Quantitative estimates of the field excited by an emitter in a narrow region between two circular inclusions
Authors:
Hyeonbae Kang and KiHyun Yun
Journal:
Quart. Appl. Math. 77 (2019), 861-873
MSC (2010):
Primary 35J25, 74C20
DOI:
https://doi.org/10.1090/qam/1551
Published electronically:
July 23, 2019
MathSciNet review:
4009335
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Abstract: A field excited by an emitter can be enhanced due to the presence of closely located inclusions. In this paper we consider such field enhancement when inclusions are disks of the same radii, and the emitter is of dipole-type and located in the narrow region between two inclusions. We derive quantitatively precise estimates of the field enhancement in the narrow region. The estimates reveal that the field is enhanced by a factor of $\epsilon ^{-1/2}$ in most areas, where $\epsilon$ is the distance between two inclusions. This factor is the same as that of gradient blow-up when there is a smooth background field, not a field excited by an emitter. The method of deriving estimates shows clearly that enhancement is due to a potential gap between two inclusions.
References
- 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
- 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
- Joseph E. Flaherty and Joseph B. Keller, Elastic behavior of composite media, Comm. Pure Appl. Math. 26 (1973), 565–580. MR 375910, DOI https://doi.org/10.1002/cpa.3160260409
- Hyeonbae Kang, Hyundae Lee, and KiHyun Yun, Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions, Math. Ann. 363 (2015), no. 3-4, 1281–1306. MR 3412359, DOI https://doi.org/10.1007/s00208-015-1203-2
- 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
- H. Kang and K. Yun, Precise estimates of the field excited by an emitter in presence of closely located inclusions of a bow-tie shape, arXiv:1810.08945.
- J. B. Keller, Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys. 34 (1963), 991–993. https://doi.org/10.1063/1.1729580
- J. B. Keller, Stresses in narrow regions, Trans. ASME J. Appl. Mech. 60 (1993), 1054–1056. https://doi.org/10.1115/1.2900977
- 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
- V. Pacheco-Penã, M. Beruete, A.I. Fernández-Domínquez, Y. Luo, and M. Navarro-Cía, Description of bow-tie nanoantennas excited by localized emitters using conformal transformation, ACS Photonics 3 (2016), 1223–1232. https://doi.org/10.1021/acsphotonics.6b00232
- 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
References
- H. Ammari, H. Kang, and M. Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann. 332 (2005), 277–286. MR2178063, https://doi.org/10.1007/s00208-004-0626-y
- 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
- E. S. Bao, Y. Li, and B. Yin, Gradient estimates for the perfect conductivity problem, Arch. Rational. Mech. Anal. 193 (2009), 195-226. MR2506075, https://doi.org/10.1007/s00205-008-0159-8
- J. E. Flaherty and J. B. Keller, Elastic behavior of composite media, Comm. Pure. Appl. Math. 26 (1973), 565–580. MR0375910, https://doi.org/10.1002/cpa.3160260409
- Hyeonbae Kang, Hyundae Lee, and KiHyun Yun, Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions, Math. Ann. 363 (2015), no. 3-4, 1281–1306. MR 3412359, DOI https://doi.org/10.1007/s00208-015-1203-2
- 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
- H. Kang and K. Yun, Precise estimates of the field excited by an emitter in presence of closely located inclusions of a bow-tie shape, arXiv:1810.08945.
- J. B. Keller, Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys. 34 (1963), 991–993. https://doi.org/10.1063/1.1729580
- J. B. Keller, Stresses in narrow regions, Trans. ASME J. Appl. Mech. 60 (1993), 1054–1056. https://doi.org/10.1115/1.2900977
- 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
- V. Pacheco-Penã, M. Beruete, A.I. Fernández-Domínquez, Y. Luo, and M. Navarro-Cía, Description of bow-tie nanoantennas excited by localized emitters using conformal transformation, ACS Photonics 3 (2016), 1223–1232. https://doi.org/10.1021/acsphotonics.6b00232
- 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
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Additional Information
Hyeonbae Kang
Affiliation:
Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212, South Korea
MR Author ID:
268781
Email:
hbkang@inha.ac.kr
KiHyun Yun
Affiliation:
Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si, Gyeonggi-do 17035, South Korea
MR Author ID:
675391
Email:
kihyun.yun@gmail.com
Received by editor(s):
March 15, 2019
Received by editor(s) in revised form:
June 22, 2019
Published electronically:
July 23, 2019
Additional Notes:
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government No. 2015R1D1A1A01059212, 2016R1A2B4011304, and 2017R1A4A1014735, and by Hankuk University of Foreign Studies Research Fund of 2019.
KiHyun Yun is the corresponding author.
Article copyright:
© Copyright 2019
Brown University
