Domínios f-extremais para o operador laplaciano

Abstract

In this dissertation, we investigate geometric properties of extremal domains, boun ded or not. The extremal domains are the domains that are critical points for the first eigenvalue functional for volume-preserving variations. We show that these domains are characterized by admitting a non-trivial solution to an overdetermined Serrin-type problem. This motivates us to define the f-extremal domains, when we use an arbitrary function f as the nonlinearity of the overdetermined problem. The main tool used is the maximum principle in the format of the moving planes method and the main result is the Ros-Sicbaldi Theorem on the proof of Berestycki-Caffarelli-Nirenberg Conjecture in dimension two when the nonlinearity grows at least as a function linear.