The course aims to give an introduction to the Calculus of Variations.

Lectures

Oral exam

Basic knowledge of Functional Analysis and Measure Theory (the main definitions and results will be given during the course).

Convex functions. Lower semicontinuous functions.

Direct method of the Calculus of Variations. Integral functionals on Lebesgue and Sobolev spaces. Relaxation. Euler-Lagrange equation. DuBois-Reymond equation. Gamma-convergence and applications.





Direct method of the Calculus of Variations: lower semicontinuity and compactness. Convex functions. Lower semicontinuous functions. Legendre transforms.

Integral functionals on Lebesgue spaces. Basic properties of Sobolev spaces. Integral functionals on Sobolev spaces. Relaxation of integral functionals. Euler-Lagrange equation. DuBois-Reymond equation.

Gamma-convergence: definition and abstract properties. Fundamental theorem of Gamma-convergence. Homogenization in dimension one.



For the updated summary see http://www-dimat.unipv.it/mora/cdv14.html

G. Buttazzo, M. Giaquinta, S. HIldebrandt

One-dimensional Variational Problems, An Introduction

Oxford University Press, 1998



B. Dacorogna

Direct Methods in the Calculus of Variations

Springer 2002, 2nd edition



A. Braides

Gamma-convergence for beginners

Oxford University Press, 2002