Estimating thermal insulating ability of anisotropic coatings via Robin eigenvalues and eigenfunctions

The problem considered in this paper is the protection from overheating of a thermal conductor $\Omega_1$ by a thin anisotropic coating $\Omega_2$ (e.g. a space shuttle painted with a nano-insulator). We assume Newton's Cooling Law, so the temperature satisfies the Robin boundary condition on the outer boundary of the coating. Since the temperature function on $\Omega=\overline{\Omega}_1\cup\Omega_2$ can be expanded in terms of the eigenvalues and eigenfunctions of the elliptic operator $u\mapsto -\nabla (A \nabla u)$ with the Robin boundary condition on $\partial\Omega$, where $A$ is the thermal tensor of $\Omega$, we propose the following means to ensure the insulating ability of $\Omega_2$: (A) as many eigenvalues as possible should be small, in particular, the first eigenvalue should be small, (B) the first normalized eigenfunction should take large values on the body $\Omega_1$; we also argue that it is helpful for the understanding of the dynamics if (C) higher normalized eigenfunctions take small absolute values on $\Omega_1$. We assume that the thermal conductivity of $\Omega_2$ is small either in all directions or at least in the direction normal to $\partial\Omega_1$ (the case of "optimally aligned coating"). We study the asymptotic behavior of Robin eigenpairs as outcome of the interplay of the thermal tensor $A$, the thickness of $\Omega_2$ and the thermal transport coefficient in the Robin boundary condition, in the singular limit when either the thermal conductivity of $\Omega_2$, or the thickness of $\Omega_2$, or the thermal transport coefficient approaches $0$. By doing so, we identify the parameter ranges in which some or all of (A)-(C) occur.