A homogenization result for planar, polygonal networks

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1. [1] D. N. ARNOLD, L. RIDGWAY SCOTT and M. VOGELIUS, Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon, Ann. Scuola Norm. Sup. Pisa, 15 (1988), pp. 169-192. Zbl0702.35208MR1007396
2. [2] H. ATTOUCH and G. BUTTAZZO, Homogenization of reinforced periodic one-codimensional structures, Ann. Scuola Norm. Sup. Pisa, 14(1987), pp. 465-484. Zbl0654.73017MR951229
3. [3] D. AZE and G. BUTTAZZO, Some remarks on the optimal design of periodically reinforced structures, RAIRO Model. Math. Anal. Numer., 23 (1989), pp. 53-61. Zbl0679.49005MR1015919
4. [4] J. BRAMBLE and M. ZLAMAL, Triangular elements in the finite element method, Math. Comp., 24 (1970), pp. 809-820. Zbl0226.65073MR282540
5. [5] G. BUTTAZZO and G. DAL MASO, Γ-limits of integral functionals, J. Analyse Math., 37 (1980), pp. 145-185. Zbl0446.49012MR583636
6. [6] E. DEGIORGI and S. SPAGNOLO, Sulla convergenza degli integrali dell'energia per operatori ellitici del secundo ordine, Boll. U.M.I., 8 (1973), pp. 391-411. Zbl0274.35002MR348255
7. [7] R. J. DUFFIN, Distributed and lumped networks, J. Math. Mech., 8 (1959), pp. 793-825. Zbl0092.20501MR106032
8. [8] R. J. DUFFIN, Extremal length of a network, J. Math. Anal. Appl., 5 (1962), pp. 200-215. Zbl0107.43604MR143468
9. [9] R. J. DUFFIN, Topology of series-parallel networks, J. Math. Anal. Appl., 10 (1965), pp. 303-318. Zbl0128.37002MR175809
10. [10] R. J. DUFFIN, Estimating Dirichlet's integral and electrical conductance for systems which are not self-adjoint. Arch. Rat. Mech. Anal., 30 (1968), pp. 90-101. Zbl0159.40604MR228229
11. [11] P. GRISVARD, Elliptic problems in Nonsmooth Domains, Pitman, Marshfield, MA, 1985. Zbl0695.35060MR775683
12. [12] P.-O. JANSSON and G. GRIMWALL, Joule heat distribution in disordered resistor networks, J. Phys. D: Appl. Phys., 18 (1985), pp. 893-900. 
13. [13] R. KÜNNEMANN, The diffusion limit for reversible jump processes on Zd with ergodic random bond conductivities. Comm. Math. Phys., 90 (1983), pp. 27-68. Zbl0523.60097MR714611
14. [14] K. A. LURIE and A. V. CHERKAEV, G-closure of a set of anisotropically conducting media in the two dimensional case. J. Opt. Th. Appl., 42 (1984), pp. 283-304. Zbl0504.73060MR737972
15. [15] K. A. LURIE and A. V. CHERKAEV, Exact estimates of the conductivity of composites formed by two materials taken in prescribed proportion. Proc. Roy. Soc. Edinburgh, 99 A (1984), pp. 71-87. Zbl0564.73079MR781086
16. [16] F. MURAT, H-convergence, Mimeographed notes, Université d'Alger, 1978. 
17. [17] F. MURAT and L. TARTAR, Calcul des variations et homogeneisation, In Les Méthodes de l'Homogénéization : Théorie et Applicationsen Physique ; proc. of summer school on homogenization, Breau-sans-Nappe, July 1983 ; Eyrolles, Paris, 1985, pp. 319-369. MR844873
18. [18] J. A. NITSCHE and A. H. SCHATZ, Interior estimates for Ritz-Galerkun methods. Math. Comp., 28 (1974), pp. 937-958. Zbl0298.65071MR373325
19. [19] G. PAPANICOLAOU and S. R. S. VARADHAN, Boundary value problems with rapidly oscillating random coefficients, Coll. Math. Societatis János Bólyai, #27, Esztergom, Hungary, pp. 835-873, North-Holland, Amsterdam, 1982. Zbl0499.60059MR712714
20. [20] M. SODERBERG, P.-O. JANSSON and G. GRIMWALL, Effective medium theory for resistor networks in checkerboard geometres. J. Phys. A. Math. Gen., 18 (1985), pp. L633-L636. 
21. [21] L. TARTAR, Estimations fines des coefficients homogénéisés, In Ennio De-Giorgi's Colloquium, P. Krée, ed., Pitman Press, London, 1985. Zbl0586.35004MR909716

1. Peter I. Kogut, Günter Leugering, Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs
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