# American Institute of Mathematical Sciences

July  2017, 16(4): 1493-1516. doi: 10.3934/cpaa.2017071

## Asymptotic behavior of Dirichlet eigenvalues on a body coated by functionally graded material

 1 Center for Partial Differential Equations, East China Normal University, 500 Dongchuan Road, Minhang 200241, Shanghai, China 2 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Received  October 2016 Revised  December 2016 Published  April 2017

We consider the physical problem of protecting a thermally conducting body from overheating by thermal barrier coatings on a bounded domain, which has two components with a thin coating surrounding the body (of metallic nature), subject to the Dirichlet boundary condition. The coating is composed of two layers, the pure ceramic part and the mixed part. The latter is assumed to be functionally graded material (FGM) that is meant to make a smooth transition from being metallic to being ceramic. The thermal tensor $A$ is isotropic on the body, and anisotropic on the coating; and the size of thermal tensor may differ significantly in these components. Eigenfunction expansion of the interior temperature function indicates that small eigenvalues of the elliptic operator $u\mapsto -\nabla\cdot \left(A\nabla u\right)$ are desirable for the insulation of the body. Therefore, we are motivated to study the asymptotic behavior of the eigenpairs of the Dirichelt eigenvalue problem, as the thickness of the coating shrinks. Our results greatly generalize those by Rosencrans and Wang [8] where the case of single coating layer is considered. In particular, we find new optimal scaling relationship between the thickness of the coating and its thermal conductivity that guarantees at least the principal eigenvalue is small for any general FGMs.

Citation: Huicong Li, Jingyu Li. Asymptotic behavior of Dirichlet eigenvalues on a body coated by functionally graded material. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1493-1516. doi: 10.3934/cpaa.2017071
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##### References:
$\Omega=\overline\Omega_1\cup \Omega_2$. The coating $\Omega_2$ is uniformly thick with thickness $\delta$
$\Omega=\overline{\Omega_1}\cup \Omega_2$. The coating $\Omega_2$ is uniformly thick with thickness $\delta$ and the mixed part $\Omega_3$ has thickness $\delta_1\in(0, \delta)$
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