arXiv subjects

Davide Donati

Publications and source records attributed to Davide Donati.

10 recordsLinked to original sources

arXiv2026

Structured deformations for energies with general surface terms

We develop a variational theory of structured deformations for energies whose surface densities satisfy general growth conditions. This requires a formulation in the generalised space ${\rm GBV}_\star$, introduced by Dal Maso and Toader, which is the natural setting for surface energies that are linear near the origin and bounded at infinity. In this framework, we prove three main results: an approximation theorem for structured deformations, an integral representation theorem for abstract lower semicontinuous functionals, and an explicit representation formula for relaxed energies. The proofs rely on new density results for functions of bounded variation and on Poincar\'e-type inequalities tailored to ${\rm GBV}_\star$. Our results extend the applicability of structured deformations to cohesive models in fracture mechanics.

math.AP↗

arXiv2026

$\Gamma$-convergence of convolution-type functionals for free discontinuity problems

We prove compactness with respect to $\Gamma$-convergence for a general class of non-local energies modelled after the ones considered in [Gobbino, CPAM (1998)]. We give an integral representation result for the limits, which are free discontinuity functionals defined on the space of generalised special functions of bounded variation. We then characterise the bulk and surface energy densities of the obtained limits by means of minimisation problems on small cubes for the approximating energies.

math.AP↗

arXiv2026

${\Gamma}$-convergence and homogenisation for free discontinuity functionals with linear growth in the space of functions with bounded deformation

We study the $\Gamma$-convergence of sequences of free discontinuity functionals with linear growth defined in the space ${\rm BD}$ of functions with bounded deformation. We prove a compactness result with respect to $\Gamma$-convergence and outline the main properties of the $\Gamma$-limits, which lead to an integral representation result. The corresponding integrands are obtained by taking limits of suitable minimisation problems on small cubes. These results are then used to study the deterministic and stochastic homogenisation problem for a large class of free discontinuity functionals defined in ${\rm BD}$.

math.AP↗

arXiv2025

A matrix-valued measure associated to the derivatives of a function of generalised bounded deformation

We associate to every function $u\in GBD(\Omega)$ a measure $\mu_u$ with values in the space of symmetric matrices, which generalises the distributional symmetric gradient $Eu$ defined for functions of bounded deformation. We show that this measure $\mu_u$ admits a decomposition as the sum of three mutually singular matrix-valued measures $\mu^a_u$, $\mu^c_u$, and $\mu^j_u$, the absolutely continuous part, the Cantor part, and the jump part, as in the case of $BD(\Omega)$ functions. We then characterise the space $GSBD(\Omega)$, originally defined only by slicing, as the space of functions $u\in GBD(\Omega)$ such that $\mu^c_u=0$.

math.FA↗

arXiv2025

Singular perturbations models in phase transitions for anisotropic higher-order materials

We discuss a model for phase transitions in which a double-well potential is singularly perturbed by possibly several terms involving different, arbitrarily high orders of derivation. We study by $\Gamma$-convergence the asymptotic behaviour as $\varepsilon\to 0$ of the functionals \begin{equation*} F_\varepsilon(u):=\int_\Omega \Bigl[\frac{1}{\varepsilon}W(u)+\sum_{\ell=1}^{k}q_\ell\varepsilon^{2\ell-1}|\nabla^{(\ell)}u|_\ell^2\Bigr]\,dx, \qquad u\in H^k(\Omega), \end{equation*} for fixed $k>1$ integer, addressing also to the case in which the coefficients $q_1,...,q_{k-1}$ are negative and $|\cdot|_\ell$ is any norm on the space of symmetric $\ell$-tensors for each $\ell\in\{1,...,k\}$. The negativity of the coefficients leads to the lack of a priori bounds on the functionals; such issue is overcome by proving a nonlinear interpolation inequality. With this inequality at our disposal, a compactness result is achieved by resorting to the recent paper [10]. A further difficulty is the presence of general tensor norms which carry anisotropies, making standard slicing arguments not suitable. We prove that the $\Gamma$-limit is finite only on sharp interfaces and that it equals an anisotropic perimeter, with a surface energy density described by a cell formula.

math.AP↗

arXiv2024

Homogenisation of vectorial free-discontinuity functionals with cohesive type surface terms

The results on $\Gamma$-limits of sequences of free-discontinuity functionals with bounded cohesive surface terms are extended to the case of vector-valued functions. In this framework, we prove an integral representation result for the $\Gamma$-limit, which is then used to study deterministic and stochastic homogenisation problems for this type of functionals.

math.AP↗

arXiv2024

Higher-order singular perturbation models for phase transitions

Variational models of phase transitions take into account double-well energies singularly perturbed by gradient terms, such as the Cahn-Hilliard free energy. The derivation by $\Gamma$-convergence of a sharp-interface limit for such energy is a classical result by Modica and Mortola. We consider a singular perturbation of a double-well energy by derivatives of order $k$, and show that we still can describe the limit as in the case $k=1$ with a suitable interfacial energy density, in accord with the case $k=1$ and with the case $k=2$ previously analyzed by Fonseca and Mantegazza. The main isssue is the derivation of an optimal-profile problem on the real line describing the interfacial energy density, which must be conveniently approximated by minimum problems on finite intervals with homogeneous condition on the derivatives at the endpoints up to order $k-1$. To that end a careful study must be carried on of sets where sequences of functions with equibounded energy are ``close to the wells'' and have ``small derivatives'', in terms of interpolation inequalities and energy estimates.

math.AP↗

arXiv2023

A new space of generalised vector-valued functions of bounded variation

In [Dal Maso and Toader, NoDEA 2022], the authors introduced the space $GBV_\star(A)$ to minimise a class of functionals whose study is motivated by fracture mechanics. In this paper, we extend the definition of $GBV_\star(A)$ to the vectorial case, introducing the space $GBV_\star(A;\mathbb{R}^k)$. We study the main properties of $GBV_\star(A;\mathbb{R}^k)$ and prove a lower semicontinuity result useful for minimisation purposes. With the Direct Method in mind, we adapt the arguments of [Dal Maso and Toader, NoDEA 2022] to show that minimising sequences in $GBV_\star(A;\mathbb{R}^k)$ can be modified to obtain a minimising sequence converging $\mathcal{L}^d$-a.e in $A$.

math.AP↗

arXiv2023

Gamma-convergence of quadratic functionals perturbed by bounded linear functionals

Given a bounded open set $Ω\subset \mathbb{R}^n$, we study sequences of quadratic functionals on the Sobolev space $H^1_0(Ω)$, perturbed by sequences of bounded linear functionals. We prove that their $Γ$-limits, in the weak topology of $H^1_0(Ω)$, can always be written as the sum of a quadratic functional, a linear functional, and a non-positive constant. The classical theory of $G$- and $H$-convergence completely characterises the quadratic and linear parts of the $Γ$-limit and shows that their coefficients do not depend on $Ω$. The constant, which instead depends on $Ω$ and will be denoted by $-ν(Ω)$, plays an important role in the study of the limit behaviour of the energies of the solutions. The main result of this paper is that, passing to a subsequence, we can prove that $ν$ coincides with a non-negative Radon measure on a sufficiently large collection of bounded open sets $Ω$. Moreover, we exhibit an example that shows that the previous result cannot be obtained for every bounded open set. The specific form of this example shows that the compactness theorem for the localisation method in $Γ$-convergence cannot be easily improved.

math.AP↗

arXiv2023

Another look at elliptic homogenization

We consider the limit of sequences of normalized $(s,2)$-Gagliardo seminorms with an oscillating coefficient as $s\to 1$. In a seminal paper by Bourgain, Brezis and Mironescu (subsequently extended by Ponce) it is proven that if the coefficient is constant then this sequence $\Gamma$-converges to a multiple of the Dirichlet integral. Here we prove that, if we denote by $\varepsilon$ the scale of the oscillations and we assume that $1-s<\!<\varepsilon^2$, this sequence converges to the homogenized functional formally obtained by separating the effects of $s$ and $\varepsilon$; that is, by the homogenization as $\varepsilon\to 0$ of the Dirichlet integral with oscillating coefficient obtained by formally letting $s\to 1$ first.

math.AP↗