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).

Direct method of the Calculus of Variations. Lower semicontinuous functions: sequential and topological definition; properties. Coercive and sequentially coercive functions. Convex functions: domain, epigraph, properties. Lower semicontinuous envelope, convex envelope. Integral functionals on Lebesgue spaces: lower semicontinuity with respect to strong and weak topology. Nemytskii operators. Riemann-Lebesgue Lemma. Convexity as a necessary and sufficient condition for weak lower semicontinuity. Sobolev spaces. Integral functionals on Sobolev spaces: lower semicontinuity with respect to strong and weak topology. Quasi-convexity, policonvexity and rank-one convexity. Quasi-convexity as a necessary and sufficient condition for weak lower semicontinuity. Relaxation. Fréchet and Gâteaux differentiability. Euler-Lagrange equation. Du Bois-Reymond equation. Regularity results for one-dimensional problems. Gamma-convergence: the fundamental theorem, stability with respect to continuous perturbations, connections with uniform and pointwise convergence, lower semicontinuity of Gamma-limits, relaxation, examples and applications.

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