|
Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body
Author(s):
Jingyu
Li;
Steve
Rosencrans;
Xuefeng
Wang;
Kaijun
Zhang
Journal:
Proc. Amer. Math. Soc.
137
(2009),
1711-1721.
MSC (2000):
Primary 35K05, 35K20, 35R05, 80A20, 80M35
Posted:
December 11, 2008
MathSciNet review:
2470829
Retrieve article in:
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.
References:
-
- 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.
- D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Third Edition, Springer-Verlag, 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), 132-166.
- 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), 942-947.
- 9.
- S. Rosencrans and X. Wang, Suppression of the Dirichlet eigenvalues of a coated body, SIAM J. Appl. Math., 66 (2006), 1895-1916; Corrigendum, SIAM J. Appl. Math., 68 (2008), 1202. MR 2262957 (2007h:35038)
- 10.
- E. Sanchez-Palencia, Problèmes de perturbations liés aux phénomènes de conduction à travers des couches minces de grande résistivité, J. Math. Pures Appl., 53 (1974), 251-269. MR 0364917 (51:1171)
- 11.
- L. Tartar, An Introduction to the Homogenization Method in Optimal Design, Lecture Notes in Mathematics 1740, Springer-Verlag, 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 nano-composite monodomains, Adv. Funct. Mat., 15 (2005), 627-638.
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2000):
35K05, 35K20, 35R05, 80A20, 80M35
Retrieve articles in all Journals with
MSC (2000):
35K05, 35K20, 35R05, 80A20, 80M35
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:
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
Posted:
December 11, 2008
Communicated by:
Walter Craig
Copyright of article:
Copyright
2008,
American Mathematical Society
|
