Synopsis for C5.2b: Calculus of Variations

Weight: Half-unit, OSS paper code 2B65

Recommended Prerequisites

C5.1a: Methods of Functional Analysis for PDEs. 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. B5.1a: Dynamics and Energy Minimization: Some knowledge of local minimizers in the 1D calculus of variations is helpful. This material will be reviewed in the course.

Overview

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.

Learning Outcomes

Students will learn rigorous results in the classical and modern one-dimensional calculus of variations and see possible behaviour and application of these results in examples. They will see some examples of multi-dimensional problems.

Synopsis

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 Du Bois-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 generalized solutions. Isoperimetric problems and Lagrange multipliers. Invariant variational problems, Noether's theorem, conservation laws.

Multi-dimensional problems, done via some examples.

1. G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-dimensional Variational Problems, Oxford Lecture Series in Mathematics, Vol. 15 (Oxford University Press, 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.

Additional Reading

1. U. Brechtken-Manderscheid, Introduction to the Calculus of Variations (Chapman & Hall, 1991).
2. H. Sagan, Introduction to the Calculus of Variations (Dover, 1992).
3. J. Troutman, Variational Calculus and Optimal Control (Springer-Verlag, 1995).
4. L. C. Evans, Partial Differential Equations (American Mathematical Society, 2010).