Ομιλητής: Jose Espinar

         National Institute for Pure and Applied Mathematics, Brasil https://www.researchgate.net/profile/Jose-Espinar

Μέρα: Τρίτη 10/6/2025  στις 14.00

Αίθουσα: Γ31

Title: Overdetermined elliptic problems on \mathbb{S}^2 and the Critical Catenoid Conjecture

Abstract: We study domains $\Omega \subset \s^2$ that support positive solutions to the overdetermined problem $ \Delta{u} + f(u, |\nabla u|) = 0 $ in $\Omega$, with the boundary conditions $u = 0$ on $\partial \Omega$ and $\abs{\nabla u}$ being locally constant along $\partial \Omega$. We refer to such domains as $f$-extremal domains and focus on those with disconnected boundaries.

We extend the moving plane method on $\s^2$ to demonstrate that if $\Omega$ is an $f$-extremal domain containing a simple closed curve of maximum points of $u$, then $\Omega$ must be either rotationally symmetric or antipodally symmetric.

As a consequence, we obtain a characterization of the critical catenoid as the only embedded free boundary minimal annulus in the unit ball whose support function has infinitely many critical points.