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

DOI:
https://doi.org/10.1090/S0002-9939-08-09766-9

Published electronically:
December 11, 2008

MathSciNet review:
2470829

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**A. Bensoussan, J.-L. Lions and G. Papanicolaou,*Asymptotic Analysis for Periodic Structures*, North-Holland, Amsterdam, New York, 1978. MR**0503330 (82h:35001)****2.**G. Buttazzo and R. Kohn,*Reinforcement by a thin layer with oscillating thickness*, Applied Math. Optimization,**16**(1987), 247-261. MR**0901816 (89a:73048)****3.**H. Brézis, L. A. Caffarelli and A. Friedman,*Reinforcement problems for elliptic equations and variational inequalities*, Ann. Mat. Pura Appl.,**123**(1980), 219-246. MR**0581931 (81m:35040)****4.**A. Friedman,*Reinforcement of the principal eigenvalue of an elliptic operator*, Arch. Rational Mech. Anal.,**73**(1980), 1-17. MR**0555579 (81c:35097)****5.**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR**1814364****6.**O. A. Ladyženskaja, J. Rivkind and N. N. Ural'ceva,*The classical solvability of diffraction problems*, Proc. Steklov Inst. Math.,**92**(1966), 132-166.**7.**O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva,*Linear and quasilinear equations of parabolic type*, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). MR**0241822****8.**G. P. Panasenko,*Asymptotics of the solutions and eigenvalues of elliptic equations with strongly varying coefficients*, Soviet Math. Dokl.,**21**(1980), 942-947.**9.**Steve Rosencrans and Xuefeng Wang,*Suppression of the Dirichlet eigenvalues of a coated body*, SIAM J. Appl. Math.**66**(2006), no. 6, 1895–1916. MR**2262957**, https://doi.org/10.1137/040621181**10.**Enrique Sánchez-Palencia,*Problèmes de perturbations liés aux phénomènes de conduction à travers des couches minces de grande résistivité*, J. Math. Pures Appl. (9)**53**(1974), 251–269 (French). MR**0364917****11.**Luc Tartar,*An introduction to the homogenization method in optimal design*, Optimal shape design (Tróia, 1998) Lecture Notes in Math., vol. 1740, Springer, Berlin, 2000, pp. 47–156. MR**1804685**, https://doi.org/10.1007/BFb0106742**12.**X. Zheng, M. G. Forest, R. Lipton, R. Zhou and Q. Wang,*Exact scaling laws for electrical conductivity properties of nematic polymer nano-composite monodomains*, Adv. Funct. Mat.,**15**(2005), 627-638.

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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:
https://doi.org/10.1090/S0002-9939-08-09766-9

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