Mathematics Seminar 04/14/23

Abstract:  In this talk I will introduce to you some interesting and easy to understand open problems proposed by De Giorgi concerning degenerate or singular second order elliptic equations. When the given data in a PDE problem are weak, one must generalize the notion of a solution to obtain an existence theory.  Such so-called weak solutions are often sought in low order Soblev spaces. Functions in these spaces are only defined up to a set of measure 0. As a result, they are not always suitable to describe certain physical quantities. It is an intereting and challenge issue to find the actual gap between weak solutions and classical solutions. Our open problems are in this area.