On a new partitioning problem

• Lia Bronsard (McMaster University)

Room: Aula Dini I will first review the classical partitioning problem, discuss the double/triple/K-bubble conjectures, and introduce a new partitioning problem that arises in the study of triblock copolymers in certain limiting regimes. I will present the setting for this new geometrical problem, discuss existence and uniqueness of locally minimizing solutions as well as answers to several conjectures. These represent joint works with Stan Alama, Silas Vriend, Mike Novack, Robin Neumayer and Anna Skorobogatova.

Some result on relaxation of polyconvex functions with linear growth

• Riccardo Scala (Università di Siena)

Room: Aula Dini We discuss the relaxation on $L^1$ of polyconvex functions with linear growth, and recall some old and new results. As prototype of this class of energies, the analysis of the area functional leads to the main example of nonlocality and non-subadditivity, actually confirming a conjecture by De Giorgi (proved by Acerbi and Dal Maso). We discuss what has been recently done to understand the behaviour of this kind of energies and we show how the situation is completely different when one considers relaxation under stronger topologies than $L^1$, as for instance the strict convergence in BV.

Linear Stability of the self-similarly shrinking lens

• Theresa Simon (Universität Münster)

Room: Aula Dini When performing a parabolic blowup analysis of singularities in 2D multiphase mean curvature flow, one is led to the notion of self-similar shrinker: Networks whose evolution by mean curvature is given by shrinking homotheties. It can be shown that they are critical points of an entropy given by the interface length functional with a suitable Gaussian weight. Furthermore, this entropy is decreased during the flow. Hence the dynamic stability of the shrinkers can be studied via their stability with respect to the entropy, a matter that is complicated by the existence of, generically, four unstable modes arising from dilation, translation, and rotation. In the talk, I will demonstrate how to perform a linear stability analysis of self-similar shrinkers for the example of the lens.

Geometric flows, monotonicity formulas, and functional inequalities.

• Alessandra Pluda (Università di Pisa)

Room: Aula Dini Taking advantage of monotone quantities along geometric flow to derive functional inequalities is a recurring scheme in geometric analysis. Recently, we have provided a unified perspective on a broad range of monotonicity formulas in both linear and nonlinear potential theory, as well as along the inverse mean curvature flow. The quantities involved in this study are generalizations and variants of the Willmore functional. In the talk I will focus on the implications of these formulas and present Willmore-type inequalities in R^n and in Riemannian manifolds with suitable bounds on the Ricci curvature. Based on joint works with Luca Benatti, Marco Pozzetta, and Stefano Mannella.

Gamma-convergence of the square sticky disk to the octagonal crystalline perimeter

• Giacomo Del Nin (MPI MiS Leipzig)

Room: Aula Dini We consider a variant of the sticky disk model for N interacting particles in the plane, where distances are evaluated by means of the supremum norm instead of the Euclidean norm. We show crystallization for minima of such an energy (for fixed N) and we prove Gamma-convergence (in the limit as N goes to infinity) of suitably rescaled energies to the anisotropic perimeter with octagonal Wulff shape. The key result to establish this is an energy decomposition for graphs in the plane that hinges upon the notion of angular defect, and that is quite flexible and potentially adaptable to other energies. The talk is based on joint work with Lucia De Luca (IAC-CNR).

Uniqueness and characteristic flow for a non strictly convex singular variational problem

• Jean-Francois Babadjian (Université Paris Saclay)

Room: Aula Dini This talk addresses the question of uniqueness of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand - whose precise form derives directly from the theory of perfect plasticity - behaves quadratically close to the origin and grows linearly once a specific threshold is reached. We make use of spatial hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector field - the Cauchy stress in the terminology of perfect plasticity - which allows us to define characteristic lines, and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape, we show that this vector field is actually continuous, save for possibly two points. The different behaviors of the energy density at zero and at infinity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data. This is a joint work with Gilles Francfort.

Dimension reduction for elastic materials with voids

• Manuel Friedrich (FAU Erlangen-Nürnberg)

Room: Aula Dini In this talk I present some recent dimension-reduction results for elastic materials with voids. We consider three-dimensional models with an elastic bulk and an interfacial energy featuring a Willmore-type curvature penalization. By Gamma-convergence we rigorously derive lower-dimensional models for rods and plates where the effective limit comprises a classical elastic bending energy and surface terms reflecting the possibility that voids can persist in the limit and that the material can be folded or broken apart into several pieces. The main ingredient for the analysis is a novel rigidity estimate in varying domains under vanishing curvature regularization. Joint work with Leonard Kreutz and Konstantinos Zemas.

Superlinear free-discontinuity models: relaxation and phase field approximation

• Flaviana Iurlano (Università di Genova)

Room: Aula Dini We develop the Direct Method in the Calculus of Variations for free-discontinuity energies whose bulk and surface densities exhibit superlinear growth, respectively for large gradients and small jump amplitudes. A distinctive feature of this kind of models is that the functionals are defined on SBV functions whose jump sets may have infinite measure. Establishing general lower semicontinuity and relaxation results in this setting requires new analytical techniques. In addition, we propose a variational approximation of certain superlinear energies via phase field models. This is a joint work with Sergio Conti and Matteo Focardi.

Curvature penalization of strongly anisotropic interface models and their phase-field approximation

• Michael Goldman (CMAP Ecole Polytechnique)

Room: Aula Dini In this talk I will present recent results obtained with J.F. Babadjian and B. Buet about the regularizing effects of curvature terms for interface models with strong anisotropy. We will consider two main (related) questions for two types of problems. The questions are lower semi-continuity of the energies and phase-field approximations. The models are isoperimetric problems on the one hand and free discontinuity problems on the other hand. Both are motivated by applications in material sciences. One of the original aspects of our work in the setting of free discontinuity problems, is the treatment of point energies which relies on a Gauss-Bonnet type result for varifolds.

Characterisation of area-strict limits of planar BV homeomorphisms

• Emanuela Radici (Università dell’Aquila)

Room: Aula Dini We present the classification of area-strict limits of planar BV homeomorphisms. This class of mappings allows for cavitations and fractures but fulfils a suitable generalization of the INV condition. As pointed out by J. Ball, these features are expected in limit configurations of elastic deformations. Few years ago, De Philippis and Pratelli introduced the no-crossing condition which characterises the Sobolev W1,p closure of planar homeomorphisms, where cavitations may occur. In our work we show that a suitable generalisation of this concept is equivalent with a map being the area-strict limit of BV homeomorphisms. In the BV setting more complicated singularities (fractures) may occur. This is a joint work with Daniel Campbell and Aapo Kauranen.

Generic level sets in mean curvature flow with and without obstacles

• Tim Laux (Universität Regensburg)

Room: Aula Dini Mean curvature flow has been a central object in geometric analysis. Weak solutions describe the evolution past singularities, but different solution concepts might lead to different behavior. In this talk, I'll present recent results on the relation between the viscosity solution and distributional solutions. I will also present extensions to the associated obstacle problem, introduce weak solution concepts and show their relation.

Elementary discrete convolution/redistancing schemes for the mean curvature flow

• Daniele De Gennaro (Bocconi University)

Room: Aula Dini In the talk I will present some recent results obtained in collaboration with A. Chambolle and M. Morini, concerning some fully discrete (in both space and time) and explicit schemes for the mean curvature flow of boundaries. These schemes are based on an elementary diffusion step and a more costly redistancing operation. We give an elementary convergence proof for the schemes under the standard CFL condition. We will then discuss extensions to more general convolution-redistancing schemes, with the aim of laying the ground of a sound mathematical explanation of the very good results produced by fully learned approaches for the mean curvature flow, recently introduced in the literature.

The epsilon-regularity theorem for Brakke flows near triple junctions

• Salvatore Stuvard (Università di Milano)

Room: Aula Dini In a pioneering paper published on JDG in 1993, Leon Simon established a powerful method to demonstrate, among other things, the validity of the following result: if a multiplicity one minimal k-surface (stationary varifold) is sufficiently close, in the unit ball and in a weak measure-theoretic sense, to the stationary cone given by the union of three k-dimensional half-planes meeting along a (k-1)-dimensional subspace and forming angles of 120 degrees with one another, then, in a smaller ball, the surface must be a C^{1,\alpha} deformation of the cone. In this talk, I will present the proof of a parabolic counterpart of this result, which applies to general classes of (possibly forced) Brakke flows. I will particularly focus on the apparent need of an assumption, which is absent in the elliptic case, and which, on the other hand, is satisfied by both canonical multi-phase Brakke flows and Brakke flows obtained by elliptic regularization with mod(3) coefficients: these are the main classes of Brakke flows for which a satisfactory existence theory is currently available and triple junction singularities are expected. This is a joint work with Yoshihiro Tonegawa (Institute of Science Tokyo).

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

• Sergio Conti (Universität Bonn)

Room: Aula Dini I will present a phase field approximation for sharp interface energies, defined on partitions, as appropriate for modeling grain boundaries in polycrystals. The label takes value in O(d)/G, where G is the point group of a lattice, the phase-field approximation fully respects the symmetry. These functionals can be used for the simulation of grain growth or for image reconstruction of grain boundaries. The talk is based on joint work with Vito Crismale, Adriana Garroni and Annalisa Malusa (Sapienza, Roma)

Minimization of fractional seminorms on the real line and applications to misfit dislocations

• Lucia De Luca (IAC-CNR, Rome)

Room: Aula Dini We prove that minimizers of fractional Gagliardo seminorms, among piecewise affine functions defined on the real line with two given - opposite - slopes (suitably prescribing the length scale of the oscillations) are periodic. We extend such a result to a less rigid setting that allows to study also the gradient flow of the energy functionals. Our analysis applies to the van der Merwe theory of misfit dislocations. We consider two elastic materials casting parallel lattices having different but very close spacing (semi-coherent interface). Identifying (small) intervals with negative derivative with the core regions of dislocations lying on semi-coherent interfaces, and describing the elastic energy on the half-planes delimited by such interface as the Gagliardo seminorm of the boundary datum, our analysis proves that the minimal energy is obtained when the dislocations are uniformly distributed. Joint work with M. Goldman, M. Ponsiglione, E. Spadaro.

Concentration Phenomena in Magnetic Lattice Models

• Marco Cicalese (TU Munich)

Room: Aula Dini We will gently review both classical and recent results concerning the variational analysis of energy concentration phenomena in magnetic models defined on lattices in the limit as the lattice spacing tends to zero. The discussion includes results on the classical XY model on the square lattice, when the magnetization takes values in S^1, along with some of its variants. We will then focus on recent advances in the analysis of magnetic skyrmion models, where the magnetization is S^2-valued, highlighting the crucial role of the topology of the target space.

Variational analysis of nematic axisymmetric films

• Giulia Bevilacqua (Università di Pisa)

Room: Aula Dini Nematic films are thin fluid structures, ideally two dimensional, endowed with an in-plane degenerate nematic order. Some variational models for nematic films have been introduced by Giomi in 2012 and by Napoli and Vergori in 2018. At equilibrium, the shape of the nematic film results from the competition between surface tension, which favors the minimization of the area, and the nematic elasticity, which instead promotes the alignment of the molecules along a common direction. The main difference between the two mentioned approaches is the way to compute the surface derivative of the nematic vector field. In this seminar I will briefly describe the physical models and I will present a complete variational analysis of the model proposed by Giomi for revolution surfaces spanning two coaxial rings. Time permitting, I will present the case of superficial gradient introduced by Napoli and Vergori in a specific geometrical setting. Joint work with Chiara Lonati (POLITO), Luca Lussardi (POLITO) and Alfredo Marzocchi (UNICATT).

Rigidity and flexibility in differential inclusions: the Tartar square

• Antonio Tribuzio (Universität Bonn)

Room: Aula Dini In recent years, the study of highly non-convex differential inclusions increased a lot, also motivated by applications to materials science. Due to the lack of convexity, according to the prescribed regularity, there may be either many (flexibility) or one (rigidity) class of solutions. After introducing and motivating the problem, we try to find information about the threshold regularity between rigidity and flexibility by studying a simplified toy-model, the so-called Tartar square, by relaxing the problem studying scaling laws of the related singularly-perturbed elastic energy. If time permits, we will also see its geometrically-linearized analogous. The results presented in this talk are in collaboration with Angkana Rüland.

Quantitative isoperimetric inequalities in capillarity problems

• Marco Pozzetta (Politecnico di Milano)

Room: Aula Dini The classical capillarity perimeter of a set in a half-space is defined as the sum of its relative perimeter inside the half-space and a constant multiple of the area of the portion of its boundary lying on the boundary of the half-space. The isoperimetric capillarity problem seeks to minimize the capillarity perimeter under a volume constraint. A classical isoperimetric inequality implies that suitable bubbles, given by truncations of balls, are minimizers of the capillarity problem. In this talk, we present a sharp strong form of the quantitative isoperimetric inequality for the capillarity problem. We consider a notion of asymmetry that quantifies how much the unit normals to the boundary of a competitor deviate from those of an optimal bubble. Hence the inequality bounds this asymmetry from above in terms of the isoperimetric deficit of the competitor. Roughly speaking, the inequality bounds an $H^1$-notion of distance of a competitor from the set of minimizers in terms of its isoperimetric deficit. The talk is based on results obtained in collaboration with Davide Carazzato (University of Vienna) and Giulio Pascale (University of Naples Federico II).

Isoperimetric planar Tilings with unequal cells

• Francesco Nobili (Università di Pisa)

Room: Aula Dini In this talk, we consider an isoperimetric problem for periodic planar Tilings allowing for unequal repeating cells. We discuss general existence and regularity results and we study classification results for double Tilings, i.e. Tilings with two repeating cells. In this case, we explicitly compute the associated energy profile and we give a complete description of the phase transitions. Based on joint works with M. Novaga and E. Paolini.

Variational models for partial defects

• Adriana Garroni (Università di Roma, La Sapienza)

Room: Aula Dini I will treat a class of sharp interface models for partial defects, in which partial defects are codimension 2 objects connected by codimension 1 objects. I will present two derivations of such energies. In dimension 2 a discrete model for crystal defects based on nearest neighbours and next to nearest neighbours interaction, via period potentials. In the asymptotic limit as the lattice spacing tends to zero, in terms of Gamma convergence, the model accounts for the formation and interaction of partial dislocations, point defects, and stacking faults, line defects connecting partials. The model falls into a class of discrete models with topological (fractional) singularities. One of the key ingredients is the characterisation of the minimisers for the core energy of the singularities, the so called one-vortex solutions. A second approach, strictly related, consists in a diffuse interface energy, which can be interprete as a semi-discrete model, à la Nabarro Peierls, which has the structure of a phase transition model with a multiple well potential and a non locale singular perturbation. This latter gives rise to the same asymptotics allowing also to treat the three dimensional case, under the assumption that the partial line dislocations lie on a single slip plane.