Past OxPDE Lunchtime Seminar

2 May 2014
12:00
Giovanni Alberti
Abstract
In this talk I will describe a multiple frequency approach to the boundary control of Helmholtz and Maxwell equations. We give boundary conditions and a finite number of frequencies such that the corresponding solutions satisfy certain non-zero constraints inside the domain. The suitable boundary conditions and frequencies are explicitly constructed and do not depend on the coefficients, in contrast to the illuminations given as traces of complex geometric optics solutions. This theory finds applications in several hybrid imaging modalities. Some examples will be discussed.
  • OxPDE Lunchtime Seminar
13 March 2014
12:00
Abstract
<p> <div>One of the main unsolved problems in the field of homogenization of multiple integrals concerns integrands which are not bounded polynomially from above. This is typically the case when incompressible (or quasi-incompressible) materials are considered, although this is still currently a major open problem.</div> <div>In this talk I will present recent progress on the stochastic homogenization of nonconvex integral functionals in view of the&nbsp;derivation of nonlinear elasticity from polymer physics, and consider integrands which satisfy very mild convex growth conditions from above.</div> <div>I will first treat convex integrands and prove homogenization by combining approximation arguments in physical space with the Fenchel duality theory in probability. In a second part I will generalize this homogenization result to the case of nonconvex integrands which can be written in the form of a convex part (with mild growth condition from above) and a nonconvex part (that satisfies a standard polynomial growth condition). This decomposition is particularly relevant for&nbsp;the&nbsp;derivation of nonlinear elasticity from polymer physics.</div> <div>This is joint work with Mitia Duerinckx (ULB).</div> </p>
  • OxPDE Lunchtime Seminar
27 February 2014
12:00
Dr. Filippo Cagnetti
Abstract
Steiner symmetrization is a very useful tool in the study of isoperimetric inequality. This is also due to the fact that the perimeter of a set is less or equal than the perimeter of its Steiner symmetral. In the same way, in the Gaussian setting, it is well known that Ehrhard symmetrization does not increase the Gaussian perimeter. We will show characterization results for equality cases in both Steiner and Ehrhard perimeter inequalities. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when all equality cases are trivially obtained by a translation of the Steiner symmetral (or, in the Gaussian setting, by a reflection of the Ehrhard symmetral). We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function for a special class of sets. These results are obtained in collaboration with Maria Colombo, Guido De Philippis, and Francesco Maggi.
  • OxPDE Lunchtime Seminar
20 February 2014
13:00
Diogo Oliveira e Silva
Abstract
This talk will focus on extremizers for a family of Fourier restriction inequalities on planar curves. It turns out that, depending on whether or not a certain geometric condition related to the curvature is satisfied, extremizing sequences of nonnegative functions may or may not have a subsequence which converges to an extremizer. We hope to describe the method of proof, which is of concentration compactness flavor, in some detail. Tools include bilinear estimates, a variational calculation, a modification of the usual method of stationary phase and several explicit computations.
  • OxPDE Lunchtime Seminar
13 February 2014
12:00
Prof. Philip Maini
Abstract
We will present three different recent applications of cell motion in biology: (i) Movement of epithelial sheets and rosette formation, (ii) neural crest cell migrations, (iii) acid-mediated cancer cell invasion. While the talk will focus primarily on the biological application, it will be shown that all of these processes can be represented by reaction-diffusion equations with nonlinear diffusion term.
  • OxPDE Lunchtime Seminar
5 December 2013
12:00
Manuel Friedrich
Abstract

We study the behavior of atomistic models under uniaxial tension and investigate the system for critical fracture loads. We rigorously prove that in the discrete-to- continuum limit the minimal energy satisfies a particular cleavage law with quadratic response to small boundary displacements followed by a sharp constant cut-off beyond some critical value. Moreover, we show that the minimal energy is attained by homogeneous elastic configurations in the subcritical case and that beyond critical loading cleavage along specific crystallographic hyperplanes is energetically favorable. We present examples of mass spring models with full nearest and next-to-nearest pair interactions and provide the limiting minimal energy and minimal configurations.

  • OxPDE Lunchtime Seminar
28 November 2013
12:00
Dr. Nicholas Katzourakis
Abstract
<p><span>Calculus of Variations for $L^{\infty}$ functionals has a successful history of 50 years, but until recently was restricted to the scalar case. Motivated by these developments, we have recently initiated the vector-valued case. In order to handle the complicated non-divergence PDE systems which arise as the analogue of the Euler-Lagrange equations, we have introduced a theory of "weak solutions" for general fully nonlinear PDE systems. This theory extends Viscosity Solutions of Crandall-Ishii-Lions to the general vector case. A central ingredient is the discovery of a vectorial notion of extremum for maps which is a vectorial substitute of the "Maximum Principle Calculus" and allows to "pass derivatives to test maps" in a duality-free fashion. In this talk we will discuss some rudimentary aspects of these recent developments.</span></p>
  • OxPDE Lunchtime Seminar

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