## Analysis Seminars 2013 - 2014

### Programme

All seminars are held at 3:10pm in Room M/2.06, Senghennydd Road, Cardiff unless stated otherwise.

Programme Organiser and Contact: Dr Federica Dragoni

##### Wednesday 11 September 2013 at 11:00-12:30 in M/2.06

Speaker: James Evans (Cardiff).

Title: Higher Order Homogenisation of Maxwell's Equations.

Abstract: For around four decades multiscale asymptotic expansions have been used for analysing the behaviour of solutions to periodic PDE with rapidly oscillating periodic coefficients. The related analysis has been exploited in a number of applied contexts, including elasticity, heat conduction and the study of electromagnetic properties of composite materials. One relatively recent related development is the rigorous derivation, via such asymptotic expansions, of a "size-effect'' in the overall behaviour of deformable composites (in the simplest scalar case of an anti-plane shear).

The term "size effect'' here stands for the deviation of the overall response of the composite from that given by the usual homogenised tensor, when the length-scales involved in the problem are not clearly separated from each other. I will discuss my recent results on the development of an analogous theory for the system of governing equations of electrodynamics (Maxwell's equations). These include some new remarkable features in comparison to the scalar case.

##### 30 September 2013

Speaker: Mathias Langer (Strathclyde).

Title: Essential spectrum of block operator matrices.

Abstract: TBC.

##### 7 October 2013

Speaker: Nicholas Katzourakis (Reading).

Title: Contact Solutions for fully nonlinear PDE systems and applications to vector-valued Calculus of Variations in $L^\infty$.

Abstract: Motivated by the successful developments of the scalar case in the last 50 years, we have recently initiated the vector-valued case of Calculus of Variations in the space $L^\infty$. In order to handle the complicated non-divergence PDE systems which arise as the analogues of the Euler-Lagrange equations, we have introduced a theory of non-differentiable weak solutions which applies to general fully nonlinear PDE systems and extends Viscosity Solutions of Crandall-Ishii-Lions to the general vector case. One central ingredient is the discovery of a vectorial notion of extremum which applies to maps and is a substitute of the "Maximum Principle Calculus" in the vector case. In this talk we will discuss some rudimentary aspects of these recent developments.

##### 14 October 2013

Speaker: Jey Sivaloganathan (Bath).

Title: Symmetrisation arguments in nonlinear elasticity.

Abstract: TBC.

##### 21 October 2013

Speaker: David Krejcirik (Academy of Sciences, Czech Republic).

Title: The Cheeger constant and why one should avoid corners.

Abstract: We give an introductory talk on a geometric minimisation problem associated with non-linear partial differential equations arising in the context of image denoising and reconstruction. We then present our results obtained for domains obtained as tubular neighbourhoods of curves in the plane.

This is joint work with Aldo Pratelli.

##### 28 October 2013

Speaker: Boguslaw Zegarlinski (Imperial College London).

Title: Smoothing and ergodicity of Markov semigroups in infinite dimensions.

Abstract: I will review recent development in the study of Markov semigroups with Hoermander type generators on infinite dimensional spaces.

##### 4 November 2013

Speaker: Michael Levitin (University of Reading).

Title: TBC.

Abstract: TBC.

##### 11 November 2013

Speaker: Jonathan Bevan (Surrey).

Title: A stability criterion for the radial cavitating map in nonlinear elasticity.

##### 18 November 2013

Speaker: Igor Wigman (King's College London).

Title: On asymptotic angular distributions of lattice points lying on circles.

Abstract: This work is joint with Par Kurlberg (KTH Stockholm). We prove a number of results on the set of "attainable'' measures that arise as limits of delta measure placed on the unit circle according to the angular distribution of Gaussian integers. Our results, in particular, imply that there  exist "unattainable" measures, an interesting result for itself.  As part of the work we had to rediscover a classical result of Riesz (1911) on the generalized moments problem.

##### 25 November 2013

Speaker: Juan Reyes (Cardiff University).

Title: Conditional stability of Calderon problem for less regular conductivities.

Abstract: I will present a recent log-type stability result with Holder norm for the Calderon problem assuming continuously di erentiable conductivities with Holder continuous rst-order derivatives in a Lipschitz domain of the Euclidean space with dimension greater than or equal to three.

This is a joint work with Pedro Caro and Andoni Garcia from the University of Helsinki. We follow the idea of decay in average used by B. Haberman and D. Tataru to obtain their uniqueness result for either continuously di erentiable conductivities or Lipschitz conductivities such that their logarythm has small gradient in a Lipschitz domain of Rn with n > 3..

##### 2 December 2013

Speaker: Mikhail Cherdantsev (Cardiff University).

Title: Homogenisation of Periodic Composite Elastic Plates.

Abstract: A periodic elastic plate is characterised by two parameters, the thickness h and the period \e of the in-plain composite structure. We consider consider a problem of homogenisation of the periodic plate as both parameters go to zero simultaneously. We start from the fully non-linear setting. In our approach we use the geometric rigidity estimate, g-convergence and the two-scale convergence. In the limit we get a quadratic functional defined on the second fundamental form of the limiting isometric surface - the "zero thickness" plate. The form of the quadratic functional depends on the relation between h and \e.

##### 9 December 2013

Speaker: Karsten Matthies (University of Bath).

Title: Travelling waves in a quasilinear plasticity model.

Abstract: We consider an exact reduction of a model of Field Dislocation Mechanics to a scalar parabolic problem in one spatial dimension and investigate the existence of static and slow, rigidly moving collections of planar screw dislocation walls in this setting. Two choices of the nonlinearity arise from assuming different drag coefficientw namely those with linear growth near the origin and those with constant or more generally sublinear growth there. A mathematical characterisation of all possible equilibria of these screw wall microstructures is given. We also prove the existence of travelling wave solutions for linear drag coefficient functions at low wave speeds and rule out the existence of nonconstant bounded travelling wave solutions for sublinear drag coefficients functions. It turns out that the appropriate concept of a solution in this scalar case is that of a viscosity solution. This is based on joint work with A.Acharya and J.Zimmer.

##### 27 January 2014

Speaker: Sanju Velani (University of York).

Title: TBC.

Abstract: TBC.

##### 10 February 2014

Speaker: Juan J. Manfredi (University of Pittsburgh).

Title: TBC.

Abstract: TBC.

##### 3 March 2014

Speaker: Eugenie Hunsicker (Loughborough University).

Title: TBC.

Abstract: TBC.

##### 17 March 2014 at 14:10

Speaker: Adriana Garrone (University La Sapienza, Rome, Italy).

Title: TBC.

Abstract: TBC.

##### 17 March 2014 at 15:10

Speaker: Andrea Braides (University Tor Vergata, Rome, Italy).

Title: TBC.

Abstract: TBC.

##### 24 March 2014

Speaker: Patrick Dondl (Durham University).

Title: TBC.

Abstract: TBC.

##### 7 April 2014

Speaker: Mariapia Palombaro (University of Sussex).

Title: TBC.

Abstract: TBC.