Calculus of variations
Additional Info
- ECTS credits: 6
- Code: DT0402
- University: Hamburg University of Technology
- Semester: 2
- Lecturer 1: T. Schmidt
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Objectives:
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. -
Topics:
Model problems (brachistochrone, Dirichlet energy, minimal surfaces, etc.), convex integrands and generalizations, existence and uniqueness of minimizers by direct methods, necessary and sufficient (PDE) conditions for minimizers, generalized minimizers (via relaxation or Young measures), problems with constraints, variational principles and applications, duality theory, outlook on regularity.
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Prerequisites:
A solid background in analysis and linear algebra is necessary.
Familiarity with functional analysis, Sobolev spaces, and PDEs can be advantageous. -
Books:
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.