** Asymptotics, Operators, and Functionals **

Leading the discussion in

**16 January** Kirill Cherednichenko (Topic: Operator-norm resolvent estimates for degenerate PDE with periodic coefficients I)

**23 January** Kirill Cherednichenko (Topic: Operator-norm resolvent estimates for degenerate PDE with periodic coefficients II)

**30 January** Mikhail Cherdantsev (Cardiff) (Topic: Superexponential decay of defect eigenfunctions in one-dimensional high-contrast media II)
** Abstract**

**6 February** Alexander Kiselev (Kyiv) (Topic: A reduction process on quantum graphs and equivalent Hamiltonian formulations with Datta -- Das Sarma junction conditions I)

Away: [**21 February** Oberseminar, Angewandte Mathematik, University of
Freiburg: "Boundary triples, Krein formula, and resolvent estimates for one-dimensional high-contrast periodic problems"]

**27 February** Alexander Kiselev (Kyiv) (Topic: A reduction process on quantum graphs and equivalent Hamiltonian formulations with Datta -- Das Sarma junction conditions II)

Away: [**6 March** APDE Seminar, University of Sussex: "Extreme localisation property for eigenfunctions of one-dimensional high-contrast periodic problems with a defect"]

**13 March** Patrick Dondl (Universität Freiburg) (Topic: An overview of crystal plasticity and some open problems: Energy estimates, relaxation, and existence for strain gradient plasticity with cross hardening) **Abstract:** We consider a variational formulation of gradient elasto-plasticity subject to a class of single-slip side conditions. Such side conditions typically render the associated boundary-value problems non- convex. We first show that, for a large class of plastic deformations, a given single-slip condition (specification of Burgers' vectors and slip planes) can be relaxed by introducing a microstructure through a two-stage process of mollification and lamination. This yields a relaxed side condition which only prescribes slip planes and allows for arbitrary slip directions. This relaxed model can be thought of as an aid to simulating macroscopic plastic behaviour without the need to resolve arbitrarily fine spatial scales. We then discuss issues of existence of solutions for the relaxed model.
Finally, we apply this relaxed model to a specific system, in order to be able to compare the analytical results with experiments. A rectangular shear sample is clamped at each end, and is subjected to a prescribed horizontal, modelled by an appropriate Dirichlet condition. We ask: how much energy is required to impose such a shear, and how does the energy depend on the aspect ratio of the sample? Assuming that just two slip systems are active, we show that there is a critical aspect ratio, above which the energy is strictly positive, and below which it is zero. Furthermore, in the respective regimes determined by the aspect ratio, we prove energy scaling bounds, expressed in terms of the amount of prescribed shear.

**20 March** Maria Korotyaeva (Besançon) (Topic: The resolvent method for shear waves spectra calculation in 2D phononic crystals)** Abstract**

Away: [**3 April** BUC-VI: Spring School on Analysis and Applications to Mathematical Physics and Materials Science, UNAM, Mexico]

Away: [**18 April** PDE Seminar UAB, Argentina: "The rotation number for the nonlinear p-Laplacian with a periodic potential and new results for the eigenvalue problem on a bounded interval"]

** 8 May** Marcus Waurick (Topic: Conference "Operator Theory and Indefinite Inner Product Spaces" in Vienna, December 2016)

**15 May ** Harsha Hutridurga (Imperial) (Topic: New results on some functional inequalities and associated sharp constants in homogenisation theory) ** Abstract:** It is a classical matter that the best constant in the Poincare inequality is simply the first positive eigenvalue. The eigenvalue homogenisation problem is now well understood. In this talk, I will be reporting on some recent results obtained in collaboration with Jean Dolbeault (Paris Dauphine). We consider a family of functional inequalities with heterogeneous weight functions. We prove some asymptotic results on the associated best constants. We employ some variational techniques and the Bakry-Emery method to prove our results. Homogenisation of logarithmic Sobolev inequality and the Poincare inequality come as a corollary to our result. In this talk, I will recall some essential details on the classical functional inequalities and some essential details on the two-scale approach to handle the associated Euler-Lagrange equation.

Away: [**22 May** International Conference on Elliptic and Parabolic Problems, Gaeta]

Away: [**5 June** PDE and Numerical Analysis Seminar, University of Zagreb]

**19 June ** Sergey Naboko (St. Petersburg) (Topic: On the rich complex spectra of Hermitian linear pencils. The functional model for maximal dissipative operators and the boundary triples IIa)

**26 June ** Sergey Naboko (St. Petersburg) (Topic: The functional model for maximal dissipative operators and the boundary triples IIb)

Away: [**3 July** 9th St. Petersburg Conference in Spectral Theory dedicated to the memory of M. Sh. Birman]

**17 July ** James Roberts

Away: [**11 September** SIAM Conference on Mathematical and Computational Issues in the Geosciences, Erlangen]

**18 September ** Sergey Mikhailov (Brunel) (Topic: Boundary-domain integral equations for nonsmooth-coefficient scalar BVPs on Lipschitz domains)

**25 September ** James Roberts (Topic: Regularity Theory of Fractional Harmonic Mappings of Riemannian Manifolds)

**2 October ** Valery Smyshlyaev (UCL) (Topic: Boundary inflection problem in high-frequency diffraction) **Abstract:** Like Airy ODE and associated Airy function are fundamental objects for describing transition from oscillatory to exponentially decaying asymptotic behaviours and so e.g. transition from "light" to "shadow" near caustics, the boundary inflection problem leads to an arguably equally fundamental boundary-value problem for a PDE, describing transition from a "modal" to a "scattered" behaviour. The associated operators have asymptotic behaviours with a discrete spectrum at one end and with a continuum spectrum at the other end, and of central interest is to find the map connecting these two asymptotics. The latter remains an open problem, and may ultimately require a hybrid of asymptotic and numerical tools. I will review the background from works of M.M. Popov starting from 1970s and will attempt to discuss possible analytic ideas for advancing the problem and possibly ultimately solving it numerically.

**9 October ** Jari Taskinen (Helsinki)
(Topic: Essential spectra of elliptic boundary problems in some nearly periodic domains) **Abstract:** The domains are some perturbations of periodic waveguides in R^d, d=2,3. I consider at least two interesting cases: one, where the essential spectrum consists of an unbounded sequence of positive real numbers and thus has infinitely many gaps, and another one having band-gap-spectrum with isolated points of the essential spectrum inside the gaps.

**16 October ** Kirill Cherednichenko (Topic: Sharp operator-norm asymptotics for linearised elastic plates with rapidly oscillating properties I)

**23 October ** Kirill Cherednichenko (Topic: Sharp operator-norm asymptotics for linearised elastic plates with rapidly oscillating properties II)

**6 November ** Amit Einav (TU Wien) (Topic: On the entropy method for Kac’s Model) **Abstract:** Kac’s model is one of the first many particle models to be explored mathematically. It was introduced by Marc Kac in 1956 in order to give a probabilistic justification to the Boltzmann equation, which arises from it by using the notion of choaticity. This notion of chaoticity is one that is used to this day in the study of most of the many elements models we investigate.
In our talk we will briefly mention the history of Kac’s model and focus our attention on known and recent results in the study of the convergence to equilibrium for the model, and whether or not we can ‘push’ this convergence to the Boltzmann equation, using the so-called entropy method. (Slides from a mini-course in a summer school)

**13 November ** Frank Rösler (Freiburg) (Topic: Norm-resolvent convergence in perforated domains)

**20 November** Chris Bowen (Mechanical Engineering, Bath) (Topic: From piezoelectrics to high-contrast porous media: a guide to functional materials)

Away: [**27 November, 4 December** BUC-XI: Advances in the Mathematics of Multiple Scales, CIMAT-Mérida, Mexico]

**11 December** Yulia Ershova (Topic: A unified operator-theoretical approach to high-contrast homogenisation and its applications to metamaterials)

**18 December** Kirill Cherednichenko (Topic: Norm-resolvent estimates for elliptic problems on periodic singular structures)