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Effective boundary conditions of the heat equation on a body coated by functionally graded material
1.  Center for Partial Dierential Equations, East China Normal University, Minhang, Shanghai, 200241, China 
References:
[1] 
J. Berger, P. Martin, V. Mantic and L. Gray, Fundamental solution for steadystate heat transfer in an exponentially graded anisotropic material,, Z. angew. Math. Phys., 56 (2005), 293. doi: 10.1007/s0003300411316. 
[2] 
H. Brezis, L. A. Caffarelli and A. Friedman, Reinforcement problem for elliptic equations and variational inequalities,, Ann. Mat. Pura Appl., 123 (1980), 219. doi: 10.1007/BF01796546. 
[3] 
H. Carslaw and J. Jaeger, Conduction of Heat in Solids,, Reprint of the second edition, (1988). 
[4] 
X, Chen, C. Pond and X. Wang, Effective boundary conditions resulting from anisotropic and optimally aligned coatings: The two dimensional case,, Arch. Ration. Mech. Anal., 206 (2012), 911. doi: 10.1007/s002050120547y. 
[5] 
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998). 
[6] 
H. Li, X. Wang and Y. Wu, The effective boundary conditions and their lifespan of the logistic diffusion equation on a coated body,, J. Differential Equations, 257 (2014), 3640. doi: 10.1016/j.jde.2014.07.004. 
[7] 
J. Li, Asymptotic behavior of solutions to elliptic equations in a coated body,, Comm. Pure Appl. Anal., 8 (2009), 1251. doi: 10.3934/cpaa.2009.8.1251. 
[8] 
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 (2009), 1711. doi: 10.1090/S0002993908097669. 
[9] 
J. Li, X. Wang, G. Zhang and K. Zhang, Asymptotic behavior of Robin problem for heat equation on a coated body,, Rocky Mountain J. Math., 42 (2012), 937. doi: 10.1216/RMJ2012423937. 
[10] 
J. Li and K. Zhang, Reinforcement of the Poisson equation by a thin layer,, Math. Models Methods Appl. Sci., 21 (2012), 1153. doi: 10.1142/S0218202511005283. 
[11] 
M. Mahamood, T. Akinlabi, M. Shukal and S. Pityana, Functionally Graded Material: An Overview,, Proceedings of the World Congress on Engineering, (2012). 
[12] 
S. Rosencrans and X. Wang, Suppression of the Dirichlet eigenvalues of a coated body,, SIAM J. Appl. Math., 66 (2006), 1895. doi: 10.1137/040621181. 
[13] 
J. Wessel, Handbook of Advanced Materials: Enabling New Design,, J. Wiley, (2004). doi: 10.1002/0471465186. 
show all references
References:
[1] 
J. Berger, P. Martin, V. Mantic and L. Gray, Fundamental solution for steadystate heat transfer in an exponentially graded anisotropic material,, Z. angew. Math. Phys., 56 (2005), 293. doi: 10.1007/s0003300411316. 
[2] 
H. Brezis, L. A. Caffarelli and A. Friedman, Reinforcement problem for elliptic equations and variational inequalities,, Ann. Mat. Pura Appl., 123 (1980), 219. doi: 10.1007/BF01796546. 
[3] 
H. Carslaw and J. Jaeger, Conduction of Heat in Solids,, Reprint of the second edition, (1988). 
[4] 
X, Chen, C. Pond and X. Wang, Effective boundary conditions resulting from anisotropic and optimally aligned coatings: The two dimensional case,, Arch. Ration. Mech. Anal., 206 (2012), 911. doi: 10.1007/s002050120547y. 
[5] 
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998). 
[6] 
H. Li, X. Wang and Y. Wu, The effective boundary conditions and their lifespan of the logistic diffusion equation on a coated body,, J. Differential Equations, 257 (2014), 3640. doi: 10.1016/j.jde.2014.07.004. 
[7] 
J. Li, Asymptotic behavior of solutions to elliptic equations in a coated body,, Comm. Pure Appl. Anal., 8 (2009), 1251. doi: 10.3934/cpaa.2009.8.1251. 
[8] 
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 (2009), 1711. doi: 10.1090/S0002993908097669. 
[9] 
J. Li, X. Wang, G. Zhang and K. Zhang, Asymptotic behavior of Robin problem for heat equation on a coated body,, Rocky Mountain J. Math., 42 (2012), 937. doi: 10.1216/RMJ2012423937. 
[10] 
J. Li and K. Zhang, Reinforcement of the Poisson equation by a thin layer,, Math. Models Methods Appl. Sci., 21 (2012), 1153. doi: 10.1142/S0218202511005283. 
[11] 
M. Mahamood, T. Akinlabi, M. Shukal and S. Pityana, Functionally Graded Material: An Overview,, Proceedings of the World Congress on Engineering, (2012). 
[12] 
S. Rosencrans and X. Wang, Suppression of the Dirichlet eigenvalues of a coated body,, SIAM J. Appl. Math., 66 (2006), 1895. doi: 10.1137/040621181. 
[13] 
J. Wessel, Handbook of Advanced Materials: Enabling New Design,, J. Wiley, (2004). doi: 10.1002/0471465186. 
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