Title: Stability of the Pohozaev obstruction and Non-existence
Speaker: Saikat Mazumdar (IIT, Bombay)
Date: 02 November 2022
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk, we will consider the issues of non-existence of solutions to a Yamabe
type equation on bounded Euclidean domains (dim>2). The leading order terms of this
equation are invariant under conformal transformations which leads to the classical
Pohozaev identity. This in turn gives non-existence of solutions to the PDE when the
domain is star-shaped with respect to the origin.
We show that this non-existence is surprisingly stable under perturbations, which
includes situations not covered by the Pohozaev obstruction, if the boundary of the
domain has a positive curvature. In particular, we show that there are no positive
variational solutions to our PDE under $C^1$-perturbations of the potential when the
domain is star-shaped with respect to the origin and the mean curvature of the boundary
at the origin is positive. The proof of our result relies on sharp blow-up analysis.
This is a joint work with Nassif Ghoussoub (UBC, Vancouver) and Frédéric Robert
(Institut Élie Cartan, Nancy).