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Mathematics Institute

MA953 - Topics in Partial Differential Equations

Lecturer: Charles Elliott

Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral exam

Timetable:

Thursday 9-11 in D1.07

Friday 10-11 in D.107

Prerequisites: Elements of Elliptic PDE theory, Sobolev spaces, Functional analysis.

For example, familiarity with topics covered in the Warwick modules:-

MA949 - Applied and Numerical Analysis for Linear PDEs

MA4A2 Advanced Partial Differential Equations. Consult lecture notes of Felix Schulze

Content:
The primary goal of this Module is to present some fundamental ideas and concepts relating to Surface and Geometric PDEs that not only arise in many physical models but are also of intrinsic importance within differential geometry. This module is primarily motivated by models from Cell Biology.
  • Elementary geometric analysis
  • PDEs on surfaces
  • PDEs on evolving domains
  • Surface energies -Area and Canham-Helfrich functionals
  • Geometric PDEs

  • Phase separation on biomembranes and Surface Cahn-Hilliard equations

  • Bulk - surface systems and Receptor Ligand dynamics in cell biology
  • Surface finite elements
  • Surface Naiver-Stokes equations

Reading:

  1. Computation of geometric partial differential equations and mean curvature flow K Deckelnick, G Dziuk, CM Elliott Acta Numerica 14, 139-232 (2005)
  2. Finite element methods for surface PDEs G Dziuk, CM Elliott Acta Numerica 22, 289-396 (2013)
  3. Curvature driven interface evolution H Garcke Jahresbericht der Deutschen Mathematiker-Vereinigung 115, 63-100 (2013)
  4. Lecture notes