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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body


Authors: Jingyu Li, Steve Rosencrans, Xuefeng Wang and Kaijun Zhang
Journal: Proc. Amer. Math. Soc. 137 (2009), 1711-1721
MSC (2000): Primary 35K05, 35K20, 35R05, 80A20, 80M35
Published electronically: December 11, 2008
MathSciNet review: 2470829
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Abstract: Of concern is the protection from overheating of an isotropically conducting body by an anisotropically conducting coating which is thin compared to the scale of the body. We assume either that the whole thermal tensor of the coating is small or that it is small in the directions normal to the body (a case we call “ optimally aligned coating”). We study the asymptotic behavior of the solution to the heat equation with Dirichlet boundary conditions on the outer surface of the coating, as the thickness of the coating shrinks. We obtain the exact scaling relations between the thermal tensor and the thickness of the coating so that the effective (limiting) condition on the boundary of the body is of Dirichlet, Robin or Neumann type, with the last condition indicating good insulation.


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Additional Information

Jingyu Li
Affiliation: School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, People’s Republic of China
Email: lijy645@yahoo.com.cn

Steve Rosencrans
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
Email: srosenc@tulane.edu

Xuefeng Wang
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
Email: xdw@math.tulane.edu

Kaijun Zhang
Affiliation: School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, People’s Republic of China
Email: zhangkj201@nenu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09766-9
PII: S 0002-9939(08)09766-9
Keywords: Overheating, thin insulator, Dirichlet problem, heat equation, asymptotic analysis, effective boundary condition
Received by editor(s): May 19, 2008
Published electronically: December 11, 2008
Communicated by: Walter Craig
Article copyright: © Copyright 2008 American Mathematical Society