MA953 - Topics in Partial Differential Equations
Lecturer: Charles Elliott
Term(s): Term 2
Commitment: 30 lectures
Assessment: Oral exam
Timetable:
Tuesday 17:00 - 18:00 Room D1.07
Wednesday 16:00 - 18:00 Room D1.07
Prerequisites:
Because of the ubiquitous nature of PDE based mathematical models in biology, advanced materials, finance, physics and engineering much of mathematical analysis is devoted to their study.
The complexity of the models means that finding formulae for solutions is impossible in most practical situations. Issues for mathematical analysis include: the formulation of well-posed problems in appropriate function spaces, regularity and qualitative information about the solution.
The purpose of this module is to provide a wide ranging introduction to selected topics in the modern analysis of PDEs selected for relevance to applications (e.g. geometry, material science, cell biology, continuum mechanics) and research timeliness.
Syllabus:
This is an indicative outline only showing the sort of topics that may be covered:
- Functional analysis and Sobolev spaces
- Review of variational analysis of elliptic equations
- Bochner spaces
- Variational analysis of parabolic equations
- Nonlinear models: Variational inequalities, Allen-Cahn and Cahn Hilliard equations
- Gradient flow
- Introduction to:- Geometric and surface partial differential equations
- Finite element approximation
References:
Illustrative bibliography:
-
Computation of Geometric PDEs and Mean Curvature Flow K.P. Deckelnick, G. Dziuk and C.M. Elliott Acta Numerica (2005) 139-232
- Finite elements III A. Ern and J. L. Guermond Texts in Applied Mathematics Vol 74 Ebook Springer
- Functional analysis, Sobolev spaces and partial differential equations, H. Brezis Universitat Ebook (2011) Springer
- Partial Differential Equations L.C. Evans, AMS Grad. Stud. Math. Vol. 1
- Computation of Geometric PDEs and Mean Curvature Flow K.P. Deckelnick, G. Dziuk and C.M. Elliott Acta Numerica (2005) 139-232
- Finite element methods for surface partial differential equations
G. Dziuk and C.M. Elliott Acta Numerica (2013) 289--396 - An introduction to variational inequalities and their applications D. Kinderleher and G. Stampacchia Academic Press (1980)