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Elvira Zappale

Publications and source records attributed to Elvira Zappale.

At least 19 recordsLinked to original sources
arXiv2026

Homogenization and integral representation of energy functionals in manifold valued Orlicz-Sobolev spaces

This paper aims to extend to Orlicz-Sobolev spaces some results of integral representation for the simultaneous homogenization and dimensional reduction of integral energies defined on fields taking values on a differentiable manifold. Since our functional framework goes beyond the classical Sobolev's spaces, we also prove, via $Γ$-convergence, a general integral representation results in the unconstrained Orlicz setting. Due to $Δ_2$ and $\nabla_2$ conditions verified by the Young function $Φ$ (which modulated the growth behaviour), we prove that the density of the $Γ$-limit is a tangential quasiconvex integrand represented by a cell formula.

math.AP↗
arXiv2026

Functions of bounded Musielak-Orlicz-type deformation and anisotropic Total Generalized Variation for image-denoising problems

In the first part of this paper we introduce the space of bounded deformation fields with generalized Orlicz growth. We establish their main properties, provide a modular representation, and characterize a decomposition of the modular into an absolutely continuous part and a singular part weighted via a recession function. A further analysis in the variable exponent case is also provided. The second part of the paper contains a notion of Musielak-Orlicz anisotropic Total Generalized Variation. We establish a duality representation, and show well-posedness of the corresponding image reconstruction problem.

math.AP↗
arXiv2026

Relaxation of quasi-convex functionals with variable exponent growth

We prove a relaxation result for a quasi-convex bulk integral functional with variable exponent growth in a suitable space of bounded variation type. A key tool is a decomposition under mild assumptions of the energy into absolutely continuous and singular parts weighted via a recession function.

math.AP↗
arXiv2026

Structured Deformations in Linearized Elasticity

We extend the theory of structured deformations to the setting of linearized elasticity by providing an integral representation for the underlying energy that features bulk and surface contributions. Our derivation is obtained both via a direct approach by means of a global method for relaxation in BD and via an approximation from nonlinear elastic energies associated to {nonsimple} materials.

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

A Comprehensive Approach via Global Relaxation to the Variational Modelling of Hierarchical Structured Deformations

The response of many materials to applied forces and boundary constraints depends upon internal geometric changes at multiple submacroscopic levels. Hierarchical structured deformations provide a mathematical setting for the description of such changes and for the variational determination of the corresponding energetic response. The research in this article provides substantial refinements and broadenings of the mathematical setting both for the underlying geometrical structure and for the variational analysis of energetic response. The mathematical tools employed in this research include the global method for relaxation and establish the equivalence of a relaxed energy obtained via relaxation under simultaneous geometrical changes at all levels and a relaxed energy obtained via iterated relaxations proceeding from the deepest submacroscopic level successively to the macroscopic level.

math.CA↗
arXiv2025

Junction in a thin multi-domain for nonsimple grade two materials in BH

We consider a thin multi-domain of $\mathbb R^N$, with $N\geq 2$, consisting of a vertical rod upon a horizontal disk. In this thin multi-domain, we introduce a bulk energy density of the kind $W(D^2U)$, where $W$ is a continuous function with linear growth at $\infty$ and $D^2U$ denotes the Hessian tensor of a vector-valued function $U$ that represents a deformation of the multi-domain. Considering suitable boundary conditions on the admissible deformations and assuming that the two volumes tend to zero with same rate, we prove that the limit model is well posed in the union of the limit domains, with dimensions $1$ and $N-1$, respectively. Moreover, we show that the limit problem is uncoupled if $N\geq 3$, and ``partially" coupled if $N=2$.

math.AP↗
arXiv2025

Lower semicontinuity of nonlocal $L^\infty$ energies on $SBV_0(I)$

We characterize the lower-semicontinuity of nonlocal one-dimensional energies of the type \[{\rm ess}\!\!\!\!\!\!\!\!\sup_{(s,t) \in I\times I} h([u](s), [u](t)),\] where $I$ is an open and bounded interval in the real line, $u \in SBV_0(I)$ and $[u](r):= u(r^+)- u(r^-)$, with $r\in I$.

math.AP↗
arXiv2025

Integral representation for a relaxed optimal design problem for non-simple grade two materials

A measure representation result for a functional modelling optimal design problems for plastic deformations, under linear growth conditions, is obtained. Departing from an energy with a bulk term depending on the deformation gradient and its derivatives, as well as a perimeter term, the functional in question corresponds to the relaxation of this energy with respect to a pair $(χ,u)$, where $χ$ is the characteristic function of a set of finite perimeter and $u$ is a function of bounded hessian.

math.AP↗
arXiv2024

Approximation of $L^\infty$ functionals with generalized Orlicz norms

The aim of this paper is to deal with the asymptotics of generalized Orlicz norms when the lower growth rate tends to infinity. $Γ$-convergence results and related representation theorems in terms of $L^\infty$ functionals are proven for sequences of generalized Orlicz energies under mild convexity assumptions. This latter hypothesis is removed in the variable exponent setting.

math.AP↗
arXiv2024

Measure structured deformations

Measure structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure structured deformation is defined via relaxation departing either from energies associated with classical deformations or from energies associated with structured deformations. A concise integral representation of the energy functional is provided both in the unconstrained case and under Dirichlet conditions on a part of the boundary.

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↗