7. References
Alvino, A., Trombetti, G. &Lions, P. L.,On optimization problems with prescribed rearrangements, Nonlinear Analysis, TAM,13, 1989, pp. 185–220.
Avellaneda, M.,Optimal bounds and microgeometries for elastic composites, SIAM J. Appl. Math.47, 1987, pp. 1216–1228.
Berryman, J. G.,Bounds on decay constants for diffusion through inhomogeneous media, J. Phys. A:Math. Gen.21, 1988, pp. 4423–4441.
Boccardo, L. &Marcellini P., Sulla convergenza delle soluzioni di disequazioni variazionali, Ann. Mat. Pura Appl.110, 1977, pp. 137–159.
Buttazzo, G. &Dal Maso, G.,An existence result for a class of shape optimization problems, Arch. Rational Mech. Anal.122, 1993, pp. 183–195.
Clark, F.,Nonsmooth Analysis and Optimization, SIAM, Philidelphia, 1990.
Cox, S.,The two phase drum with the deepest bass note, Japan J. Industrial Appl. Math.8, 1991, pp. 345–355.
Cox, S.,The generalized gradient at a multiple eigenvalue, J. Func. Anal.,133, 1995, pp. 30–40.
Cox, S. &McLaughlin, J. R.,Extremal eigenvalue problems for composite membranes I & II, Applied Math. & Optimization22, 1990, pp. 153–167, 169–187.
Cox, S. &Ross, M.,Extremal eigenvalue problems for starlike planar domains, J. Diff. Eqs.120, 1995, pp. 174–197.
Francfort, G., Murat, F. &Tartar, L. Fourth order moments of non-negative measures on S 2 and applications, Archive for Rational Mech. Anal.,131, 1995, pp. 305–333.
Jouron, C.,Sur un probléme d'optimisation ou la contrainte porte sur la fréquence fondamentale, R.A.I.R.O. Analyse numérique12, 1978, pp. 349–374.
Kohn, R. V. &Milton, G. W. On bounding the effective conductivity of anisotropic composites, in Homogenization and Effective Moduli of Materials and Media,J. Erickson et al. eds. Springer-Verlag, 1986, pp. 97–126.
Krein, M. G.,On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, Amer. Math. Soc. Translations Ser.2(1), 1955, pp. 163–187.
Lurie, K. A.,Optimal Control in Problems of Mathematical Physics (in Russian), Nauka, Moscow, 1975.
Lurie, K. A., &Cherkaev, A. V.,Non-homogeneous bar of extremal torsional rigidity (in Russian), Nonlinear Problems in Structural Mechanics; Structural optimization, Kiev, 1978, pp 64–68.
Lurie, K. A., &Cherkaev, A. V.,Exact estimates of conductivity of a binary mixture of isotropic compounds, A. F. Ioffe Physico-Technical Institute, Technical Report No. 894, Academy of Sciences of the USSR, Leningrad, 1984.
Lurie, K. A., &Cherkaev, A. V.,Effective characteristics of composite materials and the optimal design of structural elements (in Russian), Uspekhi Mekhaniki 9, Advances in Mechanics, 1986.
Mirsky, L.,On the trace of matrix products, Math. Nach20, 1959, pp. 171–174.
Murat, F. &Tartar, L.,Calcul des variations et homogénéisation, in Les methodes de l'homogénéisation: théorie et applications en physique,D. Bergman et al. eds., Eyrolles, Paris, 1985, pp. 321–369.
Rauch, J., &M. Taylor,Potential and scattering theory on wildly perturbed domains, J. Func. Anal.18, 1975, pp. 27–59.
Tahraoui, R.,Quelques remarques sur le controle des valeurs propres, in Nonlinear PDE's and their Applications: College de France Seminar, Volume 8,H. Brezis &J. L. Lions, eds., Longman, Essex, 1988, pp. 176–213.
Tartar, L.,Estimations fines de coefficients homogeneises, inEnnio DeGiorgo Colloquium,Kree,P., ed., Pitman, Boston, 1985, pp. 168–187.
Wolf, S. A., &Keller, J. B.,Range of the first two eigenvalues of the laplacian, Proc. R. Soc. Lond. A,447, 1994, pp. 397–412.
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Cox, S., Lipton, R. Extremal eigenvalue problems for two-phase conductors. Arch. Rational Mech. Anal. 136, 101–117 (1996). https://doi.org/10.1007/BF02316974
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DOI: https://doi.org/10.1007/BF02316974