Diff. Geom, Math. Phys., PDE Seminar: Azahara de la Torre
Topic
On Singular Solutions for the Fractional Yamabe Problem
Speakers
Details
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.
Additional Information
Location: ESB 2012
Azahara de la Torre, Politechic University of Catalonia
Azahara de la Torre, Politechic University of Catalonia