Working Seminar on Calculus of Variations and Gammaconvergence Gruppo di lavoro su Calcolo delle Variazioni e Gammaconvergenza 
The group meetings consist of an
uptotwohour informal seminar or brainstorming
discussion Venue: Aula B  Dipartimento di Matematica, Sapienza Università di Roma (unless otherwise stated) 
NEXT SEMINAR Monday
March 18 2019
Aula C, 11:0013:00 Gianluca
Orlando (TU, Munich)
Title:
Does the Nclock model approximate the XY model?
Abstract
In this seminar we will investigate the relationship between the Nclock model and the XY model (at zero temperature) through a Gammaconvergence analysis as both the number of particles and N diverge. The Nclock model is a twodimensional 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 Nclock 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 Nclock 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 phasetransition problems

May
11 2018 
Elena Zhizhina (Russian Academy of
Science)
Astral diffusion as a limit process for symmetric random walks in a highcontrast 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 operatorsAbstract.
In the course two types of singularly
perturbed operators will be considered.
First we will consider homogenization
problems for periodic linear
convectiondiffusion 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 secondorder 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) Largetime
behaviour of the heat kernel of a convolutiontype
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 Gammaconvergence 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) Diffeomorphic 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 diffeomorphisms
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, biSobolev and BV (actually biBV )
settings. This is a joint work with Aldo Pratelli (FAU
ErlangenNurnberg).

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
manybody 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 smallstrain 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 discretetime incremental minimum problems. The main difficulty in the passage to the continuoustime 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 selfsimilar 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. selfsimilar set (a general class of finitely ramified fractals) there exists a selfsimilar Dirichlet form (with good property). In this talk, I show a counterexample to that conjecture. 
January 30 2017 
Rufat
Badal (TU Munich) Gammaconvergence 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: FibreReinforced 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 unidirectional 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 forcedisplacement relationship. Several failure regimes, involving for instance microcracking and diffuse delamination phenomena, are highlighted through the construction of a ”damage mode map”. A numerical solution strategy is finally proposed, based on phasefield models. 
January 16 2017 
Brainstorming 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 semicoherent interfaces Abstract: We propose and analyze a simple variational model for dislocations at semicoherent 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 semidiscrete 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 LennardJones 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) Secondorder approximation of anisotropic freediscontinuity functionals Abstract: We provide an approximation result for freediscontinuity functionals whose surfaceintegrand is an arbitrary smooth Finsler metric. Our main result states that these functionals can be obtained as the Gammalimit of elliptic energies of AmbrosioTortorelli 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 pairinteraction 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^(d1). 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 pDirichlet energies from a twodimensional set into an oscillating twodimensional manifold. We show that for p<2 we obtain an effective (homogenized) energy of pgrowth finite in the whole Sobolev space W^{1,p}. For p>2 the energy density is quasiconvex and infinite on ranktwo 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 helimagnetferromagnet 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 Gammaconvergence with ModicaMortola 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 freediscontinuity 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 dimensionreduction 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 nontrivial 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) Linetension 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) Lowersemicontinuity and relaxation results for energies in capillarity problems Abstract: The free energy in capillarity problems usually consists of a surface term accounting for the liquidgas interface and a 'wetting' term accounting for the interaction at the walls of the container of theliquidsolid 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 PeronaMalik functional in 1D Abstract: Under a suitable scaling the discrete PeronaMalik scheme has been proved to converge to the MumfordShah functional by Morini and Negri. In the same scaling we analyze the asymptotic description of local minima, quasistatic evolution and the minimizingmovement 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 nexttonearest 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 positivedeterminant 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 softdisc crystallization result Abstract: We discuss a simple model for finite crystallization in the case of zero temperature, shortrange 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 twodimensional 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 rateindependent 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 