The project is concerned with the theoretical development and applications of homogenization theory. This theory studies the properties of heterogeneous materials (primarily composites), which are of critical importance for modern technology. Fundamental questions are raised in the process of modeling such materials -- primarily in the fields of PDEs and Calculus of Variations. The work addresses special classes of homogenization problems where, in addition to the homogenization parameter, another small material parameter (e.g., the inverse Ginzburg-Landau parameter) is present. The results and techniques developed for two-parameter homogenization problems essentially depend on the relation between these parameters, and several such relations are considered. These problems are studied in the context of Ginzburg-Landau models for superconducting composites. Specifically, the investigator and his collaborators study pinning by holes in small superconducting samples. Here the main mathematical issue is the homogenization limit in a nonstandard discrete/continuum nonlinear finite-dimensional variational problem with integer constraints for the unknown family of degrees of vortices. The investigator and his collaborators develop Gamma convergence techniques for such problems. It is expected that the homogenization vorticity drastically depends on the relation between the geometrical parameters (size of the holes and distances between them) and the material parameter. In particular, the investigator and his collaborators look for special scaling relations, which result in vortices with multiple degrees. Also, the investigator and his collaborators develop novel techniques of Gamma-convergence for the study of nonlinear random homogenization problems. Applications of these techniques for material science problems are considered. One of the key issues is the identification of new effects due to randomness when compared to periodically located inclusions.
The project is primarily motivated by the quest for energy-efficient materials that comprise a foundation for a new generation of superconductivity-based microelectronics. It utilizes Ginzburg-Landau models of superconductivity, describing their ability to carry electric current without power loss. In practical applications of superconductivity the passing current creates magnetic vortices that move, dissipate the energy, and destroy the superconducting state. Thus the central problem for practical application is the problem of immobilization of vortices, known among specialists as the problem of vortex pinning. The approach studied here offers an efficient tool to tackle this problem and to develop practical recipes for an artificial manufacturing pinning configuration in superconducting wires to crucially improve their performance. The project has a direct impact on the education and future careers of graduate students. They receive an interdisciplinary training through interactions with the investigator's collaborators at Argonne National Laboratory. Additionally, the investigator continues involving undergraduate students in his NSF-supported research projects.