Skip to main content
SpringerLink
Log in

Electric field concentration in the presence of an inclusion with eccentric core-shell geometry

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this paper we analyze the gradient blow-up of the solution to the conductivity problem in two dimensions in the presence of an inclusion with eccentric core-shell geometry. Assuming that the core and shell have circular boundaries that are nearly touching, we derive an asymptotic formula for the solution in terms of the single and double layer potentials with image line charges. We also deduce an integral formula with image line charges for the problem relating to two nearly touching separated conductors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Ammari, H., Ciraolo, G., Kang, H., Lee, H., Milton, G.W.: Spectral theory of a Neumann–Poincaré-type operator and analysis of cloaking due to anomalous localized resonance. Arch. Ration. Mech. Anal. 208(2), 667–692 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ammari, H., Ciraolo, G., Kang, H., Lee, H., Yun, K.: Spectral analysis of the Neumann–Poincaré operator and characterization of the stress concentration in anti-plane elasticity. Arch. Ration. Mech. Anal. 208(1), 275–304 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ammari, H., Dassios, G., Kang, H., Lim, M.: Estimates for the electric field in the presence of adjacent perfectly conducting spheres. Q. Appl. Math. 65(2), 339–356 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ammari, H., Garnier, J., Jing, W., Kang, H., Lim, M., Sølna, K., Wang, H.: Mathematical and statistical methods for multistatic imaging, vol. 2098. Springer, Berlin (2013)

    MATH  Google Scholar 

  5. Ammari, H., Kang, H.: Reconstruction of small inhomogeneities from boundary measurements. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  6. Ammari, H., Kang, H.: Polarization and moment tensors: with applications to inverse problems and effective medium theory, vol. 162. Springer, Berlin (2007)

    MATH  Google Scholar 

  7. Ammari, H., Kang, H., Lee, H., Lee, J., Lim, M.: Optimal estimates for the electric field in two dimensions. J. Math. Appl. 88(4), 307–324 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Ammari, H., Kang, H., Lee, H., Lim, M., Zribi, H.: Decomposition theorems and fine estimates for electrical fields in the presence of closely located circular inclusions. J. Differ. Equ. 247(11), 2897–2912 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ammari, H., Kang, H., Lim, M.: Gradient estimates for solutions to the conductivity problem. Math. Ann. 332(2), 277–286 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Babus̆ka, I., Andersson, B., Smith, P., Levin, K.: Damage analysis of fiber composites. I. Statistical analysis on fiber scale. Comput. Methods Appl. Mech. Engrg. 172, 27–77 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Batchelor, G.K., O’Brien, R.W.: Thermal or electrical conduction through a granular material. Proc. R. Soc. A 355, 313–333 (1977)

    Article  Google Scholar 

  12. Budiansky, B., Carrier, G.F.: High shear stresses in stiff fiber composites. J. Appl. Mech. 51, 733–735 (1984)

    Article  MATH  Google Scholar 

  13. Bao, E.S., Li, Y.Y., Yin, B.: Gradient estimates for the perfect conductivity problem. Arch. Ration. Mech. Anal. 193(1), 195–226 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bao, E.S., Li, Y.Y., Yin, B.: Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions. Commun. Partial Differ. Equ. 35(11), 1982–2006 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bao, J.G., Hongjie, J., Li, H.: Optimal boundary gradient estimates for Lamé systems with partially infinite coefficients. Adv. Math. 314, 583–629 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bao, J.G., Li, H.G., Li, Y.Y.: Gradient estimates for solutions of the Lamé system with partially infinite coefficients. Arch. Ration. Mech. Anal. 215(1), 307–351 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bao, J.G., Li, H.G., Li, Y.Y.: Gradient estimates for solutions of the Lamé system with partially infinite coefficients in dimensions greater than two. Adv. Math. 305, 298–338 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bonnetier, E., Triki, F.: Pointwise bounds on the gradient and the spectrum of the Neumann–Poincaré operator: the case of 2 discs. Contemp. Math 577, 81–92 (2012)

    Article  MATH  Google Scholar 

  19. Davis, M.H.: Two charged spherical conductors in a uniform electric field: forces and field strength. Q J Mech Appl Math 17(4), 499–511 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jeffery, G.B.: On a form of the solution of Laplace’s equation suitable for problems relating to two spheres. Proc. R. Soc. Lond. Series A, containing papers of a mathematical and physical character, 87(593):109–120 (1912)

  21. Kang, H., Lee, H., Yun, K.H.: Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions. Math. Ann. 363(3–4), 1281–1306 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kang, H., Lim, M., Yun, K.H.: Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities. J. Math. Pures Appl. 99(2), 234–249 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kang, H., Lim, M., Yun, K.H.: Characterization of the electric field concentration between two adjacent spherical perfect conductors. SIAM J. Appl. Math. 74(1), 125–146 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kang, H., Yu, S.: Quantitative characterization of stress concentration in the presence of closely spaced hard inclusions in two-dimensional linear elasticity. arXiv preprint arXiv:1707.02207 (2017)

  25. Kang, H., Yun, K.: Optimal estimates of the field enhancement in presence of a bow-tie structure of perfectly conducting inclusions in two dimensions. arXiv preprint arXiv:1707.00098 (2017)

  26. Keller, J.B.: Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders. J. Appl. Phys. 34(4), 991–993 (1963)

    Article  MATH  Google Scholar 

  27. Kellogg, O.D.: Foundations of potential theory, vol. 31. Springer, Berlin (2012)

    MATH  Google Scholar 

  28. Lassiter, J.B., Aizpurua, J., Hernandez, L.I., Brandl, D.W., Romero, I., Lal, S., Hafner, J.H., Nordlander, P., Halas, N.J.: Close encounters between two nanoshells. Nano Lett. 8(4), 1212–1218 (2008)

    Article  Google Scholar 

  29. Lekner, J.: Near approach of two conducting spheres: enhancement of external electric field. J. Electrostat. 69(6), 559–563 (2011)

    Article  Google Scholar 

  30. Li, H., Xu, L.: Optimal estimates for the perfect conductivity problem with inclusions close to the boundary. arXiv preprint. arXiv:1705.04459 (2017)

  31. Li, Y.Y., Vogelius, M.: Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients. Arch. Ration. Mech. Anal. 153(2), 91–151 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, Y., Nirenberg, L.: Estimates for elliptic systems from composite material. Commun. Pure Appl. Math. 56(7), 892–925 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lim, M., Yu, S.: Asymptotic analysis for superfocusing of the electric field in between two nearly touching metallic spheres. arXiv preprint arXiv:1412.2464 (2014)

  34. Lim, M., Sanghyeon, Y.: Asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities. J. Math. Anal. Appl. 421(1), 131–156 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lim, M., Yu, S.: Stress concentration for two nearly touching circular holes. arXiv preprint arXiv:1705.10400 (2017)

  36. Lim, M., Yun, K.: Blow-up of electric fields between closely spaced spherical perfect conductors. Commun. Partial Differ. Equ. 34(10), 1287–1315 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Maxwell, J.C.: A treatise on electricity and magnetism, vol. I, 3rd edn. Oxford University Press, Oxford (1891) [reprinted by Dover, New York (1954)]

  38. McPhedran, R.C., Movchan, A.B.: The Rayleigh multipole method for linear elasticity. J. Mech. Phys. Solids 42(5), 711–727 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. McPhedran, R.C., Poladian, L., Milton, G.W.: Asymptotic studies of closely spaced, highly conducting cylinders. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 415, 185–196 (1988)

    Article  Google Scholar 

  40. Poladian, L.: General theory of electrical images in sphere pairs. Q. J. Mech. Appl. Math. 41(3), 395–417 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  41. Poladian, L.: Asymptotic behaviour of the effective dielectric constants of composite materials. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 426, 343–359 (1989)

    Article  Google Scholar 

  42. Romero, I., Aizpurua, J., Bryant, G.W., Javier, F., de Abajo, G.: Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers. Opt. Express 14, 9988–9999 (2006)

    Article  Google Scholar 

  43. Smythe, W.B.: Static and dynamic electricity. McGraw-Hill (1950)

  44. Yu, S., Ammari, H.: Plasmonic interaction between nanospheres. SIAM Rev. 60(2), 356–385 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  45. Yu, S., Lim, M.: Shielding at a distance due to anomalous resonance. New J. Phys. 19(3), 033018 (2017)

    Article  Google Scholar 

  46. Yun, K.: Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape. SIAM J. Appl. Math. 67(3), 714–730 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  47. Yun, K.H.: Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections. J. Math. Anal. Appl. 350(1), 306–312 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Yun, K.: An optimal estimate for electric fields on the shortest line segment between two spherical insulators in three dimensions. arXiv:1504.07679

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, 34141, Korea

    Junbeom Kim & Mikyoung Lim

Authors
  1. Junbeom Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikyoung Lim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikyoung Lim.

Additional information

Communicated by Y. Giga.

This work is supported by the Korean Ministry of Science, ICT and Future Planning through NRF Grant No. 2016R1A2B4014530 (to ML and JK).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, J., Lim, M. Electric field concentration in the presence of an inclusion with eccentric core-shell geometry. Math. Ann. 373, 517–551 (2019). https://doi.org/10.1007/s00208-018-1688-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-018-1688-6

Search

Navigation