Variational methods for materials science, mechanics, and imaging

LeoniDMS-1007989 这个项目的中心主题是现代变分法在材料科学、力学和成像中的应用。 主要研究者强调以下特定主题：-材料空隙弹性固体各向异性表面能量。 在这里，他研究了空洞的平衡形状的定性特征如何取决于界面能的光滑性和凸性（或缺乏）。 - 表面扩散的演化 本文主要研究线弹性固体中第二相粒子和材料空洞模型解的短时存在性和正则性。 - 图像恢复的非本地运算符。 在这里，他研究了吉尔博亚和Osher提出的用于图像和信号处理的非局部模型的解的规律性和定性特征。 他也研究水波和微磁学的问题。 这个项目的中心主题是现代变分法在材料科学、力学和成像中的应用。 当弹性材料受到应力时，它可以拉伸并在材料内形成空隙;空隙的形状影响材料的其他特性。 首席研究员研究了形状如何取决于材料和空隙之间界面处的能量。 当界面能是高度各向异性时，通过表面扩散的界面运动还不是很清楚。研究者研究这个问题，集中在短期行为，严格的结果在这里提高了我们的理解各种行为的材料。 他还考虑了图像处理、水波和微磁学的问题。 该项目的结果和方法被纳入新的研究生课程，有些被纳入一系列的书籍，调查人员正在与我合作编写。丰塞卡研究生参与了这个项目。

LeoniDMS-1007989 The central theme of this project is the application ofmodern calculus of variations to materials science, mechanics,and imaging. The principal investigator emphasizes the followingspecific topics: - Material Voids in Elastic Solids with Anisotropic SurfaceEnergies. Here he studies how the qualitative features of theequilibrium shape of the void depend on the smoothness andconvexity (or lack thereof) of the interfacial energy. - Evolution by Surface Diffusion. The principal investigatorstudies the short-time existence and regularity of solutions of amodel of second-phase particles and material voids in a linearlyelastic solid. - Non-Local Operators for Image Recovery. Here he studiesregularity and qualitative features of solutions of a non-localmodel for image and signal processing proposed by Gilboa andOsher. He also takes up problems in water waves andmicromagnetics. The central theme of this project is the application ofmodern calculus of variations to materials science, mechanics,and imaging. When an elastic material is subject to stresses, itcan stretch and develop voids within the material; the shape ofthe voids affects other properties of the material. Theprincipal investigator studies how the shape depends on theenergy at the interface between the material and the void. Themovement of interfaces by means of surface diffusion is not wellunderstood when the interfacial energy is highly anisotropic. The investigator studies this problem, focusing on the short-timebehavior; rigorous results here improve our understanding ofvarious behaviors of materials. He also considers problems inimage processing, water waves, and micromagnetics. The resultsand methods developed in the project are incorporated in newgraduate courses, and some are included in a series of books thatthe investigator is writing in collaboration with I. Fonseca. Graduate students are involved in the project.