Variational calculus
- Code: DT0653
- Unit Coordinator: Thomas Schmidt
- ECTS Credits: 6
- Semester: 2
- Year: 1
- Campus: Hamburg University of Technology
- Language: English
- Aims:
The module introduces to variational minimization problems and/or variational methods for PDEs.
It may cover problems in a classical smooth setting as well as theory in Sobolev spaces.
- Content:
A selection out of the following:
- Model problems and examples (Dirichlet energy, isoperimetric and brachistochrone problems, minimal surfaces, Bolza and Weierstrass examples, …),
- Existence and uniqueness of minimizers by direct methods,
- Weak lower semicontinuity of (quasi)convex variational integrals,
- Necessary and sufficient (PDE) conditions for minimizers,
- Problems with constraints (obstacles, capacities, manifold and volume constraints, ...),
- Generalized minimizers (relaxation, Young measures, ...),
- Variational principles and applications,
- Duality theory,
- Outlook on regularity.
- Pre-requisites:
A solid background in analysis and linear algebra is necessary.
Familiarity with functional analysis, Sobolev spaces, and PDEs can be advantageous.
- Reading list:
- H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces, Applications to PDEs and Optimization, MOS-SIAM Series on Optimization 17, Philadelphia, 2014.
- G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-Dimensional Variational Problems, An Introduction, Oxford Lecture Series in Mathematics and its Applications 15, Clarendon Press, Oxford, 1998.
- B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, London, 2014.
- B. Dacorogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, Springer, Berlin, 2008.
- I. Ekeland, R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics 28, SIAM, Philadelphia, 1999.
- M. Giaquinta, S. Hildbrandt, Calculus of Variations 1, The Lagrangian Formalism, Grundlehren der Mathematischen Wissenschaften 310, Springer, Berlin, 1996.
- E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003.
- F. Rindler, Calculus of Variations, Universitext, Springer, Cham, 2018.
- F. Santambrogio, Optimal Transport for Applied Mathematicians, Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and Their Applications 87, Birkhäuser/Springer, Cham, 2015.
- M. Struwe, Variational Methods, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 34, Springer, Berlin, 2008.
