×

Asymptotics for the concentrated field between closely located hard inclusions in all dimensions. (English) Zbl 1478.78042

Summary: 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 H. Li et al. [Multiscale Model. Simul. 17, No. 3, 899–925 (2019; Zbl 1426.78022)] 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 \(m,n \geq 2\), where \(m\) and \(n\) are the parameters of convexity and dimension, respectively, which is only partially solved in [H. Li, SIAM J. Math. Anal. 52, No. 4, 3350–3375 (2020; Zbl 1447.35140)].

MSC:

78A48 Composite media; random media in optics and electromagnetic theory
35B44 Blow-up in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35C20 Asymptotic expansions of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI

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. · Zbl 1319.35255
[2] H. Ammari; G. Ciraolo; H. Kang; H. Lee; K. Yun, Spectral analysis of the Neumann-Poincaré operator and characterization of the stress concentration in anti-plane elasticity, Arch. Ration. Mech. Anal., 208, 275-304 (2013) · Zbl 1320.35165
[3] H. Ammari, H. Kang and M. Lim, Gradient estimates to the conductivity problem, Math. Ann. 332 (2005), 277-286. · Zbl 1129.78308
[4] H. Ammari; H. Kang; H. Lee; J. Lee; M. Lim, Optimal estimates for the electric field in two dimensions, J. Math. Pures Appl., 88, 307-324 (2007) · Zbl 1136.35095
[5] I. Babuška; B. Andersson; P. Smith; K. Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg., 172, 27-77 (1999) · Zbl 0956.74048
[6] B. Budiansky; G. F. Carrier, High shear stresses in stiff fiber composites, J. App. Mech., 51, 733-735 (1984) · Zbl 0549.73061
[7] M. Briane; Y. Capdeboscq; L. Nguyen, Interior regularity estimates in high conductivity homogenization and application, Arch. Ration. Mech. Anal., 207, 75-137 (2013) · Zbl 1269.78021
[8] J. G. Bao; H. G. Li; Y. Y. Li, Gradient estimates for solutions of the Lamé system with partially infinite coefficients, Arch. Ration. Mech. Anal., 215, 307-351 (2015) · Zbl 1309.35160
[9] J. G. Bao; H. G. Li; Y. Y. Li, Gradient estimates for solutions of the Lamé system with partially infinite coefficients in dimensions greater than two, Adv. Math., 305, 298-338 (2017) · Zbl 1353.35079
[10] E. Bao; Y. Y. Li; B. Yin, Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal., 193, 195-226 (2009) · Zbl 1173.78002
[11] E. Bao; Y. Y. Li; B. Yin, Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Commun. Partial Differ. Equ., 35, 1982-2006 (2010) · Zbl 1218.35230
[12] E. Bonnetier; F. Triki, On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D, Arch. Ration. Mech. Anal., 209, 541-567 (2013) · Zbl 1280.49029
[13] E. Bonnetier; M. Vogelius, An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section, SIAM J. Math. Anal., 31, 651-677 (2000) · Zbl 0947.35044
[14] V. M. Calo; Y. Efendiev; J. Galvis, Asymptotic expansions for high-contrast elliptic equations, Math. Models Methods Appl. Sci., 24, 465-494 (2014) · Zbl 1292.35100
[15] G. Ciraolo; A. Sciammetta, Gradient estimates for the perfect conductivity problem in anisotropic media, J. Math. Pures Appl., 127, 268-298 (2019) · Zbl 1421.35078
[16] G. Ciraolo; A. Sciammetta, Stress concentration for closely located inclusions in nonlinear perfect conductivity problems, J. Differ. Equ., 266, 6149-6178 (2019) · Zbl 1421.35166
[17] H. J. Dong; H. G. Li, Optimal estimates for the conductivity problem by Green’s function method, Arch. Ration. Mech. Anal., 231, 1427-1453 (2019) · Zbl 1412.35082
[18] Y. Gorb; A. Novikov, Blow-up of solutions to a p-Laplace equation, Multiscale Model. Simul., 10, 727-743 (2012) · Zbl 1261.35137
[19] Y. Gorb, Singular behavior of electric field of high-contrast concentrated composites, Multiscale Model. Simul., 13, 1312-1326 (2015) · Zbl 1333.35266
[20] J. B. Keller, Stresses in narrow regions, Trans. ASME J. APPl. Mech., 60 (1993) 1054-1056. · Zbl 0803.73042
[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. · Zbl 1411.74047
[22] H. Kang; M. Lim; K. Yun, Characterization of the electric field concentration between two adjacent spherical perfect conductors, SIAM J. Appl. Math., 74, 125-146 (2014) · Zbl 1293.35314
[23] H. Kang; H. Lee; K. Yun, Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions, Math. Ann., 363, 1281-1306 (2015) · Zbl 1398.35037
[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 · Zbl 1371.78033
[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. · Zbl 1298.35220
[26] H. G. Li; Y. Y. Li; Z. L. Yang, Asymptotics of the gradient of solutions to the perfect conductivity problem, Multiscale Model. Simul., 17, 899-925 (2019) · Zbl 1426.78022
[27] H. G. Li; F. Wang; L. J. Xu, Characterization of electric fields between two spherical perfect conductors with general radii in 3D, J. Differ. Equ., 267, 6644-6690 (2019) · Zbl 1428.35557
[28] H. G. Li; L. J. Xu, Optimal estimates for the perfect conductivity problem with inclusions close to the boundary, SIAM J. Math. Anal., 49, 3125-3142 (2017) · Zbl 1379.35100
[29] H. G. Li, Asymptotics for the electric field concentration in the perfect conductivity problem, SIAM J. Math. Anal., 52, 3350-3375 (2020) · Zbl 1447.35140
[30] H. G. Li; 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, 3777-3817 (2020) · Zbl 1448.74027
[31] Y. Y. Li; L. Nirenberg, Estimates for elliptic system from composite material, Commun. Pure Appl. Math., 56, 892-925 (2003) · Zbl 1125.35339
[32] Y. Y. Li; M. Vogelius, Gradient stimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal., 153, 91-151 (2000) · Zbl 0958.35060
[33] M. Lim; K. Yun, Blow-up of electric fields between closely spaced spherical perfect conductors, Commun. Partial Differ. Equ., 34, 1287-1315 (2009) · Zbl 1188.78011
[34] K. Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math., 67, 714-730 (2007) · Zbl 1189.35324
[35] K. Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal. Appl., 350, 306-312 (2009) · Zbl 1198.35042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.