Calculus of Variations and Geometric Measure Theory
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A non-compactness result on the fractional Yamabe problem in large dimensions

Monica Musso

created by malchiodi on 23 Nov 2015

2 dec 2015 -- 15:00   [open in google calendar]

Scuola Normale Superiore, Aula Mancini

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 \[P^{\gamma}[g^+,h] (u) = cu^{n+2\gamma \over n-2\gamma}, \quad u > 0 \quad \text{on } M\] where $c \in {\mathbb{R}} $ and $P^{\gamma}[g^+,h]$ is the fractional conformal Laplacian whose principal symbol is $(-\Delta)^{\gamma}$. 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 $n \ge 24$ for $\gamma \in (0, \gamma^*)$ and $n \ge 25$ for $\gamma \in [\gamma^*,1)$ where $\gamma^* \in (0, 1)$ is a certain transition exponent. The value of $\gamma^*$ turns out to be approximately 0.940197. This is a joint work with Seunghyeok Kim and Juncheng Wei.

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