Homogenization of Ginzburg-Landau and Elasticity Problems and Related Questions

Berlyand

0708324

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.