# 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 and Applied Analysis, 2017, 16 (4) : 1493-1516. doi: 10.3934/cpaa.2017071
##### References:
 [1] J. Berger, P. Martin, V. Mantic and L. Gray, Fundamental solution for steady-state heat transfer in an exponentially graded anisotropic material, Z. angew. Math. Phys., 56 (2005), 293-303.  doi: 10.1007/s00033-004-1131-6. [2] A. Friedman, Reinforcement of the principal eigenvalue of an elliptic operator, Arch. Rational Mech. Anal., 73 (1980), 1-17.  doi: 10.1007/BF00283252. [3] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Third Edition, Springer-Verlag, Berlin, 1998. [4] L. Tartar, An Introduction to the Homogenization Method in Optimal Design, Lecture Notes in Mathematics, 1740, Springer Verlag, 2000. doi: 10.1007/BFb0106742. [5] H. Li, Effective boundary conditions of the heat equation on a body coated by functionally graded material, Discrete Contin. Dyn. Syst., 36 (2016), 1415-1430.  doi: 10.3934/dcds.2016.36.1415. [6] J. Li, S. Rosencrans, X. Wang and K. Zhang, Asymptotic Analysis of a Dirichlet problem for the heat equation on a coated body, Proc. Amer. Math. Soc., 137 (2008), 1711-1721.  doi: 10.1090/S0002-9939-08-09766-9. [7] M. Mahamood, T. Akinlabi, M. Shukal and S. Pityana, Functionally Graded Material: An Overview, Proceedings of the World Congress on Engineering, WCE 2012. [8] S. Rosencrans and X. Wang, Suppression of the Dirichlet eigenvalues of a coated body, SIAM J. Appl. Math. , 66 (2006), 1895-1916; Corrigendum, SIAM J. Appl. Math. , 68 (2008), 1202. doi: 10.1137/040621181. [9] J. Wessel, Handbook of Advanced Materials: Enabling New Design, J. Wiley, New Jersey, 2004. [10] G. Zhang, S. Rosencrans, X. Wang and K. Zhang, Estimating thermal insulating ability of anisotropic coatings via Robin eigenvalues and eigenfunctions, Discrete Contin. Dyn. Sys., 25 (2009), 1061-1079.  doi: 10.3934/dcds.2009.25.1061. [11] X. Zheng, M. G. Forest, R. Lipton, R. Zhou and Q. Wang, Exact scaling laws for electrical conducting properties of nematic polymer nano-composite monodomains, Adv. Funct. Math., 15 (2005), 627-638.  doi: 10.1007/s00161-006-0032-7.

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##### References:
 [1] J. Berger, P. Martin, V. Mantic and L. Gray, Fundamental solution for steady-state heat transfer in an exponentially graded anisotropic material, Z. angew. Math. Phys., 56 (2005), 293-303.  doi: 10.1007/s00033-004-1131-6. [2] A. Friedman, Reinforcement of the principal eigenvalue of an elliptic operator, Arch. Rational Mech. Anal., 73 (1980), 1-17.  doi: 10.1007/BF00283252. [3] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Third Edition, Springer-Verlag, Berlin, 1998. [4] L. Tartar, An Introduction to the Homogenization Method in Optimal Design, Lecture Notes in Mathematics, 1740, Springer Verlag, 2000. doi: 10.1007/BFb0106742. [5] H. Li, Effective boundary conditions of the heat equation on a body coated by functionally graded material, Discrete Contin. Dyn. Syst., 36 (2016), 1415-1430.  doi: 10.3934/dcds.2016.36.1415. [6] J. Li, S. Rosencrans, X. Wang and K. Zhang, Asymptotic Analysis of a Dirichlet problem for the heat equation on a coated body, Proc. Amer. Math. Soc., 137 (2008), 1711-1721.  doi: 10.1090/S0002-9939-08-09766-9. [7] M. Mahamood, T. Akinlabi, M. Shukal and S. Pityana, Functionally Graded Material: An Overview, Proceedings of the World Congress on Engineering, WCE 2012. [8] S. Rosencrans and X. Wang, Suppression of the Dirichlet eigenvalues of a coated body, SIAM J. Appl. Math. , 66 (2006), 1895-1916; Corrigendum, SIAM J. Appl. Math. , 68 (2008), 1202. doi: 10.1137/040621181. [9] J. Wessel, Handbook of Advanced Materials: Enabling New Design, J. Wiley, New Jersey, 2004. [10] G. Zhang, S. Rosencrans, X. Wang and K. Zhang, Estimating thermal insulating ability of anisotropic coatings via Robin eigenvalues and eigenfunctions, Discrete Contin. Dyn. Sys., 25 (2009), 1061-1079.  doi: 10.3934/dcds.2009.25.1061. [11] X. Zheng, M. G. Forest, R. Lipton, R. Zhou and Q. Wang, Exact scaling laws for electrical conducting properties of nematic polymer nano-composite monodomains, Adv. Funct. Math., 15 (2005), 627-638.  doi: 10.1007/s00161-006-0032-7.
$\Omega=\overline\Omega_1\cup \Omega_2$. 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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