June  2021, 20(6): 2379-2398. doi: 10.3934/cpaa.2021086

Asymptotics for the concentrated field between closely located hard inclusions in all dimensions

a. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

b. 

Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands

* Corresponding author

Received  December 2020 Revised  April 2021 Published  May 2021

Fund Project: Z. W. Zhao was partially supported by NSFC (11971061) and BJNSF (1202013)

When hard inclusions are frequently spaced very closely, the electric field, which is the gradient of the solution to the perfect conductivity equation, may be arbitrarily large as the distance between two inclusions goes to zero. In this paper, our objectives are two-fold: first, we extend the asymptotic expansions of [26] to the higher dimensions greater than three by capturing the blow-up factors in all dimensions, which consist of some certain integrals of the solutions to the case when two inclusions are touching; second, our results answer the optimality of the blow-up rate for any $ m,n\geq2 $, where $ m $ and $ n $ are the parameters of convexity and dimension, respectively, which is only partially solved in [29].

Citation: Zhiwen Zhao, Xia Hao. Asymptotics for the concentrated field between closely located hard inclusions in all dimensions. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2379-2398. doi: 10.3934/cpaa.2021086
References:
[1]

H. Ammari, E. Bonnetier, F. Triki and M. Vogelius, Elliptic estimates in composite media with smooth inclusions: an integral equation approach, Ann. Sci. Éc. Norm. Supér., 48 (2015), 453–495. doi: 10.24033/asens. 2249.  Google Scholar

[2]

H. AmmariG. CiraoloH. KangH. Lee and K. Yun, Spectral analysis of the Neumann-Poincaré operator and characterization of the stress concentration in anti-plane elasticity, Arch. Ration. Mech. Anal., 208 (2013), 275-304.  doi: 10.1007/s00205-012-0590-8.  Google Scholar

[3]

H. Ammari, H. Kang and M. Lim, Gradient estimates to the conductivity problem, Math. Ann. 332 (2005), 277-286. doi: 10.1007/s00208-004-0626-y.  Google Scholar

[4]

H. AmmariH. KangH. LeeJ. Lee and M. Lim, Optimal estimates for the electric field in two dimensions, J. Math. Pures Appl., 88 (2007), 307-324.  doi: 10.1016/j.matpur.2007.07.005.  Google Scholar

[5]

I. BabuškaB. AnderssonP. Smith and K. Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg., 172 (1999), 27-77.  doi: 10.1016/S0045-7825(98)00225-4.  Google Scholar

[6]

B. Budiansky and G. F. Carrier, High shear stresses in stiff fiber composites, J. App. Mech., 51 (1984), 733-735.   Google Scholar

[7]

M. BrianeY. Capdeboscq and L. Nguyen, Interior regularity estimates in high conductivity homogenization and application, Arch. Ration. Mech. Anal., 207 (2013), 75-137.  doi: 10.1007/s00205-012-0553-0.  Google Scholar

[8]

J. G. BaoH. G. Li and Y. Y. Li, Gradient estimates for solutions of the Lamé system with partially infinite coefficients, Arch. Ration. Mech. Anal., 215 (2015), 307-351.  doi: 10.1007/s00205-014-0779-0.  Google Scholar

[9]

J. G. BaoH. G. Li and Y. Y. Li, Gradient estimates for solutions of the Lamé system with partially infinite coefficients in dimensions greater than two, Adv. Math., 305 (2017), 298-338.  doi: 10.1016/j.aim.2016.09.023.  Google Scholar

[10]

E. BaoY.Y. Li and B. Yin, Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal., 193 (2009), 195-226.  doi: 10.1007/s00205-008-0159-8.  Google Scholar

[11]

E. BaoY. Y. Li and B. Yin, Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Commun. Partial Differ. Equ., 35 (2010), 1982-2006.  doi: 10.1080/03605300903564000.  Google Scholar

[12]

E. Bonnetier and F. Triki, On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D, Arch. Ration. Mech. Anal., 209 (2013), 541-567.  doi: 10.1007/s00205-013-0636-6.  Google Scholar

[13]

E. Bonnetier and M. Vogelius, An elliptic regularity result for a composite medium with "touching" fibers of circular cross-section, SIAM J. Math. Anal., 31 (2000), 651-677.  doi: 10.1137/S0036141098333980.  Google Scholar

[14]

V. M. CaloY. Efendiev and J. Galvis, Asymptotic expansions for high-contrast elliptic equations, Math. Models Methods Appl. Sci., 24 (2014), 465-494.  doi: 10.1142/S0218202513500565.  Google Scholar

[15]

G. Ciraolo and A. Sciammetta, Gradient estimates for the perfect conductivity problem in anisotropic media, J. Math. Pures Appl., 127 (2019), 268-298.  doi: 10.1016/j.matpur.2018.09.006.  Google Scholar

[16]

G. Ciraolo and A. Sciammetta, Stress concentration for closely located inclusions in nonlinear perfect conductivity problems, J. Differ. Equ., 266 (2019), 6149-6178.  doi: 10.1016/j.jde.2018.10.041.  Google Scholar

[17]

H. J. Dong and H. G. Li, Optimal estimates for the conductivity problem by Green's function method, Arch. Ration. Mech. Anal., 231 (2019), 1427-1453.  doi: 10.1007/s00205-018-1301-x.  Google Scholar

[18]

Y. Gorb and A. Novikov, Blow-up of solutions to a p-Laplace equation, Multiscale Model. Simul., 10 (2012), 727-743.  doi: 10.1137/110857167.  Google Scholar

[19]

Y. Gorb, Singular behavior of electric field of high-contrast concentrated composites, Multiscale Model. Simul., 13 (2015), 1312-1326.  doi: 10.1137/140982076.  Google Scholar

[20]

J. B. Keller, Stresses in narrow regions, Trans. ASME J. APPl. Mech., 60 (1993) 1054–1056. Google Scholar

[21]

H. Kang, M. Lim and K. 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), 234–249. doi: 10.1016/j. matpur. 2012.06.013.  Google Scholar

[22]

H. KangM. Lim and K. Yun, Characterization of the electric field concentration between two adjacent spherical perfect conductors, SIAM J. Appl. Math., 74 (2014), 125-146.  doi: 10.1137/130922434.  Google Scholar

[23]

H. KangH. Lee and K. Yun, Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions, Math. Ann., 363 (2015), 1281-1306.  doi: 10.1007/s00208-015-1203-2.  Google Scholar

[24]

J. Lekner, Electrostatics of two charged conducting spheres, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 (2012), no. 2145, 2829-2848 doi: 10.1098/rspa. 2012.0133.  Google Scholar

[25]

H. G. Li, Y. Y. Li, E. S. Bao and B. Yin, Derivative estimates of solutions of elliptic systems in narrow regions, Quart. Appl. Math. 72 (2014), no. 3, 589-596. doi: 10.1090/S0033-569X-2014-01339-0.  Google Scholar

[26]

H. G. LiY. Y. Li and Z. L. Yang, Asymptotics of the gradient of solutions to the perfect conductivity problem, Multiscale Model. Simul., 17 (2019), 899-925.  doi: 10.1137/18M1214329.  Google Scholar

[27]

H. G. LiF. Wang and L. J. Xu, Characterization of electric fields between two spherical perfect conductors with general radii in 3D, J. Differ. Equ., 267 (2019), 6644-6690.  doi: 10.1016/j.jde.2019.07.007.  Google Scholar

[28]

H. G. Li and L. J. Xu, Optimal estimates for the perfect conductivity problem with inclusions close to the boundary, SIAM J. Math. Anal., 49 (2017), 3125-3142.  doi: 10.1137/16M1067858.  Google Scholar

[29]

H. G. Li, Asymptotics for the electric field concentration in the perfect conductivity problem, SIAM J. Math. Anal., 52 (2020), 3350-3375.  doi: 10.1137/19M1282623.  Google Scholar

[30]

H. G. Li and Z. W. Zhao, Boundary Blow-Up Analysis of Gradient Estimates for Lamé Systems in the Presence of m-Convex Hard Inclusions, SIAM J. Math. Anal., 52 (2020), 3777-3817.  doi: 10.1137/19M1306038.  Google Scholar

[31]

Y. Y. Li and L. Nirenberg, Estimates for elliptic system from composite material, Commun. Pure Appl. Math., 56 (2003), 892-925.  doi: 10.1002/cpa.10079.  Google Scholar

[32]

Y. Y. Li and M. Vogelius, Gradient stimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal., 153 (2000), 91-151.  doi: 10.1007/s002050000082.  Google Scholar

[33]

M. Lim and K. Yun, Blow-up of electric fields between closely spaced spherical perfect conductors, Commun. Partial Differ. Equ., 34 (2009), 1287-1315.  doi: 10.1080/03605300903079579.  Google Scholar

[34]

K. Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math., 67 (2007), 714-730.  doi: 10.1137/060648817.  Google Scholar

[35]

K. Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal. Appl., 350 (2009), 306-312.  doi: 10.1016/j.jmaa.2008.09.057.  Google Scholar

show all references

References:
[1]

H. Ammari, E. Bonnetier, F. Triki and M. Vogelius, Elliptic estimates in composite media with smooth inclusions: an integral equation approach, Ann. Sci. Éc. Norm. Supér., 48 (2015), 453–495. doi: 10.24033/asens. 2249.  Google Scholar

[2]

H. AmmariG. CiraoloH. KangH. Lee and K. Yun, Spectral analysis of the Neumann-Poincaré operator and characterization of the stress concentration in anti-plane elasticity, Arch. Ration. Mech. Anal., 208 (2013), 275-304.  doi: 10.1007/s00205-012-0590-8.  Google Scholar

[3]

H. Ammari, H. Kang and M. Lim, Gradient estimates to the conductivity problem, Math. Ann. 332 (2005), 277-286. doi: 10.1007/s00208-004-0626-y.  Google Scholar

[4]

H. AmmariH. KangH. LeeJ. Lee and M. Lim, Optimal estimates for the electric field in two dimensions, J. Math. Pures Appl., 88 (2007), 307-324.  doi: 10.1016/j.matpur.2007.07.005.  Google Scholar

[5]

I. BabuškaB. AnderssonP. Smith and K. Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg., 172 (1999), 27-77.  doi: 10.1016/S0045-7825(98)00225-4.  Google Scholar

[6]

B. Budiansky and G. F. Carrier, High shear stresses in stiff fiber composites, J. App. Mech., 51 (1984), 733-735.   Google Scholar

[7]

M. BrianeY. Capdeboscq and L. Nguyen, Interior regularity estimates in high conductivity homogenization and application, Arch. Ration. Mech. Anal., 207 (2013), 75-137.  doi: 10.1007/s00205-012-0553-0.  Google Scholar

[8]

J. G. BaoH. G. Li and Y. Y. Li, Gradient estimates for solutions of the Lamé system with partially infinite coefficients, Arch. Ration. Mech. Anal., 215 (2015), 307-351.  doi: 10.1007/s00205-014-0779-0.  Google Scholar

[9]

J. G. BaoH. G. Li and Y. Y. Li, Gradient estimates for solutions of the Lamé system with partially infinite coefficients in dimensions greater than two, Adv. Math., 305 (2017), 298-338.  doi: 10.1016/j.aim.2016.09.023.  Google Scholar

[10]

E. BaoY.Y. Li and B. Yin, Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal., 193 (2009), 195-226.  doi: 10.1007/s00205-008-0159-8.  Google Scholar

[11]

E. BaoY. Y. Li and B. Yin, Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Commun. Partial Differ. Equ., 35 (2010), 1982-2006.  doi: 10.1080/03605300903564000.  Google Scholar

[12]

E. Bonnetier and F. Triki, On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D, Arch. Ration. Mech. Anal., 209 (2013), 541-567.  doi: 10.1007/s00205-013-0636-6.  Google Scholar

[13]

E. Bonnetier and M. Vogelius, An elliptic regularity result for a composite medium with "touching" fibers of circular cross-section, SIAM J. Math. Anal., 31 (2000), 651-677.  doi: 10.1137/S0036141098333980.  Google Scholar

[14]

V. M. CaloY. Efendiev and J. Galvis, Asymptotic expansions for high-contrast elliptic equations, Math. Models Methods Appl. Sci., 24 (2014), 465-494.  doi: 10.1142/S0218202513500565.  Google Scholar

[15]

G. Ciraolo and A. Sciammetta, Gradient estimates for the perfect conductivity problem in anisotropic media, J. Math. Pures Appl., 127 (2019), 268-298.  doi: 10.1016/j.matpur.2018.09.006.  Google Scholar

[16]

G. Ciraolo and A. Sciammetta, Stress concentration for closely located inclusions in nonlinear perfect conductivity problems, J. Differ. Equ., 266 (2019), 6149-6178.  doi: 10.1016/j.jde.2018.10.041.  Google Scholar

[17]

H. J. Dong and H. G. Li, Optimal estimates for the conductivity problem by Green's function method, Arch. Ration. Mech. Anal., 231 (2019), 1427-1453.  doi: 10.1007/s00205-018-1301-x.  Google Scholar

[18]

Y. Gorb and A. Novikov, Blow-up of solutions to a p-Laplace equation, Multiscale Model. Simul., 10 (2012), 727-743.  doi: 10.1137/110857167.  Google Scholar

[19]

Y. Gorb, Singular behavior of electric field of high-contrast concentrated composites, Multiscale Model. Simul., 13 (2015), 1312-1326.  doi: 10.1137/140982076.  Google Scholar

[20]

J. B. Keller, Stresses in narrow regions, Trans. ASME J. APPl. Mech., 60 (1993) 1054–1056. Google Scholar

[21]

H. Kang, M. Lim and K. 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), 234–249. doi: 10.1016/j. matpur. 2012.06.013.  Google Scholar

[22]

H. KangM. Lim and K. Yun, Characterization of the electric field concentration between two adjacent spherical perfect conductors, SIAM J. Appl. Math., 74 (2014), 125-146.  doi: 10.1137/130922434.  Google Scholar

[23]

H. KangH. Lee and K. Yun, Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions, Math. Ann., 363 (2015), 1281-1306.  doi: 10.1007/s00208-015-1203-2.  Google Scholar

[24]

J. Lekner, Electrostatics of two charged conducting spheres, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 (2012), no. 2145, 2829-2848 doi: 10.1098/rspa. 2012.0133.  Google Scholar

[25]

H. G. Li, Y. Y. Li, E. S. Bao and B. Yin, Derivative estimates of solutions of elliptic systems in narrow regions, Quart. Appl. Math. 72 (2014), no. 3, 589-596. doi: 10.1090/S0033-569X-2014-01339-0.  Google Scholar

[26]

H. G. LiY. Y. Li and Z. L. Yang, Asymptotics of the gradient of solutions to the perfect conductivity problem, Multiscale Model. Simul., 17 (2019), 899-925.  doi: 10.1137/18M1214329.  Google Scholar

[27]

H. G. LiF. Wang and L. J. Xu, Characterization of electric fields between two spherical perfect conductors with general radii in 3D, J. Differ. Equ., 267 (2019), 6644-6690.  doi: 10.1016/j.jde.2019.07.007.  Google Scholar

[28]

H. G. Li and L. J. Xu, Optimal estimates for the perfect conductivity problem with inclusions close to the boundary, SIAM J. Math. Anal., 49 (2017), 3125-3142.  doi: 10.1137/16M1067858.  Google Scholar

[29]

H. G. Li, Asymptotics for the electric field concentration in the perfect conductivity problem, SIAM J. Math. Anal., 52 (2020), 3350-3375.  doi: 10.1137/19M1282623.  Google Scholar

[30]

H. G. Li and Z. W. Zhao, Boundary Blow-Up Analysis of Gradient Estimates for Lamé Systems in the Presence of m-Convex Hard Inclusions, SIAM J. Math. Anal., 52 (2020), 3777-3817.  doi: 10.1137/19M1306038.  Google Scholar

[31]

Y. Y. Li and L. Nirenberg, Estimates for elliptic system from composite material, Commun. Pure Appl. Math., 56 (2003), 892-925.  doi: 10.1002/cpa.10079.  Google Scholar

[32]

Y. Y. Li and M. Vogelius, Gradient stimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal., 153 (2000), 91-151.  doi: 10.1007/s002050000082.  Google Scholar

[33]

M. Lim and K. Yun, Blow-up of electric fields between closely spaced spherical perfect conductors, Commun. Partial Differ. Equ., 34 (2009), 1287-1315.  doi: 10.1080/03605300903079579.  Google Scholar

[34]

K. Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math., 67 (2007), 714-730.  doi: 10.1137/060648817.  Google Scholar

[35]

K. Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal. Appl., 350 (2009), 306-312.  doi: 10.1016/j.jmaa.2008.09.057.  Google Scholar

Figure 1.  Two close-to-touching disks
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