Pieter Blue, University of Toronto
Decay estimates and phase space analysis on some black hole manifolds.
The Schwarzschild space is a manifold arising in general relativity.
It is spherically symmetric, asymptotically flat, and contains a black
hole at the center, which provides a different topology from Euclidean
space, and contains a surface of geodesics which orbit the hole. We
consider the wave equation on this space, motivated both by relativity
and the problem of decay on manifolds, especial in the presence of closed
geodesics.
We begin by proving a Morawetz estimate, a spatially weighted $L2$
space time estimate, which shows some form of decay must occur. We find
that control of the angular derivatives near the geodesic surface is
necessary to prove decay of spatial $L^p$ norms in time. We achieve
this control through phase space analysis, making the weights in the
Morawetz multiplier depend not only on the spatial variable, but also
the angular derivatives.
Pengfei Guan, McGill University
Convexity problems in nonlinear geometric equations
The convexity is an issue of interest for a long time in PDE, it
is intimately related to the study of geometric properties of solutions
of general partial differential equations. In differential geometry,
notion of convexity can be translated to positivity of certain curvature
tensors. Quite often geometric solutions are obtained through homotopic
deformations. A pertinent question is under what structural conditions
for partial differential equations so that the "convexity"
of solution is preserved under homotopy deformation? We will describe
a recent joint work with Caffarelli and Ma on the establishment of a
general convexity principle in this direction. More specifically, we
establish a deformation lemma for a wide class of elliptic and parabolic
nonlinear equations involving symmetric curvature tensors.
A. Montero (McMaster)
Some applications of weak jacobians to the study of the Ginzburg
Landau energy.
We study the Ginzburg Landau energy in certains regime of
energy where a rectifiable singularity develops as a certain parameter
becomes small. This leads for instance to the existence of local
minimizers to this energy in domains with particular geometries.
