# American Institute of Mathematical Sciences

September  2009, 25(3): 1061-1079. doi: 10.3934/dcds.2009.25.1061

## Estimating thermal insulating ability of anisotropic coatings via Robin eigenvalues and eigenfunctions

 1 Laboratory of Nonlinear Analysis, Department of Mathematics, Central China Normal University, 430079, China 2 Mathematics Department, Tulane University, New Orleans, LA 70118, United States 3 School of Mathematics and Statistics, Northeast Normal University, 130024, China

Received  October 2008 Revised  February 2009 Published  August 2009

The problem considered in this paper is the protection from overheating of a thermal conductor $\Omega_1$ by a thin anisotropic coating $\Omega_2$ (e.g. a space shuttle painted with a nano-insulator). We assume Newton's Cooling Law, so the temperature satisfies the Robin boundary condition on the outer boundary of the coating. Since the temperature function on $\Omega=\overline{\Omega}_1\cup\Omega_2$ can be expanded in terms of the eigenvalues and eigenfunctions of the elliptic operator $u\mapsto -\nabla (A \nabla u)$ with the Robin boundary condition on $\partial\Omega$, where $A$ is the thermal tensor of $\Omega$, we propose the following means to ensure the insulating ability of $\Omega_2$: (A) as many eigenvalues as possible should be small, in particular, the first eigenvalue should be small, (B) the first normalized eigenfunction should take large values on the body $\Omega_1$; we also argue that it is helpful for the understanding of the dynamics if (C) higher normalized eigenfunctions take small absolute values on $\Omega_1$. We assume that the thermal conductivity of $\Omega_2$ is small either in all directions or at least in the direction normal to $\partial\Omega_1$ (the case of "optimally aligned coating"). We study the asymptotic behavior of Robin eigenpairs as outcome of the interplay of the thermal tensor $A$, the thickness of $\Omega_2$ and the thermal transport coefficient in the Robin boundary condition, in the singular limit when either the thermal conductivity of $\Omega_2$, or the thickness of $\Omega_2$, or the thermal transport coefficient approaches $0$. By doing so, we identify the parameter ranges in which some or all of (A)-(C) occur.
Citation: Guojing Zhang, Steve Rosencrans, Xuefeng Wang, Kaijun Zhang. Estimating thermal insulating ability of anisotropic coatings via Robin eigenvalues and eigenfunctions. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1061-1079. doi: 10.3934/dcds.2009.25.1061
 [1] Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2589-2618. doi: 10.3934/dcds.2017111 [2] VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $p(x)$-growth and generalized Robin boundary value condition. Communications on Pure and Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003 [3] Kazuaki Taira. The hypoelliptic Robin problem for quasilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1601-1618. doi: 10.3934/dcdss.2020091 [4] Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083 [5] Mihai Mihăilescu. An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue. Communications on Pure and Applied Analysis, 2011, 10 (2) : 701-708. doi: 10.3934/cpaa.2011.10.701 [6] Al-hassem Nayam. Asymptotics of an optimal compliance-network problem. Networks and Heterogeneous Media, 2013, 8 (2) : 573-589. doi: 10.3934/nhm.2013.8.573 [7] Francis Michael Russell, J. C. Eilbeck. Persistent mobile lattice excitations in a crystalline insulator. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1267-1285. doi: 10.3934/dcdss.2011.4.1267 [8] Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 [9] Giacomo Bocerani, Dimitri Mugnai. A fractional eigenvalue problem in $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 619-629. doi: 10.3934/dcdss.2016016 [10] David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems and Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004 [11] Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008 [12] Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075 [13] Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845 [14] Jean-Michel Rakotoson. Generalized eigenvalue problem for totally discontinuous operators. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 343-373. doi: 10.3934/dcds.2010.28.343 [15] Natalia P. Bondarenko, Vjacheslav A. Yurko. A new approach to the inverse discrete transmission eigenvalue problem. Inverse Problems and Imaging, 2022, 16 (4) : 739-751. doi: 10.3934/ipi.2021073 [16] Marcone C. Pereira, Ricardo P. Silva. Error estimates for a Neumann problem in highly oscillating thin domains. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 803-817. doi: 10.3934/dcds.2013.33.803 [17] Maciek Korzec, Andreas Münch, Endre Süli, Barbara Wagner. Anisotropy in wavelet-based phase field models. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1167-1187. doi: 10.3934/dcdsb.2016.21.1167 [18] Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems and Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025 [19] Jiaoxiu Ling, Zhan Zhou. Positive solutions of the discrete Robin problem with $\phi$-Laplacian. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3183-3196. doi: 10.3934/dcdss.2020338 [20] Nicolas Augier, Ugo Boscain, Mario Sigalotti. Semi-conical eigenvalue intersections and the ensemble controllability problem for quantum systems. Mathematical Control and Related Fields, 2020, 10 (4) : 877-911. doi: 10.3934/mcrf.2020023

2021 Impact Factor: 1.588