December  2020, 15(4): 581-603. doi: 10.3934/nhm.2020015

Perturbation analysis of the effective conductivity of a periodic composite

1. 

Dipartimento di Matematica 'Tullio Levi-Civita', Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy

2. 

Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca' Foscari Venezia, via Torino 155, 30172 Venezia Mestre, Italy

* Corresponding author: Paolo Musolino

Received  March 2020 Revised  June 2020 Published  July 2020

We consider the effective conductivity $ \lambda^{\mathrm{eff}} $ of a periodic two-phase composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material. Then we study the behavior of $ \lambda^{\mathrm{eff}} $ upon perturbation of the shape of the inclusions, of the periodicity structure, and of the conductivity of each material.

Citation: Paolo Luzzini, Paolo Musolino. Perturbation analysis of the effective conductivity of a periodic composite. Networks & Heterogeneous Media, 2020, 15 (4) : 581-603. doi: 10.3934/nhm.2020015
References:
[1]

G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  Google Scholar

[2]

H. Ammari and H. Kang, Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, 162, Springer, New York, 2007.  Google Scholar

[3]

H. AmmariH. Kang and K. Kim, Polarization tensors and effective properties of anisotropic composite materials, J. Differential Equations, 215 (2005), 401-428.  doi: 10.1016/j.jde.2004.09.010.  Google Scholar

[4]

H. AmmariH. Kang and M. Lim, Effective parameters of elastic composites, Indiana Univ. Math. J., 55 (2006), 903-922.  doi: 10.1512/iumj.2006.55.2681.  Google Scholar

[5]

H. AmmariH. Kang and K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials, Asymptot. Anal., 41 (2005), 119-140.   Google Scholar

[6]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

[7]

L. BerlyandD. GolovatyA. Movchan and J. Phillips, Transport properties of densely packed composites. Effect of shapes and spacings of inclusions, Quart. J. Mech. Appl. Math., 57 (2004), 495-528.  doi: 10.1093/qjmam/57.4.495.  Google Scholar

[8]

L. Berlyand and V. Mityushev, Increase and decrease of the effective conductivity of two phase composites due to polydispersity, J. Stat. Phys., 118 (2005), 481-509.  doi: 10.1007/s10955-004-8818-0.  Google Scholar

[9]

L. P. CastroD. Kapanadze and E. Pesetskaya, A heat conduction problem of 2D unbounded composites with imperfect contact conditions, ZAMM Z. Angew. Math. Mech., 95 (2015), 952-965.  doi: 10.1002/zamm.201400067.  Google Scholar

[10]

L. P. CastroD. Kapanadze and E. Pesetskaya, Effective conductivity of a composite material with stiff imperfect contact conditions, Math. Methods Appl. Sci., 38 (2015), 4638-4649.  doi: 10.1002/mma.3423.  Google Scholar

[11]

L. P. Castro and E. Pesetskaya, A composite material with inextensible-membrane-type interface, Math. Mech. Solids, 24 (2019), 499-510.  doi: 10.1177/1081286517746717.  Google Scholar

[12]

G. P. Cherepanov, Inverse problems of the plane theory of elasticity, J. Appl. Math. Mech., 38 (1974), 915–931 (1975); translated from Prikl. Mat. Meh., 38 (1974), 963–979 (Russian). doi: 10.1016/0021-8928(75)90085-4.  Google Scholar

[13]

A. Cherkaev, Variational Methods for Structural Optimization, Applied Mathematical Sciences, 140. Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1188-4.  Google Scholar

[14]

M. Dalla Riva, Stokes flow in a singularly perturbed exterior domain, Complex Var. Elliptic Equ., 58 (2013), 231-257.  doi: 10.1080/17476933.2011.575462.  Google Scholar

[15]

M. Dalla Riva and M. Lanza de Cristoforis, Weakly singular and microscopically hypersingular load perturbation for a nonlinear traction boundary value problem: A functional analytic approach, Complex Anal. Oper. Theory, 5 (2011), 811-833.  doi: 10.1007/s11785-010-0109-y.  Google Scholar

[16]

M. Dalla Riva and P. Musolino, A singularly perturbed nonideal transmission problem and application to the effective conductivity of a periodic composite, SIAM J Appl Math., 73 (2013), 24-46.  doi: 10.1137/120886637.  Google Scholar

[17]

M. Dalla RivaP. Musolino and R. Pukhtaievych, Series expansion for the effective conductivity of a periodic dilute composite with thermal resistance at the two-phase interface, Asymptot. Anal., 111 (2019), 217-250.  doi: 10.3233/ASY-181495.  Google Scholar

[18]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[19]

P. Drygaś, S. Gluzman, V. Mityushev and W. Nawalaniec, Applied Analysis of Composite Media. Analytical and Computational Results for Materials Scientists and Engineers, , Woodhead Publishing Series in Composites Science and Engineering. Woodhead Publishing, 2020. Google Scholar

[20]

P. Drygaś and V. Mityushev, Effective conductivity of unidirectional cylinders with interfacial resistance, Quart. J. Mech. Appl. Math., 62 (2009), 235-262.  doi: 10.1093/qjmam/hbp010.  Google Scholar

[21]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.  Google Scholar

[22]

L. V. Gibiansky and A. V. Cherkaev, Microstructures of composites of extremal rigidity and exact bounds on the associated energy density, in Topics in the mathematical modelling of composite materials, 273–317, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, 1997.  Google Scholar

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition, Vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[24]

S. Gluzman, V. Mityushev and W. Nawalaniec, Computational Analysis of Structured Media, Mathematical Analysis and Its Applications. Academic Press, London, 2018.  Google Scholar

[25]

Y. Gorb and L. Berlyand, Asymptotics of the effective conductivity of composites with closely spaced inclusions of optimal shape, Quart. J. Mech. Appl. Math., 58 (2005), 84-106.  doi: 10.1093/qjmamj/hbh022.  Google Scholar

[26]

S. Gryshchuk and S. Rogosin, Effective conductivity of 2D disk-ring composite material, Math. Model. Anal., 18 (2013), 386-394.  doi: 10.3846/13926292.2013.804890.  Google Scholar

[27]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

[28]

D. KapanadzeG. Mishuris and E. Pesetskaya, Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials, Complex Var. Elliptic Equ., 60 (2015), 1-23.  doi: 10.1080/17476933.2013.876418.  Google Scholar

[29]

S. M. Kozlov, Geometric aspects of averaging, Russian Math. Surveys, 44 (1989), 91-144.  doi: 10.1070/RM1989v044n02ABEH002039.  Google Scholar

[30]

P. Kurtyka and N. Rylko, Quantitative analysis of the particles distributions in reinforced composites, Composite Structures, 182 (2017), 412-419.  doi: 10.1016/j.compstruct.2017.09.048.  Google Scholar

[31]

M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces, Comput. Methods Funct. Theory, 2 (2002), 1-27.  doi: 10.1007/BF03321008.  Google Scholar

[32]

M. Lanza de Cristoforis, A domain perturbation problem for the Poisson equation, Complex Var. Theory Appl., 50 (2005), 851-867.  doi: 10.1080/02781070500136993.  Google Scholar

[33]

M. Lanza de Cristoforis, Perturbation problems in potential theory, a functional analytic approach, J. Appl. Funct. Anal., 2 (2007), 197-222.   Google Scholar

[34]

M. Lanza de Cristoforis and P. Musolino, A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients, Far East J. Math. Sci. (FJMS), 52 (2011) 75–120.  Google Scholar

[35]

M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density, J. Integral Equations Appl., 16 (2004), 137-174.  doi: 10.1216/jiea/1181075272.  Google Scholar

[36]

M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials for the Helmholtz equation upon perturbation of the support and of the density, in Analytic methods of analysis and differential equations: AMADE 2006, Camb. Sci. Publ., Cambridge, 2008,193–220.  Google Scholar

[37]

H. Lee and J. Lee, Array dependence of effective parameters of dilute periodic elastic composite, in Imaging, multi-scale and high contrast partial differential equations, 59–71, Contemp. Math., 660, Amer. Math. Soc., Providence, RI, 2016.  Google Scholar

[38]

K. A. Lurie and A. V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. Roy. Soc. Edinburgh Sect. A, 99 (1984), 71-87.  doi: 10.1017/S030821050002597X.  Google Scholar

[39]

P. LuzziniP. Musolino and R. Pukhtaievych, Shape analysis of the longitudinal flow along a periodic array of cylinders, J. Math. Anal. Appl., 477 (2019), 1369-1395.  doi: 10.1016/j.jmaa.2019.05.017.  Google Scholar

[40]

P. Luzzini, P. Musolino and R. Pukhtaievych, Real analyticity of periodic layer potentials upon perturbation of the periodicity parameters and of the support, in Proceedings of the 12th ISAAC congress (Aveiro, 2019). Research Perspectives. Birkhuser, accepted. Google Scholar

[41]

G. W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511613357.  Google Scholar

[42]

V. MityushevW. NawalaniecD. Nosov and E. Pesetskaya, Schwarz's alternating method in a matrix form and its applications to composites, Appl. Math. Comput., 356 (2019), 144-156.  doi: 10.1016/j.amc.2019.03.032.  Google Scholar

[43]

V. MityushevYu. ObnosovE. Pesetskaya and S. Rogosin, Analytical methods for heat conduction in composites, Math. Model. Anal., 13 (2008), 67-78.  doi: 10.3846/1392-6292.2008.13.67-78.  Google Scholar

[44]

V. Mityushev and N. Rylko, Optimal distribution of the nonoverlapping conducting disks, Multiscale Model. Simul., 10 (2012), 180-190.  doi: 10.1137/110823225.  Google Scholar

[45]

P. Musolino, A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach, Math. Methods Appl. Sci., 35 (2012), 334-349.  doi: 10.1002/mma.1575.  Google Scholar

[46]

A. A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-35245-4.  Google Scholar

[47]

R. Pukhtaievych, Asymptotic behavior of the solution of singularly perturbed transmission problems in a periodic domain, Math. Methods Appl. Sci., 41 (2018), 3392-3413.  doi: 10.1002/mma.4832.  Google Scholar

[48]

R. Pukhtaievych, Effective conductivity of a periodic dilute composite with perfect contact and its series expansion, Z. Angew. Math. Phys., 69 (2018), Paper no. 83, 22 pp. doi: 10.1007/s00033-018-0976-z.  Google Scholar

[49]

N. Rylko, Dipole matrix for the 2D inclusions close to circular, ZAMM Z. Angew. Math. Mech., 88 (2008), 993-999.  doi: 10.1002/zamm.200700114.  Google Scholar

[50]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, volume 120 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  Google Scholar

[2]

H. Ammari and H. Kang, Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, 162, Springer, New York, 2007.  Google Scholar

[3]

H. AmmariH. Kang and K. Kim, Polarization tensors and effective properties of anisotropic composite materials, J. Differential Equations, 215 (2005), 401-428.  doi: 10.1016/j.jde.2004.09.010.  Google Scholar

[4]

H. AmmariH. Kang and M. Lim, Effective parameters of elastic composites, Indiana Univ. Math. J., 55 (2006), 903-922.  doi: 10.1512/iumj.2006.55.2681.  Google Scholar

[5]

H. AmmariH. Kang and K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials, Asymptot. Anal., 41 (2005), 119-140.   Google Scholar

[6]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

[7]

L. BerlyandD. GolovatyA. Movchan and J. Phillips, Transport properties of densely packed composites. Effect of shapes and spacings of inclusions, Quart. J. Mech. Appl. Math., 57 (2004), 495-528.  doi: 10.1093/qjmam/57.4.495.  Google Scholar

[8]

L. Berlyand and V. Mityushev, Increase and decrease of the effective conductivity of two phase composites due to polydispersity, J. Stat. Phys., 118 (2005), 481-509.  doi: 10.1007/s10955-004-8818-0.  Google Scholar

[9]

L. P. CastroD. Kapanadze and E. Pesetskaya, A heat conduction problem of 2D unbounded composites with imperfect contact conditions, ZAMM Z. Angew. Math. Mech., 95 (2015), 952-965.  doi: 10.1002/zamm.201400067.  Google Scholar

[10]

L. P. CastroD. Kapanadze and E. Pesetskaya, Effective conductivity of a composite material with stiff imperfect contact conditions, Math. Methods Appl. Sci., 38 (2015), 4638-4649.  doi: 10.1002/mma.3423.  Google Scholar

[11]

L. P. Castro and E. Pesetskaya, A composite material with inextensible-membrane-type interface, Math. Mech. Solids, 24 (2019), 499-510.  doi: 10.1177/1081286517746717.  Google Scholar

[12]

G. P. Cherepanov, Inverse problems of the plane theory of elasticity, J. Appl. Math. Mech., 38 (1974), 915–931 (1975); translated from Prikl. Mat. Meh., 38 (1974), 963–979 (Russian). doi: 10.1016/0021-8928(75)90085-4.  Google Scholar

[13]

A. Cherkaev, Variational Methods for Structural Optimization, Applied Mathematical Sciences, 140. Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1188-4.  Google Scholar

[14]

M. Dalla Riva, Stokes flow in a singularly perturbed exterior domain, Complex Var. Elliptic Equ., 58 (2013), 231-257.  doi: 10.1080/17476933.2011.575462.  Google Scholar

[15]

M. Dalla Riva and M. Lanza de Cristoforis, Weakly singular and microscopically hypersingular load perturbation for a nonlinear traction boundary value problem: A functional analytic approach, Complex Anal. Oper. Theory, 5 (2011), 811-833.  doi: 10.1007/s11785-010-0109-y.  Google Scholar

[16]

M. Dalla Riva and P. Musolino, A singularly perturbed nonideal transmission problem and application to the effective conductivity of a periodic composite, SIAM J Appl Math., 73 (2013), 24-46.  doi: 10.1137/120886637.  Google Scholar

[17]

M. Dalla RivaP. Musolino and R. Pukhtaievych, Series expansion for the effective conductivity of a periodic dilute composite with thermal resistance at the two-phase interface, Asymptot. Anal., 111 (2019), 217-250.  doi: 10.3233/ASY-181495.  Google Scholar

[18]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[19]

P. Drygaś, S. Gluzman, V. Mityushev and W. Nawalaniec, Applied Analysis of Composite Media. Analytical and Computational Results for Materials Scientists and Engineers, , Woodhead Publishing Series in Composites Science and Engineering. Woodhead Publishing, 2020. Google Scholar

[20]

P. Drygaś and V. Mityushev, Effective conductivity of unidirectional cylinders with interfacial resistance, Quart. J. Mech. Appl. Math., 62 (2009), 235-262.  doi: 10.1093/qjmam/hbp010.  Google Scholar

[21]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.  Google Scholar

[22]

L. V. Gibiansky and A. V. Cherkaev, Microstructures of composites of extremal rigidity and exact bounds on the associated energy density, in Topics in the mathematical modelling of composite materials, 273–317, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, 1997.  Google Scholar

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition, Vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[24]

S. Gluzman, V. Mityushev and W. Nawalaniec, Computational Analysis of Structured Media, Mathematical Analysis and Its Applications. Academic Press, London, 2018.  Google Scholar

[25]

Y. Gorb and L. Berlyand, Asymptotics of the effective conductivity of composites with closely spaced inclusions of optimal shape, Quart. J. Mech. Appl. Math., 58 (2005), 84-106.  doi: 10.1093/qjmamj/hbh022.  Google Scholar

[26]

S. Gryshchuk and S. Rogosin, Effective conductivity of 2D disk-ring composite material, Math. Model. Anal., 18 (2013), 386-394.  doi: 10.3846/13926292.2013.804890.  Google Scholar

[27]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

[28]

D. KapanadzeG. Mishuris and E. Pesetskaya, Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials, Complex Var. Elliptic Equ., 60 (2015), 1-23.  doi: 10.1080/17476933.2013.876418.  Google Scholar

[29]

S. M. Kozlov, Geometric aspects of averaging, Russian Math. Surveys, 44 (1989), 91-144.  doi: 10.1070/RM1989v044n02ABEH002039.  Google Scholar

[30]

P. Kurtyka and N. Rylko, Quantitative analysis of the particles distributions in reinforced composites, Composite Structures, 182 (2017), 412-419.  doi: 10.1016/j.compstruct.2017.09.048.  Google Scholar

[31]

M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces, Comput. Methods Funct. Theory, 2 (2002), 1-27.  doi: 10.1007/BF03321008.  Google Scholar

[32]

M. Lanza de Cristoforis, A domain perturbation problem for the Poisson equation, Complex Var. Theory Appl., 50 (2005), 851-867.  doi: 10.1080/02781070500136993.  Google Scholar

[33]

M. Lanza de Cristoforis, Perturbation problems in potential theory, a functional analytic approach, J. Appl. Funct. Anal., 2 (2007), 197-222.   Google Scholar

[34]

M. Lanza de Cristoforis and P. Musolino, A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients, Far East J. Math. Sci. (FJMS), 52 (2011) 75–120.  Google Scholar

[35]

M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density, J. Integral Equations Appl., 16 (2004), 137-174.  doi: 10.1216/jiea/1181075272.  Google Scholar

[36]

M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials for the Helmholtz equation upon perturbation of the support and of the density, in Analytic methods of analysis and differential equations: AMADE 2006, Camb. Sci. Publ., Cambridge, 2008,193–220.  Google Scholar

[37]

H. Lee and J. Lee, Array dependence of effective parameters of dilute periodic elastic composite, in Imaging, multi-scale and high contrast partial differential equations, 59–71, Contemp. Math., 660, Amer. Math. Soc., Providence, RI, 2016.  Google Scholar

[38]

K. A. Lurie and A. V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. Roy. Soc. Edinburgh Sect. A, 99 (1984), 71-87.  doi: 10.1017/S030821050002597X.  Google Scholar

[39]

P. LuzziniP. Musolino and R. Pukhtaievych, Shape analysis of the longitudinal flow along a periodic array of cylinders, J. Math. Anal. Appl., 477 (2019), 1369-1395.  doi: 10.1016/j.jmaa.2019.05.017.  Google Scholar

[40]

P. Luzzini, P. Musolino and R. Pukhtaievych, Real analyticity of periodic layer potentials upon perturbation of the periodicity parameters and of the support, in Proceedings of the 12th ISAAC congress (Aveiro, 2019). Research Perspectives. Birkhuser, accepted. Google Scholar

[41]

G. W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511613357.  Google Scholar

[42]

V. MityushevW. NawalaniecD. Nosov and E. Pesetskaya, Schwarz's alternating method in a matrix form and its applications to composites, Appl. Math. Comput., 356 (2019), 144-156.  doi: 10.1016/j.amc.2019.03.032.  Google Scholar

[43]

V. MityushevYu. ObnosovE. Pesetskaya and S. Rogosin, Analytical methods for heat conduction in composites, Math. Model. Anal., 13 (2008), 67-78.  doi: 10.3846/1392-6292.2008.13.67-78.  Google Scholar

[44]

V. Mityushev and N. Rylko, Optimal distribution of the nonoverlapping conducting disks, Multiscale Model. Simul., 10 (2012), 180-190.  doi: 10.1137/110823225.  Google Scholar

[45]

P. Musolino, A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach, Math. Methods Appl. Sci., 35 (2012), 334-349.  doi: 10.1002/mma.1575.  Google Scholar

[46]

A. A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-35245-4.  Google Scholar

[47]

R. Pukhtaievych, Asymptotic behavior of the solution of singularly perturbed transmission problems in a periodic domain, Math. Methods Appl. Sci., 41 (2018), 3392-3413.  doi: 10.1002/mma.4832.  Google Scholar

[48]

R. Pukhtaievych, Effective conductivity of a periodic dilute composite with perfect contact and its series expansion, Z. Angew. Math. Phys., 69 (2018), Paper no. 83, 22 pp. doi: 10.1007/s00033-018-0976-z.  Google Scholar

[49]

N. Rylko, Dipole matrix for the 2D inclusions close to circular, ZAMM Z. Angew. Math. Mech., 88 (2008), 993-999.  doi: 10.1002/zamm.200700114.  Google Scholar

[50]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, volume 120 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

Figure 1.  The sets $ \mathbb{S}_{q}[q\mathbb{I}[\phi]]^- $, $ \mathbb{S}_{q}[q\mathbb{I}[\phi]] $, and $ q\phi(\partial\Omega) $ in case $ n = 2 $
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