Carmen Perugia

The research activity relies in the field of Functional Analysis. More specifically, it concerns the study of the asymptotic behavior of some operators and/or partial differential equations with applications to problems concerning fine structures, composite materials and transport. This study is widely based on the theory of homogenization. The precise aim of this theory is the study of the behavior of non-homogeneous materials whose physical parameters, such as conductivity and coefficient of elasticity, oscillate between different values. The study covers two levels:

- the microscopic level describing the heterogeneities, usually small compared to the global dimensions,

- the macroscopic level which describes the overall behavior of the composite.

Indicated by epsilon, the parameter representing the fineness of the microscopic structure, a good approximation of the macroscopic behavior of this material is obtained by setting epsilon to zero in the equations describing physical phenomena such as heat conduction, elasticity, etc. This convergence analysis is the natural mathematical translation of the fundamental problem of homogenization consisting in identifying a homogeneous material whose behavior is similar to that of the non-homogeneous material. This asymptotic analysis can be performed using different methods depending on the characteristics of the problem. The best known are:

- the multiscale method,

- the method of energies, also called method of Tartar,

- the Gamma-convergence method,

- the unfolding method which is the most recent.

At the same time, I leads a research in collaboration with some geologists of my department regarding the spectral analysis applied to the study of paleoclimatic data.