The central theme of this project lies in the area of nonlinear analysis (nonlinear partial differential equations, calculus of variations, geometric measure theory). It enters the general effort to bring the analysis where problems of interest to other scientists (from continuum mechanics, or physics) become mathematically accessible. Mathematical models for material instabilities, phase transitions, plasticity, fracture, or micromagnetics are such examples of equations whose analysis is the source of challenging issues for mathematicians. In particular, recent works have been devoted to the justification of effective theories (asymptotic analysis), and to the study of existence, regularity and singular behavior of solutions to nonconvex, nonlocal problems (etc.). Among this very large field, we will focus on multiscale structures in nonlinear partial differential equations. These multiscale structures may have different origins. They may appear through explicit small parameters in the model (as it is the case in homogenization or in models for micromagnetics). They may also be more subtle, and appear as the result of a competition between two terms in an energy functional (as it is the case for the bulk and surface energies in models for phase transitions). Suitable mathematical tools are usually compactness methods, Young measures, Gamma-convergence, viscosity solutions, gradient flows, etc. Some of these mathematical tools have already given very deep insights, for instance in phase transitions or micromagnetics, and still need to be further developed. Our scientific program can be split into three major parts:

Nonstandard Homogenization: First of all, we wish to study nonstandard homogenization problems, namely problems with nonstandard spatial structure assumptions (beyond the periodic setting) as well as problems with nonstandard operator structure (beyond the continuous setting). Our main goals are to study and apply the concept of homogenization structure introduced by Nguetseng, and to address the homogenization of some discrete systems in a time-dependent setting with PDE tools.

Micromagnetics: Micromagnetics, a continuum model for the behavior of ferromagnetic body essentially developed by Brown, may be considered a paradigm of multiscale problems. Indeed, ferromagnetic materials exhibit complex microstructures such as magnetic domains, domain walls (Néel walls, Bloch walls, cross-tie walls) or vortices (Bloch-lines). We plan to study variational models used to predict the morphology of a given specimen at different scales.

Defect Mechanics: We attend to analyze the evolution of damage, cracks and dislocations in various settings as quasi-static evolution or minimizing movements. Moreover we wish to stress relationships between these evolutions and other stationary concepts as damage/homogenization, fracture/Mumford-Shah, dislocations/Ginzburg-Landau vortices.

1. B. Galvao-Sousa, V. Millot: A two-gradient approach for phase transitions in thin films, submitted.
2. J.-F Babadjian: A quasistatic evolution model for the interaction between fracture and damage, Arch. Rational Mech. Anal., 200, no. 3, (2011), 945-1002.
3. R. Ignat, H. Knupfer: Vortex energy and 360° Néel walls in thin films micromagnetics, Comm. Pure Appl. Math., 63 (2010), 1677-1724.
4. N. Forcadel, C. Imbert, R. Monneau: Homogenization of overdamped Frenkel-Kontorova models with n types of particles, submitted.
5. N. Forcadel, C. Imbert, R. Monneau: Homogenization of the fully overdamped Frenkel-Kontorova models, J. Diff. Eq., 246, (2009), 1057-1097.
6. N. Forcadel, C. Imbert, R. Monneau: Homogenization of some particle systems with two-body interactions and of the dislocation dynamics, Discrete Cont. Dyn. Syst. A, 23(3), (2009), 785-826.
7. N. Forcadel, A. Monteillet: Minimizing movements for dislocation dynamics with a mean curvature term, ESAIM COCV, 15(1), (2009), 214-244.
8. A. Gloria, F. Otto: An optimal variance estimate in stochastic homogenization of discrete elliptic equations, Ann. of Probab., 39, no. 3, (2011), 779-856.
9. R. Alicandro, M. Cicalese, A. Gloria: Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity, Arch. Rational Mech. Anal., 200, no. 3, (2011), 881-943.
10. A. Gloria, F. Otto: An optimal error estimate in stochastic homogenization of discrete elliptic equations, to appear in Ann. Appl. Probab.
11. M. Barchiesi, A. Gloria: New counterexamples to the cell formula in nonconvex homogenization, Arch. Rational Mech. Anal., 195, no. 3, (2010), 991-1024.
12. A. Gloria, S. Neukamm: Commutability of homogenization and linearization at identity in finite elasticity and applications, to appear in Annales IHP.
13. J.-F Babadjian, G.A. Francfort, M.G. Mora: Quasistatic evolution in non-associative plasticity - the cap model, to appear in SIAM J. Math. Anal.
14. N. Fusco, V. Millot, M. Morini: Quantitative isoperimetric inequality for fractional perimeters, to appear in J. Funct. Anal.
15. I. Fonseca, N. Fusco, G. Leoni, V. Millot: Material voids in elastic solids with anisotropic surface energies, to appear in J. Math. Pures Appl.
16. J.-F Babadjian, A. Giacomini: Existence of strong solutions for quasi-static evolution in brittle fracture, submitted.
17. R. Ignat, R. Moser: A zigzag pattern in micromagnetics, accepted to J. Math. Pures Appl.
18. R. Ignat: Two-dimensional unit-length vector fields of vanishing divergence, J. Funct. Anal. 262 (2012), 3465-3494