Optimal gradient estimates for the perfect conductivity problem with C 1,α inclusions
Chen, Yu1 ; Li, Haigang1 ; Xu, Longjuan1
1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China
Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 953-979.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the distance between two adjacent inclusions. In this paper, we derive upper and lower bounds of the gradient of solutions to the conductivity problem where two perfectly conducting inclusions are located very close to each other. To be specific, we extend the known results of Bao-Li-Yin (ARMA 2009) in two folds: First, we weaken the smoothness of the inclusions from C2,α to C1,α. To obtain a pointwise upper bound of the gradient, we follow an iteration technique which is first used to deal with elliptic systems in a narrow domain by Li-Li-Bao-Yin (QAM 2014). However, when the inclusions are of C1,α, we can not use W2,p estimates for elliptic equations any more. In order to overcome this new difficulty, we take advantage of De Giorgi-Nash estimates and Campanato's approach to apply an adapted version of the iteration technique with respect to the energy. A lower bound in the shortest line between two inclusions is also obtained to show the optimality of the blow-up rate. Second, when two inclusions are only convex but not strictly convex, we prove that blow-up does not occur any more. The establishment of the relationship between the blow-up rate of the gradient and the order of the convexity of the inclusions reveals the mechanism of such concentration phenomenon.

DOI : 10.1016/j.anihpc.2020.09.009
Mots-clés : Perfect conductivity problem, Gradient estimates, Blow-up rates
Chen, Yu 1 ; Li, Haigang 1 ; Xu, Longjuan 1

1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China 
@article{AIHPC_2021__38_4_953_0,
     author = {Chen, Yu and Li, Haigang and Xu, Longjuan},
     title = {Optimal gradient estimates for the perfect conductivity problem with {\protect\emph{C}
}               \protect\textsuperscript{1,\protect\emph{\ensuremath{\alpha}}
               } inclusions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {953--979},
     publisher = {Elsevier},
     volume = {38},
     number = {4},
     year = {2021},
     doi = {10.1016/j.anihpc.2020.09.009},
     mrnumber = {4266231},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.009/}
}
TY  - JOUR
AU  - Chen, Yu
AU  - Li, Haigang
AU  - Xu, Longjuan
TI  - Optimal gradient estimates for the perfect conductivity problem with C
               1,α
                inclusions
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2021
SP  - 953
EP  - 979
VL  - 38
IS  - 4
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.009/
DO  - 10.1016/j.anihpc.2020.09.009
LA  - en
ID  - AIHPC_2021__38_4_953_0
ER  - 
%0 Journal Article
%A Chen, Yu
%A Li, Haigang
%A Xu, Longjuan
%T Optimal gradient estimates for the perfect conductivity problem with C
               1,α
                inclusions
%J Annales de l'I.H.P. Analyse non linéaire
%D 2021
%P 953-979
%V 38
%N 4
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.009/
%R 10.1016/j.anihpc.2020.09.009
%G en
%F AIHPC_2021__38_4_953_0
Chen, Yu; Li, Haigang; Xu, Longjuan. Optimal gradient estimates for the perfect conductivity problem with C
               1,α
                inclusions. Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 953-979. doi : 10.1016/j.anihpc.2020.09.009. http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.009/

[1] Ammari, H.; Kang, H.; Lim, M. Gradient estimates to the conductivity problem, Math. Ann., Volume 332 (2005), pp. 277-286 | DOI | MR | Zbl

[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., Volume 208 (2013), pp. 275-304 | DOI | MR | Zbl

[3] Ammari, H.; Dassios, H.; Kang, H.; Lim, M. Estimates for the electric field in the presence of adjacent perfectly conducting spheres, Q. Appl. Math., Volume 65 (2007), pp. 339-355 | DOI | MR | Zbl

[4] Ammari, H.; Kang, H.; Lee, H.; Lee, J.; Lim, M. Optimal estimates for the electrical field in two dimensions, J. Math. Pures Appl., Volume 88 (2007), pp. 307-324 | DOI | MR | Zbl

[5] 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., Volume 247 (2009), pp. 2897-2912 | DOI | MR | Zbl

[6] Babuška, I.; Andersson, B.; Smith, P.; Levin, K. Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Eng., Volume 172 (1999), pp. 27-77 | DOI | MR | Zbl

[7] Bao, E.; Li, Y.Y.; Yin, B. Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal., Volume 193 (2009), pp. 195-226 | DOI | MR | Zbl

[8] Bao, E.; Li, Y.Y.; Yin, B. Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Commun. Partial Differ. Equ., Volume 35 (2010), pp. 1982-2006 | DOI | MR | Zbl

[9] Bao, J.G.; Ju, H.J.; Li, H.G. Optimal boundary gradient estimates for Lamé systems with partially infinite coefficients, Adv. Math., Volume 314 (2017), pp. 583-629 | DOI | MR

[10] 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., Volume 215 (2015) no. 1, pp. 307-351 | DOI | MR | Zbl

[11] 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., Volume 305 (2017), pp. 298-338 | DOI | MR

[12] Bonnetier, E.; Triki, F. Pointwise bounds on the gradient and the spectrum of the Neumann-Poincaré operator: the case of 2 discs, Multi-Scale and High-Contrast PDE: From Modeling, to Mathematical Analysis, to Inversion, Contemp. Math., vol. 577, Amer. Math. Soc., Providence, RI, 2012, pp. 81-91 | DOI | MR | Zbl

[13] Bonnetier, E.; Triki, F. On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D, Arch. Ration. Mech. Anal., Volume 209 (2013), pp. 541-567 | DOI | MR | Zbl

[14] Bonnetier, E.; Vogelius, M. An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section, SIAM J. Math. Anal., Volume 31 (2000), pp. 651-677 | DOI | MR | Zbl

[15] Budiansky, B.; Carrier, G.F. High shear stresses in stiff fiber composites, J. Appl. Mech., Volume 51 (1984), pp. 733-735 | DOI | Zbl

[16] Dong, H.J. Gradient estimates for parabolic and elliptic systems from linear laminates, Arch. Ration. Mech. Anal., Volume 205 (2012), pp. 119-149 | DOI | MR | Zbl

[17] Dong, H.J.; Li, H.G. Optimal estimates for the conductivity problem by Green's function method, Arch. Ration. Mech. Anal., Volume 231 (2019), pp. 1427-1453 | DOI | MR

[18] Dong, H.J.; Kim, S. On C1, C2, and weak type-(1, 1) estimates for linear elliptic operators, Commun. Partial Differ. Equ., Volume 42 (2017), pp. 417-435 | DOI | MR

[19] Dong, H.J.; Escauriaza, L.; Kim, S. On C1, C2, and weak type-(1, 1) estimates for linear elliptic operators: Part 2, Math. Ann., Volume 370 (2018), pp. 447-489 | DOI | MR

[20] Dong, H.J.; Zhang, H. On an elliptic equation arising from composite materials, Arch. Ration. Mech. Anal., Volume 222 (2016), pp. 47-89 | DOI | MR

[21] Giaquinta, M.; Martinazzi, L. An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Springer Science Business Media, 2013

[22] Gorb, Y.; Novikov, A. Blow-up of solutions to a p -Laplace equation, Multiscale Model. Simul., Volume 10 (2012), pp. 727-743 | DOI | MR | Zbl

[23] Gutiérrez, C.E. The Monge-Ampère Equation, Progress in Nonlinear Differential Equations and Their Applications, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 2001 | MR | Zbl

[24] Gilbarg, D.; Trudinger, N.S. Elliptic Partial Differential Equations of Second Order, Springer, 1998

[25] Kang, H.; Lee, H.; Yun, K. Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions, Math. Ann., Volume 363 (2015), pp. 1281-1306 | DOI | MR

[26] Kang, H.; Lim, M.; Yun, K. Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities, J. Math. Pures Appl. (9), Volume 99 (2013), pp. 234-249 | DOI | MR | Zbl

[27] Kang, H.; Lim, M.; Yun, K. Characterization of the electric field concentration between two adjacent spherical perfect conductors, SIAM J. Appl. Math., Volume 74 (2014), pp. 125-146 | DOI | MR | Zbl

[28] Kang, H.; Yu, S. Quantitative characterization of stress concentration in the presence of closely spaced hard inclusions in two-dimensional linear elasticity, Arch. Ration. Mech. Anal., Volume 232 (2019), pp. 121-196 | DOI | MR

[29] Kang, H.; Yun, K. Optimal estimates of the field enhancement in presence of a bow-tie structure of perfectly conducting inclusions in two dimensions, J. Differ. Equ., Volume 266 (2019), pp. 5064-5094 | DOI | MR

[30] Keller, J.B. Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys., Volume 34 (1963), pp. 991-993 | DOI | Zbl

[31] Kozlov, V.A.; Maz'ya, V.G.; Rossmann, J. Elliptic Boundary Value Problems in Domains with Point Singularities, Math. Surveys Monogr., vol. 52, Amer. Math. Soc., Providence, 1997 | MR | Zbl

[32] Li, H.G. Asymptotics for the electric field concentration in the perfect conductivity problem, SIAM J. Math. Anal., Volume 52 (2020), pp. 3350-3375 | DOI | MR

[33] Li, H.G. Lower bounds of gradient's blow-up for the Lamé system with partially infinite coefficients, J. Math. Pures Appl. (2020) (in press) | DOI | MR

[34] Li, H.G.; Li, Y.Y.; Yang, Z.L. Asymptotics of the gradient of solutions to the perfect conductivity problem, Multiscale Model. Simul., Volume 17 (2019), pp. 899-925 | DOI | MR

[35] Li, H.G.; Wang, F.; Xu, L.J. Characterization of electric fields between two spherical perfect conductors with general radii in 3D, J. Differ. Equ., Volume 267 (2019), pp. 6644-6690 | DOI | MR

[36] Ju, H.J.; Li, H.G.; Xu, L.J. Estimates for elliptic systems in a narrow region arising from composite materials, Q. Appl. Math., Volume 77 (2019), pp. 177-199 | DOI | MR

[37] Li, H.G.; Li, Y.Y.; Bao, E.S.; Yin, B. Derivative estimates of solutions of elliptic systems in narrow regions, Q. Appl. Math., Volume 72 (2014), pp. 589-596 | DOI | MR | Zbl

[38] Li, H.G.; Li, Y.Y. Gradient estimates for parabolic systems from composite material, Sci. China Math., Volume 60 (2017), pp. 2011-2052 | DOI | MR

[39] Li, H.G.; Xu, L.J. Optimal estimates for the perfect conductivity problem with inclusions close to the boundary, SIAM J. Math. Anal., Volume 49 (2017), pp. 3125-3142 | DOI | MR

[40] Li, H.G.; Zhao, Z.W. Boundary blow-up analysis of gradient estimates for Lamé systems in the presence of M-convex hard inclusions, SIAM J. Math. Anal., Volume 52 (2020), pp. 3777-3817 | DOI | MR

[41] Li, Y.Y.; Nirenberg, L. Estimates for elliptic system from composite material, Commun. Pure Appl. Math., Volume 56 (2003), pp. 892-925 | DOI | MR | Zbl

[42] Li, Y.Y.; Vogelius, M. Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal., Volume 153 (2000), pp. 91-151 | DOI | MR | Zbl

[43] Lim, M.; Yu, S. Stress concentration for two nearly touching circular holes, 2017 | arXiv

[44] Lim, M.; Yun, K. Strong influence of a small fiber on shear stress in fiber-reinforced composites, J. Differ. Equ., Volume 250 (2011), pp. 2402-2439 | DOI | MR | Zbl

[45] Lim, M.; Yun, K. Blow-up of electric fields between closely spaced spherical perfect conductors, Commun. Partial Differ. Equ., Volume 34 (2009), pp. 1287-1315 | DOI | MR | Zbl

[46] Markenscoff, X. Stress amplification in vanishingly small geometries, Comput. Mech., Volume 19 (1996), pp. 77-83 | DOI | Zbl

[47] Yun, K. Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math., Volume 67 (2007), pp. 714-730 | DOI | MR | Zbl

[48] Yun, H. An optimal estimate for electric fields on the shortest line segment between two spherical insulators in three dimensions, J. Differ. Equ., Volume 261 (2016), pp. 148-188 | DOI | MR

Cité par Sources :