Variationals Methods in Imaging and in Materials

Fonseca DMS-0905778近年来，由于对偏微分方程、变分法和几何测度论的思想和技术的深入阐述，在对非线性现象的数学严谨理解方面取得了显著进展。本项目要解决的问题的共同特征是处理涉及不同维度的项(体项、表面项等)的能量、大范围的长度和时间尺度、更高阶导数和不连续的下垫场。主题包括：-在成像中，使用RGB和CB模型对受损图像进行重新着色和重建；-在薄膜结构中，了解刚性和脆性特征之间的相互作用；-在单扰动领域内，研究泡沫、共聚熔体的微相分离、多相、多组分多重积分、微磁学的松弛和均化；-在外延应变薄膜中，表面形态和扩散；-在微磁学中，从各向异性能量和交换能量项之间的竞争的小物体模型推导出大物体模型；-在变分中，多尺度理论和降维技术的发展超越了传统的(更高阶)潜在的梯度场，并可能适用于麦克斯韦类型的系统。上面概述的计划是由成像和材料科学的当代问题强烈推动的，成像和材料科学是高端技术进步的核心。这些课程包括损坏图像的重新着色，对薄结构中断裂的了解，对用于采油、洗涤剂和轻质结构材料的泡沫的研究，对外延沉积过程中与重要光学、电子和磁性有关的形态和缺陷的研究，以及对铁电、电磁和磁致伸缩材料和复合材料的行为的预测。基础模型处于传统数学理论的前沿，需要最先进的技术、新的想法和创新的数学工具的引入。有必要通过适当的理论、数值和实验方法相结合的方案，将存在的众多尺度和其他具有数学挑战性的特征联系起来。该项目侧重于这一项目的理论方面，目的是帮助确定具有国家科学重要性的问题，这些问题为将应用分析纳入研究和高级研究生和博士后研究员的教育提供了新的机会。

FonsecaDMS-0905778 In recent years there has been remarkable progress in the mathematical rigorous understanding of nonlinear phenomena, resulting from a deep articulation of ideas and techniques from partial differential equations, calculus of variations, and geometric measure theory. Common features to the problems to be addressed in this project are the treatment of energies that involve terms of different dimensionality (bulk, surface terms, etc.), a large range of length and time scales, higher order derivatives, and discontinuous underlying fields. Topics include: - in imaging, the recolorization and reconstruction of damaged images using the RGB and the CB models; - in thin structures, the understanding of the interplay between rigidity and brittle features; - within the realm of singular perturbations, the study of foams, microphase separation of copolymer melts, relaxation and homogenization of multi-phase, multi-component multiple integrals, micromagnetics; - in epitaxially strained thin films, the surface morphology and diffusion; - in micromagnetics, the derivation of a model for large bodies from the small bodies model that exhibits competition between the anisotropic energy and the exchange energy terms; - in the calculus of variations, the development of multi-scale theories and dimension reduction techniques for systems that go beyond traditional (higher order) underlying gradient fields and may apply to Maxwell-type systems. The program outlined above is strongly motivated by contemporary issues in imaging and materials science at the core of advances in high-end technology. These include recolorization of damaged images, the understanding of fracture in thin structures, the study of foams used in oil recovery, detergents, and lightweight structural materials, the study of morphology and defects in the epitaxial deposition process that are responsible for important optical, electronic, and magnetic properties, and the prediction of the behavior of ferroelectric, electromagnetic, and magnetostrictive materials and composites. The underlying models are at the forefront of traditional mathematical theories, and require state-of-the-art techniques, new ideas, and the introduction of innovative mathematical tools. It is necessary to bridge the multitude of scales present, and other mathematically challenging features, by appropriate schemes of articulated theoretical, numerical, and experimental approaches. This project is focused on the theoretical side of this venture, with the aim of 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.