Variational calculus

Unit Coordinator: Thomas Schmidt, Ingenuin Gasser,
Programme: Erasmus Mundus
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.

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