Elliptic and Parabolic PDEs

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Description

Module Overview: Elliptic and Parabolic PDEs (semester 2)

This module is on parabolic and elliptic equations.

In the first 6 weeks we will cover the classical theory of elliptic and parabolic equations, with Poisson's equation and the Heat equation occupying centerstage. We will use methods such as the energy method and the maximum principle to prove important properties of solutions, including uniqueness, well-posedness and, for parabolic equations, decay to equilibrium. We will also compute explicit solutions in some special cases (e.g. in the whole space).

The remaining lectures are on variational formulation of elliptic PDEs. The material includes an introduction to the Sobolev function spaces, Sobolev embedding and trace theorems.

• Analysis of parabolic and elliptic PDEs (including existence, uniqueness, maximum principles, energy estimates, eigenfunctions)
• Variational theory of PDEs
• Sobolev embeddings and trace theorems

Assessment

This module is assessed in two assignments.

Prerequisites

This module assumes knowledge of undergraduate level ordinary differential equations, single- and multivariable real analysis, and linear algebra.