Dirichlet and Neumann eigenvalues and their applications

本项目研究了出现在隔热层模型中的椭圆算子的谱。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特征值和特征函数来测量各向异性材料的隔热性能。主狄利克雷特征值和高狄利克雷特征值将通过仅使用材料的热张量的三个旋转不变量进行严格和数值估计。这些估计将用于研究纳米绝缘体的优化设计和反求热张量的不变量。该项目还将研究各向异性绝缘体包覆的各向同性体的罗宾特征值和特征函数。第二个领域的工作将继续发展一种理论来估计拉普拉斯函数的第一个正诺伊曼特征值，并将研究与特征值问题密切相关的逆散射问题。应用包括衍射光学技术中周期结构的优化设计。该项目的第一部分是由纳米复合材料科学中重要的工程应用驱动的。在宏观尺度上，纳米复合材料通常表现出各向异性，通过适当的设计可以创造出高效的隔热材料。对于这种材料，可以用对应椭圆算子的狄利克雷主特征值来衡量其保温效果。本项目中建立的Dirichlet特征值的用户友好估计将有助于设计更好的纳米复合隔热材料，并有助于在无法获得热张量解析公式时理解热张量。这项工作扩展了对各向异性涂层保护各向异性导体（例如航天飞机）免受过热的实际问题的研究；获得了一种易于使用的最佳涂层厚度规则。这一规则有可能广泛应用于热工领域。该项目的第二部分涉及在绝缘良好的容器中扩散强度的测量。该理论的预期应用包括在衍射光学技术周期结构优化设计中出现的逆问题的唯一性。

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.