Mathematics

Dr. Saikat Mazumdar
McGill University

In this talk we will consider a class of elliptic PDEs with Hardy weight and Sobolev critical growth, which are in general non-compact due to scale invariance. We want to arrive at suitable conditions which would ensure the compactness and this in turn will help establish the existence of solutions to these equations. We will start by describing the blow-up behaviour of a sequence of approximating solutions approaching our PDE and obtain optimal control on such a sequence. Next we will look at the interaction of the various terms in the Pohozaev identity and calculate the blow-up rates. The compactness theorems will follow from this.

We will see that the location of the singularity, be it in the interior of the domain or on its boundary, affects the analytical properties of the equation and makes the two situations quite different. When the singularity is in the interior, then a lower order perturbation suffices for high dimensions, while the curvature of the boundary plays a crucial role if the singularity is on the boundary for high dimensions.

This is a joint work with Nassif Ghoussoub(UBC) and Frederic Robert(Universite de Lorraine).