Homogenization of Ginzburg-Landau and Elasticity Problems and Related Questions

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Berlyand0708324 The investigator works on the theoretical development andapplications of homogenization theory. This theory deals withthe properties of heterogeneous materials, which are of criticalimportance for modern technology. Modeling of such materialsraises fundamental mathematical questions, primarily in partialdifferential equations and Calculus of Variations. The projectfocuses on two areas, with homogenization and multiscale analysisas their common themes. Area A. Ginzburg-Landau models: homogenization, well-posedness,and near-boundary vortices. Vortices of the minimizers of theGinzburg-Landau energy functional capture essential features ofsuperconductors and superfluids. They have many common featureswith vortices in fluids, defects in liquid crystals, dislocationsin solids, etc. The investigator studies the homogenization andrise of a special type of near-boundary vortex for theGinzburg-Landau functional in the class of maps with the degree(winding number) prescribed on the boundary of amultiply-connected domain. In this problem, he establishes novellocal minimizers that have near-boundary vortices with boundedenergy. Area B. Homogenization of an elasticity problem with manynonseparated scales and the Cauchy-Born rule. Homogenization(upscaling) in the presence of many nonseparated spatial scalesis far from understood from a mathematical standpoint and itarises in the study of turbulence, soils, biological tissues,etc. The investigator studies such a problem for elasticityequations and constructs an approximate (upscaled) solutionbelonging to a finite-dimensional functional space. Theclassical Cauchy-Born rule is a postulate that allows the passagefrom atomistic to continuum models in monoatomic crystals. Theinvestigator uses the above approximate solution to derive ageneralized Cauchy-Born rule for strongly heterogeneousmaterials. The investigator focuses on the development of noveltechniques of applied analysis to address the needs of moderntechnology. The Ginzburg-Landau equations arose in modelingsuperconductivity but have wider implications. In this project,the investigator's work on this topic has potential applicationsin the design of superconducting materials (which at certaintemperatures conduct electric current with no resistance) andferromagnetic materials. Additionally it addresses theunderstanding of vortex behavior, which is one of the majorchallenges for the science of superconductivity and itstechnological applications. For the second topic, homogenizationof elasticity problems with nonseparated scales, he aims todevelop novel, efficient computational tools suitable formodeling strongly heterogeneous (disordered) materials, bothnatural and man-made. In particular, the issue of elastic"cloaking," when a portion of a geological medium is shieldedfrom elastic waves, is addressed. Advances here could help inthe design of earthquake-proofed buildings.