Homogenization of pinning conditions on periodic networks

Laura Sigalotti; Laura Sigalotti

doi:10.3934/nhm.2012.7.543

Networks and Heterogeneous Media

2012, Volume 7, Issue 3: 543-582. doi: 10.3934/nhm.2012.7.543

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Homogenization of pinning conditions on periodic networks

Laura Sigalotti ^{1
,}

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

Received: 01 June 2011 Revised: 01 July 2012
Primary: 49J45, 82B20; Secondary: 74Q05, 49M25.

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.
- Discrete energies,
- $\Gamma$-convergence,
- periodic network,
- pinning sites,
- critical exponent
Citation: Laura Sigalotti. Homogenization of pinning conditions on periodic networks[J]. Networks and Heterogeneous Media, 2012, 7(3): 543-582. doi: 10.3934/nhm.2012.7.543

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Abstract

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.

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