August 17, 2022

January 29, 2005
Young Mathematicians Conference

held at McMaster University

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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.