Estimates for the electric field in the presence of adjacent perfectly conducting spheres
Authors:
Habib Ammari, George Dassios, Hyeonbae Kang and Mikyoung Lim
Journal:
Quart. Appl. Math. 65 (2007), 339-355
MSC (2000):
Primary 35J25; Secondary 73C40
DOI:
https://doi.org/10.1090/S0033-569X-07-01034-1
Published electronically:
January 16, 2007
MathSciNet review:
2330561
Full-text PDF Free Access
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Additional Information
Abstract: In this paper we prove that, unlike the two-dimensional case, the electric field in the presence of closely adjacent spherical perfect conductors does not blow up even though the separation distance between the conducting inclusions approaches zero.
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
- H. Ammari, H. Kang, H. Lee, J. Lee, and M. Lim, Optimal bounds on the gradient of solutions to conductivity problems, preprint.
- 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
- 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
- B. Budiansky and G.F. Carrier, High shear stresses in stiff fiber composites, J. Appl. Mech., 51 (1984), 733–735.
- A. Charalambopoulos, G. Dassios, and M. Hadjinicolaou, An analytic solution for low-frequency scattering by two soft spheres, SIAM J. Appl. Math. 58 (1998), no. 2, 370–386. MR 1617658, DOI https://doi.org/10.1137/S0036139996304081
- 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
- X. Markenscoff, Stress amplification in vanishing small geometries, Comput. Mech., 19 (1996), 77–83.
- P. Moon and D. E. Spencer, Field theory handbook, 2nd ed., Springer-Verlag, Berlin, 1988. Including coordinate systems, differential equations and their solutions. MR 947546
- H. Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, preprint, 2006.
References
- H. Ammari, H. Kang, and M. Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann., 332 (2005), 277–286. MR 2178063 (2006h:78010)
- H. Ammari, H. Kang, H. Lee, J. Lee, and M. Lim, Optimal bounds on the gradient of solutions to conductivity problems, preprint.
- I. Babus̆ka, B. Andersson, P. Smith, and K. Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg., 172 (1999), 27–77. MR 1685902 (2000a:74115)
- E. Bonnetier and M. Vogelius, An elliptic regurality result for a composite medium with touching fibers of circular cross-section, SIAM J. Math. Anal., 31 (2000), 651–677. MR 1745481 (2002a:35052)
- B. Budiansky and G.F. Carrier, High shear stresses in stiff fiber composites, J. Appl. Mech., 51 (1984), 733–735.
- A. Charalambopoulos, G. Dassios, and M. Hadjinicolaou, An analytic solution for low-frequency scattering by two soft spheres, SIAM J. Appl. Math., 58 (1998), 370–386. MR 1617658 (98m:35020)
- 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.
- Y.Y. Li and L. Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math., LVI (2003), 892–925. MR 1990481 (2004k:35097)
- Y.Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal., 153 (2000), 91–151. MR 1770682 (2001m:35083)
- X. Markenscoff, Stress amplification in vanishing small geometries, Comput. Mech., 19 (1996), 77–83.
- P. Moon and D.E. Spencer, Field Theory Handbook, 2nd Ed. Springer-Verlag, Berlin, 1988. MR 947546 (89i:00026)
- H. Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, preprint, 2006.
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Additional Information
Habib Ammari
Affiliation:
Centre de Mathématiques Appliquées, CNRS UMR 7641 and Ecole Polytechnique, 91128 Palaiseau Cedex, France
MR Author ID:
353050
Email:
ammari@cmapx.polytechnique.fr
George Dassios
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom
MR Author ID:
54715
Email:
G.Dassios@damtp.cam.ac.uk
Hyeonbae Kang
Affiliation:
Department of Mathematical Sciences and RIM, Seoul National University, Seoul 151-747, Korea
MR Author ID:
268781
Email:
hkang@math.snu.ac.kr
Mikyoung Lim
Affiliation:
Centre de Mathématiques Appliquées, CNRS UMR 7641 and Ecole Polytechnique, 91128 Palaiseau Cedex, France
MR Author ID:
689036
Email:
mklim@cmapx.polytechnique.fr
Keywords:
Electric field,
gradient estimates,
composite materials
Received by editor(s):
April 30, 2006
Published electronically:
January 16, 2007
Article copyright:
© Copyright 2007
Brown University
The copyright for this article reverts to public domain 28 years after publication.
