Calculus of Variations and Geometric Measure Theory

R. Alicandro - A. Braides - M. Cicalese

Continuum limits of discrete thin films with superlinear growth densities

created by braidesa on 06 Dec 2005
modified on 27 Sep 2008


Published Paper

Inserted: 6 dec 2005
Last Updated: 27 sep 2008

Journal: Calc. Var. Partial Diff. Eq.
Volume: 33
Pages: 267-297
Year: 2008


Object of this paper is the description of the overall behaviour of variational pair-interaction lattice systems defined on `thin' domains of $*Z*^N$; {\it i.e.} on domains consisting on a finite number $M$ of mutually interacting copies of a portion of a $N-1$-dimensional discrete lattice. On one hand we draw a parallel with the analogous theories for `continuous' thin films showing that general compactness and homogenization results can be proven by adapting the techniques commonly used for problems on Sobolev spaces; on the other hand we show that new phenomena arise due to the different nature of the microscopic interactions, and in particular that for long-range interactions a surface energy on the free surfaces of the film due to boundary layer effects renders the effective behaviour depend in a non-trivial way on the number $M$ of layers.

Keywords: Homogenization, Thin films, Discrete to continuous, Lattice systems