The investigator works on the theoretical development and applications of homogenization theory. This theory deals with the properties of heterogeneous materials, which are of critical importance for modern technology. Modeling of such materials raises fundamental mathematical questions, primarily in partial differential equations and Calculus of Variations. The project focuses on two areas, with homogenization and multiscale analysis as their common themes. Area A. Ginzburg-Landau models: homogenization, well-posedness, and near-boundary vortices. Vortices of the minimizers of the Ginzburg-Landau energy functional capture essential features of superconductors and superfluids. They have many common features with vortices in fluids, defects in liquid crystals, dislocations in solids, etc. The investigator studies the homogenization and rise of a special type of near-boundary vortex for the Ginzburg-Landau functional in the class of maps with the degree (winding number) prescribed on the boundary of a multiply-connected domain. In this problem, he establishes novel local minimizers that have near-boundary vortices with bounded energy. Area B. Homogenization of an elasticity problem with many nonseparated scales and the Cauchy-Born rule. Homogenization (upscaling) in the presence of many nonseparated spatial scales is far from understood from a mathematical standpoint and it arises in the study of turbulence, soils, biological tissues, etc. The investigator studies such a problem for elasticity equations and constructs an approximate (upscaled) solution belonging to a finite-dimensional functional space. The classical Cauchy-Born rule is a postulate that allows the passage from atomistic to continuum models in monoatomic crystals. The investigator uses the above approximate solution to derive a generalized Cauchy-Born rule for strongly heterogeneous materials.
The investigator focuses on the development of novel techniques of applied analysis to address the needs of modern technology. The Ginzburg-Landau equations arose in modeling superconductivity but have wider implications. In this project, the investigator's work on this topic has potential applications in the design of superconducting materials (which at certain temperatures conduct electric current with no resistance) and ferromagnetic materials. Additionally it addresses the understanding of vortex behavior, which is one of the major challenges for the science of superconductivity and its technological applications. For the second topic, homogenization of elasticity problems with nonseparated scales, he aims to develop novel, efficient computational tools suitable for modeling strongly heterogeneous (disordered) materials, both natural and man-made. In particular, the issue of elastic "cloaking," when a portion of a geological medium is shielded from elastic waves, is addressed. Advances here could help in the design of earthquake-proofed buildings.
