Academic Year 2023/2024 - Teacher: Salvatore Angelo MARANO

Expected Learning Outcomes

The main objective of the course is to provide students with the basic elements of the Calculus of Variations in dimension one. In particular, the course aims to make students acquire the following skills:

1) Knowledge and understanding: Knowledge of results and fundamental methods of the Calculus of Variations in dimension one. Ability to read, understand and deepen a topic of mathematical literature and propose it again clearly and accurately. Ability to understand problems and to extract their substantial elements.

2) Applying knowledge and understanding: Ability to build or solve examples or exercises and to face new theoretical problems, researching the most suitable techniques and applying them appropriately.

3) Making judgments: Being able to produce proposals capable of correctly interpreting complex problems in the context of the Calculus of Variations and its applications.

4) Communication skills: Ability to present arguments, problems, ideas and solutions, both one's own and others, in mathematical terms and their conclusions, with clarity and accuracy and in ways appropriate to the listeners to whom one is addressing, both in oral and written form. Ability to clearly motivate the choice of strategies, methods and contents, as well as the computational tools adopted.

5) Learning skills: Read and deepen a topic of literature in the context of the Calculus of Variations. To deal autonomously with the systematic study of topics not previously explored.

Course Structure

The main topics of the program will be illustrated by the teacher, with lectures, in their general aspects and with particular regard to the points in which new ideas are introduced, The insights relating to these chapters and other particular topics will be presented in the classroom by groups of students, which will be established from time to time respecting a criterion of rotation. This pursues the aim of making students acquire that degree of autonomy in the study and preparation of the exhibition which is indispensable both for those who want to enter the field of research and for future teachers.

If the teaching is given in a mixed or remote mode, the necessary changes with respect to what was previously stated may be introduced, in order to respect the program envisaged and reported in the syllabus.

PLEASE NOTE: Information for students with disabilities and / or DSA

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs.

It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or DSA) of our Department, prof. Filippo Stanco.

Required Prerequisites

The main topics of the courses of Mathematical Analysis 1 and 2 and of Functional Analysis, as well as the basic elements of  Meaure Theory.

Attendance of Lessons

Attendance at lessons is not compulsory but is strongly recommended.

Detailed Course Content

Classical problems and indirect methods. Absolutely continuous functions and Sobolev spaces. Semicontinuity and existence results.
Regularity of minimzers and Morrey spaces. Applications to boundary value problems. Direct methods.

Textbook Information

G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-dimensional Variational Problems. An Introduction, Oxford University Press, 1998.
L. Pick, A. Kufner, O. John, S. Fucik, Function Spaces, Volume 1, De Gruyter, 2013.
G. Talenti - A. Colesanti - P. Salani, Un'introduzione al calcolo delle variazioni: teoria ed esercizi, Unione Matematica Italiana, Bologna, 2016.

Course Planning

 SubjectsText References
1Testi 1) e 3).Monographs 1) and 3)
2Testo 2)Monograph 2)
3Testi 1) e 3)Monographs 1) and 3)

Learning Assessment

Learning Assessment Procedures

Oral interview.

The following criteria will normally be followed to assign the grade:

not approved: the student has not acquired the basic concepts and is not able to carry out the exercises. 
18-23: the student demonstrates minimal mastery of the basic concepts, his skills in exposition and connection of contents are modest, he is able to solve simple exercises. 24-27: the student demonstrates good mastery of the course contents, his presentation and content connection skills are good, he solves the exercises with few errors. 
28-30 cum laude: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit; she solves the exercises completely and without errors.

Examples of frequently asked questions and / or exercises

The topics contained in the "Course Programming" constitute examples of frequently asked questions.