arXiv subjects

Dario Reggiani

Publications and source records attributed to Dario Reggiani.

6 recordsLinked to original sources

arXiv2026

Fractured membranes under determinant constraints: the case of cohesive surface energies

This paper is devoted to the variational derivation of reduced models for elastic membranes with fracture under constraints on the determinant of the deformation gradient. We consider two physically relevant settings: the orientation-preserving regime, in which the deformation is required to preserve orientation locally ($\det \nabla u > 0$), and the incompressible regime, in which the deformation preserves volume ($\det \nabla u = 1$). In both cases, the surface energy density is allowed to depend on the jump amplitude, thus encompassing cohesive fracture models with activation threshold. The main technical contribution is the construction of recovery sequences that simultaneously satisfy the determinant constraint and optimize the surface energy. This is achieved through a combination of $C^\infty$ diffeomorphisms converging to the identity (which rotate the normal to the jump set so as to minimize the reduced surface energy), and a new smooth approximation result for $GSBV^p$ functions.

math.AP↗

arXiv2026

A distributional approach to nonlocal curvature flows

In \cite{CMP17} a novel distributional approach has been introduced to provide a well-posed formulation of a class of crystalline mean curvature flows. In this paper, such an approach is extended to the nonlocal setting. Applications include the fractional mean curvature flow and the Minkowski flow; i.e., the geometric flow generated by the $(N-1)$-dimensional Minkowski pre-content.

math.AP↗

arXiv2024

From discrete to continuum in the helical XY-model: emergence of chirality transitions in the $S^1$ to $S^2$ limit

We analyze the discrete-to-continuum limit of a frustrated ferromagnetic/anti-ferromagnetic $\mathbb{S}^2$-valued spin system on the lattice $λ_n\mathbb{Z}^2$ as $λ_n\to 0$. For $\mathbb{S}^2$ spin systems close to the Landau-Lifschitz point (where the helimagnetic/ferromagnetic transition occurs), it is well established that for chirality transitions emerge with vanishing energy. Inspired by recent work on the $N$-clock model, we consider a spin model where spins are constrained to $k_n$ copies of $\mathbb{S}^1$ covering $\mathbb{S}^2$ as $n\to\infty$. We identify a critical energy-scaling regime and a threshold for the divergence rate of $k_n\to+\infty$, below which the $Γ$-limit of the discrete energies capture chirality transitions while retaining an $\mathbb{S}^2$-valued energy description in the continuum limit.

math.AP↗

arXiv2023

Geometric rigidity for incompatible fields in the multi-well case and an application to strain-gradient plasticity

We derive a quantitative rigidity estimate for a multi-well problem in nonlinear elasticity with dislocations. Precisely, we show that the $L^{1^{*}}$-distance of a possibly incompatible strain field $β$ from a single well is controlled in terms of the $L^{1^{*}}$-distance from a finite set of wells, of ${\rm curl}β$, and of ${\rm div}β$. As a consequence, we derive a strain-gradient plasticity model as $Γ$-limit of a nonlinear finite dislocation model, containing a singular perturbation term accounting for the divergence of the strain field. This can also be seen as a generalization of the result of (Alicandro et al. 2018) to the case of incompatible vector fields.

math.AP↗

arXiv2023

Lower semicontinuity and relaxation for free discontinuity functionals with non-standard growth

A lower semicontinuity result and a relaxation formula for free discontinuity functionals with non-standard growth in the bulk energy are provided. Our analysis is based on a non-trivial adaptation of the blow-up (Ambrosio 1994) and of the global method for relaxation (Bouchitté et al. 1998) to the setting of generalized special function of bounded variation with Orlicz growth. Key tools developed in this paper are an integral representation result and a Poincaré inequality under non-standard growth.

math.AP↗

arXiv2022

Brittle membranes in finite elasticity

This work is devoted to the variational derivation of a reduced model for brittle membranes in finite elasticity. The main mathematical tools we develop for our analysis are: (i) a new density result in $GSBV^{p}$ of functions satisfying a maximal-rank constraint on the subgradients, which can be approximated by $C^{1}$-local immersions on regular subdomains of the cracked set, and (ii) the construction of recovery sequences by means of suitable $W^{1,\infty}$ diffeomorphisms mapping the regular subdomains onto the fractured configuration.

math.AP↗