Working Seminar on Calculus of Variations and Gamma-convergence
Gruppo di lavoro su Calcolo delle Variazioni e Gamma-convergenza

The group meetings consist of an up-to-two-hour informal seminar or brain-storming discussion
Venue: Aula B - Dipartimento di Matematica, Sapienza Università di Roma (unless otherwise stated)


Monday March 18 2019
Aula C, 11:00-13:00

Gianluca Orlando (TU, Munich) 

Title: Does the N-clock model approximate the XY model?

In this seminar we will investigate the relationship between the N-clock model and the XY model (at zero temperature) through a Gamma-convergence analysis as both the number of particles and N diverge. The N-clock model is a two-dimensional nearest neighbors ferromagnetic spin system, in which the values of the spin field are constrained to lie in a set of N equispaced points of the unit circle. For N large enough, it is usually considered as an approximation of the XY model, for which the spin field is allowed to attain all the values of the unit circle. By suitably renormalizing the energy of the N-clock model, we will illustrate how its thermodynamic limit strongly depends on the rate of divergence of N with respect to the number of particles. We shall see that the N-clock model turns out to be a good approximation of the XY model only for N sufficiently large; in other regimes of N, we will show with the aid of cartesian currents that its asymptotic behavior can be described by an energy which may concentrate on geometric objects of different dimensions. The results presented in the talk are based on a work in collaboration with M. Cicalese and M. Ruf.

Past seminars

January 23 2019
January 30 2019
February 13 2019
February 18 2019
February 20 2019

Andrey Piatnitski (The Arctic University of Norway e Russian Academy of Science)
Short course: Partial differential equations on singular or thin structures
Oct 5 2018
Stephan Luckhaus (Leipzig)
Geometric rigidity and grain boundaries in phase-transition problems

May 11 2018
Elena Zhizhina (Russian Academy of Science)
Astral diffusion as a limit process for symmetric random walks in a high-contrast periodic medium

May 8 2018
May 10 2018

Andrey Piatnitski (The Arctic University of Norway e Russian Academy of Science)
Short course: Introduction to the homogenization of singularly perturbed operators

In the course two types of singularly perturbed operators will be considered.
First we will consider homogenization problems for periodic linear convection-diffusion operators with a large convection term. Our goal is to show that these operators admit homogenization in moving coordinates. 
We will determine the effective speed and diffusion matrix and then prove convergence. 
The second part of the course will focus on homogenization of spectral problem for a symmetric second-order elliptic operator with locally periodic coefficients and a large potential. We will construct the limit spectral problem and prove the convergence of the eigenpairs.

May 4 2018
Andrey Piatnitski (Narvik and Moscow)
Large-time behaviour of the heat kernel of a convolution-type operator

April 20 2018
Paolo Cermelli (Torino)
The limit of the cut functional on dense graph sequences

Abstract. The cut functional on a finite graph is a measure of the total number of edges connecting different communities, and can be used to find optimal splittings of the graph into highly interconnected components. It also arises as a spin functional with simple pairwise interactions between the nodes. Given that large graphs have increasing importance in applications, it is important to understand the structure of the limit of the cut functional when the order of the graph goes to infinity, as well as its relation with its finite counterpart. In this work we exploit the spin functional analogy, the theory of limits of graph sequences, and Gamma-convergence to compute the limit functional on dense graph sequences, in order to elucidate the structure of the interfaces in the large graph limit.

May 29 2017
Emanuela Radici (Università dell'Aquila)
Diff eomorphic approximation of planar elastic deformations

Abstract: To study existence and regularity of solutions of some standard variational models in nonlinear elasticity, it is convenient to know whether it is possible to approximate homeomorphisms u with di ffeomorphisms u_n whose elastic energy is close to that of u. A big step in this direction would be to show the existence of an approximating sequence (u_n)_n such that u_n converges to u and, at the same time, the inverse of u_n converges to the inverse of u in a suitable sense. We briefly introduce hyperelasticity and the pure- displacement boundary problem, then we discuss the approximation of planar homeomorphisms in the Sobolev, bi-Sobolev and BV (actually bi-BV ) settings. This is a joint work with Aldo Pratelli (FAU Erlangen-Nurnberg).

April 10 2017
Augusto Gerolin (University of Jyvaskyla, Finland)
Some mathematical aspects of Density Functional Theory

Abstract: Density functional theory (DFT) is a computational method used in Physics, Chemistry and Materials Science to investigate the electronic structure of many-body systems. Although DFT is widely used in Applied Chemistry, some mathematical aspects of this theory are far to be understood. The idea of this talk is to present an informal introduction to DFT and, moreover, discuss a connection between a ground state problem for an Electronic Schrödinger Equation and optimal transport theory with Coulomb costs.

March 27 2017
Antonio Tribuzio (Roma Tor Vergata)
Perturbazioni di movimenti minimizzanti e curve di massima pendenza

Abstract: Si studiano le proprietà di alcune perturbazione dello schema dei movimenti minimizzanti usato  da Ambrosio, Gigli e Savaré per provare l'esistenza di curve di massima pendenza in spazi metrici. Tali perturbazioni corrispondono all'aggiunta di un coefficiente, dipendente dal tempo (e dalla discretizzazione temporale), anche oscillante.

March 6 2017
Vito Crismale and Gianluca Orlando (SISSA, Trieste)
A model for cohesive fracture with irreversibility and fatigue

Abstract: In this talk we will prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model in analysis is that the dissipated energy depends on the cumulated jump. This implies a fatigue phenomenon, i.e., a complete fracture may be produced by oscillation of small jumps.
The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that one lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. We will explain how, eventually, after proving the existence in this weak formulation, we are able to improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement. The result has been obtained in collaboration with Giuliano Lazzaroni.

February 20 2017 Roberto Peirone (Università di Roma "Tor Vergata")
A finitely ramified fractal with no self-similar energy.

Abstract: The starting point for many problems in analysis on fractals is the construction of a self- similar Dirchlet form (a sort of abstract version of the Dirichlet integral). An old conjecture was that on every P.C.F. self-similar set (a general class of finitely ramified fractals) there exists a self-similar Dirichlet form (with good property). In this talk, I show a counterexample to that conjecture.

January 30 2017
Rufat Badal (TU Munich)
Gamma-convergence analysis of a generalized XY model: fractional vortices and string defects

January 23 2017
Roberto Alessi (Sapienza Università di Roma)
Mechanical response of hybrid laminates with cohesive interface law

Abstract: Fibre-Reinforced Polymers are increasingly used as structural materials in several engineering fields, mostly due to their extremely high resistance to weight ratio. Nevertheless, the brittle response at failure due to complete lack of ductility, limit the advantage of the full strength of the material since great safety coefficients are imposed by technical guidelines at the design level.
A possible strategy to provide the composite with a ductile failure response is to consider novel composite architectures where fibres of different stiffness and ultimate strain values are combined (hybridisation). When loaded in a displacement controlled tensile test, the fibres with the lowest ultimate strain fail first, allowing a stress relaxation (decrease in stress with the same global mean strain). The remaining high elongation fibres are proportioned to sustain the total load up to failure.
Within this framework, we discuss an analytical model for uni-directional hybrid composites that allows the optimal material properties of the hybrid composite to be determined. The single layers are assumed to have a brittle response whereas the interface is assumed to follow a cohesive force-displacement relationship. Several failure regimes, involving for instance micro-cracking and diffuse delamination phenomena, are highlighted through the construction of a ”damage mode map”. A numerical solution strategy is finally proposed, based on phase-field models.

January 16 2017
Brain-storming session on theories of disordered elastic discrete thin films around the work of Alfie Mimum.

December 19 2016
Silvio Fanzon (University of Sussex)
Variational models for semi-coherent interfaces

Abstract: We propose and analyze a simple variational model for dislocations at semi-coherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations and a bulk elastic energy, spent to decrease the amount of dislocations needed to compensate the lattice misfit. We prove that, for minimizers, the former scales like the surface area of the interface, the latter like its diameter. The proposed continuum model is built on some explicit computations done in the framework of the semi-discrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the far field strain vanishes as the interface size increases.

December 12 2016
Andrea Braides (Roma "Tor Vergata")
A damage model as a result of the interaction between elasticity and fracture

Abstract: We examine a simple model of discrete interactions by Novak and Truskinovsky mixing elastic and Lennard-Jones potentials. We derive a continuum (semidiscrete) counterpart depending on a small parameter corresponding to the interatomic distance and compute its behaviour as this parameter tends to 0. The resulting energy is of "damage type". We compare the limit with that of Novak and Truskinovsky highlighting differences between the discrete and the semidiscrete formulation (work in progress with A.Causin and M.Solci).

November 28 2016 Annika Bach (WWU Münster)
Second-order approximation of anisotropic free-discontinuity functionals

Abstract: We provide an approximation result for free-discontinuity functionals whose surface-integrand is an arbitrary smooth Finsler metric. Our main result states that these functionals can be obtained as the Gamma-limit of elliptic energies of Ambrosio-Tortorelli type depending on the Hessian of the edge variable.

November 21 2016 Alfie Mimum (Sapienza Università di Roma)
Dimension reduction for discrete disordered elastic systems: a case study

Abstract: I describe the overall behavior of variational stochastic pair-interaction systems defined on thin domains of R^d, that is, domains consisting on a finite number of mutually interacting overlaid copies of a portion of a stochastic lattice in R^(d-1). One of the reasons for this research lies in the fact that this kind of systems can be used to model thin rubber objects.

November 14 2016
Leonard Kreutz (GSSI L'Aquila)
On the crystallinity of homogenized energy densities of periodic spin systems

Abstract: We show that the homogenized surface tension of a periodic spin system in dimension two is crystalline (i.e, its Wulff shape is a polyhedron). This question is motivated by problems in Image Processing (ongoing work with Antonin Chambolle)

November 7 2016 Andrea Braides (Università di Roma "Tor Vergata")
Quasiconvex functions from constrained Dirichlet energies

Abstract: We consider p-Dirichlet energies from a two-dimensional set into an oscillating two-dimensional manifold. We show that for p<2 we obtain an effective (homogenized) energy of p-growth finite in the whole Sobolev space W^{1,p}. For p>2 the energy density is quasiconvex and infinite on rank-two matrices. An open problem is whether the energy density, as depending on p, is continuous or not at the transition exponent p=2, which would be interesting in either cases.Ongoing work with J. Kristensen.

June 6 2016
Graeme Milton (Distinguished Professor of Mathematics, The University of Utah)
Cloaking: where Science Fiction meets Science

Abstract: Cloaking involves making an object partly or completely invisible to incoming waves such as sound waves, sea waves or seismic waves, but usually electromagnetic waves such as visible light, microwaves, infrared light, or radio waves. Camouflage and stealth technology achieve partial invisibility, but can one achieve true invisibility from such waves? This lecture will survey some of the wide variety of ideas on cloaking: these include transformation based cloaking, non Euclidean cloaking, cloaking due to anomalous resonance, cloaking by complementary media, active interior cloaking and active exterior cloaking. Beautiful mathematics is involved.
May 30 2016
Giovanni Scilla
Chirality transitions in frustrated ferromagnetic spin chains: a link with the gradient theory of phase transitions

Abstract: We study chirality transitions in frustrated ferromagnetic spin chains, in view of a possible link with the theory of Liquid Crystals. A variational approach to the study of these systems has been recently proposed by Cicalese and Solombrino (J. Nonlinear Sci.'15), focusing close to the helimagnet-ferromagnet transition point. We reformulate this problem in the framework of surface energies in nonconvex discrete systems with NN and NNN interactions and we link it to the gradient theory of phase transitions, by showing an equivalence by Gamma-convergence with Modica-Mortola type functionals. Joint work with Valerio Vallocchia.

May 10 2016
Virginia De Cicco (Sapienza Università di Roma)
Lower semicontinuity for nonautonomous surface integrals

Abstract: New lower semicontinuity results are obtained for nonautonomous surface integrals depending in a discontinuous way on the spatial variable. Surface energies of this type occur in free-discontinuity problems, as in fracture mechanics, when one considers quasistatic evolution of stratified, heterogeneous materials .

May 2 2016
Andrea Braides (Università di Roma "Tor Vergata")
Models of discrete thin films

Abstract: We discuss some results on dimension-reduction for discrete thin films, where the thickness is expressed as the number of discrete layers. The discrete framework amplifies the surface effects, resulting in a non-trivial dependence on the number of layers, and allows to treat models of quasicrystals or aperiodic lattices (e.g. Penrose lattice). We focus particularly on surface energies, with some results also in the random case and for antiferromagnetic spin systems. Joint works with R.Alicandro, A.Causin, M.Cicalese, M.Ruf and M.Solci.

April 4 2016
Adriana Garroni (Sapienza Università di Roma)
Line-tension models for dislocations

March 21 2016
Andrea Braides (Università di Roma "Tor Vergata")
Asymptotic analysis of microscopic impenetrability constraints for atomistic systems

Abstract: We consider systems of interactions in a triangular lattice which forbid change of orientations of triangles. We analyze the macroscopic effect of such a constraint on fractures and related analytical difficulties. Joint work with Maria Stella Gelli

March 14 2016
Maria Stella Gelli (Università di Pisa)
Lower-semicontinuity and relaxation results for energies in capillarity problems

Abstract: The free energy in capillarity problems usually consists of a surface term accounting for the liquid-gas interface and a 'wetting' term accounting for the interaction at the walls of the container of theliquid-solid phases. This last term plays a relevant role in the study of the equilibrium surfaces and the associated Young's law. In order to establish existence results when the container is only Lipschitz regular I will provide structure conditions on the surface density and on the adhesion coefficient that ensure the lower semicontinuity of the (total) free energy with respect to the convergence in measure of sets. I will discuss the optimality  of the bounds appearing in the l.s.c. theorem through suitable counterexamples and taking into account the 'regularity' of the surface energy density. When the structure conditions above are violated I will give a description of the relaxed functional. Eventually, I will apply the lower semicontinuity result to detect the capillarity equilibrium surfaces in case: (a) the adhesion coefficient is discontinuous; (b) the external normal to the boundary of the container is discontinuous, discussing in both cases the validity of Young's law.

March 7 2016
Valerio Vallocchia (Università di Roma Tor Vergata)
Asymptotic analyses of the Perona-Malik functional in 1D

Abstract: Under a suitable scaling the discrete Perona-Malik scheme has been proved to converge to the Mumford-Shah functional by Morini and Negri. In the same scaling we analyze the asymptotic description of local minima, quasistatic evolution and the minimizing-movement scheme.

February 22 2016
Roberto Alicandro (Università di Cassino)
Interactions beyond nearest neighbours and rigidity of discrete energies: some results and open problems

Abstract: I will analyse the rigidity of discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. My purpose is to examine the role of interactions beyond nearest neighbours in penalising changes of orientation and how, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. Following the same approach, I will also present the asymptotic analysis of a discrete model for nanowires.

January 25 2016
Lucia De Luca (TU Munich)
A soft-disc crystallization result

Abstract: We discuss a simple model for finite crystallization in the case of zero temperature, short-range pairwise interactions and two dimensions. For any fixed N, we prove that minimizers of the soft disc energy among the configurations with N points, are all the subsets of the triangular lattice which minimize the perimeter of the bond graph associated to V. Joint work with Gero Friesecke (TUM).

January 11 2016
Annalisa Malusa (Sapienza Università di Roma)
Homogenization of flat flows with oscillating forcing terms

Abstract: I will describe how the (crystalline) mean curvature motion of two-dimensional sets is modified by the effect of a finely layered forcing term. The results have been obtained in collaboration with A. Braides and M. Novaga.

December 14 2015
Emilio Turco (Università di Sassari)
Back to discrete: computational aspects of multiscale analysis

November 30 2015
Marcello Ponsiglione (Sapienza Università di Roma)
Nonlocal and crystalline curvature flows

Abstract: I will introduce a class of generalized (local and nonlocal) perimeters and curvatures. I will prove an existence and uniqueness result for the corresponding geometric flows, showing the consistency between viscosity and variational methods. Finally, I will introduce a new notion of crystalline curvature flow, providing existence and uniqueness (up to fattening) of the solution in any dimension and starting from any closed set. The results have been obtained in collaboration with A. Chambolle and M. Morini. 

November 23 2015
Roberto Alessi (Sapienza Università di Roma)
Energetic formulation for rate-independent processes: remarks on discontinuous evolutions with a simple example

November 16 2015
Andrea Braides (Università di Roma Tor Vergata)
Surface energies of systems of chiral molecules

November 9 2015
Nadia Ansini (Sapienza Università di Roma)
Minimising movements along oscillating energies at the critical regime. A case study.

November 2 2015
Leonard Kreutz (GSSI, L'Aquila)
Optimal bounds for mixtures of ferromagnetic interactions