Synopsis

The aim of the course is to give a modern treatment of the calculus of variations from a rigorous perspective, blending classical and modern approaches and applications.

No prior knowledge of the calculus of variations will be assumed. However, some familiarity with the Lebesgue integral is essential, and some knowledge of elementary functional analysis (e.g. Banach spaces and their duals, weak convergence) an advantage.

Classical and modern examples of variational problems (e.g. brachistochrone, models of phase transformations).

One-dimensional problems.

Function spaces and definitions of weak and strong relative minimizers. Necessary conditions; the Euler-Lagrange and DuBois-Reymond equations, theory of the second variation, the Weierstrass condition. Sufficient conditions; field theory and sufficiency theorems for weak and strong relative minimizers. The direct method of the calculus of variations and Tonelli s existence theorem. Regularity of minimizers. Examples of singular minimizers and the Lavrentiev phenomenon. Problems whose infimum is not attained. Relaxation and generalised solutions. Isoperimetric problems and Lagrange multipliers.

Introduction to multi-dimensional problems, done via some examples.

Reading List

• G Buttazo, M Giaquinta, S Hildebrandt, One-dimensional Variational Problems, Oxford Lecture Series in Mathematics, Vol 15, OUP (1998), Ch 1, Sections 1.1,1.2 (treated differently in course), 1.3, Ch 2 (background), Ch 3, Sections 3.1, 3.2, Ch 4, Sections 4.1, 4.3

• U Brechtken-Manderscheid, Introduction to the Calculcus of Variations, Chapman and Hall 1991

• H Sagan, Introduction to the Calculus of Variations, Dover 1992

• J Troutman, Variational Calculus and Optimal Control, Springer-Verlag, 1995