# American Institute of Mathematical Sciences

September  2012, 7(3): 543-582. doi: 10.3934/nhm.2012.7.543

## Homogenization of pinning conditions on periodic networks

 1 Dipartimento di Matematica, Università di Roma 'La Sapienza', p.le A.Moro 2, 00185 Roma, Italy

Received  June 2011 Revised  July 2012 Published  October 2012

This paper deals with the description of the overall effect of pinning conditions in discrete systems. We study a variational problem on the discrete in which pinning sites are modeled as network subsets on which concentrated forces are imposed. We want to determine the asymptotic effect of pinning conditions on a periodic lattice as its size vanishes. Our analysis is performed in the framework of $\Gamma$-convergence and highlights the analogies and differences with the corresponding continuous problem, i.e. periodically perforated domains. We derive a functional form for the limit energies which depends on the relationship between the space dimension and the growth rate of the interaction functions.
Citation: Laura Sigalotti. Homogenization of pinning conditions on periodic networks. Networks & Heterogeneous Media, 2012, 7 (3) : 543-582. doi: 10.3934/nhm.2012.7.543
##### References:
 [1] R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth, SIAM J. Math Anal., 36 (2004), 1-37. doi: 10.1137/S0036141003426471.  Google Scholar [2] R. Alicandro and M. Cicalese, Variational analysis of the asymptotics of the $XY$ model, Arch. Rat. Mech. Anal., 192 (2009), 501-536. doi: 10.1007/s00205-008-0146-0.  Google Scholar [3] N. Ansini and A. Braides, Asymptotic analysis of periodically-perforated nonlinear media, J. Math. Pures Appl., 81 (2002), 439-451; Erratum in 84 (2005), 147-148.  Google Scholar [4] A. Braides, "$\Gamma$-convergence for Beginners," Oxford University Press, Oxford, 2002.  Google Scholar [5] A. Braides, A handbook of $\Gamma$-convergence, in "Handbook of differential Equations: Stationary Partial DifferentialEquations" (eds. M. Chipot and P. Quittner), Elsevier, 3 (2006). Google Scholar [6] A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems, Arch. Ration.Mech. Anal., 135 (1996), 297-356. doi: 10.1007/BF02198476.  Google Scholar [7] A. Braides and L. Sigalotti, Models of defects in atomistic systems, Calculus of Variations and PDE, 41 (2011), 71-109. doi: 10.1007/s00526-010-0354-y.  Google Scholar [8] D. Cioranescu and F. Murat, Un term étrange venu d'ailleurs, I and II, Nonlinear Partial Differential Equations and Their Applications, Colle de FranceSeminar. Vol. II, 98-138, and Vol. III, 154-178, Res. Notes in Math., 60 and 70, Pitman, London, 1982 and 1983, translated in (A strange term coming from nowhere), Topics in the Mathematical Modelling of Composite Materials,(eds. A. V. Cherkaev and R. V. Kohn), Birkhäuser, 1994. Google Scholar [9] G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications. Birkhser Boston, Boston, 1993.  Google Scholar [10] G. Dal Maso, Asymptotic behaviour of solutions of Dirichlet problems, Boll. Unione Mat. Ital., 11A (1997), 253-277.  Google Scholar [11] G. Dal Maso and A. Defranceschi, Limits of nonlinear Dirichlet problems in varying domains, Manuscripta Math., 61 (1988), 251-278. doi: 10.1007/BF01258438.  Google Scholar [12] G. Dal Maso and A. Garroni, New results on the asymptotic analysis of Dirichlet problems in perforated domains, Math. Mod. Meth. Appl. Sci., 4 (1994), 373-407.  Google Scholar [13] G. Dal Maso, A. Garroni and I. V. Skrypnik, A capacitary method for the asymptotic analysis of Dirichlet problems for monotone operators, J. Anal. Math., 71 (1997), 263-313. doi: 10.1007/BF02788033.  Google Scholar [14] G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 293-290.  Google Scholar [15] A. Defranceschi and E. Vitali, Limits of minimum problems with convex obstacles for vector valued functions, Appl. Anal., 52 (1994), 1-33.  Google Scholar [16] A. Garroni and S. Müller, A variational model for dislocations in the line tension limit, Arch. Rat. Mech.,181 (2006), 535-578. doi: 10.1007/s00205-006-0432-7.  Google Scholar [17] A. V. Marchenko and E. Ya. Khruslov, New results in the theory of boundary value problems for regions with closed-grained boundaries, Uspekhi Math. Nauk, 33 (1978). Google Scholar [18] L. Sigalotti, Asymptotic analysis of periodically perforated nonlinear media at the critical exponent, Comm. Cont. Math., 11 (2009), 1009-1033. doi: 10.1142/S0219199709003648.  Google Scholar [19] I. V. Skrypnik, Asymptotic behaviour of solutions of nonlinear elliptic problems in perforated domains, Math. Sb., 184 (1993), 67-70. Google Scholar

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##### References:
 [1] R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth, SIAM J. Math Anal., 36 (2004), 1-37. doi: 10.1137/S0036141003426471.  Google Scholar [2] R. Alicandro and M. Cicalese, Variational analysis of the asymptotics of the $XY$ model, Arch. Rat. Mech. Anal., 192 (2009), 501-536. doi: 10.1007/s00205-008-0146-0.  Google Scholar [3] N. Ansini and A. Braides, Asymptotic analysis of periodically-perforated nonlinear media, J. Math. Pures Appl., 81 (2002), 439-451; Erratum in 84 (2005), 147-148.  Google Scholar [4] A. Braides, "$\Gamma$-convergence for Beginners," Oxford University Press, Oxford, 2002.  Google Scholar [5] A. Braides, A handbook of $\Gamma$-convergence, in "Handbook of differential Equations: Stationary Partial DifferentialEquations" (eds. M. Chipot and P. Quittner), Elsevier, 3 (2006). Google Scholar [6] A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems, Arch. Ration.Mech. Anal., 135 (1996), 297-356. doi: 10.1007/BF02198476.  Google Scholar [7] A. Braides and L. Sigalotti, Models of defects in atomistic systems, Calculus of Variations and PDE, 41 (2011), 71-109. doi: 10.1007/s00526-010-0354-y.  Google Scholar [8] D. Cioranescu and F. Murat, Un term étrange venu d'ailleurs, I and II, Nonlinear Partial Differential Equations and Their Applications, Colle de FranceSeminar. Vol. II, 98-138, and Vol. III, 154-178, Res. Notes in Math., 60 and 70, Pitman, London, 1982 and 1983, translated in (A strange term coming from nowhere), Topics in the Mathematical Modelling of Composite Materials,(eds. A. V. Cherkaev and R. V. Kohn), Birkhäuser, 1994. Google Scholar [9] G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications. Birkhser Boston, Boston, 1993.  Google Scholar [10] G. Dal Maso, Asymptotic behaviour of solutions of Dirichlet problems, Boll. Unione Mat. Ital., 11A (1997), 253-277.  Google Scholar [11] G. Dal Maso and A. Defranceschi, Limits of nonlinear Dirichlet problems in varying domains, Manuscripta Math., 61 (1988), 251-278. doi: 10.1007/BF01258438.  Google Scholar [12] G. Dal Maso and A. Garroni, New results on the asymptotic analysis of Dirichlet problems in perforated domains, Math. Mod. Meth. Appl. Sci., 4 (1994), 373-407.  Google Scholar [13] G. Dal Maso, A. Garroni and I. V. Skrypnik, A capacitary method for the asymptotic analysis of Dirichlet problems for monotone operators, J. Anal. Math., 71 (1997), 263-313. doi: 10.1007/BF02788033.  Google Scholar [14] G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 293-290.  Google Scholar [15] A. Defranceschi and E. Vitali, Limits of minimum problems with convex obstacles for vector valued functions, Appl. Anal., 52 (1994), 1-33.  Google Scholar [16] A. Garroni and S. Müller, A variational model for dislocations in the line tension limit, Arch. Rat. Mech.,181 (2006), 535-578. doi: 10.1007/s00205-006-0432-7.  Google Scholar [17] A. V. Marchenko and E. Ya. Khruslov, New results in the theory of boundary value problems for regions with closed-grained boundaries, Uspekhi Math. Nauk, 33 (1978). Google Scholar [18] L. Sigalotti, Asymptotic analysis of periodically perforated nonlinear media at the critical exponent, Comm. Cont. Math., 11 (2009), 1009-1033. doi: 10.1142/S0219199709003648.  Google Scholar [19] I. V. Skrypnik, Asymptotic behaviour of solutions of nonlinear elliptic problems in perforated domains, Math. Sb., 184 (1993), 67-70. Google Scholar
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