Asymptotics, operators and functionals

Leading the discussion in 2017:

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 University) (Topic: (Topic: Superexponential decay of defect eigenfunctions in one-dimensional high-contrast media II) Abstract

6 February Alexander Kiselev (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 (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)

Next year (2018):

15 January Freddy Symons (now at Bath) (Topic: Uniqueness and cloaking in a boundary-singular two-dimensional inverse​ problem) Abstract: I will present a topic from my recently finished doctoral thesis. The area is uniqueness in inverse recovery for the Schrödinger boundary-value problem in a bounded domain. The crucial new progress is in the situation of a singular Robin boundary condition, of a type first defined by Berry and Dennis in 2008, attached to this set-up. Firstly I will present a rigorous definition of this problem and derive some useful properties, then I will state and explain some new uniqueness theorems. The proof methods combine existing results on the two-dimensional inverse uniqueness problem (with Dirichlet conditions) with some new ideas involving the asymptotics of the negative eigen-values arising from the Berry--Dennis condition.

29 January Matthew Lewis (Cardiff)

5 February Igor Velčić (Zagreb)

12 February Will Graham (Topic: Generalised Leontovich condition and surface waves in lossy dielectrics)

5 March Lucia Scardia (Topic: Stochastic homogenisation of free-discontinuity problems)

12 March Serena D'Onofrio

19 March David Bird

Away: [26 March University of Helsinki

Away: [23 April Universidad de Chile, Santiago]

14 May Gennady Mishuris (Aberystwyth)


Leading the discussion in 2016:

8 February Yulia Ershova (Topic: The inverse topology problem for quantum graphs)

22 February Shane Cooper (Topic: Results, shortcomings and recent developments in high-contrast periodic spectral problems I)

29 February Marcus Waurick (Topic: Invertibility of operators in space-time, and a new approach to the homogenisation of evolutionary equations) Abstract: We present a concept of solving partial differential equations via a space-time Hilbert space approach. Within the approach developed, we provide an operator-theoretic perspective to homogenisation theory and exemplify our findings with a 1+1-dimensional problem of mixed type. In this example, the change of type from hyperbolic to elliptic will be highly oscillatory and the effective equation is studied.

7 March Alexander Kiselev (Kyiv) (Topic: Uniqueness of reconstruction of the topology of a quantum graph from its spectrum, via the analysis of the behaviour of the M-matrix at zero I)

Away: [14 March Conference "Spectral Theory and Applications", Stockholm University, Sweden]

4 April Mikhail Cherdantsev (Cardiff) (Topic: Superexponential decay of defect eigenfunctions in one-dimensional high-contrast media I)

11 April Yulia Ershova (Topic: The inverse topology problem for quantum graphs. Part II: Boundary triples and M-matrix)

Away: [18 April Conference "Spectral Theory of Novel Materials" , CIRM, Marseille, France]

25 April Sergey Naboko (St. Petersburg) The functional model for maximal dissipative operators and the boundary triples I

Away: [ 9 May SIAM Conference on Mathematical Aspects of Materials Science , Philadelphia, USA]

Away: [ 16 May Conference "Operators, Operator Families and Asymptotics", University of Bath]

Away: [ 22 May Course High-Velocity Estimates, Inverse Scattering and Topological Effects by Ricardo Weder (IIMAS-UNAM, Mexico), at the University of Bath]

Away: [ 6 June Conference "Computational and Analytic Problems in Spectral Theory", Cardiff University, UK]

17 June (Friday) Fritz Gesztesy (Missouri) (Topic: A bound for the eigenvalue counting function for Krein-von Neumann extensions for elliptic second-order partial differential operators on bounded sets) Abstract

Away: [20 June Conference "New Trends in Nonlinear PDEs: from Theory to Applications", Cardiff University, UK]

27 June Kirill Cherednichenko (Topic: Microscale isometry constraint for thin heterogeneous plates with finite bending energy)

4 July Yulia Ershova (Topic: Theorem of Kurasov-Kostrykin-Schrader in the inverse topology problem for graphs, and its shortcomings)

Away: [11, 21 July EPSRC Durham Symposium Mathematical and Computational Aspects of Maxwell's Equations, Durham University, UK]

19 September Alexander Kiselev (Kyiv) (Topic: Uniqueness of reconstruction of the topology of a quantum graph from its spectrum, via the analysis of the behaviour of the M-matrix at zero II)

26 September Igor Velčić (Zagreb) (Topic: Homogenization of thin structures in nonlinear elasticity, periodic and non-periodic) Abstract: We will give the results on the models of thin plates and rods in nonlinear elasticity by doing simultaneous homogenization and dimensional reduction. In the case of bending plate we are able to obtain the models only under periodicity assumption and assuming some special relation between the periodicity of the material and thickness of the body. In the von Kármán regime of rods and plates and in the bending regime of rods we are able to obtain the models in the general non-periodic setting. In this talk we will focus on the derivation of the rod model in the bending regime without any assumption on periodicity.

3 October Kirill Cherednichenko (Topic: Asymptotic expansions as a heuristic and rigorous tool in the analysis of multiscale differential equations)

10 October Yulia Ershova (Topic: Homogenisation: the Krein resolvent formula and operator-norm convergence estimates)

17 October Davit Harutyunyan (EPFL) (Topic: Recent progress in the shell buckling theory) Abstract

31 October Yulia Meshkova (St. Petersburg) (Topic: Homogenization of elliptic and parabolic Dirichlet problems in a bounded domain) Abstract

14 November Yuri Antipov (Luisiana State) (Topic: Singular integral equations with two fixed singularities: theory and applications) Abstract: Motivated by model problems of fracture, the talk focuses on the solution of SIEs with the Cauchy kernel and kernels with fixed singularities at the endpoints. The method developed is based on solving an associated vector Riemann-Hilbert problem with a piece-wise constant matrix coefficient having three discontinuities. In a symmetric case, the number of the discontinuities reduces to two, and the general solution of the SIE is significantly simplified. For a specific right hand-side the solution is expressed in terms of some polynomials with weights, and a spectral relation for the singular integral operator with two fixed singularities is derived. This relation is employed for an approximate solution of the corresponding complete SIE. Applications to model problems of fracture and the theory of composites are discussed.

21 November Harsha Hutridurga (Imperial) (Topic: A new approach to study strong advection problems) Abstract

28 November Shane Cooper (Topic: Results, shortcomings and recent developments in high-contrast periodic spectral problems II)

5 December Marcus Waurick (Topic: Invertibility of operators in space-time, and a new approach to the homogenisation of evolutionary equations II)

Away: [12, 19 December Conference "Mathematical and Numerical Modelling in Optics", IMA, Minnesota]