## 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.

**Abstract: **Click here to download.

##### 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 dierentiable 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 dierentiable 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: **Bad on curves is winning.

**Abstract: **I will discuss recent results on a problem of Davenport from the sixties regarding badly approximable points on the parabola. The strongest of these states that the intersection of any simultaneously badly approximable set with a non-degenerate planar curve is winning in the sense of Schmidt games.

##### Wednesday 29th January 2014 at 15:00

**Speaker:** Vicentiu Radulescu (Mathematics Institute of the Romanian Academy).

**Title: ** Anisotropic phenomena described by nonhomogeneous differential operators.

**Abstract: **We develop some qualitative results in the analysis of differential operators with variable exponent and we point out the differences with respect
to the standard case corresponding to the Laplace operator. The mathematical treatment is based on variational and topological arguments and the study is inspired by the contributions of Zhikov on Lavrentiev's phenomenon, in relationship with function spaces with variable exponent.

##### 3 February 2014

**Speaker:** Marcello Seri (UCL).

**Title: ** Resonances in the two centers Coulomb system.

**Abstract: **We investigate the existence of resonances for two- centers Coulomb systems with arbitrary charges in two and three dimensions, defining them in terms of generalized complex eigenvalues of a non-selfadjoint deformation of the two-center Schrödinger operator. In the talk we will restrict ourselves to the two dimensional problem and focus on why we expect those resonances to appear and what are the main technical obstacles to their analysis.

##### 10 February 2014

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

**Title: **On the Horizontal mean curvature flow in the Heisenberg group.

**Abstract: **We study the horizontal mean curvature flow in the Heisenberg group by using the level set method. We prove the uniqueness, existence and stability of axisymmetric viscosity solutions of the level-set equation. An explicit solution is given for the motion starting from a subelliptic sphere. We also give several properties of the level-set method and the mean curvature flow in the Heisenberg group.

##### 17 February 2014

**Speaker:** Matthew Lettington (Cardiff University).

**Title: **Rational convergents to the cosine of rational multiples of Pi.

**Abstract: **The ability to approximate a real number alpha, by a sequence of rational numbers which converges to alpha has long been of interest. In this talk we give a possibly new standpoint on rational approximations to conjugate sums of the n-th roots of unity, i.e. twice the cosine of rational multiples of Pi. These rational approximations arise by considering a non-standard perspective of the Lucas and Fibonacci numbers, resulting in non-intuitive higher-dimensional analogues of these numbers. The methods involved might be considered as that of the continued-fraction algorithm in higher dimensions and rely on binomial sums and the divisibility of binomial coefficients.

##### 24 February 2014

**Speaker:** Marco Di Francesco (University of Bath).

**Title: **Nonlocal transport equations and systems: from particle description to large time asymptotics.

**Abstract: ** Aggregation phenomena in microbiology, animal biology, and social sciences, can be often described by PDEs of “transport” type, with a “nonlocal” velocity field. I shall quickly provide a formal derivation of those PDEs from particle-based ODEs. I shall then present their variational structure, which often leads to well-posedness in a probability-measure sense. I will recall the basic theory from Ambrosio-Gigli-Savaré and its extension in a paper in collaboration withCarrillo-Figalli-Laurent-Slepcev. A major issue is providing a mathematical description of the emergence (or not) of collective behaviour, or “multiple” behaviour in the large-time asymptotics, depending on the choice of the initial conditions or other parameters. I will briefly describe a recent work on existence, uniqueness, finite time blow up, and “multiple collapse” for a model with two species of agents (in collaboration with S. Fagioli, PhD student from L’Aquila). Then, I shall focus on the derivation of a “mildly” singular repulsive model as “large particle limit” of discrete ODE systems in one space dimension (in collaboration with G. A. Bonaschi, J. A. Carrillo, and M. Peletier), and its interplay with the theory of entropy solutions for scalar conservation laws. Finally, I will consider the case with quadratic diffusion (collaboration with M. Burger), which leads to either a diffusive behaviour or to the existence and uniqueness of steady states in the large times.

##### 3 March 2014

**Speaker:** Eugenie Hunsicker (Loughborough University).

**Title: ** TBC.

**Abstract: **TBC.

##### 10 March 2014

**Speaker:** Marco Marletta (Cardiff University).

**Title: ** The Finite Section Method for Dissipative Jacobi and Schrödinger Operators.

**Abstract: ** We show that for self-adjoint Jacobi matrices and Schrödinger operators, peturbed by dissipative potentials which are relative trace class, the finite section method does not omit any points of the spectrum. The methods of proof are based on Titchmarsh-Weyl functions, uniform bounds on certain families of Blaschke products and, in the Schrödinger case, an unexpected appeal to a 20-year-old result of Chernyavskaya and Shuster concerning the Green's function on the diagonal.

This is joint work with Sergey Naboko.

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

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

**Title: **Metastability and dynamics of discrete topological singularities via Gamma-convergence: Application to dislocations.

**Abstract: **TBC.

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

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

**Title: **Variational methods for ferromagnetic and antiferromagnetic spin systems.

**Abstract: **TBC.

##### 24 March 2014

**Speaker:** Patrick Dondl (Durham University).

**Title: **A phase field model for the optimization of the Willmore energy in the class of connected surfaces.

**Abstract: **We consider the problem of minimizing the Willmore energy on confined and connected surfaces with prescribed surface area. To this end, we approximate the surface by a level set function $u$ admitting the value $+1$ on the inside of the surface and $-1$ on its outside. The confinement of the surface is now simply given by the domain of definition of $u$. A diffuse interface approximation for the area functional, as well as for the Willmore energy are well known. We address the main difficulty, namely the topological constraint of connectedness by a nested minimization of two phase fields, the second one being used to identify connected components of the surface. In this article, we provide a proof of Gamma-convergence of our model to the sharp interface limit. This is joint work with Matthias Röger (TU Dortmund) and Luca Mugnai (MPI Leipzig)..

##### 31 March 2014

**Speaker:** Stephan Luckhaus (University of Leipzig/Isaac Newton Institute).

**Title: **Gradient Gibbs measures and nonlinear elasticity.

**Abstract: **In a joint work with R.Kotecky we investigate the large scale limit of a stochastic particle system.We start from equilibrium statistical mechanics and by adapting tools from the calculus of variations as lower semicontinuity of quasiconvex functionals,two scale convergence,and Young measures,we are able to pass to a limit characterizing macroscopic behaviour and fluctuations at once.

##### 7 April 2014

**Speaker:** Martin Huxley (Cardiff University).

**Title: **A sideways approach to the circle problem abstract.

**Abstract: **The average of the sum-of-two-squares function r(n) can be read as counting integer points in a circle with centre at the origin. Moving the centre, or changing the shape, hasn't helped yet, but it leads to some interesting problems.

##### 12 May 2014

**Speaker:** Paleksey Kostenko? (University of Vienna).

**Title: ** Indefinite Sturm-Liouville spectral problems and the HELP inequality.

**Abstract: **We study two problems. The first one is the similarity problem for indefinite Sturm–Liouville operators. The second object is the so-called HELP inequality, a version of the classical Hardy–Littlewood inequality proposed by W.N. Everitt in 1971.

Both problems are well understood when the corresponding Sturm-Liouville differential expression is regular. Our main main objective is to give criteria for both the validity of the HELP inequality and the similarity to a self-adjoint operator in the singular case. Namely, we establish new criteria formulated in terms of the behavior of the corresponding Weyl–Titchmarsh m–functions at zero and at infinity. In particular, we show that both problems are closely connected.