Asymptotics, Operators, and Functionals

Leading the discussion in 2018:

15 January Freddy Symons (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.

22 January Aaron Pim (Topic: Asymptotic analysis of minimisation problems from the theory of liquid crystals)

29 January Matthew Lewis (Cardiff) (Topic: On the spectral problem for the p-Laplacian operator with periodic coefficients)

5 February Igor Velčić (Zagreb) (Topic: Regularity of W^{2,2} convex manifolds and homogenization of bending theory of shells) Abstract

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

23 February (Friday) Kirill Cherednichenko (Topic: Two-scale series expansions for travelling wave packets in one-dimensional periodic media)

Away: [26 February Conference "Liquid Crystals, Metamaterials, Transformation Optics, Photonic Crystals, and Solar Cells", IMA, Minnesota]

5 March Yulia Ershova (Topic: A new approach to the analysis of quantum graphs and its applications to periodic homogenisation)

12 March (Joint session with BUC-XII) Serena D'Onofrio (Topic: Operator-norm convergence estimates for elliptic homogenisation problems on periodic singular structures)

Away: [22 March University of Helsinki]

Away: [4 April Mathematics and Mathematical Physics Seminar, St.Petersburg]

Away: [23 April Center for Mathematical Modeling, Universidad de Chile, Santiago

            24 April Pontificia Universidad Católica,Santiago de Chile]

30 April Alexander Kiselev (St. Petersburg) (Topic: Asymptotic analysis of Dirichlet-to-Neumann maps for multidimensional media with critical contrast and effective theories for metamaterials I)

Tuesday 8 May Yulia Meshkova (St. Petersburg) (Topic: Operator error estimates for homogenisation of periodic hyperbolic systems) Abstract

                   Andrew Comech (Texas A&M) (Topic: Limiting absorption principle and resonances)

14 May Gennady Mishuris (Aberystwyth) (Topic: Waves and fracture in discrete structures. Slepyan's method: its advantages and limitations.) Abstract

Wednesday 23 May Duvan Henao (Pontificia Universidad Católica de Chile) (Topic: Lower bound for the coalescence load in 2D neoHookean materials) Abstract: This work (joint with V. Cañulef-Aguilar) forms part of the variational analysis of cavitation (the nucleation and sudden burst of voids) in elastomers, made possible by Gent & Lindley '59, Ball '82, Muller & Spector '95, and Sivaloganathan & Spector '00. It is known (Sivaloganathan & Spector '10; H. & Serfaty '13) that the energetically most favourable cavities are spherical; nonetheless, for large external loads it is impossible to create only spherical cavities and at the same time satisfy the constraint of incompressiblity. This raises the question of determining the largest load for which incompressible deformations of an hyperelastic body still exist that open only spherical cavities, itself a lower bound for the load at which cavities start to interact and coalesce. We give a simple geometric answer in a simplified 2D setting, using the flow of Dacorogna & Moser '90.

Tuesday 29 May Kirill Cherednichenko (Topic: New developments in the operator-norm resolvent analysis of thin-plate problems)

4 June Euan Spence (Topic: Frequency‐uniform coercive boundary integral equation for acoustic scattering)

11 June Alexander Kiselev (St. Petersburg) (Topic: Asymptotic analysis of Dirichlet-to-Neumann maps for multidimensional media with critical contrast and effective theories for metamaterials II)

18 June Matthias Täufer (Dortmund) (Topic: Control cost for heat-like equations in the (de-)homogenization limit) Abstract

25 June Kirill Cherednichenko (Topic: Asymptotic analysis of Dirichlet-to-Neumann maps and dispersion relations for composite media I)

2 July Xavier Pellet (Topic: Homogenization of high-contrast Mumford-Shah energies)

9 July Kirill Cherednichenko (Topic: Asymptotic analysis of Dirichlet-to-Neumann maps and dispersion relations for composite media II)

Away: [16 July IMSE2018, University of Brighton]

Away: [3 September Workshop on Calculus of Variations and Applications, University of Zagreb]

10 September David Bird (Topic: Guidance of light in hollow-core optical fibres) Abstract: Conventional optical fibres guide light in a core region that has a higher refractive index than the surrounding cladding. Trapping and guiding light in a low refractive index core (for example, air) is much more challenging, but potential technological applications have driven a sustained research effort in this field. Cladding structures that support a photonic band gap were introduced about twenty years ago, but this has largely been superseded by a surge of recent interest in new designs of hollow-core fibres that have surprisingly simple cladding structures with no band gap, but which nevertheless support core-guided modes with a remarkably low loss. I will discuss the "anti-resonance" guiding mechanism that is believed to operate, and present analytic and numerical results on simple model structures that provide insights into what is currently understood and not understood about these fibres.

Away: [17 September BUC-XV: "Function spaces meet materials science: recent developments in spectral theory and scattering", CIMAT, Guanajuato, México]

24 September Matthias Langer (Strathclyde) (Topic: Quasi boundary triples and extensions of symmetric operators) Abstract: In this talk I will discuss the concept of quasi boundary triples, which can be used to describe extensions of symmetric operators and obtain information about the location of the spectrum of such extensions. In particular, this concept can be applied to elliptic operators with non-local and/or non-self-adjoint boundary conditions and to Schroedinger operators with potentials supported on hypersurfaces.

1 October Aaron Pim (Topic: The analysis of singularities in nematic liquid crystals and some approaches to their regularisation) Abstract: I shall start by introducing the the physical concept of a liquid crystal. I shall then discuss the Oseen-Frank model and solve it asymptotically on a domain with a small circular exclusion. I shall also talk about how the energy of the system changes with respect to the radius and position of the exclusion and what the optimal place for the exclusion is, in terms of energy minimisation. I shall then move onto the Landau-de Gennes model and discuss how it is an advancement of the Oseen-Frank approach. I shall conclude by providing the results of a numerical investigation into the behaviour of the corresponding minimisers (local and global), with respect to the Landau-de Gennes penalisation parameter.

Away: [8 October Workshop "Dynamic Phenomena in Media with Microstructure", Tel Aviv University]

15 October Harry Rainbird (Topic: Conservation laws for systems of differential equations) Abstract: I will start by discussing where conservation laws of differential equations come from and how they are derived. I will then explain how Noether’s theorem is applied to differential equations and show how conservation laws can be constructed without using the theory of Lie groups. I will conclude by deriving conservation laws for some simple examples.

22 October Julius Kaplunov (Keele) (Topic: Floquet-Bloch and Rayleigh-Lamb spectra: comparative asymptotic analysis of thin and periodic structures)

29 October Luis Silva (IIMAS-UNAM, Bath) (Topic: Oversampling, undersampling, and their applications to the approximation of functions in Paley-Wiener and de Branges spaces)

5 November Anton Souslov (Topic: Topological mechanics and acoustics) Abstract: The 2019 Breakthrough Prize in Fundamental Physics was awarded to Charles Kane and Eugene Mele ‘for new ideas about topology and symmetry in physics, leading to the prediction of a new class of materials that conduct electricity only on their surface.’ Over the last few years, the reach of these ideas about topological insulators has been extended to photonics as well as the focus of this talk, mechanics and acoustics. After connecting deformations and waves in materials to the mathematics of topology, I will discuss three different examples of topological mechanics. In the first example, time-reversal symmetry is preserved and the rigidity of an elastic network is characterized by a topological invariant called the polarization. Materials with a uniform polarization display a dramatic range of edge softnesses depending on the orientation of the polarization relative to the terminating surface. I will discuss the design of a 3D material in which topological soft modes are localized in interior regions via defects called dislocation lines. In the last two examples, time-reversal symmetry is broken by including active fluids composed of self-driven components inside the material. In that case, I will present two designs for topological states: one using periodic confinement and another using a bulk fluid without periodic order. In a periodic lattice, the geometry of confinement controls the structure of topological waves. Without periodic order, topological edge waves can arise in a fluid of self-spinning particles undergoing spontaneous active rotation. This can occur because a fluid undergoing rotation experiences a Coriolis force that breaks Galilean invariance and opens a gap at low frequency. I will discuss how all of these examples are connected via the bulk-boundary correspondence principle, which shows how a topological invariant characterizing the bulk of the material has profound consequences on the material’s surface.

12 November David Lafontaine (Topic: Scattering for non-linear waves equations in non-trapping and unstable trapping geometries) Abstract: We are concerned with non-linear wave equations in exterior domains. When solutions exist for all times, it is natural to wonder what they look like in large times. In particular, when they behave linearly in such times, we say that the solutions scatter. The intuition, supported by rigorous criterions, is that such a behavior should occur when all the energy of the wave goes away of the obstacle in infinite time. Thus, we expect scattering in geometries where all the rays of geometrical optics go to infinity: the so called non-trapping geometries, and in weakly trapping geometries as well. This is however a very open problem. We will present some methods and results using arguments inspired by Morawetz in the non-trapping case, and explain how these arguments, combined with more recent ones inspired by Kenig and Merle, should lead to the scattering in some unstable trapping geometries.

19 November Luis Silva (IIMAS-UNAM, Bath) (Topic: Subordinacy theory and the asymptotic analysis of the spectrum of the Schrödinger operator on the half line)

Away: [21 November University of Lyon and University of Saint-Étienne]

Away: [26 November Mathematical Analysis Seminar, Cardiff University]

3 December Kirill Cherednichenko (Topic: Stochastic homogenisation of high-contrast media)

10 December Serena D'Onofrio (Topic: Operator-norm homogenisation estimates for the system of Maxwell equations on periodic singular structures)

17 December Jonathan Bevan (Surrey) (Topic: Estimating critical cavitation loads in nonlinear elasticity)


2019:

18 February Elaine Crooks (Swansea)

22 April Grégoire Allaire (École Polytechnique)

13 May Günter Stolz (Alabama at Birmingham)

10 June Graeme Milton (Utah)


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) (Topic: (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)


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]