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 & 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, (2007). [2] H. Ammari, H. Kang and K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials,, \emph{Asymptot. Anal.}, 41 (2005), 119. [3] J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity,, \emph{Philos. Trans. Roy. Soc. London Ser. A}, 306 (1982), 557. doi: 10.1098/rsta.1982.0095. [4] R. Böhme and F. Tomi, Zur Struktur der Lösungsmenge des Plateauproblems,, \emph{Math. Z.}, 133 (1973), 1. [5] V. Bonnaillie-Noël, M. Dambrine, S. Tordeux and G. Vial, Interactions between moderately close inclusions for the Laplace equation,, \emph{Math. Models Methods Appl. Sci.}, 19 (2009), 1853. doi: 10.1142/S021820250900398X. [6] L. P. Castro and E. Pesetskaya, A transmission problem with imperfect contact for an unbounded multiply connected domain,, \emph{Math. Methods Appl. Sci.}, 33 (2010), 517. 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,, \emph{Complex Var. Elliptic Equ.}, 54 (2009), 1085. 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,, \emph{J. Appl. Funct. Anal.}, 5 (2010), 10. [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,, \emph{Complex Var. Elliptic Equ.}, 55 (2010), 771. 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,, \emph{Complex Anal. Oper. Theory}, 5 (2011), 811. 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,, \emph{J. Differential Equations}, 252 (2012), 6337. 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,, \emph{SIAM J. Appl. Math.}, 73 (2013), 24. 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,, \emph{Ann. Inst. H. Poincaré Anal. Non Linéaire}, 21 (2004), 445. doi: 10.1016/j.anihpc.2003.05.001. [14] K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (1985). doi: 10.1007/978-3-662-00547-7. [15] P. Drygas and V. Mityushev, Effective conductivity of unidirectional cylinders with interfacial resistance,, \emph{Quart. J. Mech. Appl. Math.}, 62 (2009), 235. doi: 10.1093/qjmam/hbp010. [16] G. B. Folland, Introduction to Partial Differential Equations,, Princeton University Press, (1995). [17] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer Verlag, (1983). doi: 10.1007/978-3-642-61798-0. [18] D. Henry, Topics in Nonlinear Analysis,, Trabalho de Matem\'atica, (1982). [19] M. Iguernane, S. A. Nazarov, J. R. Roche, J. Sokolowski and K. Szulc, Topological derivatives for semilinear elliptic equations,, \emph{Int. J. Appl. Math. Comput. Sci.}, 19 (2009), 191. doi: 10.2478/v10006-009-0016-4. [20] A. Kirsch, Surface gradients and continuity properties for some integral operators in classical scattering theory,, \emph{Math. Methods Appl. Sci.}, 11 (1989), 789. doi: 10.1002/mma.1670110605. [21] M. Lanza de Cristoforis, Properties and pathologies of the composition and inversion operators in Schauder spaces,, \emph{Acc. Naz. delle Sci. detta dei XL}, 15 (1991), 93. [22] M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces,, \emph{Comput. Methods Funct. Theory}, 2 (2002), 1. 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,, \emph{Complex Var. Elliptic Equ.}, 52 (2007), 945. 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,, \emph{Analysis (Munich)}, 28 (2008), 63. 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,, \emph{Complex Var. Elliptic Equ.}, 55 (2010), 269. 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,, \emph{Far East J. Math. Sci. (FJMS)}, 52 (2011), 75. [27] M. Lanza de Cristoforis and P. Musolino, A real analyticity result for a nonlinear integral operator,, \emph{J. Integral Equations Appl.}, 25 (2013), 21. 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,, \emph{Complex Var. Elliptic Equ.}, 58 (2013), 511. 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,, \emph{J. Integral Equations Appl.}, 16 (2004), 137. 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). 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, (2000). [33] C. Miranda, Sulle proprietà di regolarità di certe trasformazioni integrali,, \emph{Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I.}, 7 (1965), 303. [34] V. V. Mityushev, Transport properties of double-periodic arrays of circular cylinders,, \emph{Z. Angew. Math. Mech.}, 77 (1997), 115. doi: 10.1002/zamm.19970770209. [35] V. Mityushev, Transport properties of doubly periodic arrays of circular cylinders and optimal design problems,, \emph{Appl. Math. Optim.}, 44 (2001), 17. 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, (2000). [37] P. Musolino, A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach,, \emph{Math. Methods Appl. Sci.}, 35 (2012), 334. 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 \emph{Advances in Harmonic Analysis and Operator Theory, (2013), 269. doi: 10.1007/978-3-0348-0516-2_15. [39] S. A. Nazarov and J. Sokołowski, Asymptotic analysis of shape functionals,, \emph{J. Math. Pures Appl.}, 82 (2003), 125. doi: 10.1016/S0021-7824(03)00004-7. [40] J. Schauder, Potentialtheoretische Untersuchungen,, \emph{Math. Z.}, 33 (1931), 602. doi: 10.1007/BF01174371. [41] J. Schauder, Bemerkung zu meiner Arbeit "Potentialtheoretische Untersuchungen I (Anhang)'',, \emph{Math. Z.}, 35 (1932), 536. doi: 10.1007/BF01186569. [42] J. Sivaloganathan, S. J. Spector and V. Tilakraj, The convergence of regularized minimizers for cavitation problems in nonlinear elasticity,, \emph{SIAM J. Appl. Math.}, 66 (2006), 736. 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,, \emph{Canad. Appl. Math. Quart.}, 7 (1999), 313. [44] T. Valent, Boundary Value Problems of Finite Elasticity. Local Theorems on Existence, Uniqueness and Analytic Dependence on Data,, Springer-Verlag, (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,, \emph{SIAM J. Appl. Math.}, 53 (1993), 799. doi: 10.1137/0153039. [46] M. J. Ward and J. B. Keller, Nonlinear eigenvalue problems under strong localized perturbations with applications to chemical reactors,, \emph{Stud. Appl. Math.}, 85 (1991), 1. [47] M. J. Ward and J. B. Keller, Strong localized perturbations of eigenvalue problems,, \emph{SIAM J. Appl. Math.}, 53 (1993), 770. doi: 10.1137/0153038.

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References:
 [1] H. Ammari and H. Kang, Polarization and Moment Tensors,, Applied Mathematical Sciences, (2007). [2] H. Ammari, H. Kang and K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials,, \emph{Asymptot. Anal.}, 41 (2005), 119. [3] J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity,, \emph{Philos. Trans. Roy. Soc. London Ser. A}, 306 (1982), 557. doi: 10.1098/rsta.1982.0095. [4] R. Böhme and F. Tomi, Zur Struktur der Lösungsmenge des Plateauproblems,, \emph{Math. Z.}, 133 (1973), 1. [5] V. Bonnaillie-Noël, M. Dambrine, S. Tordeux and G. Vial, Interactions between moderately close inclusions for the Laplace equation,, \emph{Math. Models Methods Appl. Sci.}, 19 (2009), 1853. doi: 10.1142/S021820250900398X. [6] L. P. Castro and E. Pesetskaya, A transmission problem with imperfect contact for an unbounded multiply connected domain,, \emph{Math. Methods Appl. Sci.}, 33 (2010), 517. 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,, \emph{Complex Var. Elliptic Equ.}, 54 (2009), 1085. 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,, \emph{J. Appl. Funct. Anal.}, 5 (2010), 10. [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,, \emph{Complex Var. Elliptic Equ.}, 55 (2010), 771. 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,, \emph{Complex Anal. Oper. Theory}, 5 (2011), 811. 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,, \emph{J. Differential Equations}, 252 (2012), 6337. 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,, \emph{SIAM J. Appl. Math.}, 73 (2013), 24. 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,, \emph{Ann. Inst. H. Poincaré Anal. Non Linéaire}, 21 (2004), 445. doi: 10.1016/j.anihpc.2003.05.001. [14] K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (1985). doi: 10.1007/978-3-662-00547-7. [15] P. Drygas and V. Mityushev, Effective conductivity of unidirectional cylinders with interfacial resistance,, \emph{Quart. J. Mech. Appl. Math.}, 62 (2009), 235. doi: 10.1093/qjmam/hbp010. [16] G. B. Folland, Introduction to Partial Differential Equations,, Princeton University Press, (1995). [17] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer Verlag, (1983). doi: 10.1007/978-3-642-61798-0. [18] D. Henry, Topics in Nonlinear Analysis,, Trabalho de Matem\'atica, (1982). [19] M. Iguernane, S. A. Nazarov, J. R. Roche, J. Sokolowski and K. Szulc, Topological derivatives for semilinear elliptic equations,, \emph{Int. J. Appl. Math. Comput. Sci.}, 19 (2009), 191. doi: 10.2478/v10006-009-0016-4. [20] A. Kirsch, Surface gradients and continuity properties for some integral operators in classical scattering theory,, \emph{Math. Methods Appl. Sci.}, 11 (1989), 789. doi: 10.1002/mma.1670110605. [21] M. Lanza de Cristoforis, Properties and pathologies of the composition and inversion operators in Schauder spaces,, \emph{Acc. Naz. delle Sci. detta dei XL}, 15 (1991), 93. [22] M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces,, \emph{Comput. Methods Funct. Theory}, 2 (2002), 1. 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,, \emph{Complex Var. Elliptic Equ.}, 52 (2007), 945. 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,, \emph{Analysis (Munich)}, 28 (2008), 63. 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,, \emph{Complex Var. Elliptic Equ.}, 55 (2010), 269. 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,, \emph{Far East J. Math. Sci. (FJMS)}, 52 (2011), 75. [27] M. Lanza de Cristoforis and P. Musolino, A real analyticity result for a nonlinear integral operator,, \emph{J. Integral Equations Appl.}, 25 (2013), 21. 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,, \emph{Complex Var. Elliptic Equ.}, 58 (2013), 511. 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,, \emph{J. Integral Equations Appl.}, 16 (2004), 137. 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). 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, (2000). [33] C. Miranda, Sulle proprietà di regolarità di certe trasformazioni integrali,, \emph{Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I.}, 7 (1965), 303. [34] V. V. Mityushev, Transport properties of double-periodic arrays of circular cylinders,, \emph{Z. Angew. Math. Mech.}, 77 (1997), 115. doi: 10.1002/zamm.19970770209. [35] V. Mityushev, Transport properties of doubly periodic arrays of circular cylinders and optimal design problems,, \emph{Appl. Math. Optim.}, 44 (2001), 17. 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, (2000). [37] P. Musolino, A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach,, \emph{Math. Methods Appl. Sci.}, 35 (2012), 334. 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 \emph{Advances in Harmonic Analysis and Operator Theory, (2013), 269. doi: 10.1007/978-3-0348-0516-2_15. [39] S. A. Nazarov and J. Sokołowski, Asymptotic analysis of shape functionals,, \emph{J. Math. Pures Appl.}, 82 (2003), 125. doi: 10.1016/S0021-7824(03)00004-7. [40] J. Schauder, Potentialtheoretische Untersuchungen,, \emph{Math. Z.}, 33 (1931), 602. doi: 10.1007/BF01174371. [41] J. Schauder, Bemerkung zu meiner Arbeit "Potentialtheoretische Untersuchungen I (Anhang)'',, \emph{Math. Z.}, 35 (1932), 536. doi: 10.1007/BF01186569. [42] J. Sivaloganathan, S. J. Spector and V. Tilakraj, The convergence of regularized minimizers for cavitation problems in nonlinear elasticity,, \emph{SIAM J. Appl. Math.}, 66 (2006), 736. 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,, \emph{Canad. Appl. Math. Quart.}, 7 (1999), 313. [44] T. Valent, Boundary Value Problems of Finite Elasticity. Local Theorems on Existence, Uniqueness and Analytic Dependence on Data,, Springer-Verlag, (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,, \emph{SIAM J. Appl. Math.}, 53 (1993), 799. doi: 10.1137/0153039. [46] M. J. Ward and J. B. Keller, Nonlinear eigenvalue problems under strong localized perturbations with applications to chemical reactors,, \emph{Stud. Appl. Math.}, 85 (1991), 1. [47] M. J. Ward and J. B. Keller, Strong localized perturbations of eigenvalue problems,, \emph{SIAM J. Appl. Math.}, 53 (1993), 770. doi: 10.1137/0153038.
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