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), 17111721
MSC (2000):
Primary 35K05, 35K20, 35R05, 80A20, 80M35
Published electronically:
December 11, 2008
MathSciNet review:
2470829
Fulltext PDF Free Access
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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.
 1.
Alain
Bensoussan, JacquesLouis
Lions, and George
Papanicolaou, Asymptotic analysis for periodic structures,
Studies in Mathematics and its Applications, vol. 5, NorthHolland
Publishing Co., AmsterdamNew York, 1978. MR 503330
(82h:35001)
 2.
Giuseppe
Buttazzo and Robert
V. Kohn, Reinforcement by a thin layer with oscillating
thickness, Appl. Math. Optim. 16 (1987), no. 3,
247–261. MR
901816 (89a:73048), http://dx.doi.org/10.1007/BF01442194
 3.
Haïm
Brézis, Luis
A. Caffarelli, and Avner
Friedman, Reinforcement problems for elliptic equations and
variational inequalities, Ann. Mat. Pura Appl. (4)
123 (1980), 219–246. MR 581931
(81m:35040), http://dx.doi.org/10.1007/BF01796546
 4.
Avner
Friedman, Reinforcement of the principal eigenvalue of an elliptic
operator, Arch. Rational Mech. Anal. 73 (1980),
no. 1, 1–17. MR 555579
(81c:35097), http://dx.doi.org/10.1007/BF00283252
 5.
David
Gilbarg and Neil
S. Trudinger, Elliptic partial differential equations of second
order, Classics in Mathematics, SpringerVerlag, Berlin, 2001. Reprint
of the 1998 edition. MR 1814364
(2001k:35004)
 6.
O. A. Ladyženskaja, J. Rivkind and N. N. Ural'ceva, The classical solvability of diffraction problems, Proc. Steklov Inst. Math., 92 (1966), 132166.
 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
(39 #3159b)
 8.
G. P. Panasenko, Asymptotics of the solutions and eigenvalues of elliptic equations with strongly varying coefficients, Soviet Math. Dokl., 21 (1980), 942947.
 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 (electronic). MR 2262957
(2007h:35038), http://dx.doi.org/10.1137/040621181
 10.
Enrique
SánchezPalencia, 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
(51 #1171)
 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, http://dx.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 nanocomposite monodomains, Adv. Funct. Mat., 15 (2005), 627638.
 1.
 A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, NorthHolland, 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), 247261. 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), 219246. MR 0581931 (81m:35040)
 4.
 A. Friedman, Reinforcement of the principal eigenvalue of an elliptic operator, Arch. Rational Mech. Anal., 73 (1980), 117. MR 0555579 (81c:35097)
 5.
 D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Third Edition, SpringerVerlag, Berlin, 1998. MR 1814364 (2001k:35004)
 6.
 O. A. Ladyženskaja, J. Rivkind and N. N. Ural'ceva, The classical solvability of diffraction problems, Proc. Steklov Inst. Math., 92 (1966), 132166.
 7.
 O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs 23, Amer. Math. Soc., Providence, Rhode Island, 1967. MR 0241822 (39:3159b)
 8.
 G. P. Panasenko, Asymptotics of the solutions and eigenvalues of elliptic equations with strongly varying coefficients, Soviet Math. Dokl., 21 (1980), 942947.
 9.
 S. Rosencrans and X. Wang, Suppression of the Dirichlet eigenvalues of a coated body, SIAM J. Appl. Math., 66 (2006), 18951916; Corrigendum, SIAM J. Appl. Math., 68 (2008), 1202. MR 2262957 (2007h:35038)
 10.
 E. SanchezPalencia, Problèmes de perturbations liés aux phénomènes de conduction à travers des couches minces de grande résistivité, J. Math. Pures Appl., 53 (1974), 251269. MR 0364917 (51:1171)
 11.
 L. Tartar, An Introduction to the Homogenization Method in Optimal Design, Lecture Notes in Mathematics 1740, SpringerVerlag, Berlin, 2000. MR 1804685
 12.
 X. Zheng, M. G. Forest, R. Lipton, R. Zhou and Q. Wang, Exact scaling laws for electrical conductivity properties of nematic polymer nanocomposite monodomains, Adv. Funct. Mat., 15 (2005), 627638.
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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/S0002993908097669
PII:
S 00029939(08)097669
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
