Dottorato: Modelli e Metodi Matematici per la Tecnologia e la Società

Ph D course: Direct Methods in the Calculus of Variations.

(Metodi diretti nel calcolo delle variazioni) (prof. Andrea Braides)

February-May 2011

Exam page

Program

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.