arXiv subjects

Paolo Albano

Publications and source records attributed to Paolo Albano.

14 recordsLinked to original sources

arXiv2026

Long-time behavior of generalized gradient flows of solutions to Hamilton-Jacobi equations

We study the long-time behavior of the generalized gradient flow associated with solutions of the critical Hamilton-Jacobi equation for mechanical Hamiltonians on the flat torus. For any semiconcave function, we show that its critical set -- points whose superdifferential contains the zero vector -- acts as an approximate attractor for the flow. When the function is a solution of the critical equation, the critical set decomposes into regular and singular parts, and we establish a dichotomy describing which part trajectories approach as $t \to \infty$. Our analysis uses limiting occupational measures, a class of invariant measures capturing the asymptotic distribution of the flow. An essential ingredient is a complete proof of the global invariance of the singular set, a result previously announced by Albano (2016) but not fully established.

math.AP↗

arXiv2026

Obstacles and Singularities of Riemannian Distance Functions

We study the distance function from a point target in the complement of a compact obstacle endowed with a smooth Riemannian metric. We prove that the obstacle necessarily generates singularities of the distance function: every sufficiently high level set contains a singular point. We also show that every singular point outside the obstacle belongs to a nontrivial Lipschitz arc of singularities, thereby extending to the constrained setting classical propagation results for Hamilton--Jacobi equations. Finally, we provide examples showing that these results are essentially sharp, including a nonconvex obstacle for which the distance function is differentiable at every boundary point.

math.AP↗

arXiv2025

Evalita-LLM: Benchmarking Large Language Models on Italian

We describe Evalita-LLM, a new benchmark designed to evaluate Large Language Models (LLMs) on Italian tasks. The distinguishing and innovative features of Evalita-LLM are the following: (i) all tasks are native Italian, avoiding issues of translating from Italian and potential cultural biases; (ii) in addition to well established multiple-choice tasks, the benchmark includes generative tasks, enabling more natural interaction with LLMs; (iii) all tasks are evaluated against multiple prompts, this way mitigating the model sensitivity to specific prompts and allowing a fairer and objective evaluation. We propose an iterative methodology, where candidate tasks and candidate prompts are validated against a set of LLMs used for development. We report experimental results from the benchmark's development phase, and provide performance statistics for several state-of-the-art LLMs.

cs.CL↗

arXiv2022

Partial Lipschitz regularity of the minimum time function for sub-Riemannian control systems

In Euclidean space of dimension 2 or 3, we study a minimum time problem associated with a system of real-analytic vector fields satisfying Hörmander's bracket generating condition, where the target is a nonempty closed set. We show that, in dimension 2, the minimum time function is locally Lipschitz continuous while, in dimension 3, it is Lipschitz continuous in the complement of a set of measure zero. In particular, in both cases, the minimum time function is a.e. differentiable on the complement of the target. In dimension 3, in general, there is no hope to have the same regularity result as in dimension 2. Indeed, examples are known where the minimum time function fails to be locally Lipschitz continuous.

math.OC↗

arXiv2022

An interpolation problem in the Denjoy-Carleman classes

Inspired by some iterative algorithms useful for proving the real analyticity (or the Gevrey regularity) of a solution of a linear partial differential equation with real-analytic coefficients, we consider the following question. Given a smooth function defined on $[a,b]\subset\mathbb{R}$ and given an increasing divergent sequence $d_n$ of positive integers such that the derivative of order $d_n$ of $f$ has a growth of the type $M_{d_n}$, when can we deduce that $f$ is a function in the Denjoy-Carleman class $C^M([a,b])$? We provide a positive result, and we show that a suitable condition on the gaps between the terms of the sequence $d_n$ is needed.

math.AP↗

arXiv2021

On the extension problem for semiconcave functions with fractional modulus

Consider a locally Lipschitz function $u$ on the closure of a possibly unbounded open subset $Ω$ of $\mathbb{R}^n$ with $C^{1,1}$ boundary. Suppose $u$ is semiconcave on $\overline Ω$ with a fractional semiconcavity modulus. Is it possible to extend $u$ in a neighborhood of any boundary point retaining the same semiconcavity modulus? We show that this is indeed the case and we give two applications of this extension property. First, we derive an approximation result for semiconcave functions on closed domains. Then, we use the above extension property to study the propagation of singularities of semiconcave functions at boundary points.

math.AP↗

arXiv2021

The distance function in the presence of an obstacle

We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a fractional modulus of order one half and that, near the obstacle, this regularity is optimal. Then, in the Euclidean setting, we prove that the singularities of the distance function propagate, in the sense that each singular point belongs to a nontrivial singular continuum. Finally, we investigate the lack of differentiability of the distance function when a convex obstacle is present.

math.OC↗

arXiv2019

Generation of singularities from the initial datum for Hamilton-Jacobi equations

We study the generation of singularities from the initial datum for a solution of the Cauchy problem for a class of Hamilton-Jacobi equations of evolution. For such equations, we give conditions for the existence of singular generalized characteristics starting at the initial time from a given point of the domain, depending on the properties of the proximal subdifferential of the initial datum in a neighbourhood of that point.

math.AP↗

arXiv2018

Partial regularity of minimizers for real-analytic sub-Riemannian metrics

In the real-analytic setting, we show that all sub-Riemannian minimizers (parametrized by the arc-length) are real-analytic everywhere except an at most countable non-dense set. In particular, non-analyticity may occur only on a set of measure zero of the domain of definition of a sub-Riemannian minimizer. Furthermore, we provide a geometrical condition which implies the absence of the so called strictly abnormal minimizers. In particular, under such condition, all sub-Riemannian minimizers (parametrized by the arc-length) are real-analytic.

math.MG↗

arXiv2018

Partial regularity for solutions to subelliptic eikonal equations

On a bounded domain $Ω$ in euclidean space $\mathbb{R}^n$, we study the homogeneous Dirichlet problem for the eikonal equation associated with a system of smooth vector fields, which satisfies Hörmander's bracket generating condition. We prove that the solution is smooth in the complement of a closed set of Lebesgue measure zero.

math.OC↗

arXiv2017

Regularity results for the minimum time function with Hörmander vector fields

In a bounded domain of $\mathbb{R}^n$ with smooth boundary, we study the regularity of the viscosity solution, $T$, of the Dirichlet problem for the eikonal equation associated with a family of smooth vector fields $\{X_1,\ldots ,X_N\}$, subject to Hörmander's bracket generating condition. Due to the presence of characteristic boundary points, singular trajectories may occur in this case. We characterize such trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. We then prove that the local Lipschitz continuity of $T$, the local semiconcavity of $T$, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied when the characteristic set of $\{X_1,\ldots ,X_N\}$ is a symplectic manifold. We apply our results to Heisenberg's and Martinet's vector fields.

math.OC↗

arXiv2016

Analytic Hypoellipticity for Sums of Squares and the Treves Conjecture

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson-Treves stratification are symplectic. We produce a model operator, $ P_{1} $, having a single symplectic stratum and prove that it is Gevrey $ s_{0} $ hypoelliptic and not better. See Theorem \ref{th:1} for a definition of $ s_{0} $. We also show that this phenomenon has a microlocal character. We point out explicitly that this is a counterexample to the sufficient part of Treves conjecture and not to the necessary part, which is still an open problem.

math.AP↗

arXiv2011

Singular gradient flow of the distance function and homotopy equivalence

It is a generally shared opinion that significant information about the topology of a bounded domain $Ω$ of a riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partialΩ}$, %, $d:Ω\rightarrow [0,\infty [$, from the boundary of $Ω$. To confirm such an idea we propose an approach based on the invariance of the singular set of the distance function with respect to the generalized gradient flow of of $d_{\partialΩ}$. As an application, we deduce that such a singular set has the same homotopy type as $Ω$.

math.AP↗

arXiv2006

Analytic Hypoellipticity in the Presence of Lower Order Terms

We consider a second order operator with analytic coefficients whose principal symbol vanishes exactly to order two on a symplectic real analytic manifold. We assume that the first (non degenerate) eigenvalue vanishes on a symplectic submanifold of the characteristic manifold. In the $C^\infty$ framework this situation would mean a loss of 3/2 derivatives. We prove that this operator is analytic hypoelliptic. The main tool is the FBI transform. A case in which $C^\infty$ hypoellipticity fails is also discussed.

math.AP↗