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. 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. 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. Google Scholar

[4]

A. Braides, "$\Gamma$-convergence for Beginners,", Oxford University Press, (2002). Google Scholar

[5]

A. Braides, A handbook of $\Gamma$-convergence,, in, 3 (2006). Google Scholar

[6]

A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems,, Arch. Ration.Mech. Anal., 135 (1996), 297. 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. 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, 60 (1982), 98. Google Scholar

[9]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Progress in Nonlinear Differential Equations and their Applications. Birkhser Boston, (1993). Google Scholar

[10]

G. Dal Maso, Asymptotic behaviour of solutions of Dirichlet problems,, Boll. Unione Mat. Ital., 11A (1997), 253. Google Scholar

[11]

G. Dal Maso and A. Defranceschi, Limits of nonlinear Dirichlet problems in varying domains,, Manuscripta Math., 61 (1988), 251. 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. 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. 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. Google Scholar

[15]

A. Defranceschi and E. Vitali, Limits of minimum problems with convex obstacles for vector valued functions,, Appl. Anal., 52 (1994), 1. Google Scholar

[16]

A. Garroni and S. Müller, A variational model for dislocations in the line tension limit,, Arch. Rat. Mech., 181 (2006), 535. 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. 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. Google Scholar

show all references

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. 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. 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. Google Scholar

[4]

A. Braides, "$\Gamma$-convergence for Beginners,", Oxford University Press, (2002). Google Scholar

[5]

A. Braides, A handbook of $\Gamma$-convergence,, in, 3 (2006). Google Scholar

[6]

A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems,, Arch. Ration.Mech. Anal., 135 (1996), 297. 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. 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, 60 (1982), 98. Google Scholar

[9]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Progress in Nonlinear Differential Equations and their Applications. Birkhser Boston, (1993). Google Scholar

[10]

G. Dal Maso, Asymptotic behaviour of solutions of Dirichlet problems,, Boll. Unione Mat. Ital., 11A (1997), 253. Google Scholar

[11]

G. Dal Maso and A. Defranceschi, Limits of nonlinear Dirichlet problems in varying domains,, Manuscripta Math., 61 (1988), 251. 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. 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. 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. Google Scholar

[15]

A. Defranceschi and E. Vitali, Limits of minimum problems with convex obstacles for vector valued functions,, Appl. Anal., 52 (1994), 1. Google Scholar

[16]

A. Garroni and S. Müller, A variational model for dislocations in the line tension limit,, Arch. Rat. Mech., 181 (2006), 535. 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. 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. Google Scholar

[1]

Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143

[2]

Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017

[3]

Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679

[4]

Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357

[5]

Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427

[6]

Lucia Scardia, Anja Schlömerkemper, Chiara Zanini. Towards uniformly $\Gamma$-equivalent theories for nonconvex discrete systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 661-686. doi: 10.3934/dcdsb.2012.17.661

[7]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231

[8]

Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351

[9]

Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104

[10]

Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071

[11]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[12]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773

[13]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025

[14]

Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921

[15]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103

[16]

T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875

[17]

Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307

[18]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179

[19]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527

[20]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]