PDE CDT Lunchtime Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Past events in this series
30 January 2020
12:00
José Manuel Palacios
Abstract

The Camassa-Holm (CH) equation is a nonlinear nonlocal dispersive equation which arises as a model for the propagation of unidirectional shallow water waves over a flat bottom. One of the most important features of the CH equation is the existence of peaked travelling waves, also called peakons. The aim of this talk is to review some asymptotic stability result for peakon solutions for CH-type equations as well as to present some new result for higher-order generalization of the CH equation.

  • PDE CDT Lunchtime Seminar
6 February 2020
12:00
Katie Gittins
Abstract


Let $\Omega \subset \mathbb{R}^n$, $n \geq 2$, be a bounded, connected, open set with Lipschitz boundary.
Let $u$ be an eigenfunction of the Laplacian on $\Omega$ with either a Dirichlet, Neumann or Robin boundary condition.
If an eigenfunction $u$ associated with the $k$--th eigenvalue has exactly $k$ nodal domains, then we call it a Courant-sharp eigenfunction. In this case, we call the corresponding eigenvalue a Courant-sharp eigenvalue.

We first discuss some known results for the Courant-sharp Dirichlet and Neumann eigenvalues of the Laplacian on Euclidean domains.

We then discuss whether the Robin eigenvalues of the Laplacian on the square are Courant-sharp.

This is based on joint work with B. Helffer (Université de Nantes).
 

  • PDE CDT Lunchtime Seminar
13 February 2020
12:00
Tristan Giron / Craig Roberston
Abstract

TBA

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Teichmüller harmonic map flow is a geometric flow designed to evolve combinations of maps and metrics on a surface into minimal surfaces in a Riemannian manifold. I will introduce the flow and describe known existence results, and discuss recent joint work with M. Rupflin that demonstrates how singularities can develop in the metric component in finite time.

 

  • PDE CDT Lunchtime Seminar
20 February 2020
12:00
Cintia Pacchiano
Abstract

In this talk I will discuss some aspects of the potential theory, fine properties and boundary behaviour of the solutions to the Total Variation Flow. Instead of the classical Euclidean setting, we intend to work mostly in the general setting of metric measure spaces. During the past two decades, a theory of Sobolev functions and BV functions has been developed in this abstract setting.  A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as weighted Euclidean spaces, Riemannian manifolds, Heisenberg groups, graphs, etc.

The total variation flow can be understood as a process diminishing the total variation using the gradient descent method.  This idea can be reformulated using parabolic minimizers, and it gives rise to a definition of variational solutions.  The advantages of the approach using a minimization formulation include much better convergence and stability properties.  This is a very essential advantage as the solutions naturally lie only in the space of BV functions. Our main goal is to give a necessary and sufficient condition for continuity at a given point for proper solutions to the total variation flow in metric spaces. This is joint work with Vito Buffa and Juha Kinnunen.

  • PDE CDT Lunchtime Seminar
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