Nathanael Ong | PDEs, Weak Derivatives, and Sobolev Spaces I | Tuesday, February 6 @ 3:30 pm

Abstract: I will introduce models of first and second order partial differential equations and their general solutions. In generalizing solutions to PDEs, one often cares about so-called "weak solutions." This leads to the notion of a "weak derivative" which we'll define. Then we'll talk about two types of function spaces (vector spaces of functions) where solutions to second order equations can be found: Lp spaces and Sobolev spaces. We are mainly concerned with the Sobolev embedding theorems which provide a powerful tool for studying solutions to elliptic PDEs. Some analysis background would be helpful (compactness, completeness and uniform convergence). But it is not necessary. This is the first talk in a two-part series.