Uniform approximability of almost isometric maps

• Sergio Conti

Room: Aula Dini Given a Sobolev map from a Lipschitz subset U of an oriented, compact manifold M into another oriented, compact manifold N of the same dimension, we construct an approximation by a C^{1,\alpha} map with a uniform bound on its W^{2,m} norm, for every m>1. The W^{1,p}-distance between the approximation and the original map is controlled in terms of the L^p-deviation from being an isometry, with optimal scaling. This approximation result is closely connected to geometric rigidity properties, which have played a central role in many developments in nonlinear elasticity over recent decades. The talk is based on joint work with Georg Dolzmann (Regensburg) and Stefan Müller (Bonn).

Topological and geometric defects in discrete systems: Fractional vortices and partial dislocations" (Part I)

• Marco Cicalese (TU Munich)

Room: Aula Dini

Topological and geometric defects in discrete systems: Fractional vortices and partial dislocations" (Part II)

• Annika Bach (Eindhoven University of Technology)

Room: Aula Dini

Gamma-convergence and stochastic homogenization for functionals in the A-free setting

• Gianni Dal Maso (SISSA)

Room: Aula Dini We obtain a compactness result for Gamma-convergence of integral functionals defined on A-free vector fields. This is used to study homogenization problems for these functionals without periodicity assumptions. More precisely, we prove that the homogenized integrand can be obtained by taking limits of minimum values of suitable minimization problems on large cubes, when the side length of these cubes tends to infinity, assuming that these limit values do not depend on the center of the cube. Under the usual stochastic periodicity assumptions, this result is then used to solve the stochastic homogenization problem by means of the subadditive ergodic theorem.

Wrinkling in the Lamé problem: a Γ-convergence approach

• Roberta Marziani (Università degli studi di Siena)

Room: Aula Dini In this talk I will discuss the formation of wrinkling patterns in a thin elastic annulus subjected to radial stretching within the framework of the Föppl–von Kármán theory. In the limit of vanishing thickness, azimuthal compression develops in an inner region of the sheet, leading to highly oscillatory wrinkle patterns. After subtracting the relaxed membrane energy and performing a suitable rescaling, we study the next-order variational problem governing the distribution of wrinkle frequencies. Using a Fourier decomposition and a measure-theoretic formulation, we prove a Γ-convergence result for the rescaled energies toward a convex functional defined on measures satisfying a marginal constraint. The limiting problem captures the effective mechanism selecting wrinkle patterns beyond scaling laws.

Quasistatic cohesive-type crack growth

• Vito Crismale (Sapienza Università di Roma)

Room: Aula Dini With Manuel Friedrich (JKU Linz), we proved the existence of globally stable quasistatic evolutions for cohesive-type crack energies in the SBVp framework. The crack path is not prescribed a priori nor subject to any topological restriction, and the result is valid in any dimension. This generalizes to a cohesive setting the existence result by Francfort and Larsen in 2003.

Skyrmionic bubbles in ultrathin ferromagnetic films

• Cyrill Muratov (Università di Pisa)

Room: Aula Dini Magnetic skyrmions are tiny whirls of magnetization, in which the magnetization vector locally reverses its preferred orientation and whose stability is due to a delicate balance of forces acting at nanoscale. In this talk, I will discuss the situation in which these magnetization states take the shape of skyrmionic bubbles in ultrathin ferromagnetic films, whereby the skyrmion consists of a compact region of magnetization opposite to the magnetization in the bulk of the film, with a narrow transition layer in the form of a chiral domain wall. Starting with the reduced two-dimensional micromagnetic energy, I will derive the energy of these bubbles via Gamma-convergence in finite samples of increasing size, resulting in a geometric variational problem describing the shape of skyrmionic bubbles in the presence of an applied magnetic field. I will finally touch upon radially symmetric bubbles as the DMI strength approaches critical. This is joint work with M. Piccinini, M. Novaga and V. Slastikov.

Currents with coefficients in a group and branched transport problems

• Annalisa Massaccesi (Università degli studi di Padova)

Room: Aula Dini In this seminar I will review the theory of flat G-chains, as they were introduced by H. W. Fleming in 1966, and currents with coefficients in groups with the aim of showing some recent applications to variants of the branched optimal transport. One development of the theory concerns its application to the Steiner tree problem and other minimal network problems which are related with a Eulerian formulation of the branched optimal transport. Starting from a 2016 paper by A. Marchese and myself, I will show how these problems and their variants are equivalent to a mass-minimization problem in the framework of currents with coefficients in a (suitably chosen) normed group. The variants I'm referring to include the multicommodity flow, the mailing problem and new models for robust and resilient traffic plans, as shown in a recent paper in collaboration with L. De Masi and A. Marchese.

On the Minimisation of Nonlocal Interaction Energies

• Maria Giovanna Mora (Università di Pavia)

Room: Aula Dini Nonlocal interaction energies play a central role in describing the collective behaviour of large particle systems in a wide range of applications. In this lecture we will focus on interactions that are short-range repulsive and long-range attractive. We will review the key results on the existence and uniqueness of minimisers, and present their explicit characterisation in the classical case of isotropic kernels with Riesz-type repulsion and quadratic attraction. We will then show how a complete characterisation can be given for a broad class of anisotropic variants of the repulsive kernel. If time permits, we will conclude with a discussion of open problems and future directions. This talk is based on joint work with several collaborators: R. Frank, J. Mateu, L. Rondi, L. Scardia, E.G. Tolotti, and J. Verdera.

Is brittle fracture broken?

• Gilles Francfort (Flatiron Institue)

Room: Aula Dini After nearly thirty years, the ``variational approach to fracture" is very much alive both in the mechanics and mathematical communities. Yet a host of issues are emerging that obfuscate the elegance of the original theory. I will review a few of those, show the shortcomings of the existing sharp theories and the additional hurdles introduced by the phase field approximations. I will comment on what I now view as unwarranted departures, even if mathematically sound, like cohesive fracture or the various kinds of models that introduce viscosity as a dissipation mechanism during jumps. I will hopefully propose a nascent brittle theory which unfortunately would require a complete revisiting of the existing body of mathematical results.

Stochastic homogenization of fractional obstacle problems

• Matteo Focardi (Università degli studi di Firenze)

Room: Aula Dini

Homogenisation of droplets in liquid-liquid phase-transitions

• Caterina Ida Zeppieri (Universität Münster)

Room: Aula Dini We rigorously derive a sharp-interface model for the coexistence of multiple liquid phases in a domain containing a random distribution of small droplets, within which a prescribed phase is enforced. Starting from a diffuse-interface model of Modica–Mortola type, we prove that, under very general assumptions on the droplet distribution (modelled probabilistically by a stationary marked point process) and at a critical scaling for the droplet radii, an additional stochastic bulk term of capacitary type emerges in the limit functional.

Asymptotic expansion of the relative s-fractional perimeter as s->0.

• Lucia De Luca (CNR, Roma)

Room: Aula Dini We prove a first-order Gamma-convergence result for the relative s-fractional perimeter, as s->0, with respect to a given bounded set and prescribed exterior data. We investigate the robustness of our approach under perturbations of the exterior data and in the presence of volume constraints. We then use the Gamma-convergence expansion to study the asymptotic behavior of s-minimal sets as s->0. Joint work with G. Palatucci, M. Piccinini, and M. Ponsiglione.

Beating filaments: from eukaryotic cilia to photo-responsive rods

• Antonio De Simone (Scuola Superiore Sant'Anna)

Room: Aula Dini Spontaneous oscillations are observed across many natural and artificial systems. We present a comparative analysis of the chemo-mechanical mechanisms that drive spontaneous oscillations in two distinct active filamentous structures: photo-chemically deformable liquid crystal elastomer (LCE) rods and ATP-powered eukaryotic cilia. Using the unifying framework of active planar rods, we develop simplified mathematical models for both systems. We reduce the governing partial differential equations to one-degree-of-freedom (1-DOF) nonlinear oscillators, each undergoing a supercritical Hopf bifurcation. For these reduced models, we obtain explicit analytical expressions for the onset and characteristics of self-sustained oscillations. Despite the common mathematical structure, the underlying physical mechanisms are fundamentally different. For LCEs, self-oscillation is an inertial phenomenon driven by an elastic-inertial feedback over the timescale of the photochemical reaction. In contrast, eukaryotic cilia live in the inertia-less (low Reynolds number) regime, and the instability is driven by a negative effective damping - a motive force that arises from the mechano-chemistry of molecular motors. The analytical predictions for critical activation thresholds, frequencies, and amplitudes agree with full nonlinear simulations, providing quantitative insight into the dynamics of these complex self-oscillating systems.

Convergence of entropic transport energies

• Antonin Chambolle (Université Paris Dauphine-PSL)

Room: Aula Dini

Quasiconvexity in the Riemannian setting

• Carlo Mantegazza (Università di Napoli Federico II)

Room: Aula Dini

The structure of normal currents and regularity of transport operators

• Giovanni Alberti (Università di Pisa)

Room: Aula Dini