Derivative estimates of solutions of elliptic systems in narrow regions
Authors:
HaiGang Li, YanYan Li, Ellen ShiTing Bao and Biao Yin
Journal:
Quart. Appl. Math. 72 (2014), 589-596
MSC (2010):
Primary 35J25, 35Q74, 74Bxx
DOI:
https://doi.org/10.1090/S0033-569X-2014-01339-0
Published electronically:
June 10, 2014
MathSciNet review:
3237564
Full-text PDF Free Access
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Additional Information
Abstract: In this paper, we establish $C^k$ estimates for a class of elliptic systems, including linear systems of elasticity, in a narrow region. The problem arises from studies of fiber-reinforced elastic composite materials.
References
- H. Ammari, G. Ciraolo, H. Kang, H. Lee and K. Yun, Spectral analysis of the Neumann-Poincaré operator and characterization of the gradient blow-up, arXiv:1206.2074.
- Habib Ammari, George Dassios, Hyeonbae Kang, and Mikyoung Lim, Estimates for the electric field in the presence of adjacent perfectly conducting spheres, Quart. Appl. Math. 65 (2007), no. 2, 339–355. MR 2330561, DOI https://doi.org/10.1090/S0033-569X-07-01034-1
- Habib Ammari, Hyeonbae Kang, and Mikyoung Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann. 332 (2005), no. 2, 277–286. MR 2178063, DOI https://doi.org/10.1007/s00208-004-0626-y
- Habib Ammari, Hyeonbae Kang, Hyundae Lee, Jungwook Lee, and Mikyoung Lim, Optimal estimates for the electric field in two dimensions, J. Math. Pures Appl. (9) 88 (2007), no. 4, 307–324 (English, with English and French summaries). MR 2384571, DOI https://doi.org/10.1016/j.matpur.2007.07.005
- Habib Ammari, Hyeonbae Kang, Hyundae Lee, Mikyoung Lim, and Habib Zribi, Decomposition theorems and fine estimates for electrical fields in the presence of closely located circular inclusions, J. Differential Equations 247 (2009), no. 11, 2897–2912. MR 2569851, DOI https://doi.org/10.1016/j.jde.2009.08.005
- Ivo Babuška, Börje Andersson, Paul J. Smith, and Klas Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg. 172 (1999), no. 1-4, 27–77. MR 1685902, DOI https://doi.org/10.1016/S0045-7825%2898%2900225-4
- Ellen Shiting Bao, Yan Yan Li, and Biao Yin, Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal. 193 (2009), no. 1, 195–226. MR 2506075, DOI https://doi.org/10.1007/s00205-008-0159-8
- Ellen Shiting Bao, Yan Yan Li, and Biao Yin, Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Comm. Partial Differential Equations 35 (2010), no. 11, 1982–2006. MR 2754076, DOI https://doi.org/10.1080/03605300903564000
- Eric Bonnetier and Michael Vogelius, An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section, SIAM J. Math. Anal. 31 (2000), no. 3, 651–677. MR 1745481, DOI https://doi.org/10.1137/S0036141098333980
- Eric Bonnetier and Faouzi Triki, Asymptotics in the presence of inclusions of small volume for a conduction equation: a case with a non-smooth reference potential, Imaging microstructures, Contemp. Math., vol. 494, Amer. Math. Soc., Providence, RI, 2009, pp. 95–111. MR 2581768, DOI https://doi.org/10.1090/conm/494/09645
- B. Budiansky and G.F. Carrier, High shear stresses in stiff fiber composites, J. Appl. Mech., 51 (1984), 733-735.
- Hyeonbae Kang, Mikyoung Lim, and KiHyun Yun, Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities, J. Math. Pures Appl. (9) 99 (2013), no. 2, 234–249. MR 3007847, DOI https://doi.org/10.1016/j.matpur.2012.06.013
- J.B. Keller, Stresses in narrow regions, Trans. ASME J. Appl. Mech., 60 (1993), 1054-1056.
- J.B. Keller, Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys., 3 (1963), 991-993.
- Yanyan Li and Louis Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math. 56 (2003), no. 7, 892–925. Dedicated to the memory of Jürgen K. Moser. MR 1990481, DOI https://doi.org/10.1002/cpa.10079
- Yan Yan Li and Michael Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal. 153 (2000), no. 2, 91–151. MR 1770682, DOI https://doi.org/10.1007/s002050000082
- Mikyoung Lim and Kihyun Yun, Blow-up of electric fields between closely spaced spherical perfect conductors, Comm. Partial Differential Equations 34 (2009), no. 10-12, 1287–1315. MR 2581974, DOI https://doi.org/10.1080/03605300903079579
- Kihyun Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math. 67 (2007), no. 3, 714–730. MR 2300307, DOI https://doi.org/10.1137/060648817
- KiHyun Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal. Appl. 350 (2009), no. 1, 306–312. MR 2476915, DOI https://doi.org/10.1016/j.jmaa.2008.09.057
- O. A. Oleĭnik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, vol. 26, North-Holland Publishing Co., Amsterdam, 1992. MR 1195131
References
- H. Ammari, G. Ciraolo, H. Kang, H. Lee and K. Yun, Spectral analysis of the Neumann-Poincaré operator and characterization of the gradient blow-up, arXiv:1206.2074.
- Habib Ammari, George Dassios, Hyeonbae Kang, and Mikyoung Lim, Estimates for the electric field in the presence of adjacent perfectly conducting spheres, Quart. Appl. Math. 65 (2007), no. 2, 339–355. MR 2330561 (2008i:35036)
- Habib Ammari, Hyeonbae Kang, and Mikyoung Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann. 332 (2005), no. 2, 277–286. MR 2178063 (2006h:78010), DOI https://doi.org/10.1007/s00208-004-0626-y
- Habib Ammari, Hyeonbae Kang, Hyundae Lee, Jungwook Lee, and Mikyoung Lim, Optimal estimates for the electric field in two dimensions, J. Math. Pures Appl. (9) 88 (2007), no. 4, 307–324 (English, with English and French summaries). MR 2384571 (2009a:35043), DOI https://doi.org/10.1016/j.matpur.2007.07.005
- Habib Ammari, Hyeonbae Kang, Hyundae Lee, Mikyoung Lim, and Habib Zribi, Decomposition theorems and fine estimates for electrical fields in the presence of closely located circular inclusions, J. Differential Equations 247 (2009), no. 11, 2897–2912. MR 2569851 (2010j:35102), DOI https://doi.org/10.1016/j.jde.2009.08.005
- Ivo Babuška, Börje Andersson, Paul J. Smith, and Klas Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg. 172 (1999), no. 1-4, 27–77. MR 1685902 (2000a:74115), DOI https://doi.org/10.1016/S0045-7825%2898%2900225-4
- Ellen Shiting Bao, Yan Yan Li, and Biao Yin, Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal. 193 (2009), no. 1, 195–226. MR 2506075 (2010k:35093), DOI https://doi.org/10.1007/s00205-008-0159-8
- Ellen Shiting Bao, Yan Yan Li, and Biao Yin, Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Comm. Partial Differential Equations 35 (2010), no. 11, 1982–2006. MR 2754076 (2011m:35065), DOI https://doi.org/10.1080/03605300903564000
- Eric Bonnetier and Michael Vogelius, An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section, SIAM J. Math. Anal. 31 (2000), no. 3, 651–677. MR 1745481 (2002a:35052), DOI https://doi.org/10.1137/S0036141098333980
- Eric Bonnetier and Faouzi Triki, Asymptotics in the presence of inclusions of small volume for a conduction equation: a case with a non-smooth reference potential, Imaging microstructures, Contemp. Math., vol. 494, Amer. Math. Soc., Providence, RI, 2009, pp. 95–111. MR 2581768 (2011j:35044), DOI https://doi.org/10.1090/conm/494/09645
- B. Budiansky and G.F. Carrier, High shear stresses in stiff fiber composites, J. Appl. Mech., 51 (1984), 733-735.
- Hyeonbae Kang, Mikyoung Lim, and KiHyun Yun, Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities, J. Math. Pures Appl. (9) 99 (2013), no. 2, 234–249. MR 3007847, DOI https://doi.org/10.1016/j.matpur.2012.06.013
- J.B. Keller, Stresses in narrow regions, Trans. ASME J. Appl. Mech., 60 (1993), 1054-1056.
- J.B. Keller, Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys., 3 (1963), 991-993.
- Yanyan Li and Louis Nirenberg, Estimates for elliptic systems from composite material. Dedicated to the memory of Jürgen K. Moser, Comm. Pure Appl. Math. 56 (2003), no. 7, 892–925. MR 1990481 (2004k:35097), DOI https://doi.org/10.1002/cpa.10079
- Yan Yan Li and Michael Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal. 153 (2000), no. 2, 91–151. MR 1770682 (2001m:35083), DOI https://doi.org/10.1007/s002050000082
- Mikyoung Lim and Kihyun Yun, Blow-up of electric fields between closely spaced spherical perfect conductors, Comm. Partial Differential Equations 34 (2009), no. 10-12, 1287–1315. MR 2581974 (2011a:35078), DOI https://doi.org/10.1080/03605300903079579
- Kihyun Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math. 67 (2007), no. 3, 714–730 (electronic). MR 2300307 (2008a:74039), DOI https://doi.org/10.1137/060648817
- KiHyun Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal. Appl. 350 (2009), no. 1, 306–312. MR 2476915, DOI https://doi.org/10.1016/j.jmaa.2008.09.057
- O. A. Oleĭnik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, vol. 26, North-Holland Publishing Co., Amsterdam, 1992. MR 1195131 (93k:35025)
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Additional Information
HaiGang Li
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
MR Author ID:
928017
Email:
hgli@bnu.edu.cn
YanYan Li
Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, New Jersey 08854
Email:
yyli@math.rutgers.edu
Ellen ShiTing Bao
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Biao Yin
Affiliation:
Quantitative Advisory Services, Ernst and Young LLP, 5 Times Square, New York, New York 10036
Received by editor(s):
July 24, 2012
Published electronically:
June 10, 2014
Article copyright:
© Copyright 2014
Brown University