## Diff. Geom, Math. Phys., PDE Seminar: Azahara de la Torre

- Date: 10/13/2015
- Time: 15:30

Lecturer(s):
Azahara de la Torre,

*Politechic University of Catalonia*
Location:

University of British Columbia

Topic:

On Singular Solutions for the Fractional Yamabe Problem

Description:

We construct some ODE solutions for the fractional Yamabe problem in conformal geometry. The fractional curvature, a generalization of the usual scalar curvature, is defined from the conformal fractional Laplacian, which is a non-local operator constructed on the conformal infinity of a conformally compact Einstein manifold.

These ODE solutions are a generalization of the usual Delaunay and, in particular, solve the fractional Yamabe problem $$ (-\Delta)^\gamma u= c_{n, {\gamma}}u^{\frac{n+2\gamma}{n-2\gamma}}, u>0 \ \mbox{in} \ \r^n \backslash \{0\},$$ with an isolated singularity at the origin.

This is a fractional order ODE for which new tools need to be developed. The key of the proof is the computation of the fractional Laplacian in polar coordinates.

These ODE solutions are a generalization of the usual Delaunay and, in particular, solve the fractional Yamabe problem $$ (-\Delta)^\gamma u= c_{n, {\gamma}}u^{\frac{n+2\gamma}{n-2\gamma}}, u>0 \ \mbox{in} \ \r^n \backslash \{0\},$$ with an isolated singularity at the origin.

This is a fractional order ODE for which new tools need to be developed. The key of the proof is the computation of the fractional Laplacian in polar coordinates.

Other Information:

Location: ESB 2012