Jean-François Babadjian (Université Paris-Saclay, France):
Curvature penalization of strongly anisotropic sharp interface models and their phase field approximation
Abstract: In this talk, I will present recent results obtained with B. Buet and M. Goldman about the regularizing effects of curvature terms for sharp interface models with strong anisotropy. When the surface tension is a convex function of the normal to the interface, the anisotropy is said to be weak. This usually ensures the lower semicontinuity of the associated energy. If, however, the surface tension depends on the normal in a nonconvex way, this so-called strong anisotropy may lead to instabilities related to the lack of lower semicontinuity of the functional. We investigate the regularizing effects of adding a higher order term of Willmore type to the energy. We consider two types of problems. The first one is an anisotropic nonconvex generalization of the perimeter, and the second one is an anisotropic nonconvex Mumford-Shah free discontinuity functional. In both cases, lower semicontinuity properties of the energies are established, as well as a phase field Gamma-convergence approximation. 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.
Filippo Cagnetti (Università di Parma, Italy):
Best approximation of Lebesgue measure by discrete lattice measures in the Wasserstein metric
Abstract: We will discuss the problem of the best approximation in the Wasserstein metric of the three-dimensional Lebesgue measure, by an atomic measure supported on a Bravais lattice. This is work in collaboration with David Bourne (Heriot-Watt University) and Steven Roper (University of Glasgow).
Matteo Carducci (SNS Pisa, Italy)
Free boundary regularity for a tumor growth model with obstacle
Abstract: In this talk, we discuss a geometric free boundary problem arising in tumor growth modeling. Specifically, the tumor is contained within a bounded region, part of which is inaccessible and acts as an obstacle. We investigate the existence of viscosity solutions, as well as the interior and boundary regularity of the associated free boundary.
These results are obtained in a joint work with G. Bevilacqua.
Marco Cicalese (TU München, Germany):
Concentration Phenomena in the Variational Analysis of Magnetic Lattice Models
Abstract: We provide a concise review of both classical and recent results on the variational analysis of energy concentration phenomena in magnetic lattice models, focusing on the limit as the lattice spacing tends to zero. We draw parallels with well-established results in the continuum setting. Our discussion includes the classical XY model on the square lattice, when the magnetization takes values in S^1, as well as some of its variants. We then highlight recent advances in the variational analysis of magnetic skyrmion models, where the magnetization is S^2-valued, emphasizing the crucial role of the topology of the target space.
Vito Crismale (Sapienza Università di Roma):
Phase-field approximation of sharp-interface energies accounting for lattice symmetry
Abstract: The talk concerns 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 limiting surface energy behaves for small angles as s|log s|, according to the Read and Shockley law. These functionals can be used for image reconstruction of grain boundaries.
Joint work with S. Conti (HCM Bonn), A. Garroni, A. Malusa (Sapienza).
Carolin Kreisbeck (Katholische Universität Eichstätt-Ingolstadt, Germany):
Nonlocal gradients in variational problems: Heterogeneous horizons and local boundary conditions
Abstract: Building on recent advances in nonlocal hyperelasticity, we discuss a class of variational problems involving integral functionals with nonlocal gradients. Specific to our set-up is a space-dependent interaction range that vanishes at the boundary of the reference domain. This ensures that the operator depends only on values within the domain and localizes to the classical gradient at the boundary, which allows for a seamless integration of nonlocal modeling with local boundary values.
Our main contribution is a comprehensive study of the associated Sobolev spaces, including the analysis of a trace operator and the proof of a Poincaré inequality. A central aspect of our technical approach lies in exploiting connections with pseudo-differential operator theory.
As an application, we establish the existence of minimizers for functionals with quasiconvex or polyconvex integrands depending on heterogeneous nonlocal gradients, subject to local Dirichlet-, Neumann- or mixed-type boundary conditions.
This is joint work with Hidde Schönberger (TU Vienna).
Anna Kubin (TU Wien, Austria):
Stability of the nonlocal Mullins–Sekerka flow via a geometric stability inequality
Abstract: In this talk, I will present results on the stability of the $H^{-1/2}$-gradient flow of a nonlocal perimeter energy, known as the modified Mullins-Sekerka flow.
In particular, I will introduce a new geometric inequality for this energy, which takes the form of a quantitative Alexandrov theorem. I will then explain how this inequality can be used to establish the stability of this geometric flow. Specifically, if the initial datum is sufficiently close to a stable set for the energy, then the flow exists for all times and converges, as the time tends to infinity, to a suitable translate of the stable set exponentially fast.
This work is part of an ongoing collaboration with Daniele De Gennaro (Bocconi University).
Alice Marveggio (HCM Bonn, Germany):
The Verigin problem with phase transition as Wasserstein flow
Abstract: We study the modeling of a compressible two-phase flow in a porous medium. The governing PDE system is known as the Verigin problem with phase transition, which is the compressible analog to the Muskat problem. Our aim is to prove the convergence of an implicit time discretization scheme using the Wasserstein distance, obtaining distributional solutions in the limit that satisfy an optimal energy-dissipation rate.
This talk is based on a joint work with Anna Kubin and Tim Laux.
Roberta Marziani (Università di Siena, Italy):
Ambrosio-Tortorelli approach to topological singularities and connections with jump minimizing liftings
Abstract: We study the Gamma-convergence of Ambrosio-Tortorelli-type functionals, for maps u defined on an open bounded set Ω ⊂ R^n and taking values in the unit circle S^1 ⊂ R^2. Depending on the domain of the functional, two different Gamma-limits are possible, one of which is nonlocal, and related to the notion of jump minimizing lifting, i.e., a lifting of a map u whose measure of the jump set is minimal. The latter requires ad hoc compactness results for sequences of liftings which, besides being interesting by themselves, also allow to deduce existence of a jump minimizing lifting.
This is based on a joint work with Giovanni Bellettini and Riccardo Scala.
Gianluca Orlando (Università di Bari, Italy):
Stacking faults in the discrete-to-continuum limit of a model for partial edge dislocations
Abstract: Dislocations and stacking faults are two distinct types of defects in the arrangement of atoms arising during crystal growth in the lattice structure of a material. They are among the microscopic phenomena that explain permanent plastic deformations. With the aim of gaining insight on the relation between dislocations and stacking faults, in this talk we will present the analysis for a simple discrete lattice model able to capture these kinds of defects. The model is characterised by an energy with nearest neighbours interactions and next-to-nearest neighbours interactions designed in such a way that two distinct lattices are ground states for the system. Through the variational limit as the lattice spacing tends to zero, we are able to understand the interplay between these interactions and how they capture the presence of partial dislocations and the corresponding stacking faults that resolve singularity tension in the lattice structure.
The contents of the talk are based on a work in collaboration with Annika Bach, Marco Cicalese, and Adriana Garroni.
Lorenzo Portinale (Università di Milano, Italy):
Regularity by duality for minimising movements with nonlinear mobility
Abstract: In this talk, we will talk about conservation laws that can be written as gradient flows with respect to a Wasserstein distance with nonlinear mobility. In particular, we discuss some ideas on how to infer regularity estimates on time-discretisation schemes, via two important tools: (dynamical) duality and comparison principles.
Emanuela Radici (Università dell'Aquila, Italy):
Entropy solutions of nonlocal transport equations with congestion
Abstract: In this talk we consider a class of scalar nonlinear models describing crowd dynamics. The congestion term appears in the transport equation in the form of a compactly supported nonlinear mobility function, thus making standard weak-type compactness arguments and uniqueness of weak solutions fail. We introduce two different approaches to the problem and discuss their connections with the wellposedness of entropy solutions of the target pde in the sense of Kružkov: a deterministic particle approach relying on suitable generalisations of the Follow-the-leader scheme, which can be interpreted as the Lagrangian discretisations of the problem; and a variational approach in the spirit of a minimising movement scheme exploiting the gradient flow structure of the evolution in a suitable metric framework.
Matthias Ruf (Universität Augsburg, Germany):
On the multi-cell formula in homogenization
Abstract: Homogenization is by now a classical method to simplify models featuring fine microstructure. In the case of integral functionals on Sobolev spaces, the form of simplification depends drastically on whether the integrand is convex or not. For non-convex integrands, the homogenized integrand is given by a complicated multi-cell involving a minimization problem on larger and larger cubes. In this talk, we present a method that allows us to restrict the minimization problem to smaller classes (e.g., more regular functions beyond density in energy). We then illustrate how this strategy can help in a variational analysis.
This is based on joint work with Annika Bach (TU Eindhoven).
Marita Thomas (WIAS Berlin, Germany):
Analysis of visco-elastoplastic two-phase flows in geodynamics
Abstract: A model for incompressible fluids of both viscoelastic and plastic behavior is revisited, which is used in geodynamics, e.g., to describe the evolution of fault systems in the lithosphere on geological time scales. The Cauchy stress of this fluid is composed of a viscoelastic Stokes-like contribution and of an additional internal stress. The model thus couples the momentum balance with the evolution law of this extra stress, which features the Zaremba-Jaumann time-derivative and a non-smooth plastic dissipation mechanism.
Phase separation of a two-component mixture is taken into account by means of a Cahn-Hilliard-type evolution law. Suitable concepts of weak solutions are discussed for the coupled model in the sense of dissipative and energy-variational solutions. Different choices of potentials are addressed.
This is joint work with Fan Cheng (FU Berlin) and Robert Lasarzik (WIAS) within project B09 "Materials with discontinuities on many scales" of CRC 1114 "Scaling Cascades in Complex Systems" funded by the German Research Foundation.