3:15 - 4:00pm Tim Laux (University of Bonn)

Title: Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

Abstract: Anisotropic mean curvature flow is a simple geometric evolution equation that models microstructure in complex materials. In this talk, I will show how it arises as the sharp-interface limit of the anisotropic Allen-Cahn equation. The proof relies on distributional solution concepts for both the diffuse and sharp interface models, and a suitable relative energy. With the same relative energy, one can prove a weak-strong uniqueness result, which relies on the construction of gradient flow calibrations for the anisotropic energy functionals. If time permits, I will also mention a few open problems. This is joint work with Kerrek Stinson and Clemens Ullrich.

3:30 - 4:30pm John Ball  (Heriot-Watt University and Maxwell Institute for Mathematical Sciences, Edinburgh)

Title: Monodromy and approach to equilibrium for viscoelastic models allowing microstructure

Abstract: For certain models of one-dimensional viscoelasticity, there are infinitely many equilibria representing phase mixtures. In order to prove convergence as time tends to infinity of solutions to a single equilibrium, it seems necessary to impose a nondegeneracy condition on the constitutive equation for the stress. The talk will explain this, and show how in some cases the nondegeneracy condition can be proved using the monodromy group of a holomorphic function. This is joint work with Inna Capdeboscq and Yasemin  Şengül.

3:00 - 4:00pm Elio Marconi (Università di Padova)

Title: Stability of the vortex in line-energy models

15:00 - 16:00 Marta Lewicka (University of Pittsburgh)

Title: The Monge-Ampere system: convex integration in arbitrary dimension and codimension

15:00 - 16:00 Mickaël Nahon (MPI Leipzig)

Title: A free discontinuity approach to optimal profiles in Stokes flows

Abstract: We consider an incompressible Stokes fluid contained in a box B

that flows around an obstacle K⊂B with a Navier boundary condition on ∂K. I will present existence and partial regularity results for the minimization of the drag of K among all profiles with certain constraints on the measure and perimeter of K, based on techniques that were developed for Griffith's fracture model in brittle materials. This is a joint work with Dorin Bucur, Antonin Chambolle and Alessandro Giacomini.

15:00 - 16:00  Roberta Marziani (TU Dortmund)

Title: 3D variational models for dislocations

Abstract: In this talk we give a brief introduction on theory of dislocations in the context of continuum elasticity. Afterwards we will introduce a 3D variational model for dislocations. We then show that the asymptotics via Gamma convergence is independent of the specific choice of the energy and of the regularization procedure.
This result is based on a joint work with Sergio Conti and Adriana Garroni.

3:00 - 4:0pm André Guerra (ETH Zürich)

Title: Quasiconvexity and nonlinear Elasticity

Abstract: Quasiconvexity is the fundamental existence condition for variational problems, yet it is poorly understood. Two outstanding problems remain:

• 1) does rank-one convexity, a simple necessary condition, imply quasiconvexity in two dimensions?

• 2) can one prove existence theorems for quasiconvex energies in the context of nonlinear Elasticity?

In this talk we show that both problems have a positive answer in a special class of isotropic energies. Our proof combines complex analysis with the theory of gradient Young measures. On the way to the main result, we establish quasiconvexity inequalities for the Burkholder function which yield, in particular, many sharp higher integrability results.
The talk is based on joint work with Kari Astala, Daniel Faraco, Aleksis Koski and Jan Kristensen.

4:30 - 5:30pm Antonio Tribuzio (Uni Heidelberg)

Title: Energy scaling of singular-perturbation models involving higher-order laminates

Abstract: Motivated by the appearance of complex microstructures in the modelling of shape-memory alloys, we study the energy scaling behaviour of some N-well problems with surface energy given by a singular higher-order term. In the case of absence of gauge invariances (e.g. with respect to the action of SO(n) or Skew(n)), we provide an ansatz-free lower bound which relies on a bootstrap argument in Fourier space and gives evidence of the higher order of lamination involved. The upper bound is provided by iterated branching constructions.
In the end, we show how a similar approach can be used in the determination of a lower bound for a more realistic model, namely the geometrically linearized cubic-to-tetragonal phase transition, in which a second order lamination is forced by the presence of affine boundary conditions. This is a joint work with Angkana Rüland.

3:00 - 4:00pm Illya M. Karabash (IAM, the University of Bonn)

Title: Composite structures with defects in the spectral optimization of leaky optical microresonators

Abstract: The optical engineering fabrication and numerical experiments for high-Q cavities led to a series of new analytical and computational problems related to optimization of resonances. The talk is devoted to the problem how to design an open  resonator that has a dissipation eigenvalue as close as possible to the real line under certain fabrication constraints. It is planned to explain the rigorous analytical background for such problems and, in particular, why the Pareto optimization settings are natural. Then we concentrate on the recent optimal control reformulation developed jointly with Herbert Koch and Ievgen Verbytskyi, as well as on resulting Hamilton-Jacobi-Bellman PDEs and extremal synthesis.

3:00 - 4:00pm Kostantinos Zemas (Universität Münster)

Title: Homogenization of nonlinear randomly perforated materials under minimal assumptions on the geometry

Abstract: In this work we combine and generalize earlier works of Giunti-Höfer-Velazquez (on the homogenization of the Poisson equation in random critically perforated domains) and Ansini-Braides (on a variational approach for the more general nonlinear vectorial problem in the periodic setting), each one in the direction of the other.
Namely, we show that under similar general assumptions on the geometry of the random perforations as the ones posed in the work of Giunti, Höfer and Velazquez, the stochastic analogue of the result of Ansini-Braides holds true, with an average deterministic nonlinear capacitary-term appearing in the Γ-limit.
This is joint work with Caterina Zeppieri and Lucia Scardia.