CALCULUS OF VARIATIONS
Stampa
Enrollment year
2015/2016
Academic year
2016/2017
Regulations
DM270
Academic discipline
MAT/05 (MATHEMATICAL ANALYSIS)
Department
DEPARTMENT OF MATHEMATICS "FELICE CASORATI"
Course
MATHEMATICS
Curriculum
PERCORSO COMUNE
Year of study
Period
1st semester (03/10/2016 - 13/01/2017)
ECTS
6
Lesson hours
48 lesson hours
Language
Italian
Activity type
ORAL TEST
Teacher
MORA MARIA GIOVANNA (titolare) - 6 ECTS
Prerequisites
Basic knowledge of Functional Analysis and Measure Theory (the main definitions and results will be given during the course).
Learning outcomes
The course aims at giving an introduction to the Calculus of Variations.
Course contents
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 topologies. 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 topologies. 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.
Teaching methods
Lectures
Reccomended or required readings
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
Assessment methods
Oral exam.
Further information
Oral exam.
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