Materials Instabilities: Regularity Theory and the Calculus of Variations

这个项目的总体目标是发展变分和偏微分方程组的正则性理论的技术。指导这项工作的应用主要来自液体和固体的连续介质力学领域。几何测度论是一种研究曲面泛化的理论，它的工具将发挥重要作用。这一分析将针对与时间相关的问题，包括在某些情况下忽略惯性效应的某些理想化问题，主要集中在寻找复杂成分材料的有效描述，研究体解和界面解的正则性，以及预测它们随着时间的演变而出现的振荡或滞后行为。这项研究将对材料科学中发生的一系列现象进行数学描述和预测。这包括(液体和固体之间的)相变、这种相的结构、形核和生长，以及固体中的断裂和缺陷。这项工作的一个重要部分将是研究新型人造材料的数学模型，如形状记忆合金、铁电、磁性和磁致伸缩材料、复合材料、液晶和薄膜。虽然出乎意料，但数学理论也为计算机视觉中的图像分割提供了新的方法。数学上的挑战在于描述微观结构的动力学和演化，以及在非常不同的时间或空间尺度上发生的现象。它们需要最近开发的数学工具和引入新的数学技术。

The general objective of this project is the development of techniques in the calculus of variations and in the regularity theory for systems of partial differential equations. The applications that guide this work come mainly fromthe areas of continuum mechanics of liquids and solids. Tools from geometric measure theory, a theory for the study of generalizations of surfaces, will play animportant role. The analysis will be undertaken for time-dependent problems, including in some cases certain idealizations where inertial effects are neglected.The main focus will be on the search for the effective description of materials with complicated composition, on the study of regularity properties of solutions in the bulk and at interfaces, and on the prediction of their oscillatory or hysteretic behavior as time evolves. The research will be on the mathematical description and prediction of a range ofphenomena that occur in material science. This includes phase transformations (between liquid and solid phases), the structure, nucleation and growth of such phases, and fracture and defects in solids. An important part of this work willbe the study of mathematical models for novel man-made materials such as shape memory alloys, ferroelectric, magnetic and magnetostrictive materials, composites, liquid crystals, and thin films. Although unexpected, the mathematicaltheory also provides new methods for image segmentation in computer vision. The mathematical challenges lie in the description of the dynamics and evolution of microscopic structures and of phenomena that occur at vastly different temporal or spatialscales. They require recently developed mathematical tools and the introduction of new mathematical techniques.