1. Minimum problems. Compactness and lower semicontinuity
properties. Relaxation.
2. Minimum problems for integral functionals in Lebesgue spaces. Weak
convergence. Convexity.
3. Relaxation for integral functionals in Lebesgue spaces. Convex
and lsc envelopes. Young-measure solutions.
4. Relaxation on spaces of measures. The blow-up method.
5. Non convex problems on spaces of measures. Subadditivity.
6. Problems in Sobolev spaces. Quasiconvexity and polyconvexity.
7. Problems in spaces of functions with bounded variation and for sets
of
finite perimeter.
8. Gamma-convergence. Homogenization. Limits of Riemannian metrics.
9. Approximate solutions. Finite-difference
approximations. Vanishing-viscosity approximation.
References
A. Braides and A. Defranceschi. Homogenization of Multiple Integrals.
Oxford UP, 1998. (Part
I)
A. Braides. Gamma-convergence for Beginners. Oxford UP, 2002.
B. Dacorogna. Direct Methods in the Calculus of Variations (Second
Edition). Springer, 2008.
I. Fonseca and G.Leoni. Modern Methods in the Calculus of Variations:
L^p Spaces. Springer, 2007.