Geometry Seminar: Jialing Zhang (U British Columbia)

GeoTop Geometry Seminar (Geometric Analysis)

Time: 17:15-18:15, Tuesday, January 14th, 2025

Speaker: Jialing Zhang (U British Columbia)

Title: Harnack type inequality and Liouville theorem for subcritical fully nonlinear equations. 

Abstract: We consider this equation:

$$\sigma_{k}(A^{u}) = u^{\left(p - \frac{n+2}{n-2}\right)k},$$
where $n \geq 3$ and $p \in \left( \frac{n}{n-2}, \frac{n+2}{n-2} \right)$. Here, $\sigma_{k}$ denotes the $k$-th elementary symmetric function of the eigenvalues of $A^u$, and

$$A^{u} = -\frac{2}{n-2} u^{-\frac{n+2}{n-2}} D^2 u + \frac{2n}{(n-2)^2} u^{-\frac{2n}{n-2}} \nabla u \otimes \nabla u - \frac{2}{(n-2)^2} u^{-\frac{2n}{n-2}} |\nabla u|^2 I,$$
where $\nabla u$ denotes the gradient of $u$ and $D^2 u$ denotes the Hessian of $u$. This equation has fruitful backgrounds in geometry.
We then obtain Schoen's Harnack type inequality in Euclidean balls and the asymptotic behavior of an entire solution.
Based on the asymptotic behavior, we give another proof of the Liouville theorem obtained by A. Li and Y. Y. Li (2005).

Zoom: Please ask Niels Martin Møller for a link.