Elliptic and Parabolic PDEs
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Module Overview: Elliptic and Parabolic PDEs (semester 2)
This module is on parabolic and elliptic equations. Useful methods to study solutions such as travelling wave and similarity solutions are developed and nonexistence and multiplicity results for solutions of nonlinear elliptic problems are studied by a variety of methods. Different concepts and tools are developed such as maximum principles, blow up and comparison theorems. One striking result that will be proved is that positive solutions of autonomous semilinear elliptic problems on a ball must be radially symmetric.
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, monotone iteration, regularity, eigenfunctions)
- Variational theory of PDEs
- Sobolev embeddings and trace theorems
This module is assessed in two assignments.
This module assumes knowledge of undergraduate level ordinary differential equations, single- and multivariable real analysis, and linear algebra.