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Introduction to Gamma-convergence

Schedule

Thursday, 2 January 2020 to Thursday, 14 May 2020
Total hours: 20
Hours of lectures: 20

Examination procedure

  • Report or seminar

Prerequisites

Basic knowledge of the theory of Sobolev spaces and Functional Analysis

Syllabus

1. General theory

2. Localization tools

3. Local integral functionals on Sobolev spaces

4. Homogenization

5. Perforated domains and relaxed Dirichlet problems

6. Phase transition and concentration problems

7. Free discontinuity problems

 

Educational goals: The theory of Gamma-convergence, introduced by E.De Giorgi in the 70', provides the natural language to study limits of variational problems. By now it is a mature theory, with a broad range of applications (homogenization, phase transitions, discrete-to-continuum approximations). The course will be based mainly on the discussion of paradigmatic examples and applications, after a preliminary discussion of the basic aspects of the theory.

 

Bibliographical references

[1] Braides A., “Approximation of free-discontinuity problems”, Lecture Notes in Mathematics, 1694. Springer-Verlag, Berlin, 1998.
[2] Braides A., “Γ-convergence for beginners”, Oxford Lecture Series in Mathematics and its Applications 22, Oxford University Press, Oxford, 2002.
[3] Braides A., “Handbook of Γ-convergence”, in “Handbook of Differential Equations: Stationary PDEs”, ed. M. Chipot, North Holland (to appear).
[4] Braides A. - Defranceschi A., “Homogenization of Multiple integrals”, Oxford University Press, Oxford, 1998.
[5] Dacorogna B., “Introduction to the calculus of variations”, Translated from the 1992 French original. Imperial College Press, London, 2004.
[6] Dal Maso G., “An Introduction to Γ-convergence”, Birkha ̈user, Boston, 1993.
[7] De Giorgi E., Franzoni, T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend.nCl. Sci. Fis. Mat. Natur., 68 (1975), 842–850.
[7] Giusti E., “Direct methods in the calculus of variations”, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
[8] Modical L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal., 98 (1987), 123–142.