Harmonic Analysis and PDE

调和分析和拟研究的P D E摘要Sagun Chanillo这笔拨款是为了支持一些涉及偏微分方程的问题的研究，以及几何、物理和材料科学的主题。其中一类问题涉及如何设计一个具有指定形状和质量的物体，并且在密度上有一定的界，并且具有最小的第一本征值。这与某些型号的复合材料有关。在前面的工作中，我们已经证明了这个问题解的某些性质，我将继续研究溶液中不同材料之间的界面性质。这些都是自由边界问题，现有的关于这类问题的正则性理论不能推广到这些系统。我们打算与卡洛斯·凯尼格一起研究一些例子，并对这些模型进行相当详细的分析。我们正在研究的第二个问题是常平均曲率曲面的平台问题。这是与A.Malchiodi的一个联合项目，是我们早先分析狄里克莱特情况的工作的继续。现在我们正试图了解这种共形情况。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，这些材料可用于医学、航空以及任何需要使用轻质和高强度材料的领域。我们早期的工作表明，有时人们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，要构建无处不在的垫圈，人们需要非对称地排列复合材料。对称性的丧失直接关系到材料界面的规律性，可以说是自由边界，这一点有望在目前的研究调查中揭示出来。

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.