Calculus of Variations (MAGIC095)
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In undergraduate calculus we learn to maximise or minimise functions of one variable, finding optimal points. The Calculus of Variations is concerned with finding optimal functions and the properties of these optimisers. Famous examples include surfaces of minimal area or the shortest or quickest paths between given points. In the classical indirect method, the optimisers are found as solutions of the Euler-Lagrange differential equations. In the modern direct method, one uses abstract means to find optimisers, which often yields existence results for solutions of differential equations.
Autumn 2019 (Monday, October 7 to Friday, December 13)
Mostly undergraduate calculus, but we will also use the implicit function theorem and some existence and uniqueness theory for ODEs. Knowledge of Lebesgue integration is helpful but not required.
In this 10-lecture series, elements of both the classical and the modern theory are presented. A tentative plan is as follows:
1. Introduction. One-dimensional variational problems: Fundamental lemma, Euler-Lagrange equations
2.-3. Convexity and existence and regularity issues
4.-5. Second variations and necessary conditions for optimality
6. Variational problems with constraints
7. Problems involving multiple integrals
8. Direct method: coercivity and lower semicontinuity
9. Sobolev spaces, weak convergence and compactness
10. The direct method for integral functionals and existence of solutions for some nonlinear PDEs
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