RESUMEN

We consider the problem of protecting from overheating the interiors of anisotropically heat-conducting bodies whose boundaries are maintained at a high temperature. The bodies are composites consisting of a thin anisotropic insulating coating surrounding an isotropically conducting interior (e.g., a space shuttle painted with an insulator). This anisotropy is a common feature of the nanocomposite materials used as insulators. Denote by $A$ the thermal tensor (matrix) of the coated body and consider the Dirichlet eigenvalues of the elliptic operator $u\mapsto -\nabla\cdot\left(A\nabla u\right)$ on the coated body. The eigenfuction expansion of the interior temperature shows that small eigenvalues favor insulation of the interior. This is the motivation for studying the idealized mathematical problem of suppression of the Dirichlet eigenvalues. Suppose $A$ is a constant matrix $\overline{A}$ on the coating. The focus of this paper is estimation of the elliptic eigenvalues and qualitative description of the eigenfunctions using only the eigenvalues of $\overline{A}$, the scalar conductivity of the uncoated body, and certain scalar characteristics of the geometry of the uncoated body. We study the effect of small matrix eigenvalues, small thickness of the coating, and their interplay. If the thermal tensor of the coating is spatially varying and optimally configured so that the minimum eigenvalue has eigenvector normal to the body at all boundary points of the body (and remains equal to that normal vector at each point in the coating on the straight line in that normal direction), only that minimum eigenvalue need be small. A by-product is a new characterization of the first positive Neumann eigenvalue in terms of a sequence of second Dirichlet eigenvalues.