms. We overcome the difficulty given by the singular terms through an approximation scheme and controlling the approximated sequences of solutions with suitable barriers from above and from below. Besides this, in some particular case, we show that the unique energy solution of the singular system is a saddle point of a suitable functional.

math.AP↗

arXiv2019

Gradient estimates for nonlinear elliptic equations with first order terms

We study existence and Lorentz regularity of distributional solutions to elliptic equations with either a convection or a drift first order term. The presence of such a term makes the problem not coercive. The main tools are pointwise estimates of the rearrangements of both the solution and its gradient.

math.AP↗