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

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1. [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. Zbl0809.35070MR1289138
2. [2] G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings. Arch. Rational Mech. Anal. 158 (2001) 155-171. Zbl0977.31006MR1838656
3. [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). MR2001070
4. [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. Zbl0913.49008MR1629349
5. [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. Zbl0958.49008MR1693645
6. [6] K. Astala, Area distortion of quasiconformal mappings. Acta Math. 173 (1994) 37-60. Zbl0815.30015MR1294669
7. [7] K. Astala and M. Miettinen, On quasiconformal mappings and $2$-dimensional $G$-closure problems. Arch. Rational Mech. Anal. 143 (1998) 207-240. Zbl0912.65106MR1649998
8. [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). Zbl1106.74052MR2020365
9. [9] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. Zbl0629.49020MR906132
10. [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). Zbl1330.35121MR1871388
11. [11] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North Holland, Amsterdam (1978). Zbl0404.35001MR503330
12. [12] L. Bers, F. John and M. Schechter, Partial Differential Equations. Interscience, New York (1964). Zbl0126.00207MR163043
13. [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. Zbl0067.32503MR76981
14. [14] A. Cherkaev, Necessary conditions technique in optimization of structures. J. Mech. Phys. Solids (accepted). 
15. [15] A. Cherkaev, Variational methods for structural optimization. Springer-Verlag, Berlin, Appl. Math. Sci. 140 (2000). Zbl0956.74001MR1763123
16. [16] A. Cherkaev and L.V. Gibiansky, Extremal structures of multiphase heat conducting composites. Int. J. Solids Struct. 18 (1996) 2609-2618. Zbl0901.73050
17. [17] A.M. Dykhne, Conductivity of a two dimensional two-phase system. Soviet Phys. JETP 32 (1971) 63-65. 
18. [18] A. Eremenko and D.H. Hamilton, The area distortion by quasiconformal mappings. Proc. Amer. Math. Soc. 123 (1995) 2793-2797. Zbl0841.30013MR1283548
19. [19] G. Francfort and G.W. Milton, Optimal bounds for conduction in two-dimensional, multiphase polycrystalline media. J. Stat. Phys. 46 (1987) 161-177. MR887243
20. [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. Zbl0668.73018MR926503
21. [21] L.V. Gibiansky and O. Sigmund, Multiphase composites with extremal bulk modulus. J. Mech. Phys. Solids 48 (2000) 461-498. Zbl0989.74060MR1737888
22. [22] Y. Grabovsky, The $G$-closure of two well ordered anisotropic conductors. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 423-432. Zbl0785.49015MR1226610
23. [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. Zbl0111.41401
24. [24] J. Keller, A theorem on the conductivity of a composite medium. J. Math. Phys. 5 (1964) 548-549. Zbl0129.44001MR161559
25. [25] W. Kohler and G. Papanicolaou, Bounds for the effective conductivity of random media. Springer, Lecture Notes in Phys. 154, p. 111. Zbl0496.73002MR674963
26. [26] R.V. Kohn, The relaxation of a double energy. Continuum Mech. Thermodyn. 3 (1991) 193-236. Zbl0825.73029MR1122017
27. [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. Zbl0631.73012MR859413
28. [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. Zbl0621.49008MR820342
29. [29] O. Lehto and K.I. Virtanen, Quasiconformal Mappings in the Plane. Springer, Berlin (1973). Zbl0267.30016MR344463
30. [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. Zbl0564.73079MR781086
31. [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. Zbl0504.73060MR737972
32. [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. Zbl0545.73005
33. [33] K.S. Mendelson, Effective conductivity of a two-phase material with cylindrical phase boundaries. J. Appl. Phys. 46 (1975) 917. 
34. [34] G.W. Milton, Concerning bounds on the transport and mechanical properties of multicomponent composite materials. Appl. Phys. A 26 (1981) 125-130. 
35. [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. Zbl0751.73041MR1024190
36. [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. Zbl0672.73012MR969257
37. [37] G.W. Milton and V. Nesi, Optimal $G$-closure bounds via stability under lamination. Arch. Rational Mech. Anal. 150 (1999) 191-207. Zbl0941.74012MR1738117
38. [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). Zbl0968.74050MR1731640
39. [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. Zbl0464.46034MR616901
40. [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. [41] V. Nesi, Using quasiconvex functionals to bound the effective conductivity of composite materials. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 633-679. Zbl0791.49042MR1237607
42. [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. Zbl0852.35016MR1363001
43. [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. Zbl0854.30015MR1392308
44. [44] K. Schulgasser, Sphere assemblage model for polycrystal and symmetric materials. J. Appl. Phys. 54 (1982) 1380-1382. 
45. [45] K. Schulgasser, A reciprocal theorem in two dimensional heat transfer and its implications. Internat. Commun. Heat Mass Transfer 19 (1992) 497-515. 
46. [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. Zbl0153.42103
47. [47] S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa (3) 22 (1968) 571-597. Zbl0174.42101MR240443
48. [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. Zbl0920.35018MR540123
49. [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. Zbl0586.35004
50. [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. Zbl0437.35004
51. [51] L. Tonelli, Fondamenti di calcolo delle variazioni. Zanichelli, Bologna (1921). JFM48.0581.09
52. [52] I.N. Vekua, Generalized Analytic Functions. Pergamon, Oxford (1962). Zbl0100.07603MR150320
53. [53] V.V. Zhikov, Estimates for the averaged matrix and the averaged tensor. Russian Math. Surveys (46) 3 (1991) 65-136. Zbl0751.15014MR1134090

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