Suppression of the Dirichlet Eigenvalues of a Coated Body

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 eig...