Harmonic Analysis and PDE

调和分析和偏微分方程

• 批准号: 0600971
• 负责人: Sagun Chanillo
• 金额: $ 11万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600971&HistoricalAwards=false
• 关键词: Harmonic; Analysis; PDE

Harmonic Analysis and P D E Abstract of Proposed ResearchSagun Chanillo This grant is to support research on some problems involving partial differential equations and topics in geometry, physics and material science. One class of problems concerns how to design a body of prescribed shape and mass, with certain bounds on the density, that has least first eigenvalue. This is related to certain models of composite materials. In earlier work we have proved certain properties of the solutions of this problem and I will continue studies on the properties of the interfaces between the different materials in the solution. These are free boundary problems and the existing regularity theory for such problems does not extend to these systems. Jointly with Carlos Kenig, we intend to study some examples and analyze these models in considerable detail. A second problem we are studying is the Plateau problem for constant mean curvature surfaces. This is a joint project with A. Malchiodi and continuation of our earlier work where the Dirichlet situation was analysed. Now we are trying to understand the conformal situation.A clear understanding of the composite membrane problem will lead to an understanding and design of useful composite materials with important properties for use in areas such as medicine, aeronautics and wherever light-weight and strong materials are to be used. Our earlier work showed that one sometimes has to build these materials by breaking the natural symmetry of the desired body, for example to build the ubiquitous washer one needs to arrange the composite materials non-symmetrically. The loss of symmetry has a direct bearing on the regularity of the interface of the materials, the free boundary so to say and this will be hopefully revealed in the current research investigation.

谐波分析和P D E摘要建议ResearchSagun Chanillo该补助金是支持研究涉及偏微分方程和几何，物理和材料科学的一些问题。一类问题涉及如何设计具有规定形状和质量、密度有一定界限且具有最小第一本征值的物体。这与复合材料的某些模型有关。在早期的工作中，我们已经证明了这个问题的解决方案的某些属性，我将继续研究解决方案中不同材料之间的界面的属性。这些都是自由边界问题和现有的正则性理论，这样的问题并不延伸到这些系统。与卡洛斯凯尼格，我们打算研究一些例子，并分析这些模型相当详细。 我们正在研究的第二个问题是常平均曲率曲面的高原问题。这是一个与A的合作项目。Malchiodi和继续我们以前的工作，狄利克雷的情况进行了分析。现在我们正试图了解保形的情况。对复合膜问题的清楚理解将导致对具有重要性能的有用复合材料的理解和设计，这些材料用于医学、航空等领域以及任何需要使用轻质和坚固材料的领域。我们早期的工作表明，人们有时必须通过打破所需物体的自然对称性来构建这些材料，例如，为了构建无处不在的垫圈，人们需要非对称地布置复合材料。对称性的丧失直接影响到材料界面的规则性，也就是自由边界，这一点有望在目前的研究中得到揭示。