Calculus of Variations
A.Y. 2026/2027
Learning objectives
The course introduces the main methods of modern Calculus of Variations, with particular emphasis on direct methods and geometric applications.
After a brief introduction to the classical differential approach based on Euler-Lagrange equations, we will discuss the role of compactness, coercivity, lower semicontinuity, and relaxation in the existence theory for minimizers of variational integrals. The fundamentals of the vectorial case will be also presented.
A central part will be devoted to BV functions and sets of finite perimeter as the natural framework for the variational study of interfaces and geometric problems of area type. Finally, we will introduce the language of Γ-convergence to discuss the modern theory of phase transitions.
After a brief introduction to the classical differential approach based on Euler-Lagrange equations, we will discuss the role of compactness, coercivity, lower semicontinuity, and relaxation in the existence theory for minimizers of variational integrals. The fundamentals of the vectorial case will be also presented.
A central part will be devoted to BV functions and sets of finite perimeter as the natural framework for the variational study of interfaces and geometric problems of area type. Finally, we will introduce the language of Γ-convergence to discuss the modern theory of phase transitions.
Expected learning outcomes
By the end of the course, students are expected to know the fundamental principles of the direct method and to be able to apply them to model existence problems for integral functionals on Sobolev spaces. They should also be able to formally derive Euler-Lagrange equations and natural boundary conditions in simple examples.
Students should understand the role of convexity and of its generalizations in lower semicontinuity and relaxation of integral functionals. They should also know the basic properties of BV functions and sets of finite perimeter.
Finally, students should be able to use these tools in the study of geometric variational problems of the area type, with particular focus on Plateau's problem and the isoperimetric problem.
Students should understand the role of convexity and of its generalizations in lower semicontinuity and relaxation of integral functionals. They should also know the basic properties of BV functions and sets of finite perimeter.
Finally, students should be able to use these tools in the study of geometric variational problems of the area type, with particular focus on Plateau's problem and the isoperimetric problem.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
* Motivations and model problems in the calculus of variations.
* Classical indirect methods: first variations, Euler-Lagrange equations, natural boundary conditions; an introduction to the Hamiltonian formulation and to the Hamilton-Jacobi equation.
* Direct methods: compactness, coercivity, lower semicontinuity, closed constraints, and existence of minimizers.
* The vectorial case: rank-one convexity, quasiconvexity, and polyconvexity; their role in the lower semicontinuity of integral functionals.
* An introduction to relaxation theory.
* BV functions and sets of finite perimeter: compactness, lower semicontinuity, the coarea formula, reduced boundary, and De Giorgi's structure theorem.
* Geometric variational problems: Plateau's problem and minimal surfaces, the isoperimetric inequality, free-boundary problems, and capillarity functionals.
* The language of Γ-convergence: phase transitions and the Modica-Mortola theorem.
* Classical indirect methods: first variations, Euler-Lagrange equations, natural boundary conditions; an introduction to the Hamiltonian formulation and to the Hamilton-Jacobi equation.
* Direct methods: compactness, coercivity, lower semicontinuity, closed constraints, and existence of minimizers.
* The vectorial case: rank-one convexity, quasiconvexity, and polyconvexity; their role in the lower semicontinuity of integral functionals.
* An introduction to relaxation theory.
* BV functions and sets of finite perimeter: compactness, lower semicontinuity, the coarea formula, reduced boundary, and De Giorgi's structure theorem.
* Geometric variational problems: Plateau's problem and minimal surfaces, the isoperimetric inequality, free-boundary problems, and capillarity functionals.
* The language of Γ-convergence: phase transitions and the Modica-Mortola theorem.
Prerequisites for admission
The prerequisites are selected topics from Real Analysis, Functional Analysis, and Partial Differential Equations.
Teaching methods
The course is delivered via standard lectures.
Attendance is strongly recommended.
Attendance is strongly recommended.
Teaching Resources
Main references:
- B. Dacorogna. Direct Methods in the Calculus of Variations. Springer.
- I. Fonseca, G. Leoni. Modern Methods in the Calculus of Variations: L^p Spaces. Springer.
- F. Maggi. Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory. Cambridge University Press.
- L. Ambrosio, N. Fusco, D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press.
- G. Dal Maso. An Introduction to Γ-Convergence. Birkhäuser.
Additional references:
- L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press.
- L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations. AMS.
- B. Dacorogna. Direct Methods in the Calculus of Variations. Springer.
- I. Fonseca, G. Leoni. Modern Methods in the Calculus of Variations: L^p Spaces. Springer.
- F. Maggi. Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory. Cambridge University Press.
- L. Ambrosio, N. Fusco, D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press.
- G. Dal Maso. An Introduction to Γ-Convergence. Birkhäuser.
Additional references:
- L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press.
- L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations. AMS.
Assessment methods and Criteria
The exam consists of an oral discussion on the topics presented during the lectures. The purpose of the discussion is to verify that the student knows and understands the content of the lectures, can establish relationships among the various topics, and can effectively apply the techniques presented in concrete situations.
Few homework problems may be suggested during the lectures, with the purpose of facilitating the study of the theoretical material with the help of concrete examples. Solving such problems is not mandatory; nonetheless, similar problems may be proposed during the oral examination.
The exam is passed upon successful completion of the oral examination. A final mark in the range 0-30 (with 18 being the minimum passing grade) is given and communicated immediately at the end of the oral examination.
Few homework problems may be suggested during the lectures, with the purpose of facilitating the study of the theoretical material with the help of concrete examples. Solving such problems is not mandatory; nonetheless, similar problems may be proposed during the oral examination.
The exam is passed upon successful completion of the oral examination. A final mark in the range 0-30 (with 18 being the minimum passing grade) is given and communicated immediately at the end of the oral examination.
MATH-03/A - Mathematical Analysis - University credits: 6
Lessons: 42 hours
Professor:
Stuvard Salvatore
Shifts:
Turno
Professor:
Stuvard SalvatoreProfessor(s)
Reception:
Please, request an appointment via email
Room 1041, Department of Mathematics, Via Cesare Saldini 50, first floor or via Zoom conference call