MA9N4 - Topics in Analysis

Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral exam

Timetable: Tu 2-3pm in D1.07, Th 12-1pm in B3.02, Fri 11am-12pm in B3.02

Prerequisites: Sobolev spaces, functional analysis, measure theory, aspects of elliptic PDE theory.

For example, it suffices to be familiar with the contents of the following Warwick modules:

Content: This module covers topics in the theory of elliptic and parabolic partial differential equations. We will focus on the regularity theory of solutions with an emphasis on modern techniques. Informally, many of the results take the form, "If the coefficients and inhomogeneity of an elliptic PDE have k derivatives, then the solution has k+2 derivatives." This module shows how this meta-theorem can hold in remarkably low regularity settings, which yields applications to nonlinear equations and variational problems. Topics include:

the DeGiorgi-Nash-Moser Theorem
the Harnack inequality
Holder, Morrey, and Campanato spaces
Cacciopoli's inequality
Schauder theory
Applications to nonlinear equations and variational problems
Equations of mean curvature type
Parabolic equations
- existence and uniqueness of weak solutions
- energy estimates & the regularity of solutions

References:

Lawrence C. Evans, Partial Differential Equations
David Gilbarg & Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order
Luigi Ambrosio, Lecture Notes on Elliptic Partial Differential Equations
Lecture notes