Geometric PDE and Applied Analysis Seminar (September 26, 2024)

Talk 1: 15:00-16:00

Speaker: Prof. Mitsuru Sugimoto (Nagoya University)

Title: On a global inverse function theorem for homogeneous map and its application

Abstract: The purpose of this talk is to explain a global inverse function theorem for homogeneous mappings on \(\mathbb{R}^n\), along with the topological fact behind it. The main difficulty lies in the fact that such mappings are not smooth at the origin, making the known global inverse function theorems on \(\mathbb{R}^n\) not readily applicable. The primary motivation for studying such inverses is their applications to the global invertibility of Hamiltonian flows, and further applications to the global-in-time construction of solutions to hyperbolic equations. 

Talk 2: 16:15-17:15

Speaker: Prof. Futoshi Takahashi (Osaka Metroplitan University)

Title: The Hardy inequality on bounded domains for mean zero functions

Abstract: In this talk, we study a minimization problem with two weight parameters, for admissible functions whose weighted \(L^2\) are 1 and weighted mean are zero. The minimum value is the best constant of the weighted version of Neumann Hardy's inequality. We find conditions of parameters and dimension, for which the best constant is positive and attained for a given domain. This talk is based on a joint work with Jaeyoung Byeon (KAIST) and Eunchan Jeon (KAIST).