A General Integral Representation Result for Continuum Limits of Discrete Energies with Superlinear Growth

Abstract

We study the asymptotic behavior, as the mesh size ε tends to zero, of a general class of discrete energies defined on functions u : α ∈ εZ N ∩ Ω → u(α) ∈ R d of the formand satisfying superlinear growth conditions. We show that all the possible variational limits are defined on W 1,p (Ω; R d ) of the local typeWe show that, in general, f may be a quasi-convex nonconvex function even if very simple interactions are considered. We also treat the case of homogenization, giving a general asymptotic formula that can be simplified in many situations (e.g., in the case of nearest neighbor interactions or under convexity hypotheses).

Effective Cohesive Behavior of Layers of Interatomic Planes

“…A sketch of P N ( √ Nδ) for a particular value of N is shown in the top figure. Also shown is a graphical representation of the solution of equation (1). Given a value of δ/ √ N , the values of δ N 1 and δ N 2 are determined by finding the horizontal line that leaves the point(s) in the dashed curve at a distance δ/ √ N from the δ = 0 axis.…”

“…-limits of functionals of the form (98) and their multidimensional analogs when φ N n depend on z 1 + · · · + z n have been extensively studied in recent times (see, e.g., [7,[4][5][6]1]). The choice of notation in terms of difference quotients is made in order to facilitate usage of-and comparison with-those results.…”

Overall Properties of a Discrete Membrane with Randomly Distributed Defects

“…It suffices to remark that for all p ∈ [0, 1], ω ∈ Ω, and u we have F b,ω ε (u) E ε (u), where E ε is as in Example 1 (2). It suffices then to use the results recalled in items (2) and (3) in Example 1.…”

“…The application of Γ -convergence to describe continuum limits of discrete systems has been used in different frameworks in various degrees of generality (see, for example, [7,13,14,9,11,2]). Here we will deal with a simple situation of nearest-neighbor energies in R 2 .…”

“…In this case, the knowledge of averaged quantities such as the percentage of strong and weak springs only does not allow any accurate description of the limit energy; in particular, under proper choices of the local geometry, we can obtain both energies (1) and (2). In fact, we may consider the extreme periodic geometries of period N where all weak connections are placed on the boundary of the periodicity cell.…”