Calculus of Variations

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Description

Time: Mondays, 11.00-13.00

Location for Maxwell Institute students: Bayes Centre (room 5.46)

Module Leader: John Ball (Heriot-Watt University)

Module Summary

The course will provide a survey of the calculus of variations in one and higher dimensions.

Lectures 1-6:The one-dimensional calculus of variations

Examples of nonexistence of minimizers. The Tonelli existence theorem. Weak and strong local minimizers. Necessary and sufficient conditions for local minimizers. The Lavrentiev phenomenon. Relaxation.

Lectures 7-14: The multi-dimensional calculus of variations

Quasiconvexity as a necessary and sufficient condition for weak lower semicontinuity. Rank-one convexity and polyconvexity. Young measures. Examples and counterexamples. Survey of open problems.

Lectures 15-20: Miscellaneous topics

A selection of applications of the calculus of variations to current research, taking into account the interests of those taking the course. Possibilities include nonlinear elasticity and the microstructure of alloys, liquid crystals, computer vision, Gamma convergence, free-discontinuity problems and the relation of the calculus of variations to dynamics and stability theory. 

 Course material

Guiseppe Buttazzo, Mariano Giaquinta & Stefan Hildebrandt, One-dimensional Variational Problems, Oxford University Press, 1998.

Irene Fonseca & Giovanni Leoni, Modern methods in the calculus of variations: Lp spaces, Springer, 2007.

Filip Rindler, Calculus of Variations, 2nd Edition, Springer, 2026.

A. Braides, Gamma-convergence for beginners, Oxford University Press, 2002.

L. Ambrosio, N. Fusco & D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, 2000.

Assessment: by short presentations at the end of the course on topics expanding or complementing course content, intended also as giving experience in giving such talks.