Variational Problems and their Applications

本建议在变分法、偏微分方程和几何测量理论等领域中，探讨多尺度应用分析的几个方面。这项工作将涉及松弛和下半连续性结果，高阶理论的发展，长度和时间尺度的多尺度均匀化以及降维问题。研究者和她的合作者将研究的问题包括：-在微磁学中，从展示各向异性能量和交换能量项之间竞争的小体模型推导出大体模型，多磁性不同颗粒材料的优化设计和多尺度均匀化；-在薄膜中，随试样厚度渐近能量的识别消失，这取决于宏观变形和弯矩，以及由缩放薄膜梯度产生的杨氏测度的表征。将研究薄膜在衬底上异质外延生长（湿式和干式）的不稳定性，特别是零接触角岛的形成；-共聚物熔体的微相分离和通过γ -收敛技术识别相邻单体对之间的界面能；-在Monge-Kantorovich设置下，当网络长度趋于无穷时，最优城市网络的渐近行为；——表面活性剂影响下泡沫的稳定性，以及表面活性剂在高原边界成核中的作用。上述研究计划的强烈动机是需要发展创新的应用数学，以应对高端技术的挑战。新颖的人造材料经常在传统数学理论的前沿展示潜在的模型。这些包括形状记忆合金、铁电、电磁和磁致伸缩材料、复合材料、液晶、用于采油的泡沫、洗涤剂和轻质结构材料以及薄膜。通常存在大范围的长度和时间尺度，多尺度方面产生了从块状材料到纳米结构的应用。为了理解并能够预测这些材料的行为，并最终将这些知识应用于工业和技术，有必要通过适当的理论、数值和实验方法来连接这众多的尺度。该提案侧重于该项目的理论方面，旨在帮助确定具有国家科学重要性的问题，为将应用分析整合到研究和高级研究生和博士后的教育中提供新的机会。

This proposal addresses several aspects of multiscale applied analysis within the areas of the Calculus of Variations, Partial Differential Equations, and Geometric Measure Theory. The work will entail relaxation and lower-semicontinuity results, development of higher order theories, multiple scale homogenization of length and time scales, and dimension reduction problems. The questions to be studied by the investigator and her collaborators include: - in micromagnetics, the derivation of a model for large bodies from the small bodies model which exhibits competition between the anisotropic energy and the exchange energy terms, optimal design and multiscale homogenization of poly-magnetic, different grains materials; - in thin films, the identification of the asymptotic energy as the thickness of the sample vanishes as depending on the macroscopic deformation and on the bending moment, and the characterization of Young measures generated by scaled thin film gradients. Instabilities in the heteroepitaxial growth of thin films over substrates (wet and dry wetting), and in particular the formation of islands with zero contact angle, will be studied; - the microphase separation of copolymer melts and the identification via Gamma-convergence techniques of the interfacial energy between pairs of adjacent monomers; - the asymptotic behavior of optimal urban networks as the length of the network goes to infinity, within the Monge-Kantorovich setting; - the stability of foams under the influence of surfactants, and the role of surfactants in the nucleation of Plateau borders. The research program outlined above is strongly motivated by the need to develop innovative applied mathematics capable to respond to challenges in high-end technology. Novel man-made materials often exhibit underlying models at the forefront of traditional mathematical theories. These include shape memory alloys, ferroelectric, electromagnetic and magnetostrictive materials, composites, liquid crystals, foams used in oil recovery, detergents and lightweight structural materials, and thin films. A large range of length and time scales is usually present, and multiscale aspects yield applications from bulk materials to nanostructures. In order to understand and be able to predict the behavior of such materials, and to ultimately put this knowledge to use in industry and technology, it is necessary to bridge this multitude of scales by appropriate schemes of articulated theoretical, numerical, and experimental approaches. This proposal is focused on the theoretical side of this venture, with the aim to contributing to the identification of problems of national scientific importance that offer new opportunities for the integration of applied analysis in research and in the education of advanced graduate students and postdoctoral fellows.