Textbook:

Gamma Convergence for Beginners, Andrea Braides (on reserve in Math. Library)

A handbook of Gamma-convergence, Andrea Braides.

Topics on Concentration Phenomena and Problems with Multiple Scales, Andrea Braides, Valeria Chiado Piat (eds.) (available online, from Purdue)

Course Announcement:

Extra information about the course will be posted graduatlly.

Introductory Chapter (Gamma Convergence for Beginners, Braides)

Weal Convergence:
Weak Convergence (excerpts from Chapter One of Weak Convergence Methods for Nonlinear Partial Differential Equations, Evans)
Lower Semicontinuity and Weak Convergence (Chapters One and Two of Homogenization of Multiple Integrals, Braides and Defranceshi)

Convexity, quasiconvexity and lower semi-continuity:
Chapters Two and Three, Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations
Chapters Four-Six, Braides-Defranceschi, Homogenization of Multiple Integrals
Acerbi-Fusco, Lower Semicontinuity (paper)
Dacorogna, Direct Methods in the Calculus of Variations (available online from Purdue library page)
Braides, Beginner's Guide, Chapter 12
Fonseca, Variational Methods in Materials Science (paper)

Homogenization:
Muller, Homogenization and Calculus of Variations (paper)
E, Homogenization and Calculus of Variations (paper)
Braides, Homogenization of Almost Periodic Functionals (paper)
Braides, Beginner's Guide, Chapter 16, Method by Localization

Homogenization in Perforated Domains:
Cioranescu-Murat, Strange Term from Nowhere (paper)
Braides, Beginner's Guide, Chapter 13
Braides, Handbook, Section 6
Ansini-Braides, Asymptotic Analysis of Periodic Perforated Media, (paper)
Ansini-Braides, Separation of Scales and Perforated Domain (paper)

Segmentation and Free Discontinuity Problems:
Evans-Gariepy, BV Funtions, Chapter Five of Measure Theory and Fine Properties of Functions
Braides, Approximation and Free-Discontinuity Problems, Lecture Notes in Mathematics, #1694 (available online through Purdue Library page);
Defranceschi, Relaxation for Bulk and Interfacal Energies, Chapter from Topics on Concentration Phenomena and Problems with Multiple Scales (available online through Purdue Library page).

Final Presentations:
Please prepare a 30-40 minutes presentation for the final exam. The following topics are just some suggestions. Feel free to find your own (relevant) presentation topics. Please send me your TITLE and ABSTRACT (by e-mail) by Apr 15. The topics are on a first-come-first-serve basis (though for longer topics, a joint presentation is possible). Your presentation must include a proof of some statement(s). Computer slides are not required but you are encouraged to use at least some transparencies for convenience so as to include more information and also speed up the process.

The presentations will be held during the last week of class and also the final exam week. A sign-up sheet for time slots will be posted soon. I intend the presentations to be open to anyone from the class. But attendance is voluntary.

Some suggested topics for presentation.

[G] = Beginner's Guide (on reserve in library)
[M] = Homogenization of Multiple Integrals
[C] = Topics on Concentration Phenomena and Problems with Multiple Scales (available online through Purdue library page)

Homogenization

(TAKEN) [G] Chapter Two (Integral Problems): Theorem 2.20 (L^p), Theorem 2.35 (W^(1,p)), Appendix B

(TAKEN) [G] 3.3 Homogenization Limits of Riemannian Metrics

[G] 3.4 Homogenization of Hamilton-Jacobi Equations
(see also: Evans, PDEs, Chapter Ten, Hamilton-Jacobi Equations
and E Homogenization and Calculus of Variations (paper));

(TAKEN) [M] Chapters Eleven and Twelve (Fundamental Estimates and Its Consequence)

(TAKEN) [M] Chapter Twenty-Two Iterated Homogenization

Braides-Maslennikov-Sigalotti: Homogenization by Blow-Up (paper);

Braides-Ansini: Separation of Scales and Almost-Periodic Effects in Perorated Media (paper)

From Discrete to Continuum

[G] Chapter Four (From Discrete to Integral Functionals)

[G] Chapter Eleven (From Discrete to Free-Discontinuity Problems)

[C] From Discrete System to Continuous Variational Problems: An Introduction

Iosifescu-Licht-Michaille: Variational Limit of a One-Dim Discrete and Statistically Homogeneous System (long paper and summary);

Dimensional Reduction

[G] Chapter Fourteen (Dimension Reduction Problems)

Braides-Ansini: Homogenization of Oscillating Bounbaries and Applications to Thin Films (paper);

Braides-Fonseca-Francfort: 3D-2D Asymptotic Analysis for Inhomogeneous Thin Films (paper);

Bhattacharya-Braides: Thin Films with Many Small Cracks (paper);

Interaction between Homogenizationd and Phase Transitions

[G] Chapter Nine: interaction between homogenization and phase transitions

Braides-Zeppieri: Multiscale Analysis of Interaction between Microstructure and Surface Energy (paper);

Ansini-Braides-Chiado Piat, a survey conference paper, proof in one-dimensional case;

Ansini-Braides-Chiado Piat, higher dimensional case.

Gamma Convergence and Critical Points (other than minimizers)

Kohn-Sternberg: Local Minimizers and Singular Perturbations (paper);

Jerrard-Sternberg: Gamma Convergence and Critical Points (paper);

Gamma Convergence and Gradient Flows

[C] Gamma-Convergence of Gradient Flows and Applications to Ginzburg-Landau Vortex Dynamics

Sandier-Serfaty: Gamma Convergence and Gradient Flows (paper);

Serfaty, a survey paper, (paper);

Miscellaneous Topics and Applications

Fonseca-Mantegazza: Second Order Singular Perturbation Models for Phase Transitions (paper);

Braides-Causin-Solci: Interfacial Energies on Quasi-crystals (paper);

Ghisi-Gobbino: The Monopolist's Problem: Existence, Relaxation and Approximation (paper);

Truskinovsky: Fracture as a Phase Transition (paper);

Del Piero-Truskinovsky: Macro- and Micro-Cracking in One Dimensional Elasticity (paper);

[G] Chapter Ten (Interaction between Elliptic and Segmentation Problems)

[C] Relaxation for Bulk and Interfacial Energies

[C] Gamma-Convergence for Concentrated Problems

[C] PDE Analysis of Concentrating Energies for Ginzburg-Landau Equation


Final Presentation Schedule:
(Attendance is voluntary but highly encouraged.)

Apr 23 (Tue):
10-11, MATH 431, Tom, Capacity and Sobolev Functions
12-1, REC 123 (regular class meeting room), Agnid, Iterated Homogenization

Apr 25 (Thur):
10-11, MATH 431, Wenhui, Fundamental Estimates and Their Consequences
12-1, REC 123 (regular class meeting room), Mariana, Gamma Convergence of Integral Functionals

Apr 29 (Mon), all in REC 121
9-10, Liang, Gamma Convergence and Gradient Flows - I
10-11, Drew, Gamma Convergence and Gradient Flows - II
12-1, Bumsik, Hamilton-Jacobi Equations - Homogenization

Apr 30 (Tue), all in REC 121
9-10, Kevin, Homogenization of Periodic Riemannian Metrics on R^n
10-11, Lidia, Gamma Convergence and Gradient Flows - III
12-1, Heejun, Hamilton-Jacobi Equations - Viscosity Solutions and Dynamic Programming