Quasiconvexity and its applications - An introduction to quasiconvexity by John M. Ball

Lecturer: John M. Ball (Oxford/IAS)

Date: 12-13 November 2002

This short course of six lectures immediately precedes the conference Quasiconvexity and its Applications to be held 14-16 November in Princeton, which celebrates the 50th anniversary of Morrey's landmark paper on quasiconvexity.

Location and times

The lectures will be held in Fine Hall, Princeton University:
Tuesday 12 November
Lecture 1. 2:00-3:00, Lecture 2. 3:00-4:00, Room 110

Wednesday 13 November
Lecture 3. 9:00-10:00, Lecture 4. 11:00-12:00, Room 1001
Lecture 5. 1:30-2:30, Lecture 6. 3:00-4:00, Room 322.


Quasiconvexity is the central convexity condition of the multi-dimensional calculus of variations, but because of the lack of an adequate characterization of quasiconvex functions it remains somewhat mysterious. The lectures will cover the following topics:

  • Definition and examples of quasiconvex functions. Null Lagrangians and polyconvexity. Rank-one convexity.
  • Lower semicontinuity and existence of minimizers. Applications to elasticity. Quasiconvexity in the interior and at the boundary as necessary conditions for a minimizer. Partial regularity. Relaxation.
  • Quasiconvexity and gradient Young measures. Quasiconvex sets of matrices. Relaxation and the passage from microscales to macroscales. Examples involving finitely many matrices or energy wells. Martensitic microstructure.
  • Extensions and applications: higher-order problems, homogenization, quasiregular maps...


The lectures will be self-contained and assume as little background knowledge as possible. However, some familiarity with Sobolev spaces and weak convergence in Lp spaces will be an advantage.