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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:

Tuesday 17:00 - 18:00 Room D1.07

Wednesday 16:00 - 18:00 Room D1.07

Prerequisites:

You will need some knowledge of functional analysis, measure theory and partial differential equations. See, for example, the Warwick modules MA3G7 Functional analysis, MA359 Measure Theory, MA3G1 Theory of PDEs, MA4A2 Advanced PDEs and the first term CDT module MA949 -Applied and numerical analysis for linear PDEs.
Companion CDT modules are
MA9M3 Topics in applied mathematics
and
MA947 Graduate real analysis.
Content:

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)