Univalent $\sigma$-harmonic mappings : applications to composites

Alessandrini, Giovanni; Nesi, Vincenzo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 379-406.

This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all the two dimensional $G$-closure problems in conductivity.

[1] G. Alessandrini and R. Magnanini, Elliptic equation in divergence form, geometric critical points of solutions and Stekloff eigenfunctions. SIAM J. Math. Anal. 25 (1994) 1259-1268. | MR | Zbl

[2] G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings. Arch. Rational Mech. Anal. 158 (2001) 155-171. | MR | Zbl

[3] G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings: Connections with quasiconformal mappings, Quaderni Matematici II serie, 510 Novembre 2001. Dipartimento di Scienze Matematiche, Trieste. J. Anal. Math. (to appear). | MR

[4] G. Allaire and G. Francfort, Existence of minimizers for non-quasiconvex functionals arising in optimal design. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 301-339. | Numdam | MR | Zbl

[5] G. Allaire and V. Lods, Minimizers for a double-well problem with affine boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 439-466. | MR | Zbl

[6] K. Astala, Area distortion of quasiconformal mappings. Acta Math. 173 (1994) 37-60. | MR | Zbl

[7] K. Astala and M. Miettinen, On quasiconformal mappings and $2$-dimensional $G$-closure problems. Arch. Rational Mech. Anal. 143 (1998) 207-240. | MR | Zbl

[8] K. Astala and V. Nesi, Composites and quasiconformal mappings: New optimal bounds. University of Jyväskylä, Department of Mathematics, Preprint 233, Ottobre 2000, Jyväskylä, Finland. Calc. Var. Partial Differential Equations (to appear). | MR | Zbl

[9] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. | MR | Zbl

[10] P. Bauman, A. Marini and V. Nesi, Univalent solutions of an elliptic system of partial differential equations arising in homogenization. Indiana Univ. Math. J. 50 (2001). | MR

[11] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North Holland, Amsterdam (1978). | MR | Zbl

[12] L. Bers, F. John and M. Schechter, Partial Differential Equations. Interscience, New York (1964). | MR | Zbl

[13] L. Bers and L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications, in Convegno Internazionale sulle Equazioni alle Derivate Parziali. Cremonese, Roma (1955) 111-138. | MR | Zbl

[14] A. Cherkaev, Necessary conditions technique in optimization of structures. J. Mech. Phys. Solids (accepted).

[15] A. Cherkaev, Variational methods for structural optimization. Springer-Verlag, Berlin, Appl. Math. Sci. 140 (2000). | MR | Zbl

[16] A. Cherkaev and L.V. Gibiansky, Extremal structures of multiphase heat conducting composites. Int. J. Solids Struct. 18 (1996) 2609-2618. | Zbl

[17] A.M. Dykhne, Conductivity of a two dimensional two-phase system. Soviet Phys. JETP 32 (1971) 63-65.

[18] A. Eremenko and D.H. Hamilton, The area distortion by quasiconformal mappings. Proc. Amer. Math. Soc. 123 (1995) 2793-2797. | MR | Zbl

[19] G. Francfort and G.W. Milton, Optimal bounds for conduction in two-dimensional, multiphase polycrystalline media. J. Stat. Phys. 46 (1987) 161-177. | MR

[20] G. Francfort and F. Murat, Optimal bounds for conduction in two-dimensional, two phase, anisotropic media, in Non-classical continuum mechanics, edited by R.J. Knops and A.A. Lacey. Cambridge, London Math. Soc. Lecture Note Ser. 122 (1987) 197-212. | MR | Zbl

[21] L.V. Gibiansky and O. Sigmund, Multiphase composites with extremal bulk modulus. J. Mech. Phys. Solids 48 (2000) 461-498. | MR | Zbl

[22] Y. Grabovsky, The $G$-closure of two well ordered anisotropic conductors. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 423-432. | MR | Zbl

[23] Z. Hashin and S. Shtrikman, A variational approach to the theory of effective magnetic permeability of multiphase materials. J. Appl. Phys. 33 (1962) 3125-3131. | Zbl

[24] J. Keller, A theorem on the conductivity of a composite medium. J. Math. Phys. 5 (1964) 548-549. | MR | Zbl

[25] W. Kohler and G. Papanicolaou, Bounds for the effective conductivity of random media. Springer, Lecture Notes in Phys. 154, p. 111. | MR | Zbl

[26] R.V. Kohn, The relaxation of a double energy. Continuum Mech. Thermodyn. 3 (1991) 193-236. | MR | Zbl

[27] R.V. Kohn and G.W. Milton, On bounding the effective conductivity of anisotropic composites, in Homogenization and effective moduli of materials and media, edited by J.L. Ericksen, D. Kinderlehrer, R. Kohn and J.-L. Lions. Springer, New York (1986) 97-125. | MR | Zbl

[28] R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems I, II, III. Comm. Pure Appl. Math. 39 (1986) 113-137, 139-182, 353-377. | MR | Zbl

[29] O. Lehto and K.I. Virtanen, Quasiconformal Mappings in the Plane. Springer, Berlin (1973). | MR | Zbl

[30] K.A. Lurie and A.V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportions. Proc. Roy. Soc. Edinburgh Sect. A 99 (1984) 71-87. | MR | Zbl

[31] K.A. Lurie and A.V. Cherkaev, $G$-closure of a set of anisotropically conducting media in the two-dimensional case. J. Optim. Theory Appl. 42 (1984) 283-304. | MR | Zbl

[32] K.A. Lurie and A.V. Cherkaev, The problem of formation of an optimal isotropic multicomponent composite. J. Optim. Theory Appl. 46 (1985) 571-589. | Zbl

[33] K.S. Mendelson, Effective conductivity of a two-phase material with cylindrical phase boundaries. J. Appl. Phys. 46 (1975) 917.

[34] G.W. Milton, Concerning bounds on the transport and mechanical properties of multicomponent composite materials. Appl. Phys. A 26 (1981) 125-130.

[35] G.W. Milton, On characterizing the set of possible effective tensors of composites: The variational method and the translation method. Comm. Pure Appl. Math. 43 (1990) 63-125. | MR | Zbl

[36] G.W. Milton and R.V. Kohn, Variational bounds on the effective moduli of anisotropic composites. J. Mech. Phys. Solids 36 (1988) 597-629. | MR | Zbl

[37] G.W. Milton and V. Nesi, Optimal $G$-closure bounds via stability under lamination. Arch. Rational Mech. Anal. 150 (1999) 191-207. | MR | Zbl

[38] S. Müller, Variational models for microstructure and phase transitions, Calculus of variations and geometric evolution problems. Cetraro (1996) 85-210. Springer, Berlin, Lecture Notes in Math. 1713 (1999). | MR | Zbl

[39] F. Murat, Compacité par compensation : condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981) 69-102. | Numdam | MR | Zbl

[40] F. Murat and L. Tartar, Calcul des variations et homogénéisation, in Les Méthodes de L'Homogénéisation : Théorie et Applications en Physique. Eyrolles (1985) 319-369.

[41] V. Nesi, Using quasiconvex functionals to bound the effective conductivity of composite materials. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 633-679. | MR | Zbl

[42] V. Nesi, Bounds on the effective conductivity of $2d$ composites made of $n\ge 3$ isotropic phases in prescribed volume fractions: The weighted translation method. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 1219-1239. | MR | Zbl

[43] V. Nesi, Quasiconformal mappings as a tool to study the effective conductivity of two dimensional composites made of $n\ge 2$ anisotropic phases in prescribed volume fraction. Arch. Rational Mech. Anal. 134 (1996) 17-51. | MR | Zbl

[44] K. Schulgasser, Sphere assemblage model for polycrystal and symmetric materials. J. Appl. Phys. 54 (1982) 1380-1382.

[45] K. Schulgasser, A reciprocal theorem in two dimensional heat transfer and its implications. Internat. Commun. Heat Mass Transfer 19 (1992) 497-515.

[46] S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore. Ann. Scuola Norm. Sup. Pisa (3) 21 (1967) 657-699. | Numdam | Zbl

[47] S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa (3) 22 (1968) 571-597. | Numdam | MR | Zbl

[48] L. Tartar, Estimations de coefficients homogénéisés. Springer, Berlin, Lecture Notes in Math. 704 (1978) 364-373. English translation: Estimations of homogenized coefficients, in Topics in the mathematical modelling of composite materials, 9-20. Birkhäuser, Progr. Nonlinear Differential Equations Appl. 31. | MR | Zbl

[49] L. Tartar, Estimations fines des coefficients homogénéisés, in Ennio De Giorgi's Colloquium (Paris 1983), edited by P. Kree. Pitman, Boston (1985) 168-187. | Zbl

[50] L. Tartar, Compensated compactness and applications to p.d.e. in nonlinear analysis and mechanics, in Heriot-Watt Symposium, Vol. IV, edited by R.J. Knops. Pitman, Boston (1979) 136-212. | Zbl

[51] L. Tonelli, Fondamenti di calcolo delle variazioni. Zanichelli, Bologna (1921). | JFM

[52] I.N. Vekua, Generalized Analytic Functions. Pergamon, Oxford (1962). | MR | Zbl

[53] V.V. Zhikov, Estimates for the averaged matrix and the averaged tensor. Russian Math. Surveys (46) 3 (1991) 65-136. | MR | Zbl