Multiscale analysis for nonlinear variational problems arising from discrete systems

The object of this thesis is the study of high density discrete systems as variational limit of low density discrete energies indexed by the number of nodes of the system itself. In this context the term discreteness should be understood rather broadly as inferring to different scales from crystal lattice to grain structure while the low-to-high density limit refers to the discrete-to-continuum passage in the energetic description of the system. Within this framework we focus our attention on central lattice systems; i.e., systems where the reference positions of the interacting points lie on a prescribed lattice, whose parameters change as the number of points changes and where all the interactions are pair interactions. In more precise terms, we consider an open set Ω ⊂ R N and take as reference lattice Z ε = εZ N ∩ Ω. The general form of a pair-potential energy in a central system is E ε (u) = i,j∈Z ε ψ ε ij (u(i), u(j)), (0.0.1) where u : Z ε → R d. The analysis of energies of the form (0.0.1) has been performed under various hypotheses on ψ ε ij. The first natural assumption is the invariance under translations (in the target space); that is, ψ ε ij (u, v) = g ε ij (u − v). Furthermore, an important class of pair potentials is that of homogeneous interactions (i.e., invariant under translations in the reference space); this can be expressed as ψ ε ij (u, v) = g ε (i−j)/ε (u, v). If both conditions are satisfied, then the energies E ε above may be rewritten in the form E ε (u) = k∈Z n i,j∈Z ε ,i−j=εk ε n f ε k u(i) − u(j) ε , where f ε k (ξ) = ε −n g ε k (εξ). In this way we can highlight the dependence of the potentials on the (discrete) difference quotients of the function u. Upon identifying i each function u with its piecewise-constant interpolation, we can consider E ε as defined on (a subset of) L p (Ω; R d), and hence consider the Γ-limit as ε tends to zero with respect to the L p-topology. Under some coerciveness conditions the computation of the Γ-limit will give a continuous approximate description of the behaviour of minimum problems involving the energies E ε for ε small (see Chapter 1, and [13] for a quick introduction to …