# MA9N4 - Topics in Analysis

**Lecturer: **Maxwell Stolarski

**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:

- MA949 Applied and Numerical Analysis for Linear PDEs
- MA4A2 Advanced Partial Differential Equations
- MA359 Measure Theory

**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