Asymptotic behavior of solutions to elliptic equations in a coated body
Jingyu Li · Communications on Pure & Applied Analysis · 2009
We consider the Dirichlet boundary-value problem for a class ofelliptic equations in a domain surrounded by a thin coating with thethickness $\delta$ and the thermal conductivity $\sigma$. By virtueof a new method we further investigate the results of Brezis,Caffarelli and Friedman [3] in three respects. If theintegral of the source term on the interior domain is zero, we studythe 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 theoptimal blow-up rate that was not given in [3]. Finally, inthe case of the so-called 'optimally aligned coating',i.e., if the thermal tensor matrix of the coating is spatiallyvarying and its smallest eigenvalue has an eigenvector normal to thebody at all boundary points, we obtain the asymptotic behavior ofthe solution by assuming only the smallest eigenvalue is of the sameorder as $\sigma$.