Content:

• 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 (Γ convergence).

If time permits:

• Other variational principles (Ekeland etc.).
• Functions of bounded variations and applications.

Aims:
The Calculus of Variations is both old and new. Starting from Euler's work up to very recent discoveries, this sub-field of Mathematical Analysis has proven to be very successful in the analysis of physical, technological and economical systems. This is due to the fact that many such systems incorporate some kind of variational (minimum, maximum, extremum) principle and understanding this structure is paramount to proving meaningful results about them. Applications range from material sciences over geometry to optimal control theory. The aim of this course is to give a thoroughly modern introduction and to lead from the basics to sophisticated recent results.

Objectives:
By the end of the module the student 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.

Books:
B. Dacorogna: Introduction to the Calculus of Variations. Imperial College Press 2004.
B. Dacorogna: Direct Methods in the Calculus of Variations. 2nd edition. Springer, 2008.
L. C. Evans: Partial Differential Equations. 2nd edition. AMS, 2010 (some chapters).
I. Fonseca and G. Leoni: Modern Methods in the Calculus of Variations: Lp -spaces. Springer, 2007.
E. Giusti: Direct Methods in the Calculus of Variations. World Scientific, 2002.