Homogenization of Ginzburg-Landau and Elasticity Problems and Related Questions

Berlyand0708324研究员致力于均匀化理论的理论发展和应用。这一理论涉及对现代技术至关重要的非均质材料的性质。对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式和变分中。该项目侧重于两个领域，同质化和多尺度分析是它们的共同主题。A区金兹堡-朗道模式：均化、适定性和近界面涡。Ginzburg-Landau能量泛函极小器的涡旋捕捉了超导体和超流体的基本特征。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。研究人员研究了在多连通区域边界上给定次数(缠绕数)的映射类中Ginzburg-Landau泛函的一种特殊类型的近边界涡旋的均匀化和上升。在这个问题中，他建立了具有能量有界的近边界涡旋的新的局部极小化方法。区域B具有多个不可分离尺度的弹性问题的齐次化和柯西-伯恩法则。在许多不分离的空间尺度下的齐次化(升标)远远不能从数学的角度来理解，它出现在湍流、土壤、生物组织等的研究中。研究者研究了弹性方程的这样一个问题，并构造了一个属于有限维函数空间的近似(升标)解。经典的柯西-伯恩规则是一个假设，它允许在单原子晶体中从原子模型到连续模型。研究人员利用上述近似解导出了强非均质材料的广义柯西-伯恩定律。研究人员专注于应用分析新技术的发展，以满足现代技术的需要。金兹堡-朗道方程是在超导模型中出现的，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(在一定温度下传导电流而没有电阻)和铁磁材料的设计中具有潜在的应用价值。此外，它还解决了对涡旋行为的理解，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，具有连续尺度的弹性问题的均匀化，他的目标是开发新的、高效的计算工具，适用于模拟强非均质(无序)材料，包括天然和人造材料。特别是，讨论了弹性“隐身”问题，即当地质介质的一部分被弹性波屏蔽时。这方面的进步可能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.