Asymptotic behavior of solutions to elliptic equations in a coated body

We consider the Dirichlet boundary-value problem for a class of elliptic equations in a domain surrounded by a thin coating with the thickness $\delta$ and the thermal conductivity $\sigma$. By virtue of a new method we further investigate the results of Brezis, Caffarelli and Friedman [3] in three respects. If the integral of the source term on the interior domain is zero, we study the asymptotic behavior of the solution in the case of $\delta^2$»$\sigma$, $\delta^2$~$\sigma$ and $\delta^2$«$\sigma$ as $\delta$ and $\sigma$ tend to zero, respectively. Also we derive the optimal blow-up rate that was not given in [3]. Finally, in the case of the so-called "optimally aligned coating", i.e., if the thermal tensor matrix of the coating is spatially varying and its smallest eigenvalue has an eigenvector normal to the body at all boundary points, we obtain the asymptotic behavior of the solution by assuming only the smallest eigenvalue is of the same order as $\sigma$.