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Dr.Sc., University of Paris
My research concerns the development of mathematical tools for studying the oscillating solutions of the nonlinear partial differential equations of continuum mechanics. For physical phenomena described at a microscopic level you need to understand what equations should be used at a macroscopic level, and the mathematical model used is based on different notions of weak convergence.
One aspect, called homogenization, is related to effective properties of mixtures. One problem is to obtain sharp bounds for effective coefficients while another is to understand how waves are damped in composite materials.
A second aspect is related to the phenomenon of relaxation in nonconvex optimization problems.
A third aspect concerns hyperbolic conservation laws and admissibility conditions for shocks, as well as propagation and interaction of oscillations for semilinear hyperbolic systems.
A fourth aspect concerns the adequation of these methods for questions like fluid turbulence, quantum mechanics and statistical mechanics.