
Abstract
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
nanoinsulator). 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.
Mathematics Subject Classification: Primary: 35J05, 35J20; Secondary: 80A20, 80M30.
\begin{equation} \\ \end{equation}

Access History
