Homogenization of Ginzburg-Landau and Elasticity Problems and Related Questions

Berlyand0708324 研究者致力于均匀化理论的理论发展和应用。 这一理论涉及非均匀材料的性质，这些性质对现代技术至关重要。 这种材料的建模提出了基本的数学问题，主要是在偏微分方程和变分法。 该项目侧重于两个领域，均质化和多尺度分析是其共同的主题。 A区。 Ginzburg-Landau模型：均匀化，适定性和近边界涡。 Ginzburg-Landau能量泛函极小子的涡旋捕捉了超导体和超流体的本质特征。 它们与流体中的涡旋、液晶中的缺陷、固体中的位错等有许多共同的特征.本文研究了一类特殊的近边界涡旋在多连通区域边界上具有规定的度（缠绕数）的映射类中的Ginzburg-Landau泛函的均匀化过程. 在这个问题中，他建立了一个新的局部极小值，它具有带边界能量的近边界涡。 区域B。 多尺度弹性力学问题的均匀化和Cauchy-Born规则。 均匀化（upscaling）的存在下，许多nonseparated空间scalesis远离理解从数学的角度来看，它出现在湍流，土壤，生物组织等研究的调查研究这样一个问题的elasticityequations和构造一个近似（upscaled）solutionbelonging到一个有限维的功能空间。 经典柯西-玻恩规则是一个假设，它允许单原子晶体从原子模型过渡到连续模型。 研究者利用上述近似解导出了强非均匀材料的广义Cauchy-Born规则。 研究者专注于应用分析的新技术的发展，以满足现代技术的需求。 金斯堡-朗道方程起源于超导模型，但具有更广泛的含义。 在这个项目中，研究人员在这一主题上的工作在超导材料（在一定温度下无电阻地传导电流）和铁磁材料的设计中具有潜在的应用。 此外，它解决了理解涡旋行为，这是超导科学及其技术应用的主要挑战之一。 对于第二个主题，均匀化的弹性问题与非分离尺度，他的目标是开发新的，有效的计算工具，适用于建模强烈异质（无序）材料，bothnatural和人造。 特别是，弹性的问题“隐身”，当一部分的地质介质是shieldedfrom弹性波，是解决。 这方面的进展有助于抗震建筑的设计。

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.