18 jan 2016 -- 14:30
Aula di Consiglio, Dip. Matematica, Università "La Sapienza", Roma
Abstract.
Let $(X^{n+1},g_+)$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n,h)$. The fractional Yamabe problem addresses to solve $Ps[g+,h](u)=cu^{n+2s/n-2s}$, $u>0$ on $M$ where $c$ is in $\mathbb{R}$ and $Ps[g_+,h]$ is the fractional conformal Laplacian whose principal symbol is $(-\Delta)s$. In this paper, we construct a metric on the half space $X=\mathbb{R}^{n+1}_+$, which is conformally equivalent to the unit ball, for which the solution set of the fractional Yamabe equation is non-compact provided that $n≥24$ for $s∈(0,s*)$ and $n≥25$ for $s∈[s^*,1)$ where $s^*∈(0,1)$ is a certain transition exponent. The value of $s^*$ turns out to be approximately 0.940197. This is a joint work with S. Kim and J. Wei.