Dirichlet and Neumann eigenvalues and their applications

本计画研究绝热层模型中椭圆算子的频谱。 分析和数值研究将在两个领域进行。 第一个领域的工作将使用狄利克雷本征值和本征函数来测量各向异性材料的隔热性能。主要和更高的狄利克雷本征值将严格估计和数值仅使用三个旋转不变量的热张量的材料。这些估计将被用来研究纳米绝缘体的最佳设计，并反向找到热张量的不变量。 该项目还将研究由各向异性绝缘体覆盖的各向同性体的Robin本征值和本征函数。 在第二个领域的工作将继续发展的理论，以估计第一个积极的诺依曼本征值的拉普拉斯算子，并将调查逆散射问题，其中一个密切相关的本征值问题的出现。 应用包括衍射光学技术中周期性结构的优化设计。该项目的第一部分是由纳米复合材料科学中的重要工程应用所激发的。 有效的热绝缘体可以通过适当设计的纳米复合材料，在宏观尺度上，通常表现出各向异性。 对于这样的材料，相应的椭圆算子的主Dirichlet本征值可以用来衡量隔热效果。 该项目中开发的Dirichlet特征值的用户友好估计将有助于设计更好的纳米复合材料绝热体，并在其解析公式不可用时理解热张量。这项工作扩展了对保护各向同性导体的实际问题的研究（例如，航天飞机）通过各向异性涂层防止过热;已经获得了所需涂层的最佳厚度的易于使用的规则。 这一规律在热工领域具有广泛的应用前景。 该项目的第二部分涉及在良好绝缘容器中测量扩散强度。 该理论的预期应用包括衍射光学技术中周期性结构的优化设计中出现的反问题的唯一性。

This project investigates the spectra of elliptic operators that arise in models of thermal insulator coatings. Analytical and numerical research will be carried out in two areas. Work in the first area will use Dirichlet eigenvalues and eigenfunctions to measure the thermal insulation properties of an anisotropic material. The principal and higher Dirichlet eigenvalues will be estimated rigorously and numerically by using only the three rotational invariants of the thermal tensor of the material. These estimates will be used to study the optimal design of nano insulators and to inversely find the invariants of the thermal tensor. The project will also investigate the Robin eigenvalues and eigenfunctions of an isotropic body coated by an anisotropic insulator. Work in the second area will continue development of a theory to estimate the first positive Neumann eigenvalue of the Laplacian and will investigate inverse scattering problems in which a closely related eigenvalue problem arises. Applications include optimal design of periodic structures in diffractive optics technology.The first part of the project is motivated by significant engineering applications in nano-composite materials science. Efficient thermal insulators can be created by proper designing of nano-composite materials, which, at the macroscale, commonly exhibit anisotropy. For such a material, the principal Dirichlet eigenvalue of the corresponding elliptic operator can be used to measure the thermal insulation effectiveness. The user-friendly estimates developed in this project of Dirichlet eigenvalues will be helpful in designing better nano-composite thermal insulators and in understanding the thermal tensor when its analytic formula is not available. This work extends studies of the practical problem of protecting an isotropic conductor (e.g., a space shuttle) from overheating by an anisotropic coating; an easy-to-use rule for the optimal thickness of the coating required has been obtained. This rule is potentially widely applicable in thermal engineering. The second part of the project addresses measurement of the strength of diffusion in a well-insulated container. The expected applications of this theory include the uniqueness of the inverse problem that arises in the optimal design of periodic structures in diffractive optics technology.