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Variational methods for PDEs

Materials related to this course will be available at In particular John Ball's lecture notes,


SMSTC Semester 2

This module is taught as part of the MIGSAA CDT programme and offered across SMSTC.

Format: compact course of 8 lectures in Week 1, then 2 lectures per week, additional student lectures

Timetable for Week 1 (taught by John M. Ball): all lectures in the ICMS lecture theatre, room 5.10.

January 7: 10.45-12.45, January 8: 10.45-12.45, January 9: 9.00-10.45, January 10: 9.00-10.45  

Lectures in Week 1 are not broadcast via SMSTC. Lectures from Week 2 are offered across SMSTC and are accessible with a solid background in Sobolev spaces. 

Aims: This course aims to give an introduction to the basics and modern applications of variational approaches to the pure and numerical analysis of partial differential equations. We particularly focus on nonlinear elliptic problems from the physical sciences and geometry and include recent developments for convex integration.

Prerequisites: a previous course in either PDEs or their numerical analysis

Contents: We cover some essential basic and advanced topics in calculus of variations and nonlinear PDEs, as well as their applications and their rigorous numerical approximation. After the course the student should know key ideas in a broad range of topics, as they are relevant in their research or in relevant pure/numerical analysis talks.

In particular, we expect to cover the following topics:

  1. A users guide to Sobolev spaces (definitions, statement of embedding theorems, some proofs in 1D) and weak convergence of sequences. (4  lectures)   
  2. The 1D calculus of variations: Existence and nonexistence of minimizers, Tonellis existence theorem, necessary and sufficient conditions for local minimizers, the Lavrentiev phenomenon. (4 lectures)
  3. Introduction to the higher-dimensional calculus of variations: lower semicontinuity, notions of convexity, compensated compactness.
  4. Applications: nonlinear elliptic PDEs and the obstacle problem, equations of continuum mechanics, minimal surfaces.
  5. Numerical discretisation: Finite element approximation of nonlinear PDEs, obstacle problem and harmonic maps.
  6. Saddle point problems: Palais-Smale condition, Pohozaev’s non-existence result, harmonic maps of Riemann surfaces.
  7. Convex integration and non-uniqueness of weak solutions of low regularity.

Some references:
* J. M. Ball, lecture notes,
* S. Bartels, Numerical Methods for Nonlinear Partial Differential Equations, Springer, 2015.
* C. De Lellis, L. Szekelyhidi Jr., John Nash’s nonlinear iteration, to appear in memorial volume for John Nash.
* I. Ekeland and N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc. 39 (2002), 207-265.
* M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, 2008.

Student talks: Interested students will give a 60-minute lecture on a topic of their choice, ideally a topic related to their research interests.

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