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Lorenza D'Elia

Publications and source records attributed to Lorenza D'Elia.

10 recordsLinked to original sources
arXiv2025

Homogenization and linearization in magnetoelasticity under small elastic response

We perform a simultaneous homogenization and linearization analysis for a magnetoelastic energy functional featuring a mixed Eulerian-Lagrangian structure. Neglecting Zeeman and anisotropic contributions, we characterize the asymptotic behavior in the sense of Gamma-convergence for the sum of a nonlinear magnetoelastic energy, a symmetric exchange term defined on the actual configuration, and for the associated magnetostatic self-energy. After establishing compactness of displacements and magnetizations with equibounded energy, we identify the limiting energy functional as the sum of a quadratic homogenized magnetoelastic contribution with a limiting homogenized exchange and magnetostatic term. This is, to the authors' knowledge, the first homogenization result for manifold-valued mixed Eulerian-Lagrangian energies.

math.AP↗
arXiv2025

Relaxation of variational problems in the space of functions with bounded $\mathcal{B}$-variation: interaction with measures and lack of concentration phenomena

We prove an integral representation result for variational functionals in the space $BV^{\mathcal{B}}$ of functions with bounded $\mathcal{B}$-variation where $\mathcal{B}$ denotes a $k$-th order, $\mathbb{C}$-elliptic, linear homogeneous differential operator. This result has been used as a key tool to get an explicit representation of relaxed energies with linear growth which lead to limiting generic measures. According to the space dimension and the order of the operator, concentration phenomena appear and an explicit interaction is featured. These results are complemented also with Sobolev-type counterparts. As a further application, a lower semicontinuity result in the space of fields with $p(\cdot)$-bounded $\mathcal{B}$-variation has also been obtained.

math.AP↗
arXiv2025

First-order homogenization

We provide a first-order homogenization result for quadratic functionals. In particular, we identify the scaling of the energy and the explicit form of the limiting functional in terms of the first-order correctors. The main novelty of the paper is the use of the dual correspondence between quadratic functionals and PDEs, combined with a refinement of the classical Riemann-Lebesgue Lemma.

math.AP↗
arXiv2024

Homogenization of supremal functionals in the vectorial case (via $L^p$-approximation)

We propose a homogenized supremal functional rigorously derived via $L^p$-approximation by functionals of the type $\underset{x\inΩ}{\mbox{ess-sup}}\hspace{0.03cm} f\left(\frac{x}{\varepsilon}, Du\right)$, when $Ω$ is a bounded open set of $\mathbb R^n$ and $u\in W^{1,\infty}(Ω;\mathbb R^d)$. The homogenized functional is also deduced directly in the case where the sublevel sets of $f(x,\cdot)$ satisfy suitable convexity properties, as a corollary of homogenization results dealing with pointwise gradient constrained integral functionals.

math.AP↗
arXiv2023

Stochastic homogenization of micromagnetic energies and emergence of magnetic skyrmions

We perform a stochastic-homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional defined on magnetizations taking value in the unit sphere, and including both symmetric and antisymmetric exchange contributions. This Gamma-limit corresponds to a micromagnetic energy functional with homogeneous coefficients. We provide explicit formulas for the effective magnetic properties of the composite material in terms of homogenization correctors. Additionally, the variational analysis of the two exchange energy terms is performed in the more general setting of functionals defined on manifold-valued maps with Sobolev regularity, in the case in which the target manifold is a bounded, orientable smooth surface with tubular neighborhood of uniform thickness. Eventually, we present an explicit characterization of minimizers of the effective exchange in the case of magnetic multilayers, providing quantitative evidence of Dzyaloshinskii's predictions on the emergence of helical structures in composite ferromagnetic materials with stochastic microstructure.

math.AP↗
arXiv2021

Gamma-convergence of quadratic functionals with non uniformly elliptic conductivity matrices

We investigate the homogenization through Gamma-convergence for the L^2(Ω)-weak topology of the conductivity functional with a zero-order term where the matrix-valued conductivity is assumed to be non strongly elliptic. Under proper assumptions, we show that the homogenized matrix A^\ast is provided by the classical homogenization formula. We also give algebraic conditions for two and three dimensional 1-periodic rank-one laminates such that the homogenization result holds. For this class of laminates, an explicit expression of A^\ast is provided which is a generalization of the classical laminate formula. We construct a two-dimensional counter-example which shows an anomalous asymptotic behaviour of the conductivity functional.

math.AP↗
arXiv2021

Homogenization of discrete thin structures

We consider graphs parameterized on a portion $X\subset\mathbb Z^d\times \{1,\ldots, M\}^k$ of a cylindrical subset of the lattice $\mathbb Z^d\times \mathbb Z^k$, and perform a discrete-to-continuum dimension-reduction process for energies defined on $X$ of quadratic type. Our only assumptions are that $X$ be connected as a graph and periodic in the first $d$-directions. We show that, upon scaling of the domain and of the energies by a small parameter $\varepsilon$, the scaled energies converge to a $d$-dimensional limit energy. The main technical points are a dimension-lowering coarse-graining process and a discrete version of the $p$-connectedness approach by Zhikov.

math.AP↗
arXiv2018

Norm Estimates for $τ$-Pseudodifferential Operators in Wiener Amalgam and Modulation Spaces

We study continuity properties on modulation spaces for $τ$-pseudodifferential operators with symbols $a$ in Wiener amalgam spaces. We obtain boundedness results for $τ\in (0,1)$ whereas, in the end-points $τ=0$ and $τ=1$, the corresponding operators are in general unbounded. Furthermore, for $τ\in (0,1)$, we exhibit a function of $τ$ which is an upper bound for the operator norm. The continuity properties of $τ$-pseudodifferential operators, for any $τ\in [0,1]$, with symbols $a$ in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter $τ\in [0,1]$, as expected. Key ingredients are uniform continuity estimates for $τ$-Wigner distributions.

math.FA↗