MA4G6 Calculus of Variations

Introductory description

The module provides a modern introduction to the calculus of variations, focusing on finding the extrema (minima or maxima) of functionals—mappings from functions to real numbers. It moves beyond classical techniques to explore powerful methods used to investigate nonlinear problems in physics, materials science, and geometry.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Various models leading to variational problems.

Sobolev spaces.

The Direct Method of the Calculus of Variations and lower semicontinuity.

Convexity and aspects of Convex Analysis (duality).

Existence of solutions for scalar problems.

Polyconvexity and existence of solutions semicontinuity for vector-valued problems.

Regularity theory for minimisation problems.

Optimal control theory and Young measures.

Quasiconvexity, laminates and microstructure.

Variational convergence of functionals (.-convergence).

If time permits: Other variational principles (Ekeland etc.).

If time permits: Functions of bounded variations and applications.

Learning outcomes

By the end of the module, students should be able to:

• Understand why variational problems are important.
• See several examples of variational problems in physics and other sciences.
• Appreciate that (and why) some problems have “classical” solutions and some do not.
• Be able to prove the existence of solutions to convex variational problems.
• Know which kinds of problems are not convex and why convexity is often an unrealistic assumption for vector-valued problems.
• Have an insight into generalised convexity conditions, such a quasiconvexity and polyconvexity and their applications.
• Be able to prove existence of solutions to quasiconvex/polyconvex variational problems.
• Have seen simple optimal control problems and can understand them as a special case of general variational problems.
• Know what microstructure is, why it forms, and what its physical significance is.
• Have seen how regularised functionals converge to a limit functional as the regularisation parameter tends to zero.

Indicative reading list

Reading lists can be found in Talis

Subject specific skills

At the conclusion of the module the student should be able to set up and solve various minimisation problems with and without constraints, to derive Euler-Lagrange equations and appreciate how the laws of mechanics and geometrical optics, as well as some geometrical problems involving least length and least area, fit into this framework.

Transferable skills

1. Precise Formulation: The capacity to restate fundamental laws of physics into precise mathematical frameworks.
2. Abstract Reasoning: Acquiring a mature understanding of abstract mathematical ideas and their real-world applications.