Modern methods in the Calculus of Variations with applications to materials science and hydrodynamics

这个项目的中心主题是现代变分在材料科学和流体力学中的应用。研究人员承担了以下具体项目：-异质外延薄膜。研究人员从变分的角度研究了在较厚的衬底上外延沉积薄膜的数学模型，这是由于两种晶体的晶格参数不匹配造成的。-应变梯度塑性。研究人员从平面区域中有限数目位错平衡问题的变分形式出发，以极其严格的方式导出了塑性应变梯度理论。--水波。对于规则水波和最高高度的斯托克斯波的存在，采用了另一种方法。其他项目包括图像分割问题和微磁学。这个项目的中心主题是将现代数学技术应用于材料科学、力学和流体力学中的各种问题。当非晶固体薄膜沉积在衬底上时，如果薄膜和衬底的晶格结构不匹配，就会出现困难。调查员研究了这个问题。材料中的一个基本问题是材料微观结构中的位错对更大尺度行为的影响。经典塑性理论没有很好地解决这个问题。研究人员发展了一种应变梯度塑性理论，该理论包含了位错及其相互作用。他还考虑了水波、图像处理和微磁学方面的问题。该项目开发的成果和方法将纳入新的研究生课程，其中一些将纳入调查员与I.Fonseca合作编写的一系列书籍中，

Leoni0708039 The central theme of this project is the application ofmodern Calculus of Variations to materials science andhydrodynamics. The investigator undertakes the followingspecific projects: -Heteroepitaxial thin films. The investigator studies from thevariational point of view a mathematical model for the epitaxialdeposition of a film onto a relatively thick substrate in thecase of a mismatch between the lattice parameters of the twocrystalline solids. -Strain gradient plasticity. The investigator derives, in amathematically rigorous way, a strain gradient theory ofplasticity starting from a variational formulation of theequilibrium problem for a finite number of dislocations in aplane domain. -Water waves. An alternative approach for the existence ofregular water waves and of Stokes wave of greatest height isundertaken. Other projects include image segmentation problems andmicromagnetics. The central theme of this project is the application ofmodern techniques in mathematics to various problems in materialsscience, mechanics, and hydrodynamics. When a thin film of acrystalline solid is deposited on a substrate, a commonindustrial process, difficulties can arise if the crystal latticestructures of the film and the substrate do not match. Theinvestigator studies this problem. A fundamental question inmaterials is the effect of dislocations in the materialmicrostructure on larger-scale behavior. Classical plasticitytheory does not address this issue well. The investigatordevelops a theory of strain gradient plasticity that incorporatesdislocations and their interactions. He also considers problemsin water waves, image processing, and micromagnetics. Theresults and methods developed in the project will be incorporatedin new graduate courses, and some will be included in a series ofbooks that the investigator is writing in collaboration with I.Fonseca,