arXiv subjects

Giovanni Franzina

Publications and source records attributed to Giovanni Franzina.

At least 19 recordsLinked to original sources

arXiv2026

Essential spectrum for the $p-$Laplacian

We introduce a variational notion of essential spectrum for the Dirichlet $p-$Laplacian. We then extend the classical Persson Theorem to this nonlinear setting. This result provides a geometric characterization of the bottom of the essential spectrum, in terms of the sharp $L^p$ Poincaré constant ``at infinity''. We also show that in the case $p=2$ our construction of the essential spectrum is perfectly consistent with the classical theory. Finally, as an example, we compute the full spectrum of the Dirichlet $p-$Laplacian on a rectilinear strip: it is purely essential, with no embedded eigenvalues. The arguments of the proofs are elementary and new already for the linear case $p=2$.

math.AP↗

arXiv2026

Posterior Bayesian Neural Networks with Dependent Weights

We consider fully connected and feedforward deep neural networks with dependent and possibly heavy-tailed weights, as introduced in [26], to address limitations of the standard Gaussian prior. It has been proved in [26] that, as the number of nodes in the hidden layers grows large, according to a sequential and ordered limit, the law of the output converges weakly to a Gaussian mixture. In this paper, we study the neural network through the lens of the posterior distribution with a Gaussian likelihood. If the random covariance matrix of the infinite-width limit is positive definite under the prior, we identify the posterior distribution of the output in the wide-width limit according to a sequential regime. Remarkably, we provide mild sufficient conditions to ensure the aforementioned invertibility of the random covariance matrix under the prior, thereby extending the results in [8]. Among our results, we present sufficient conditions on some model parameters (the activation function and the associated Lévy measures) which ensure that the sequential limits are independent of the order. We illustrate our findings with examples and numerical simulations.

stat.ML↗

arXiv2025

Numerical computation of generalized Wasserstein distances with applications to traffic model analysis

Generalized Wasserstein distances allow to quantitatively compare two continuous or atomic mass distributions with equal or different total mass. In this paper, we propose four numerical methods for the approximation of three different generalized Wasserstein distances introduced in the last years, giving some insights about their physical meaning. After that, we explore their usage in the context of the sensitivity analysis of differential models for traffic flow. The quantification of models sensitivity is obtained by computing the generalized Wasserstein distances between two (numerical) solutions corresponding to different inputs, including different boundary conditions.

math.AP↗

arXiv2024

Boundary vorticity of incompressible 2D flows

For a homogeneous incompressible 2D fluid confined within a bounded Lipschitz simply connected domain, homogeneous Neumann pressure boundary conditions are equivalent to a constant boundary vorticity. We investigate the rigidity of such conditions.

math.AP↗

arXiv2024

An overdetermined problem in 2D linearised hydrostatics

In two spatial dimensions, we discuss the relation between the solvability of Schiffer's overdetermined problem and the optimality, among sets of prescribed area, of the first eigenvalue in the buckling problem for a clamped plate and that of the first eigenvalue of the Stokes operator. For the latter, we deduce that the minimisers under area constraint that are smooth and simply connected must be discs from the fact that a pressureless velocity is a necessary condition of optimality.

math.AP↗

arXiv2023

Normal approximation of Random Gaussian Neural Networks

In this paper we provide explicit upper bounds on some distances between the (law of the) output of a random Gaussian NN and (the law of) a random Gaussian vector. Our results concern both shallow random Gaussian neural networks with univariate output and fully connected and deep random Gaussian neural networks, with a rather general activation function. The upper bounds show how the widths of the layers, the activation functions and other architecture parameters affect the Gaussian approximation of the ouput. Our techniques, relying on Stein's method and integration by parts formulas for the Gaussian law, yield estimates on distances which are indeed integral probability metrics, and include the total variation and the convex distances. These latter metrics are defined by testing against indicator functions of suitable measurable sets, and so allow for accurate estimates of the probability that the output is localized in some region of the space. Such estimates have a significant interest both from a practitioner's and a theorist's perspective.

math.PR↗

arXiv2023

Large time behavior of fractional porous media equation

Following the methodology of [Brasco and Volzone, Adv. Math. 2022], we study the long-time behavior for the signed Fractional Porous Medium Equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane-Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.

math.AP↗

arXiv2022

A non-local semilinear eigenvalue problem

For a non-local semilinear eigenvalue problem, we prove simplicity and isolation of the first eigenvalue with homogeneous Dirichlet boundary conditions on open sets supporting a suitable compact Sobolev embedding.

math.AP↗

arXiv2020

Existence and regularity for eddy currents system with non-smooth conductivity

We discuss the well-posedness of the 'transient eddy current' magneto-quasistatic approximation of Maxwell's initial value problem with bounded and measurable conductivity, with sources, on a domain. We prove existence and uniqueness of weak solutions, and we provide global Hoelder estimates for the magnetic part.

math.AP↗

arXiv2019

Positive solutions to the sublinear Lane-Emden equation are isolated

We prove that on a smooth bounded set, the positive least energy solution of the Lane-Emden equation with sublinear power is isolated. As a corollary, we obtain that the first $q-$eigenvalue of the Dirichlet-Laplacian is not an accumulation point of the $q-$spectrum, on a smooth bounded set. Our results extend to a suitable class of Lipschitz domains, as well.

math.AP↗

arXiv2019

An overview on constrained critical points of Dirichlet integrals

We consider a natural generalization of the eigenvalue problem for the Laplacian with homogeneous Dirichlet boundary conditions. This corresponds to look for the critical values of the Dirichlet integral, constrained to the unit $L^q$ sphere. We collect some results, present some counter-examples and compile a list of open problems.

math.AP↗

arXiv2018

Transmission conditions obtained by homogenisation

Given a bounded open set in $\mathbb{R}^n$, $n\ge 2$, and a sequence $(K_j)$ of compact sets converging to an $(n-1)$-dimensional manifold $M$, we study the asymptotic behaviour of the solutions to some minimum problems for integral functionals on $Ω\setminus K_j$, with Neumann boundary conditions on $\partial(Ω\setminus K_j)$. We prove that the limit of these solutions is a minimiser of the same functional on $Ω\setminus M$ subjected to a transmission condition on $M$, which can be expressed through a measure $μ$ supported on $M$. The class of all measures that can be obtained in this way is characterised, and the link between the measure $μ$ and the sequence $(K_j)$ is expressed by means of suitable local minimum problems.

math.AP↗

arXiv2018

Non-local Torsion functions and Embeddings

Given $s \in (0,1)$, we discuss the embedding of $\mathcal D^{s,p}_0(Ω)$ in $L^q(Ω)$. In particular, for $1\le q < p$ we deduce its compactness on all open sets $Ω\subset \mathbb R^N$ on which it is continuous. We then relate, for all q up the fractional Sobolev conjugate exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in $Ω$ in a suitable weak sense, for every open set $Ω$. The proofs make use of a non-local Hardy-type inequality in $\mathcal D^{s,p}_0(Ω)$, involving the fractional torsion function as a weight.

math.AP↗

arXiv2017

Schrödinger operators with negative potentials and Lane-Emden densities

We consider the Schrödinger operator $-Δ+V$ for negative potentials $V$, on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We show that the spectrum of $-Δ+V$ is positive, provided that $V$ is greater than a negative multiple of the logarithmic gradient of the solution to the Lane-Emden equation $-Δu=u^{q-1}$ (for some $1\le q< 2$). In this case, the ground state energy of $-Δ+V$ is greater than the first eigenvalue of the Dirichlet-Laplacian, up to an explicit multiplicative factor. This is achieved by means of suitable Hardy-type inequalities, that we prove in this paper.

math.AP↗

arXiv2017

A pathological example in Nonlinear Spectral Theory

We construct an open set $Ω\subset\mathbb{R}^N$ on which an eigenvalue problem for the $p-$Laplacian has not isolated first eigenvalue and the spectrum is not discrete. The same example shows that the usual Lusternik-Schnirelmann minimax construction does not exhaust the whole spectrum of this eigenvalue problem.

math.AP↗

arXiv2014

Existence of isoperimetric sets with densities "converging from below" in $\mathbb{R}^N$

In this paper, we consider the isoperimetric problem in the space $\mathbb{R}^N$ with density. Our result states that, if the density f is l.s.c. and converges to a positive limit at infinity, being smaller than this limit far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities positively answers a conjecture made in [10].

math.AP↗

arXiv2014

Convexity Properties of Dirichlet Integrals and Picone-type Inequalities

We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions.

math.AP↗

arXiv2013

Fractional p-eigenvalues

We discuss some basic properties of the eigenfunctions of a class of nonlocal operators whose model is the fractional p-Laplacian.

math.AP↗