"A user-friendly introduction to Γ-convergence"

Abstract: The Γ-convergence, introduced by the De Giorgi in the mid '70s, is nowadays the commonly- recognized notion of convergence for variational problems. The aim of this course is to give a basic introduction to the topic and to discuss its application to the variational approximation of some problems in Materials Science in their simplied one-dimensional versions.

Course requirements: Basic notions of measure theory, topology and metric spaces, Lebesgue and Sobolev spaces.

Aim of the course: To provide a simplied introduction to Γ-convergence and its many applications through one-dimensional toy-models.

Period: 10 two-hours lectures in the period October-December 2024, Department of Mathematics and Applications "R. Caccioppoli"

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Program of the course:

1. Some basic facts on Γ-convergence (Lectures 1-5: 03/10/2024 - 07/10/2024 - 14/10/2024 - 21/10/2024 - 28/10/2024). Introduction and motivation. The definitions of Γ-convergence. Convergence of minima. Upper and lower Γ-limits. Lower semicontinuity of Γ-limits, the Direct Method in the Calculus of Variations. Some properties of Γ-limits: Γ-limits of monotone sequences, compactness of Γ-convergence, Γ-convergence by subsequences, Γ-limits indexed by a continuous parameter. Development by Γ-convergence: the Anzellotti-Baldo's approach. 

2. Integral problems on Lebesgue (and Sobolev) spaces (Lecture 6: 07/11/2024). Some examples. Weak convergence, weak lower semicontinuity. Convex and lower semicontinuous envelopes. Relaxation and Γ-convergence on Lebesgue and Sobolev spaces.

3. Gradient theory of phase transitions in dimension one (Lectures 7-8: 14/11/2024 - 21/11/2024). The Van der Waals-Cahn-Hilliard model of liquid-liquid phase transitions. Gradient theory for phase-transition problems: the Modica-Mortola functional. The space of piecewise-constant functions. Gradient theory as a development by Γ-convergence.

4. Approximation of free-discontinuity problems in dimension one (Lecture 9: 28/11/2024). Free-discontinuity problems: the Mumford-Shah functional. SBV functions of one variable and piecewise-Sobolev functions. Approximation of free-discontinuity problems: the Ambrosio Tortorelli approximation.

5. An insight into the N-dimensional setting: the slicing method (Lecture 10: 05/12/2024). The slicing method. Application to the Modica-Mortola functional: a lower inequality by the slicing method, an upper inequality by density.

References

[1] A. Braides, Approximation of Free-Discontinuity Problems. Springer, London (1998)

[2] A. Braides, Γ-convergence for Beginners. Oxford University Press, Oxford (2002).

[3] G. Dal Maso, An introduction to Γ-convergence. Birkhäuser, Boston Basel Berlin 1993.