# Trimester Seminar Series

**March 15, 2023 (CEST)**

3:00 - 4:00pm Maria Carme Calderer (University of Minnesota)

Title: Mathematical Problems at the Interface of Materials Sciences and Biology

Abstract: It is well known in liquid crystal research that nuclei of ordered materials emerging from the isotropic state usually show a shape topologically equivalent to a sphere, such as in the case of nematic liquid crystal droplets. In this presentation, we analyze a new type of free boundary shapes, in the form of tori, that consist of chromonic liquid crystals in the hexagonal phase. We show that such shapes are minimizers of a multi-constraint liquid crystal bulk energy plus an anisotropic surface contribution, the latter being responsible for the faceted shapes observed in the experiments. Let us recall that chromonic liquid crystals consist of flat, plank-like molecules found, for instance, in food dies and antiasthmatic drugs that form liquid crystal phases when dissolved in water. Due to, both, the hydrophilic and hydrophobic groups present in such molecules, they form columnar stacks that tend to align themselves in a parallel fashion, forming increasingly ordered liquid crystal phases with rising concentration. From a different point of view, condensed DNA also forms chromonic liquid crystal phases, with the same optical properties as the die and drug compounds. In vitro DNA, in a gel with condensing agents, also forms toroidal droplets but with a typical size of about one-millionth that of its chromonic counterparts. In vivo, such ordered arrangements are encountered in the quiescent state of encapsidated, double stranded-DNA viruses that infect bacteria, known as bacteriophages, the liquid crystal organization being a key part of their biological function to ensure an optimal infection. Mathematically, we represent a virus as a vector field-line pair that minimize an energy, the sum of the bending and twist components of the DNA center curve plus its electrostatic energy and that of the environmental ions. We conclude with some remarks on the dynamics of the packaging and infection processes.

- Click here for the abstract (with references)

4:30 - 5:30pm Douglas Arnold (University of Minnesota)

Title: Wave localization and its landscape

Abstract: The puzzling phenomenon of wave localization refers to unexpected confinement of waves triggered by disorder in the propagating media. Localization arises in many physical and mathematical systems and has many important implications and applications. A particularly important case is the Schrödinger equation of quantum mechanics, for which the localization behavior is crucial to the electrical properties of materials. Mathematically it is tied to exponential decay of eigenfunctions of operators instead of their expected extension throughout the domain. Although localization has been studied by physicists and mathematicians for the better part of a century, many aspects remain mysterious. In particular, the sort of deterministic quantitative results needed to predict, control, and exploit localization have remained elusive. This talk will focus on major strides made in recent years based on the introduction of the landscape function and its partner, the effective potential. We will describe these developments from the viewpoint of a computational mathematician who sees the landscape theory as a completely unorthodox sort of a numerical method for computing spectra.

**March 14, 2023 (CEST)**

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.

**March 8, 2023 (CEST)**

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.

**March 7, 2023 (CEST)**

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.

**February 28, 2023 (CEST)**

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.

**February 14, 2023 (CEST)**

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 $K\subset B$ with a Navier boundary condition on $\partial K$. 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.

**February 7, 2023 (CEST)**

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

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

Abstract: In this talk, we will be concerned with flexibility of weak solutions to the Monge-Ampere system via convex integration. This system is a natural extension of the Monge-Ampere equation $det\, D^2v =f$, in the contexts of: (i) isometric immersions and (ii) nonlinear elasticity. The main technical ingredient consists of the ``stage'' construction, in which we achieve the Holder regularity $\mathcal{C}^{1,\alpha}$ of the approximating fields, for all $\alpha<\frac{1}{1+ d(d+1)/k}$ where $d$ is an arbitrary dimension and $k\geq 1$ is an arbitrary codimension. When $d=2$ and $k=1$, we recover the previous result by Lewicka-Pakzad for the Monge-Ampere equation. Our construction can be translated to the isometric immersion problem, where for $k=1$ we recover the result by Conti-Delellis-Szekelyhidi, and for large $k$ we quantify the result by Kallen.

**January 31, 2023 (CEST)**

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

Title: Stability of the vortex in line-energy models

Abstract: PDF-File

**January 24, 2023 (CEST)**

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.

**January 17, 2023 (CEST)**

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.