# American Institute of Mathematical Sciences

November  2014, 13(6): 2509-2542. doi: 10.3934/cpaa.2014.13.2509

## A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach

 1 Dipartimento di Matematica, Universitá degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy, Italy

Received  January 2014 Revised  June 2014 Published  July 2014

We consider a heat transmission problem for a composite material which fills the $n$-dimensional Euclidean space. The composite has a periodic structure and consists of two materials. In each periodicity cell one material occupies a cavity of size $\epsilon$, and the second material fills the remaining part of the cell. We assume that the thermal conductivities of the materials depend nonlinearly upon the temperature. We show that for $\epsilon$ small enough the problem has a solution, \textit{i.e.}, a pair of functions which determine the temperature distribution in the two materials. Then we analyze the behavior of such a solution as $\epsilon$ approaches $0$ by an approach which is alternative to those of asymptotic analysis. In particular we prove that if $n\geq 3$, the temperature can be expanded into a convergent series expansion of powers of $\epsilon$ and that if $n=2$ the temperature can be expanded into a convergent double series expansion of powers of $\epsilon$ and $\epsilon \log \epsilon$.
Citation: Massimo Lanza de Cristoforis, aolo Musolino. A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2509-2542. doi: 10.3934/cpaa.2014.13.2509
##### References:
 [1] H. Ammari and H. Kang, Polarization and Moment Tensors, Applied Mathematical Sciences, 162, Springer, New York, 2007. [2] H. Ammari, H. Kang and K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials, Asymptot. Anal., 41 (2005), 119-140. [3] J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A, 306 (1982), 557-611. doi: 10.1098/rsta.1982.0095. [4] R. Böhme and F. Tomi, Zur Struktur der Lösungsmenge des Plateauproblems, Math. Z., 133 (1973), 1-29. [5] V. Bonnaillie-Noël, M. Dambrine, S. Tordeux and G. Vial, Interactions between moderately close inclusions for the Laplace equation, Math. Models Methods Appl. Sci., 19 (2009), 1853-1882. doi: 10.1142/S021820250900398X. [6] L. P. Castro and E. Pesetskaya, A transmission problem with imperfect contact for an unbounded multiply connected domain, Math. Methods Appl. Sci., 33 (2010), 517-526. doi: 10.1002/mma.1217. [7] L. P. Castro, E. Pesetskaya and S. V. Rogosin, Effective conductivity of a composite material with non-ideal contact conditions, Complex Var. Elliptic Equ., 54 (2009), 1085-1100. doi: 10.1080/17476930903275995. [8] M. Dalla Riva and M. Lanza de Cristoforis, A perturbation result for the layer potentials of general second order differential operators with constant coefficients, J. Appl. Funct. Anal., 5 (2010), 10-30. [9] M. Dalla Riva and M. Lanza de Cristoforis, Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach, Complex Var. Elliptic Equ., 55 (2010), 771-794. doi: 10.1080/17476931003628216. [10] 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. [11] M. Dalla Riva and P. Musolino, Real analytic families of harmonic functions in a domain with a small hole, J. Differential Equations, 252 (2012), 6337-6355. doi: 10.1016/j.jde.2012.03.007. [12] 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. [13] G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 445-486. doi: 10.1016/j.anihpc.2003.05.001. [14] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7. [15] P. Drygas 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. [16] G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, NJ, second edition, 1995. [17] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [18] D. Henry, Topics in Nonlinear Analysis, Trabalho de Matemática, 192, Brasilia, 1982. [19] M. Iguernane, S. A. Nazarov, J. R. Roche, J. Sokolowski and K. Szulc, Topological derivatives for semilinear elliptic equations, Int. J. Appl. Math. Comput. Sci., 19 (2009), 191-205. doi: 10.2478/v10006-009-0016-4. [20] A. Kirsch, Surface gradients and continuity properties for some integral operators in classical scattering theory, Math. Methods Appl. Sci., 11 (1989), 789-804. doi: 10.1002/mma.1670110605. [21] M. Lanza de Cristoforis, Properties and pathologies of the composition and inversion operators in Schauder spaces, Acc. Naz. delle Sci. detta dei XL, 15 (1991), 93-109. [22] 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. [23] M. Lanza de Cristoforis, Asymptotic behavior of the solutions of a nonlinear Robin problem for the Laplace operator in a domain with a small hole: a functional analytic approach, Complex Var. Elliptic Equ., 52 (2007), 945-977. doi: 10.1080/17476930701485630. [24] M. Lanza de Cristoforis, Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach, Analysis (Munich), 28 (2008), 63-93. doi: 10.1524/anly.2008.0903. [25] M. Lanza de Cristoforis, Asymptotic behaviour of the solutions of a nonlinear transmission problem for the Laplace operator in a domain with a small hole. A functional analytic approach, Complex Var. Elliptic Equ., 55 (2010), 269-303. doi: 10.1080/17476930902999058. [26] 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. [27] M. Lanza de Cristoforis and P. Musolino, A real analyticity result for a nonlinear integral operator, J. Integral Equations Appl., 25 (2013), 21-46. doi: 10.1216/JIE-2013-25-1-21. [28] M. Lanza de Cristoforis and P. Musolino, A singularly perturbed nonlinear Robin problem in a periodically perforated domain: a functional analytic approach, Complex Var. Elliptic Equ., 58 (2013), 511-536. doi: 10.1080/17476933.2011.638716. [29] M. Lanza de Cristoforis and P. Musolino, A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach, Submitted, 2014. [30] 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. [31] V. Maz'ya, A. Movchan and M. Nieves, Green's Kernels and Meso-scale Approximations in Perforated Domains, Lecture Notes in Mathematics, 2077, Springer, Berlin, 2013. doi: 10.1007/978-3-319-00357-3. [32] V. Maz'ya, S. Nazarov and B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vols. I, II, Operator Theory: Advances and Applications, 111, 112, Birkhäuser Verlag, Basel, 2000. [33] C. Miranda, Sulle proprietà di regolarità di certe trasformazioni integrali, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I., 7 (1965), 303-336. [34] V. V. Mityushev, Transport properties of double-periodic arrays of circular cylinders, Z. Angew. Math. Mech., 77 (1997), 115-120. doi: 10.1002/zamm.19970770209. [35] V. Mityushev, Transport properties of doubly periodic arrays of circular cylinders and optimal design problems, Appl. Math. Optim., 44 (2001), 17-31. doi: 10.1007/s00245-001-0013-y. [36] V. V. Mityushev and S. V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 108, Chapman & Hall/CRC, Boca Raton, FL, 2000. [37] 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. [38] P. Musolino, A singularly perturbed Dirichlet problem for the Poisson equation in a periodically perforated domain. A functional analytic approach, in Advances in Harmonic Analysis and Operator Theory, The Stefan Samko Anniversary Volume (eds. A. Almeida, L. Castro and F.-O. Speck), Operator Theory: Advances and Applications 229, Birkhäuser Verlag, (2013), 269-289. doi: 10.1007/978-3-0348-0516-2_15. [39] S. A. Nazarov and J. Sokołowski, Asymptotic analysis of shape functionals, J. Math. Pures Appl., 82 (2003), 125-196. doi: 10.1016/S0021-7824(03)00004-7. [40] J. Schauder, Potentialtheoretische Untersuchungen, Math. Z., 33 (1931), 602-640. doi: 10.1007/BF01174371. [41] J. Schauder, Bemerkung zu meiner Arbeit "Potentialtheoretische Untersuchungen I (Anhang)'', Math. Z., 35 (1932), 536-538. doi: 10.1007/BF01186569. [42] J. Sivaloganathan, S. J. Spector and V. Tilakraj, The convergence of regularized minimizers for cavitation problems in nonlinear elasticity, SIAM J. Appl. Math., 66 (2006), 736-757. doi: 10.1137/040618965. [43] M. S. Titcombe and M. J. Ward, Summing logarithmic expansions for elliptic equations in multiply-connected domains with small holes, Canad. Appl. Math. Quart., 7 (1999), 313-343. [44] T. Valent, Boundary Value Problems of Finite Elasticity. Local Theorems on Existence, Uniqueness and Analytic Dependence on Data, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-3736-5. [45] M. J. Ward, W. D. Henshaw and J. B. Keller, Summing logarithmic expansions for singularly perturbed eigenvalue problems, SIAM J. Appl. Math., 53 (1993), 799-828. doi: 10.1137/0153039. [46] M. J. Ward and J. B. Keller, Nonlinear eigenvalue problems under strong localized perturbations with applications to chemical reactors, Stud. Appl. Math., 85 (1991), 1-28. [47] M. J. Ward and J. B. Keller, Strong localized perturbations of eigenvalue problems, SIAM J. Appl. Math., 53 (1993), 770-798. doi: 10.1137/0153038.

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##### References:
 [1] H. Ammari and H. Kang, Polarization and Moment Tensors, Applied Mathematical Sciences, 162, Springer, New York, 2007. [2] H. Ammari, H. Kang and K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials, Asymptot. Anal., 41 (2005), 119-140. [3] J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A, 306 (1982), 557-611. doi: 10.1098/rsta.1982.0095. [4] R. Böhme and F. Tomi, Zur Struktur der Lösungsmenge des Plateauproblems, Math. Z., 133 (1973), 1-29. [5] V. Bonnaillie-Noël, M. Dambrine, S. Tordeux and G. Vial, Interactions between moderately close inclusions for the Laplace equation, Math. Models Methods Appl. Sci., 19 (2009), 1853-1882. doi: 10.1142/S021820250900398X. [6] L. P. Castro and E. Pesetskaya, A transmission problem with imperfect contact for an unbounded multiply connected domain, Math. Methods Appl. Sci., 33 (2010), 517-526. doi: 10.1002/mma.1217. [7] L. P. Castro, E. Pesetskaya and S. V. Rogosin, Effective conductivity of a composite material with non-ideal contact conditions, Complex Var. Elliptic Equ., 54 (2009), 1085-1100. doi: 10.1080/17476930903275995. [8] M. Dalla Riva and M. Lanza de Cristoforis, A perturbation result for the layer potentials of general second order differential operators with constant coefficients, J. Appl. Funct. Anal., 5 (2010), 10-30. [9] M. Dalla Riva and M. Lanza de Cristoforis, Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach, Complex Var. Elliptic Equ., 55 (2010), 771-794. doi: 10.1080/17476931003628216. [10] 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. [11] M. Dalla Riva and P. Musolino, Real analytic families of harmonic functions in a domain with a small hole, J. Differential Equations, 252 (2012), 6337-6355. doi: 10.1016/j.jde.2012.03.007. [12] 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. [13] G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 445-486. doi: 10.1016/j.anihpc.2003.05.001. [14] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7. [15] P. Drygas 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. [16] G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, NJ, second edition, 1995. [17] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [18] D. Henry, Topics in Nonlinear Analysis, Trabalho de Matemática, 192, Brasilia, 1982. [19] M. Iguernane, S. A. Nazarov, J. R. Roche, J. Sokolowski and K. Szulc, Topological derivatives for semilinear elliptic equations, Int. J. Appl. Math. Comput. Sci., 19 (2009), 191-205. doi: 10.2478/v10006-009-0016-4. [20] A. Kirsch, Surface gradients and continuity properties for some integral operators in classical scattering theory, Math. Methods Appl. Sci., 11 (1989), 789-804. doi: 10.1002/mma.1670110605. [21] M. Lanza de Cristoforis, Properties and pathologies of the composition and inversion operators in Schauder spaces, Acc. Naz. delle Sci. detta dei XL, 15 (1991), 93-109. [22] 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. [23] M. Lanza de Cristoforis, Asymptotic behavior of the solutions of a nonlinear Robin problem for the Laplace operator in a domain with a small hole: a functional analytic approach, Complex Var. Elliptic Equ., 52 (2007), 945-977. doi: 10.1080/17476930701485630. [24] M. Lanza de Cristoforis, Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach, Analysis (Munich), 28 (2008), 63-93. doi: 10.1524/anly.2008.0903. [25] M. Lanza de Cristoforis, Asymptotic behaviour of the solutions of a nonlinear transmission problem for the Laplace operator in a domain with a small hole. A functional analytic approach, Complex Var. Elliptic Equ., 55 (2010), 269-303. doi: 10.1080/17476930902999058. [26] 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. [27] M. Lanza de Cristoforis and P. Musolino, A real analyticity result for a nonlinear integral operator, J. Integral Equations Appl., 25 (2013), 21-46. doi: 10.1216/JIE-2013-25-1-21. [28] M. Lanza de Cristoforis and P. Musolino, A singularly perturbed nonlinear Robin problem in a periodically perforated domain: a functional analytic approach, Complex Var. Elliptic Equ., 58 (2013), 511-536. doi: 10.1080/17476933.2011.638716. [29] M. Lanza de Cristoforis and P. Musolino, A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach, Submitted, 2014. [30] 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. [31] V. Maz'ya, A. Movchan and M. Nieves, Green's Kernels and Meso-scale Approximations in Perforated Domains, Lecture Notes in Mathematics, 2077, Springer, Berlin, 2013. doi: 10.1007/978-3-319-00357-3. [32] V. Maz'ya, S. Nazarov and B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vols. I, II, Operator Theory: Advances and Applications, 111, 112, Birkhäuser Verlag, Basel, 2000. [33] C. Miranda, Sulle proprietà di regolarità di certe trasformazioni integrali, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I., 7 (1965), 303-336. [34] V. V. Mityushev, Transport properties of double-periodic arrays of circular cylinders, Z. Angew. Math. Mech., 77 (1997), 115-120. doi: 10.1002/zamm.19970770209. [35] V. Mityushev, Transport properties of doubly periodic arrays of circular cylinders and optimal design problems, Appl. Math. Optim., 44 (2001), 17-31. doi: 10.1007/s00245-001-0013-y. [36] V. V. Mityushev and S. V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 108, Chapman & Hall/CRC, Boca Raton, FL, 2000. [37] 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. [38] P. Musolino, A singularly perturbed Dirichlet problem for the Poisson equation in a periodically perforated domain. A functional analytic approach, in Advances in Harmonic Analysis and Operator Theory, The Stefan Samko Anniversary Volume (eds. A. Almeida, L. Castro and F.-O. Speck), Operator Theory: Advances and Applications 229, Birkhäuser Verlag, (2013), 269-289. doi: 10.1007/978-3-0348-0516-2_15. [39] S. A. Nazarov and J. Sokołowski, Asymptotic analysis of shape functionals, J. Math. Pures Appl., 82 (2003), 125-196. doi: 10.1016/S0021-7824(03)00004-7. [40] J. Schauder, Potentialtheoretische Untersuchungen, Math. Z., 33 (1931), 602-640. doi: 10.1007/BF01174371. [41] J. Schauder, Bemerkung zu meiner Arbeit "Potentialtheoretische Untersuchungen I (Anhang)'', Math. Z., 35 (1932), 536-538. doi: 10.1007/BF01186569. [42] J. Sivaloganathan, S. J. Spector and V. Tilakraj, The convergence of regularized minimizers for cavitation problems in nonlinear elasticity, SIAM J. Appl. Math., 66 (2006), 736-757. doi: 10.1137/040618965. [43] M. S. Titcombe and M. J. Ward, Summing logarithmic expansions for elliptic equations in multiply-connected domains with small holes, Canad. Appl. Math. Quart., 7 (1999), 313-343. [44] T. Valent, Boundary Value Problems of Finite Elasticity. Local Theorems on Existence, Uniqueness and Analytic Dependence on Data, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-3736-5. [45] M. J. Ward, W. D. Henshaw and J. B. Keller, Summing logarithmic expansions for singularly perturbed eigenvalue problems, SIAM J. Appl. Math., 53 (1993), 799-828. doi: 10.1137/0153039. [46] M. J. Ward and J. B. Keller, Nonlinear eigenvalue problems under strong localized perturbations with applications to chemical reactors, Stud. Appl. Math., 85 (1991), 1-28. [47] M. J. Ward and J. B. Keller, Strong localized perturbations of eigenvalue problems, SIAM J. Appl. Math., 53 (1993), 770-798. doi: 10.1137/0153038.
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