# Part III - Analysis of Partial Differential Equations

## Lectured by C. Warnick, Michaelmas 2017

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# Contents

- V Full version
- 1 Introduction
- 2 The Cauchy–Kovalevskaya theorem
- 3 Function spaces
- 3.1 The Holder spaces
- 3.2 Sobolev spaces
- 3.3 Approximation of functions in Sobolev spaces
- 3.4 Extensions and traces
- 3.5 Sobolev inequalities
- 4 Elliptic boundary value problems
- 4.1 Existence of weak solutions
- 4.2 The Fredholm alternative
- 4.3 The spectrum of elliptic operators
- 4.4 Elliptic regularity
- 5 Hyperbolic equations