arXiv subjects

Emanuele Tasso

Publications and source records attributed to Emanuele Tasso.

11 recordsLinked to original sources

arXiv2026

Approximation of symmetric total variation on point clouds

The paper investigates the approximation of the symmetric Total Variation functional on graphs. Such an approximation is given in terms of a discrete and symmetric finite difference model defined on point clouds obtained by randomly sampling a reference probability measure. We identify suitable scalings of the point distribution that guarantee an almost surely $Γ$-convergence to an anisotropic weighted symmetric Total Variation.

math.AP↗

arXiv2025

Besicovitch-Federer projection theorem for measures

In this paper we establish a Besicovitch-Federer type projection theorem for general measures. Specifically, let $μ$ be a finite Borel measure on $\mathbb{R}^n$ and let $0 < m < n$ be an integer. We show that, under the sole assumption that the slice $μ\cap W$ is atomic for a typical $(n-m)$-plane $W \subset \mathbb{R}^n$, pure unrectifiability can be characterized simultaneously by the $μ$-almost everywhere injectivity of the orthogonal projection $π_V \colon \mathbb{R}^n \to V$ and by the singularity of the projected measure for a typical $m$-plane $V$. In particular, no assumption on $π_Vμ$ is required a priori. This yields a new rectifiability criterion via slicing for Radon measures. The result is new even in the classical setting of Hausdorff measures, and it further extends to arbitrary locally compact metric spaces endowed with a generalized family of projections.

math.CA↗

arXiv2025

On De Giorgi's Conjecture of Nonlocal approximations for free-discontinuity problems: The symmetric gradient case

We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class of continuous finite-difference approximants, we show the compactness of deformations with equibounded energies, as well as their Gamma-convergence. The compactness analysis is a crucial hurdle, which we overcome by generalizing a Fréchet-Kolmogorov approach previously introduced by two of the authors. A second essential difficulty is the identification of the limiting space of admissible deformations, since a control on the directional variations is, a priori, only available in average. A limiting representation in GSBD is eventually established via a novel characterization of this space.

math.AP↗

arXiv2025

Effective theories for incompressible magnetoelastic shallow shells

We characterize the asymptotic behaviour, in the sense of $Γ$-convergence, of a thin magnetoelastic shallow shell. The compactness is achieved up to rigid motions. For deformations, it relies on an approximation by rigid movements, whereas for magnetizations it is based on a careful consideration of the geometry of the deformed domain. The result is obtained by a combination of variational methods ($Γ$-convergence) with degree theory, fixed-point and geometrical arguments. The proof strategy relies on an adaptation of an analogous result for incompressible magnetoelastic plates from M. Bresciani in arXiv:2007.14122 and an application of results by I.Velcic on elastic shallow shells in arXiv:1102.2647.

math.AP↗

arXiv2025

Rectifiability of a class of integralgeometric measures and applications

We resolve a long-standing open problem posed by Federer concerning the rectifiability of the integral geometric measure with exponent p >1, thereby settling a question that has persisted since its formulation. While the main theorem is unchanged from previous versions, the exposition and applications have been substantially revised to highlight the result's consequences for Vitushkin's conjecture on analytic capacity and removability in the complex plane. As an application, we establish two novel results related to Vitushkin's conjecture: in a multi-scale setting, we provide an affirmative answer for sets with finite integral geometric measure within regimes of Favard length behavior at small scales not previously addressed; and in a single-scale framework, we extend the Besicovitch-Federer projection theorem beyond the classical sigma-finite setting, namely for planar sets intersecting a typical line in finitely many points. The rectifiability criterion for general Radon measures via slicing, included in earlier versions, has been removed and will appear in separate work.

math.MG↗

arXiv2025

Non-local non-homogeneous phase transitions: regularity of optimal profiles and sharp-interface limit

We provide a novel sharp-interface analysis via Gamma-convergence for a non-local and non-homogeneous diffuse-interface model for phase transitions, featuring an interplay between a non-local interaction kernel and a spatially dependent double-well potential. This interaction requires the development of new strategies both for the Gamma-liminf inequality and for the construction of recovery sequences. A key element of our approach is an asymptotic calibration, used to establish the Gamma-liminf lower bound. The study of the optimality of the lower bound hinges upon a novel analysis of the regularity dependence of one-dimensional optimal profiles on a family of parameters. In particular, we show how such regularity is influenced by the singularity of the interaction kernel at the origin, providing a precise and previously unexplored link between the two. Our results rely solely on the assumption of Hölder continuity for the moving wells, and also account for the compactness of sequences with equibounded energies.

math.AP↗

arXiv2023

Generalized bounded deformation in non-Euclidean settings

We introduce a new space of generalized functions of bounded deformation $GBD_{F}$, made of functions u whose one-dimensional slice $u(γ) \cdot \dotγ$ has bounded variation in a generalized sense for all curves $γ$ solution of the second order ODE $\ddotγ = F(γ, \dotγ)$ for a fixed field F. For $u \in GBD_{F}$ we study the structure of the jump set in connection its slices and prove the existence of a curvilinear approximate symmetric gradient. With a particular choice of F in terms of the Christoffel symbols of a Riemannian manifold M, we are able to define and recover similar properties for a space of 1-forms on M which have generalized bounded deformation in a suitable sense.

math.AP↗

arXiv2022

A general criterion for jump set slicing and applications

In this paper a novel criterion for the slicing of the jump set of a function is provided, which bypasses the codimension-one and the parallelogram law techniques developed in $BD$-spaces. The approach builds upon a recent rectifiability result of integralgeometric measures and is further applied to the study of the structure of the jump set of functions with generalized bounded deformation in a Riemannian setting.

math.AP↗

arXiv2021

A new proof of compactness in G(S)BD

We prove a compactness result in GBD which also provides a new proof of the compactness theorem in GSBD, due to Chambolle and Crismale [5, Theorem 1.1]. Our proof is based on a Fréchet-Kolmogorov compactness criterion and does not rely on Korn or Poincaré-Korn inequalities.

math.AP↗

arXiv2021

Brittle fracture in linearly elastic plates

In this work we derive by Gamma-convergence techniques a model for brittle fracture linearly elastic plates. Precisely, we start from a brittle linearly elastic thin film with positive thickness $ρ$ and study the limit as $ρ$ tends to 0. The analysis is performed with no a priori restrictions on the admissible displacements and on the geometry of the fracture set. The limit model is characterized by a Kirchhoff-Love type of structure.

math.AP↗

arXiv2019

Energy-dissipation balance of a smooth moving crack

In this paper we provide necessary and sufficient conditions in order to guarantee the energy-dissipation balance of a Mode III crack, growing on a prescribed smooth path. Moreover, we characterize the singularity of the displacement near the crack tip, generalizing the result in [S.Nicaise, A.M.Sandig - \textit{J. Math. Anal. Appl.} 2007] valid for straight fractures.

math.AP↗