Calculus of Variations and Dynamics
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This is a previous supplementary module, not currently scheduled to run this session.
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Description
The course will cover elements of the calculus of variations and connections to dynamics.
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.
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.
Connections to dynamics: Semiflows with a Lyapunov function. Omega limit sets and approach to equilibrium. Stability of equilibria. Global attractors.
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.
Jack K. Hale, Asymptotic behavior of dissipative systems, AMS, 1988.
George R. Sell & Yuncheng You, Dynamics of evolutionary equations, Springer, 2002.