Abstract
A recent theorem due to Astala establishes the best exponent for the area distortion of planar K-quasiconformal mappings. We use a refinement of Astala's theorem due to Eremenko and Hamilton to prove new bounds on the effective conductivity of two-dimensional composites. The bounds are valid for composites made of an arbitrary finite number n of possibly anisotropic phases in prescribed volume fractions. For n= 2 we prove the optimality of the bounds under certain additional assumptions on the G-closure parameters.
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References
Astala K., Area distortion of quasiconformal mappings, Acta Math., 173 (1994), 37–60.
Avellaneda, M., Cherkaev, A. V., Lurie, K. A., & Milton, G. W., On the effective conductivity of polycrystals and a three dimensional phase-interchange inequality, J. Appl. Phys. 63 (1988), 4989–5003.
Bojarski, B., Homeomorphic solutions of Beltrami system, Dokl. Akad. Nauk. SSSR 102 (1955), 661–664.
Bojarski, B., Generalized solutions of a system of differential equations of first order and elliptic type with discontinuous coefficients, Mat. Sb. 43(85) (1957), 451–503.
Bauman, P., Marini A., & Nesi V., Univalent solutions of an elliptic system of partial differential equations arising in homogenization, preprint.
Ball, J. M., & Murat, F., W 1,p-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal. 58 (1984), 225–253.
Dacorogna, B., & Marcellini, P., Existence of minimizers for non quasiconvex integrals, Arch. Rational Mech. Anal. 131 (1995), 359–399.
Eremenko, A., On Astala's proof of the area distortion theorem for quasiconformal mappings, preprint, December 1992.
Eremenko, A., & Hamilton, D. H., The area distortion by quasiconformal mappings, Proc. Amer. Math. Soc., 123 (1995), 2793–2797.
Francfort, G., & Milton, G. W., Optimal bounds for conduction in two-dimensional, multiphase polycrystalline media, J. Stat. Phys. 46 (1987), 161–177.
Francfort, G., & Murat, F., Optimal bounds for conduction in two-dimensional, two phase, anisotropic media, in Non-classical continuum mechanics, R. J. Knops and A. A. Lacey, eds., London Mathematical Society Lecture Note Series 122, Cambridge Univ. Press, 1987, 197–212.
Grabovsky, Y., The G-closure of two well ordered anisotropic conductors. Proc. Roy. Soc. Edinburgh A 123 (1993), 423–432.
Hamilton, D., Area distortion of quasiconformal mappings, preprint, December 1992.
Hashin, Z., & Shtrikman, S., A variational approach to the theory of effective magnetic permeability of multiphase materials, J. Appl. Phys. 33 (1962), 3125–3131.
Hashin, Z., & Shtrikman, S., Conductivity of polycrystals, Phys. Rev. 130 (1963), 129–133.
Kohn, R. V., The relaxation of a double energy, Continuum Mech. Thermodyn. 3 (1991), 193–236.
Kohn, R. V., & Milton, G.W., On bounding the effective conductivity of anisotropic composites, in Homogenization and effective moduli of materials and media, Ericksen, J. L., Kinderlehrer, D., Kohn, R., & Lions, J. L., eds., Springer-Verlag, New York, 1986, 97–125.
Kohn, R. V., & Strang, G., Optimal design and relaxation of variational problems I, II, III, Comm. Pure Appl. Math. 39 (1986), 113–137, 139–182, 353–377.
Iwaniec, T., & Kosecki, R., Sharp estimates for complex potentials and quasiconformal mappings, preprint of Syracuse University, 1–69.
Iwaniec, T., & Martin, G., Quasiregular mappings in even dimensions, Acta Math. 170 (1993), 29–81.
Lehto, O., Univalent functions and Teichmüller spaces, Springer-Verlag, Berlin, 1987.
Letho, O., & Virtanen, K. I., Quasiconformal mappings in the plane, Springer-Verlag, Berlin, 1973.
Lurie, K. A., & Cherkaev, A. V., Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportions, Proc. Roy. Soc. Edinburgh A 99 (1984), 71–87.
Lurie, K. A., & Cherkaev, A. V., G-closure of a set of anisotropically conducting media in the two-dimensional case, J. Opt. Th. Appl. 42 (1984), 283–304.
Meyers, N., An L p-estimate for the gradient of solutions of second order elliptic divergence form equations. Ann. Sc. Superiore di Pisa Serie III, 17 (1963), 189–206.
Milton, G. W., & Kohn, R. V., Variational bounds on the effective moduli of anisotropic composites, J. Mech. Phys. Solids, 36 (1988), 597–629.
Murat, F., & Tartar, L., Calcul des variations et homogénéisation, in Les méthodes de l'homogénéisation: Théorie et Applications en Physique, Eyrolles, 1985, 319–369.
Nesi, V., Using quasiconvex functionals to bound the effective conductivity of composite materials, Proc. Roy. Soc. Edinburgh A 123 (1993), 633–679.
Nesi, V., Bounds on the effective conductivity of 2d composite made of n ≳= 3 isotropie phases: the weighted translation method, Proc. Roy. Soc. Edinburgh A 125 (1995), 1219–1239.
Schulgasser, K., Sphere assemblage model for polycrystal and symmetric materials, J. Appl. Phys. 54 (1982), 1380–1382.
Tartar, L., Estimation fines des coefficient homogénéisés, in Ennio De Giorgi's Colloquium, P. Kree, ed., Pitman Research Notes in Mathematics 125, London, 1985, pp. 168–187.
Tartar, L., Compensated compactness and applications to partial differential equations in Nonlinear analysis and mechanics, Heriot-Watt Symposium, Vo1. IV, R. J. Knops, ed., Pitman Research Notes in mathematics 39, Boston, 1979, 136–212.
Zhikov, V. V., Estimates for the averaged matrix and the averaged tensor, Russian Math. Surveys 46 (1991), 65–136.
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Communicated by R. V. Kohn
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Nesi, V. Quasiconformal mappings as a tool to study certain two-dimensional G-closure problems. Arch. Rational Mech. Anal. 134, 17–51 (1996). https://doi.org/10.1007/BF00376254
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DOI: https://doi.org/10.1007/BF00376254