Subject: CALCULUS OF VARIATIONS (A.A. 2021/2022)
Unit Calcolo delle variazioni
Related or Additional Studies (lesson)
At the end of the course, the student will have to know the main tools for discussing existence, uniqueness and regularity of the solution in a problem from Calculus of Variation.
Basic knowledge of Mathematical Analysis, Lebesgue's integral, Lp and Sobolev spaces.
Direct method of the Calculus of Variations. Lower semicontinuous functions; convex functions; coercive functions; weak compactness in Banach spaces.
Integral functionals: necessary and sufficient conditions for weak lower semicontinuity; coercitivity.
Fréchet and Gâteaux differentiability; Euler-Lagrange equations; some regularity results.
The student is required to present a 40 minutes talk devoted to course topics or in-depth analysis of close subjects. During the oral exam, questions on the whole course contents may be asked.
1) Knowledge and understanding: the student will have to know the main tools for discussing existence, uniqueness and regularity of solutions in a problem from Calculus of Variations.
2) Applying knowledge and understanding: the student will need to be able to identify the conditions suitable to ensure existence, uniqueness and regularity of the minimizers.
3) Making judgements: Attitude towards a methodological approach that leads to verify claims and methods through rigorous arguments.
4) Communicating skills: at the end of the course, the student will be able to present and describe the topics he learned during the course, with an appropriate language and a correct mathematical formalism.
5) Learning skills: the tools which are provided during the course should allow to deepen in autonomy the main topics of the course itself, together with closely related topics.
G. Buttazzo, M. Giaquinta, S. HIldebrandt, One-dimensional Variational Problems, An Introduction
Oxford University Press, 1998.
B. Dacorogna, Introduction to the Calculus of Variations, Imperial College, 2004.
B. Dacorogna, Direct Methods in the Calculus of Variations, Springer 2002.
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific 2003.