Conférencier : Yannick Sire, Johns Hopkins

Résumé : I will describe recent results on the “size” of various sets associated to solutions of some elliptic PDEs whose coefficients are degenerate or singular. In the case of eigenfunctions of the associated second order operators, these estimates are related to some famous conjectures by Yau (formulated in a more classical setting).Degenerate PDEs appear in a lot of different contexts like  conical spaces, realizations of Dirichlet-to-Neumann maps, Poincare-Einstein manifolds in conformal geometry, etc…I will describe several strategies to get these estimates, leading for some of them to sharp bounds. Along the way, I will also describe some eigenfunction and cluster estimates, which are very much related to this topic.