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

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.

Synopsis

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...

Prerequisites

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.