Calculus of Variations

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This is a previous supplementary module, not currently scheduled to run this session.

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Description

Brief description

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

Including introductions to Gamma convergence and free-discontinuity problems.

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, Springer, 2018.

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 on topics expanding or complementing course content.