2024
Calculus of Variations
Name: Calculus of Variations
Code: MAT11701D
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese, English
Regime de Frequência: B-learning
Sustainable Development Goals
Learning Goals
Objectives:
Basic formation in calculus of variations with the aim of developing the knowledge of students in this area
or its use in other areas of mathematics or physics or economics.
Competencies:
- Develop abstract thought as a means of solving, with both greater generality & simplicity, specific
problems of other areas, e.g. economics, engineering, biology, mechanics, optics, etc.
- Abstraction skills, creative intuition, model construction and spirit of criticism.
- Skills for explaining the obtained results, both orally and written.
Basic formation in calculus of variations with the aim of developing the knowledge of students in this area
or its use in other areas of mathematics or physics or economics.
Competencies:
- Develop abstract thought as a means of solving, with both greater generality & simplicity, specific
problems of other areas, e.g. economics, engineering, biology, mechanics, optics, etc.
- Abstraction skills, creative intuition, model construction and spirit of criticism.
- Skills for explaining the obtained results, both orally and written.
Contents
1. Introduction.
2. Classic problems and indirect methods.
2.1. The Euler-Lagrange differential equation and other necessary conditions for minimizers.
2.2. Calibrators and sufficient conditions for existence of minimizers.
3. The direct method for single integrals.
3.1. Sobolev spaces in dimension 1.
3.2. Absolutely continuous functions.
3.3. Lower semicontinuity implies convexity.
3.4. Convexity implies lower semicontinuity.
3.5 Existence of minimizers in Sobolev spaces.
3.6. Introduction to minimizers regularity theory.
3.7. The DuBois-Reymond differential equation under minimal hypotheses.
3.8. Linear growth integrals and positive homogeneity.
3.9. Parametric integrals.
4. Vectorial integrals: Q-, P-, R-convexity.
4.1. The Euler-Lagrange differential equation.
4.2. Lower semicontinuity in the scalar case implies convexity.
4.3. Q-, P- and R-convexity
4.4. Q-convexity implies R-convexity.
4.5. Lower semicontinuity implies Q-convexity.
2. Classic problems and indirect methods.
2.1. The Euler-Lagrange differential equation and other necessary conditions for minimizers.
2.2. Calibrators and sufficient conditions for existence of minimizers.
3. The direct method for single integrals.
3.1. Sobolev spaces in dimension 1.
3.2. Absolutely continuous functions.
3.3. Lower semicontinuity implies convexity.
3.4. Convexity implies lower semicontinuity.
3.5 Existence of minimizers in Sobolev spaces.
3.6. Introduction to minimizers regularity theory.
3.7. The DuBois-Reymond differential equation under minimal hypotheses.
3.8. Linear growth integrals and positive homogeneity.
3.9. Parametric integrals.
4. Vectorial integrals: Q-, P-, R-convexity.
4.1. The Euler-Lagrange differential equation.
4.2. Lower semicontinuity in the scalar case implies convexity.
4.3. Q-, P- and R-convexity
4.4. Q-convexity implies R-convexity.
4.5. Lower semicontinuity implies Q-convexity.
Teaching Methods
Teaching methodologies:
Structured exposition, examples with emphasis on applications and on solving exercises.
To stimulate students initiative, so that classes become essentially centered on students activities,
guided by their teacher; instead of on teachers activities, copied by students. Particularly in what
concerns submission of questions and / or suggestions of application and / or description of contents, the
solving of exercises, participation in discussions, etc.
Evaluation:
One written test and one written work elaborated by the student; or else one written exam.
Structured exposition, examples with emphasis on applications and on solving exercises.
To stimulate students initiative, so that classes become essentially centered on students activities,
guided by their teacher; instead of on teachers activities, copied by students. Particularly in what
concerns submission of questions and / or suggestions of application and / or description of contents, the
solving of exercises, participation in discussions, etc.
Evaluation:
One written test and one written work elaborated by the student; or else one written exam.
Teaching Staff
- Luís Miguel Zorro Bandeira [responsible]
Certificações
