Abstracts

An approach to topological singularities through Mumford- Shah type functionals

Vito Crismale

The talk concerns an ongoing work with L. De Luca and R. Scala about the asymptotic variational equivalence, at any logarithmic scaling regime, between Ginzburg-Landau energies (then Core Radius energies, as well) and suitable Mumford-Shah type energies.

This extends a recent work by De Luca-Scala-Van Goethem which first employs such Mumford-Shah type functionals to approximate the energy of finitely many dislocations in a simplified topological setting.

Stochastic homogenization in micromagnetics

Lorenza D’Elia

Many key properties and applications of magnetic materials are strongly intertwined with the spatial distribution of magnetic moments inside the corresponding specimens. In addition to classical magnetic structures, magnetic skyrmions have raised interest in spintronics as carriers of information for future storage devices. In this talk, we present an advance in the mathematical modeling of magnetic skyrmions by analyzing the interplay of stochastic microstructures and chirality. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional, including the Dzyaloshinkskii-Moriya contribution. Eventually, we present an explicit characterization of minimizers of the effective model in the case of magnetic multilayers. This talk is based on a joint work with E. Davoli and J. Ingmanns.

A variational perspective on auxetic metamaterials of checkerboard type

Dominik Engl

Auxetic metamaterials have the counterintuitive property that they expand perpendicular to applied forces under stretching. In this talk, we discuss homogenization via Gamma-convergence for elastic materials with stiff checkerboard-type heterogeneities under the assumption of non-self-interpenetration. Our result rigorously confirms these structures as auxetic. The challenging part of the proof is determining the admissible macroscopic deformation behavior, or in other words, characterizing the weak Sobolev limits of deformation maps whose gradients are locally close to rotations on the stiff components. To this end, we establish an asymptotic rigidity result showing that, under suitable scaling assumptions, the attainable macroscopic deformations are affine conformal contractions. Our strategy is to tackle first an idealized model with full rigidity on the stiff tiles and then transfer the findings to a model with diverging elastic constants. The latter requires a new quantitative geometric rigidity estimate for non-connected touching squares and a tailored Poincaré-type inequality for checkerboard structures. This is joint work with Wolf-Patrick Düll (University of Stuttgart) and Carolin Kreisbeck (KU Eichstätt-Ingolstadt).

Local minimisers in higher order Calculus of Variations in L∞ : existence, uniqueness and characterisation

Nikos Katzourakis

Higher order problems are very novel in the Calculus of Variations in L^∞, and exhibit a strikingly different behaviour compared to first order problems, for which there exists an established theory, pioneered by Aronsson in 1960s. In this talk I will discuss how a complete theory can be developed for second order functionals. Under appropriate conditions, “localised” minimisers can be characterised as solutions to a nonlinear system of PDEs, which is different from the corresponding Aronsson equation; the latter is only a necessary, but not a sufficient condition for minimality. I will also discuss the existence and uniqueness of localised minimisers subject to Dirichlet boundary conditions, and also their partial regularity outside a singular set of codimension one, which may be non-empty even in 1D. The talk will not assume any previous knowledge on the topic, and is based on recent work (arXiv:2403.12625v1) with Roger Moser (University of Bath, UK).

A weighted anisotropic spectral optimization problem arising in population dynamic

Giovanni Pisante

The plan of the talk is to discuss some recent results concerning a weighted eigenvalue problem with anisotropic diffusion in bounded Lipschitz domains under Robin boundary conditions. The model aims to describe the dispersal of a population in a heterogeneous environment Ω, triggered by a motion law so that each individual moves with different probabilities depending on the direction of the movement. The heterogeneity of the habitat is modelled by representing Ω as the union of patches, favourable and hostile zones, corresponding respectively to the positivity and negativity set of a variable weight function. In this context, the positive principal eigenvalue λ, associated with the differential problem, turns out to be a threshold for the survival of the population. Minimizing λ, with respect to the weight or to other features of the model, endorses the chances of survival. The model generalizes the classical one that describes the dispersal of a population triggered by a Brownian motion law so that each individual moves in every direction with the same probability, described by the Laplace operator. When the population adopts different diffusion strategies, one is naturally led to consider different differential operators in the model. Here, we are focused on the so-called anisotropic p-Laplace operator. The optimization problem is, by now, fully understood only in one dimension with homogeneous Dirichlet or Neumann boundary conditions. Even in this simpler setting, new phenomena arise induced by the presence of anisotropic diffusion. The analogous study in higher dimension is still widely open in its generality even in the case of the Laplacian diffusion. The presentation is based on a joint work with B. Pellacci and D. Schiera.

Manifold-constrained free discontinuity problems and Sobolev approximation

Bianca Stroffolini

We discuss a recent result, obtained in collaboration with Federico Luigi Dipasquale, on the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps and a variable-exponent growth in the energy functional. To be more precise, we work in 2D domains, with sphere-valued special functions with bounded variation, and the energy functional we consider is the sum of the integral of the p(·)-power of their approximate gradient and of the H^1-measure of their jump set.

The approach we follow is remiscent of the one devised by Conti, Focardi, and Iurlano to prove existence of strong minimisers for the Griffith energy and it is divided in two steps. As a first step, we extend to this setting the Sobolev approximation results for special functions with bounded deformation and small jump set

originally proven by Conti, Focardi, and Iurlano. In second place, we use this extension and a suitable adaptation of the classical blow-up technique due to De Giorgi, Carriero and Leaci to prove the announced partial regularity theorem, avoiding truncation techniques.

Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data

Ida de Bonis

We study the periodic homogenization of a reaction-diffusion problem with large non-linear drift and Robin boundary condition posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. We are interested in deriving rigorously the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to an exploding nonlinear drift. The main mathematical difficulty lies in proving the two-scale compactness and strong convergence results needed for the passage to the homogenization limit. We will use the concept of two-scale compactness with drift, which is similar to the more classical two-scale compactness result but it is defined now in moving coordinates. The results presented are contained in the paper [1] in collaboration with E.N.M. Cirillo (Sapienza University), A. Muntean (Karlstad University) and V.Raveendran (Karlstad University).

References

[1] E.N.M. Cirillo, I. de Bonis, A. Muntean, V. Raveendran, Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data, accepted on Quarterly of Applied Mathematics.

Homogenization of supremal functionals in vectorial setting (via power-law approximation)

Michela Eleuteri

The aim of this talk is to propose a homogenized supremal functional rigorously derived via power-law approximation departing by supremal inohomogeneous functionals of the type ess-sup_{x∈Ω} f( x/ε, Du) , where Ω is a bounded open set of R^n and u ∈ W^{1,∞}(Ω; R^d ). The homogenized functional is also deduced directly in the case where the sublevel sets of f(x, ·) satisfy suitable convexity properties, as a corollary of homogenization results dealing with pointwise gradient constrained integral functionals. This is a joint project with Lorenza D’Elia (TU Wien) and Elvira Zappale (Sapienza - Università di Roma).

Variational models for grain boundaries in polycrystals

Adriana Garroni

I will present a recent result obtained in collaboration with M. Fortuna and E. Spadaro. We consider a “semi-discrete” model proposed by Lauteri and Luckhaus for the analysis polycrystals. By means of Gamma convergence we deduce a sharp interface model for grain boundaries which also shows the behaviour for small angle grain boundaries known as the Read-Shockley law.

Variational analysis of the Canham-Helfrich energy functional

Luca Lussardi

The Canham-Helfrich energy is widely used to describe the elastic properties of biological membranes at the subcellular level. The membranes are modeled as surfaces in the three-dimensional space and their shape minimizes an energy which penalizes the curvatures of the surface. A suitable multiphase energy permits to model also heterogeneous biological membranes. I will discuss existence of single and multiphase minimizers under area and enclosed volume constraints in the varifolds setting (joint work with K. Brazda and U. Stefanelli) and in the current setting (joint work with A. Kubin and M. Morandotti).

Geometric rigidity for incompatible fields in the multi-well case and applications

Francesco Solombrino

The topic of this talk is the derivation of a quantitative rigidity estimate for incompatible fields in a multi-well setting. Precisely, the L^1*-distance of a possibly incompatible strain field β from a single well is controlled in terms of the following three quantities: the L^1*-distance from a finite set of wells, of curlβ, and of divβ. As an application, the derivation of 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, will be discussed. This is a joint work with S.Almi (Università di Napoli) and D. Reggiani (Scuola Superiore Meridionale).

Asymptotic analysis for a class of non-local problems in composites with imperfect interfaces

Claudia Timofte

In this talk, we shall present some recent homogenization results for a class of non-local problems in a composite material made up of a hosting medium in which a periodic array of perfect heat conductors is inserted. We address the case in which we have two types of conductive fillers, having dissimilar thermal features. The temperature in the hosting medium is governed by a standard heat equation, while, inside each type of inclusion, the temperature depends only on time and satisfies a non-standard ordinary differential equation, involving a non-local condition. The imperfect transmission conditions at the interfaces between the conductive regions of the composite involve the microscopic geometry of the problem through the characteristic length ε of the inclusions and two amplitude factors, D1 and D2, respectively, taking into account the magnitude of the interfacial heat exchange between the hosting medium and the conductive fillers. We analyze the overall conductivity of the composite medium and we perform the limits in ε and in the two parameters D1 and D2 in different orders, with the aim of comparing the final results and identifying the cases in which the two limits commute. This is a joint work with Micol Amar and Daniele Andreucci (Sapienza - Universit`a di Roma).

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