Partial Differential Equations (MAST90133)

This subject offers a wide ranging introduction to the modern theory of partial differential equations (PDEs) in pure mathematics. Thus we will study questions of existence, uniqueness, regularity, and long time behaviour (e.g.\ energy dispersion) for solutions to PDEs. We will discuss these questions first for the classical equations (Laplace's equation, the heat equation, and the wave equation) which will lead us to the broader theory of elliptic, parabolic, and hyperbolic equations. The course covers mostly linear equations, but exposes the student also to some of the most interesting non-linear equations arising in physics and geometry.

Further topics may include: Calculus of variations, Hamilton-Jacobi equations, Systems of Conservation laws; Non-linear elliptic equations, Schauder theory; Quasi-linear hyperbolic equations, propagation of singularities, blow up phenomena.