# MA6A2 Advanced Partial Differential Equations

**Lecturer: Prof. Felix Schulze**

**Term(s):** Term 1

**Commitment: **30 lectures

**Assessment:** Oral Exam.

**Assumed knowledge:**

- Lebesgue integration
- Fubini’s Theorem
- Dominated Convergence Theorem
- Divergence Theorem
- Riesz Representation Theorem

- Differentiable functions
- Partial derivatives
- Chain rule
- Implicit and Inverse Function Theorem

**Useful background:**

- MA3G1 Theory of Partial Differential Equations
- MA3G7 Functional Analysis I
- MA3G8 Functional Analysis II

**Synergies: **This module fits well with MA3G1 Theory of PDEs and leads naturally to MA4J0 Advanced Real Analysis and MA4M2 Mathematics of Inverse Problems. Essential for research in much of geometry, analysis, probability and applied mathematics etc.

**Leads to: **The following modules have this module listed as **assumed knowledge** or **useful background:**

**Content**: Partial differential equations have always been fundamental to applied mathematics, and arise throughout the sciences, particularly in physics. More recently they have become fundamental to pure mathematics and have been at the core of many of the biggest breakthroughs in geometry and topology in particular. This course covers some of the main material behind the most common 'elliptic' PDE. In particular, we'll understand how analysis techniques help find solutions to second order PDE of this type, and determine their behaviour. Along the way we will develop a detailed understanding of Sobolev spaces.

This course is most suitable for people who have liked the analysis courses in earlier years. It will be useful for many who intend to do a PhD, and essential for others. There are not too many prerequisites, although you will need some functional analysis, and some facts from Measure Theory will be recalled and used (particularly the theory of Lp spaces, maybe Fubini's theorem and the Dominated Convergence theorem etc.). It would make sense to combine with MA3G1 Theory of PDEs, in particular the parts about Laplace's equation, in order to see the relevant context for this course, although this is not essential.

**Aims**: To introduce the rigorous, abstract theory of partial differential equations.