Mathematics

Advanced Partial Differential Equations (L.6)

Module code: G5215
Level 6
15 credits in spring semester
Teaching method: Lecture
Assessment modes: Coursework, Computer based exam

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. You will then turn to the study of nonlinear PDE, focusing on calculus of variation.

Module learning outcomes

  • Understand the theory of Sobolev spaces and various properties of Sobolev functions (approximation, extension, trace, Sobolev inequalities, compact embeddings).
  • Apply existence and uniqueness theorems for weak solutions of elliptic equations, evaluate regularity (interior) and use maximum principle.
  • Apply the theory of parabolic and hyperbolic equations of second order; use eigenfunction expansions, derive and use energy inequalities.
  • Apply the theory of calculus of variations, in particular explain the concept of a minimiser and discuss the regularity of minimisers.