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Annalisa Malusa

Publications and source records attributed to Annalisa Malusa.

14 recordsLinked to original sources
arXiv2026

On the divergence of the composition of irregular fields with BV functions

We introduce a family of (nonlinear) pairing measures that ensure the validity of the divergence rule for composite functions $\boldsymbol{B}(x,u(x))$, where $\boldsymbol{B}(\cdot,t)$ is a bounded divergence-measure vector field, and $u$ is a scalar function of bounded variation. The elements of the family depend on the choice of the pointwise representative of $u$ on its jump set. Beyond the standard properties, such as the Coarea and Gauss-Green formulas on sets of finite perimeter, this flexibility allows us to characterize the pairings that ensure the lower semicontinuity of the corresponding functionals along sequences converging in $L^1$ with controlled precise values. We show that these lower semicontinuous pairings arise as the relaxation of integral functionals defined in Sobolev spaces.

math.FA↗
arXiv2026

Phase-field approximation of sharp-interface energies accounting for lattice symmetry

We present a phase-field approximation of sharp-interface energies defined on partitions, designed for modeling grain boundaries in polycrystals. The independent variable takes values in the orthogonal group $\mathrm{O}(d)$ modulo a lattice point group $\mathcal{G}$, reflecting the crystallographic symmetries of the underlying lattice. In the sharp-interface limit, the surface energy exhibits a Read-Shockley-type behavior for small misorientation angles, scaling as $θ|\logθ|$. The regularized functionals are applicable to grain growth simulation and the reconstruction of grain boundaries from imaging data.

math.FA↗
arXiv2025

On a Differential Model for Sandpiles Growing in a Silo

We discuss some features of a boundary value problem for a system of PDEs that describes the growth of a sandpile in a container under the action of a vertical source. In particular, we characterize the long-term behavior of the profiles, and we provide a sufficient condition on the vertical source that guarantees the convergence to the equilibrium in a finite time. We show by counterexamples that a stable configuration may not be reached in a finite time, in general, even if the source is time-independent. Finally, we provide a complete characterization of the equilibrium profiles.

math.AP↗
arXiv2023

Representation formulas for pairings between divergence-measure fields and $BV$ functions

The purpose of this paper is to find pointwise representation formulas for the density of the pairing between divergence-measure fields and BV functions, in this way continuing the research started in [17,20]. In particular, we extend a representation formula from an unpublished paper of Anzellotti [7] involving the limit of cylindrical averages for normal traces, and we exploit a result of [35] in order to derive another representation in terms of limits of averages in half balls.

math.FA↗
arXiv2020

Duality arguments for linear elasticity problems with incompatible deformation fields

We prove existence and uniqueness for solutions to equilibrium problems for free-standing, traction-free, non homogeneous crystals in the presence of plastic slips. Moreover we prove that this class of problems is closed under G-convergence of the operators. In particular the homogenization procedure, valid for elliptic systems in linear elasticity, depicts the macroscopic features of a composite material in the presence of plastic deformation.

math.AP↗
arXiv2019

Pairings between bounded divergence-measure vector fields and BV functions

We introduce a family of pairings between a bounded divergence-measure vector field and a function $u$ of bounded variation, depending on the choice of the pointwise representative of $u$. We prove that these pairings inherit from the standard one, introduced in [6,10], all the main properties and features (e.g. coarea, Leibniz and Gauss--Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in $BV$. We remark that the standard pairing in general does not share this property.

math.AP↗
arXiv2019

Non-coercive radially symmetric variational problems: Existence, symmetry and convexity of minimizers

We prove existence of radially symmetric solutions and validity of Euler-Lagrange necessary conditions for a class of variational problems such that neither direct methods nor indirect methods of Calculus of Variations apply. We obtain existence and qualitative properties of the solutions by means of ad-hoc superlinear perturbations of the functional having the same minimizers of the original one.

math.OC↗
arXiv2017

Crystalline Evolutions in Chessboard-like Microstructures

We describe the macroscopic behavior of evolutions by crystalline curvature of planar sets in a chessboard--like medium, modeled by a periodic forcing term. We show that the underlying microstructure may produce both pinning and confinement effects on the geometric motion.

math.AP↗
arXiv2017

Crystalline evolutions with rapidly oscillating forcing terms

We consider the evolution by crystalline curvature of a planar set in a stratified medium, modeled by a periodic forcing term. We characterize the limit evolution law as the period of the oscillations tends to zero. Even if the model is very simple, the limit evolution problem is quite rich, and we discuss some properties such as uniqueness, comparison principle and pinning/depinning phenomena.

math.AP↗
arXiv2011

A nonhomogeneous boundary value problem in mass transfer theory

We prove a uniqueness result of solutions for a system of PDEs of Monge-Kantorovich type arising in problems of mass transfer theory. The results are obtained under very mild regularity assumptions both on the reference set $Ω\subset\mathbf{R}^n$, and on the (possibly asymmetric) norm defined in $Ω$. In the special case when $Ω$ is endowed with the Euclidean metric, our results provide a complete description of the stationary solutions to the tray table problem in granular matter theory.

math.AP↗
arXiv2008

On the Finsler metrics obtained as limits of chessboard structures

We study the geodesics in a planar chessboard structure with two values 1 and $β>1$. The results for a fixed structure allow us to infer the properties of the Finsler metrics obtained, with an homogenization procedure, as limit of oscillating chessboard structures.

math.AP↗
arXiv1994

Existence results for non-coercive variational problems

The aim of this paper is to give an existence result for a class of one-dimensional, non-convex, non-coercive problems in the Calculus of Variations. The main tools for the proof are an existence theorem in the convex case and the closure of the convex hull of the epigraph of functions strictly convex at infinity.

funct-an↗
arXiv1993

Approximation of Relaxed Dirichlet Problems by Boundary Value problems in perforated domains

Given an elliptic operator~$L$ on a bounded domain~$Ω\subseteq {\bf R}^n$, and a positive Radon measure~$μ$ on~$Ω$, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains~$Ω_h \subseteq Ω$ with the following property: for every~$f\in H^{-1}(Ω)$ the sequence~$u_h$ of the solutions of the Dirichlet problems~$L\, u_h=f$ in~$Ω_h$, $u_h=0$ on~$\partial Ω_h$, extended to 0 in~$Ω\setminus Ω_h$, converges to the solution of the \lq\lq relaxed Dirichlet problem\rq\rq\ $L\,u+μu=f$ in~$Ω$, $u=0$ on~$\partial Ω$.

funct-an↗