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Calculus of Variations

Academic Year 2023/2024 - Teacher: Maria Alessandra RAGUSA

Expected Learning Outcomes

The main objective of the course is to provide students with the basic elements on Sobolev's and Morrey's Theory of Spaces, on questions of existence and regularity of minima of functionals.

 The course aims to make students acquire the following skills:

1) Knowledge and understanding: Knowledge of results and fundamental methods of Real Analysis and Calculation of Variations. 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. Being able to autonomously formulate pertinent judgments on the applicability of models of the Calculus of Variations to theoretical and / or concrete situations.

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 the literature in the context of the Theory of Controls. To deal autonomously with the systematic study of topics of Calculus of Variations not previously explored.

Course Structure

Direct Instruction.

The lessons are integrated with exercises related to the topics covered by the course and will take place in the classroom. It should also be noted that there are 14 hours of lessons (typically, these are theory) and 12 hours of other activities (typically, these are exercises).

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the program planned and outlined in the Syllabus.

NOTE: Information for students with disabilities and/or SLD

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 SLD) of our Department, prof. Filippo Stanco).

Required Prerequisites

The main topics of the courses of Mathematical Analysis 1 and 2 as well as the basic elements of measurement theory.

Attendance of Lessons

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

[1]G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-dimensional Variational Problems. An Introduction, Oxford University Press, 1998.
[2] L. Pick, A. Kufner, O. John, S. Fucik, Function Spaces, Volume 1, De Gruyter, 2013.

Learning Assessment

Learning Assessment Procedures

At the end of the course there will be a final oral exam.

Examples of frequently asked questions and / or exercises

Sobolev spaces. Existence of minima of functionals. Regularity of minima of functionals. Spaces of Morrey. Boundary value problems.