arXiv subjects

Francesco Paolo Maiale

Publications and source records attributed to Francesco Paolo Maiale.

13 recordsLinked to original sources

arXiv2026

A note on harmonic polynomials on Heisenberg and Carnot groups

We study harmonic polynomial traces on Korányi spheres in the Heisenberg groups and polynomial trace filtrations on general Carnot groups. The homogeneous harmonic trace spaces are algebraically direct and complete on the Heisenberg sphere, but different homogeneous degrees are not generally orthogonal. In fact, on every $\mathbb H^n$ the degree-one and degree-three spaces fail to be orthogonal for every finite positive full-support surface measure. The decomposition thus uses the orthogonal increments of the cumulative harmonic-degree filtration. These increments give a complete Hilbert sum, have dimension $\binom{m+2n-1}{2n-1}$, and retain a $U(n)$-equivariant refinement. A constructive triangular recursion produces all homogeneous harmonic polynomials and their dimensions. On an arbitrary Carnot group, the same harmonic filtration always decomposes its closed span; equality with the full spherical $L^2$ space is identified as a separate harmonic-density condition. By contrast, the filtration by all polynomial traces is unconditionally complete by Stone--Weierstrass. We describe its dimensions through the vanishing ideal of the gauge sphere and compute them explicitly for every $\mathbb H^n$. Our main algebraic result is a Fischer-type decomposition. With $η_+^2(z,t)=|z|^2+4t$, we prove that $P_m(\mathbb H)=H_m(\mathbb H)\oplus η_+^2P_{m-2}(\mathbb H)$.

math.AP↗

arXiv2026

Perturbation from symmetry for semiconvex lower-order partial-wave nonlinear Dirac equations

We study strongly indefinite nonlinear Dirac functionals in the lowest partial-wave channel with radial coefficients in $\mathbb R^3$. The symmetric model has the exact power nonlinearity $F_0(x,ψ)=b(|x|)|ψ|^p/p$, where $2<p<3$ and the radial profile $b$ is bounded, continuous, and uniformly positive. Compactness in this finite-angular-mode spinor channel and radial approximation-number estimates give quadratic growth of the symmetric minimax levels. For a non-even localized perturbation of order $1<τ<p/2$, we impose a semiconvexity condition whose negative curvature is strictly smaller than the spectral gap. Together with the negative quadratic part and the convex exact-power core, this makes every negative spectral fiber uniformly strongly concave. Maximizing along that fiber reduces the problem to a $C^1$ path on the positive spectral space. The polynomial Chambers--Ghoussoub--Bolle deformation theorem then yields infinitely many high-energy critical points of the restricted partial-wave functional; under the channel-invariance hypothesis these are weak solutions of the full Dirac equation. Convex perturbing primitives are covered without a smallness restriction on their amplitude. The same reduction applies after an even Hermitian quadratic term is absorbed into a renormalized Dirac operator, provided the renormalized operator has a gap at zero and the remaining perturbation satisfies the corresponding semiconvexity bound.

math.AP↗

arXiv2026

Probabilistic Analysis of the Random Spectral Radius for a Matrix Family

We investigate joint spectral characteristics of a family of matrices $\mathcal F $, associated with products in the semigroup generated by $\mathcal F$. In the literature, extremal measures such as the well-known joint spectral radius and the lower spectral radius have been extensively studied. However, these measures fail to capture the typical growth rate of matrix products, focusing instead on the worst and best-case scenarios. Nevertheless, when examining, for instance, a switching dynamical system, a probabilistic rate of growth, which characterizes typical trajectories, emerges as a highly intriguing and significant measure. In this article, we study the random spectral radius, defined as the spectral radius of a length-$n$ product sampled at random from the semigroup according to a given probability measure. We establish asymptotic results, namely a Law of Large Numbers and a Central Limit Theorem, for diagonal (equivalently, commuting), upper- or lower-triangular, and small perturbations of diagonal matrices. Beyond recovering the correct scaling, we obtain exact closed-form expressions for the limiting value and variance, which may be useful in numerical applications requiring precise constants. In the coalescence regime, where the leading eigenvalues merge, the limiting distribution is non-Gaussian: it is given by the maximum of a correlated Gaussian vector with explicit covariance structure. This phenomenon governs phase transitions between distinct growth regimes in switching systems.

math.DS↗

arXiv2025

Approximation bounds for norm constrained deep neural networks

This paper studies the approximation capacity of neural networks with an arbitrary activation function and with norm constraint on the weights. Upper and lower bounds on the approximation error of these networks are computed for smooth function classes. The upper bound is proven by first approximating high-degree monomials and then generalizing it to functions via a partition of unity and Taylor expansion. The lower bound is derived through the Rademacher complexity of neural networks. A probabilistic version of the upper bound is also provided by considering neural networks with randomly sampled weights and biases. Finally, it is shown that the assumption on the regularity of the activation function can be significantly weakened without worsening the approximation error, and the approximation upper bound is validated with numerical experiments.

math.NA↗

arXiv2024

Gripenberg-like algorithm for the lower spectral radius

This article presents an extended algorithm for computing the lower spectral radius of finite, non-negative matrix sets. Given a set of matrices $\mathcal{F} = \{A_1, \ldots, A_m\}$, the lower spectral radius represents the minimal growth rate of sequences in the product semigroup generated by $\mathcal{F}$. This quantity is crucial for characterizing optimal stable trajectories in discrete dynamical systems of the form $x_{k+1} = A_{i_k} x_k$, where $A_{i_k} \in \mathcal{F}$ for all $k \ge 0$. For the well-known joint spectral radius (which represents the highest growth rate), a famous algorithm providing suitable lower and upper bounds and able to approximate the joint spectral radius with arbitrary accuracy was proposed by Gripenberg in 1996. For the lower spectral radius, where a lower bound is not directly available (contrarily to the joint spectral radius), this computation appears more challenging. Our work extends Gripenberg's approach to the lower spectral radius computation for non-negative matrix families. The proposed algorithm employs a time-varying antinorm and demonstrates rapid convergence. Its success is related to the property that the lower spectral radius can be obtained as a Gelfand limit, which was recently proved in Guglielmi and Zennaro (2020). Additionally, we propose an improvement to the classical Gripenberg algorithm for approximating the joint spectral radius of arbitrary matrix sets.

math.NA↗

arXiv2024

The boundary Harnack principle on optimal domains

We give a short and self-contained proof of the Boundary Harnack inequality for a class of domains satisfying some geometric conditions given in terms of a state function that behaves as the distance function to the boundary, is subharmonic inside the domain and satisfies some suitable estimates on the measure of its level sets. We also discuss the applications of this result to some shape optimization and free boundary problems.

math.AP↗

arXiv2023

$C^{1,α}$-rectifiability in low codimension in Heisenberg groups

A natural notion of higher order rectifiability is introduced for subsets of Heisenberg groups $\mathbb{H}^n$ in terms of covering a set almost everywhere by a countable union of $(\mathbf{C}_H^{1,α},\mathbb{H})$-regular surfaces, for some $0 < α\leq 1$. We prove that a sufficient condition for $C^{1,α}$-rectifiability of low-codimensional subsets in Heisenberg groups is the almost everywhere existence of suitable approximate tangent paraboloids.

math.MG↗

arXiv2022

Regularity of the optimal sets for a class of integral shape functionals

We prove {the first} regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain $Ω$ is obtained as the integral of a cost function $j(u,x)$ depending on the solution $u$ of a certain PDE problem on $Ω$. The main feature of these functionals is that the minimality of a domain $Ω$ cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions $j(u,x)=-g(x)u+Q(x)$, where $u$ is the solution of the PDE $-Δu=f$ with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal $u$ from the inwards/outwards optimality of $Ω$ and then we use the stability of $Ω$ with respect to variations with smooth vector fields in order to study the blow-up limits of the state function $u$. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose $\partialΩ$ into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of $\partialΩ$ we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove $C^\infty$ regularity of the regular part of the free boundary when the data are smooth.

math.AP↗

arXiv2021

Epsilon-regularity for the solutions of a free boundary system

This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions $u$ and $v$, and a domain $Ω$; with $u$ and $v$ being both positive in $Ω$, vanishing simultaneously on $\partialΩ$ and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on $\partialΩ$. Precisely, we consider solutions $u, v \in C(B_1)$ of $$-Δu= f \quad\text{and} \quad-Δv=g\quad\text{in}\quad Ω=\{u>0\}=\{v>0\}\ ,\qquad \frac{\partial u}{\partial n}\frac{\partial v}{\partial n}=Q\quad\text{on}\quad \partialΩ\cap B_1.$$ Our main result is an epsilon-regularity theorem for viscosity solutions of this free boundary system. We prove a partial Harnack inequality near flat points for the couple of auxiliary functions $\sqrt{uv}$ and $\frac12(u+v)$. Then, we use the gained space near the free boundary to transfer the improved flatness to the original solutions. Finally, using the partial Harnack inequality, we obtain an improvement-of-flatness result, which allows to conclude that flatness implies $C^{1,α}$ regularity.

math.AP↗

arXiv2021

Shape optimization problems in control form

We consider a shape optimization problem written in the optimal control form: the governing operator is the $p$-Laplacian in the Euclidean space $\R^d$, the cost is of an integral type, and the control variable is the domain of the state equation. Conditions that guarantee the existence of an optimal domain will be discussed in various situations. It is proved that the optimal domains have a finite perimeter and, under some suitable assumptions, that they are open sets. A crucial difference is between the case $p>d$, where the existence occurs under very mild conditions, and the case $p\le d$, where additional assumptions have to be made on the data.

math.OC↗

arXiv2020

Characterizations of $k$-rectifiability in homogenous groups

A well known notion of $k$-rectifiable set can be formulated in any metric space using Lipschitz images of subsets of $\mathbb{R}^k$. We prove some characterizations of $k$-rectifiability, when the metric space is an arbitrary homogeneous group. In particular, we show that the a.e. existence of the $(k,\mathbb{G})$-approximate tangent group implies $k$-rectifiability.

math.MG↗

arXiv2020

Shape optimization problems for functionals with a boundary integral

We consider shape optimization problems for general integral functionals of the calculus of variations that may contain a boundary term. In particular, this class includes optimization problems governed by elliptic equations with a Robin condition on the free boundary. We show the existence of an optimal domain under rather general assumptions and we study the cases when the optimal domains are open sets and have a finite perimeter.

math.OC↗

arXiv2020

Optimal one-dimensional structures for the principal eigenvalue of two-dimensional domains

A shape optimization problem arising from the optimal reinforcement of a membrane by means of one-dimensional stiffeners or from the fastest cooling of a two-dimensional object by means of ``conducting wires'' is considered. The criterion we consider is the maximization of the first eigenvalue and the admissible classes of choices are the one of one-dimensional sets with prescribed total length, or the one where the constraint of being connected (or with an a priori bounded number of connected components) is added. The corresponding relaxed problems and the related existence results are described.

math.AP↗