Mathematics

Advanced Partial Differential Equations

Module code: 866G1
Level 7 (Masters)
15 credits in spring teaching
Teaching method: Lecture
Assessment modes: Unseen examination, Coursework

You will be introduced to modern theory of linear and nonlinear Partial Differential Equations. Starting from the theory of Sobolev spaces and relevant concepts in linear operator theory, which provides the functional analytic framework, you will treat the linear second-order elliptic, parabolic, and hyperbolic equations (Lax-Milgram theorem, existence of weak solutions, regularity, maximum principles), e.g., the potential, diffusion, and wave equations that arise in inhomogeneous media.

The emphasis will be on the solvability of equations with different initial/boundary conditions, as well as the general qualitative properties of their solutions. They then turn to the study of nonlinear PDE, focusing on calculus of variation.

Module learning outcomes

  • Have a systematic and comprehensive understanding of Sobolev spaces and central properties of Sobolev functions (approximation, extension, trace, Sobolev inequalities, compactness)
  • Investigate elliptic equations of the second order by applying existence and uniqueness theorems of weak solutions, evaluate regularity and maximum principle
  • Critically evaluate and discuss the theory of parabolic and hyperbolic equations of second order
  • Critically evaluate and interpret the theory of calculus of variations, in particular explain the concept of a minimiser and discuss the regularity of minimisers