  Physically motivated variational problems involving non-convex energies are
often formulated in a discrete setting and contain boundary conditions. The
long-range interactions in such problems, combined with constraints imposed by
lattice discreteness, can give rise to the phenomenon of geometric frustration
even in a one-dimensional setting. While non-convexity entails the formation of
microstructures, incompatibility between interactions operating at different
scales can produce nontrivial mixing effects which are exacerbated in the case
of incommensuration between the optimal microstructures and the scale of the
underlying lattice. Unraveling the intricacies of the underlying interplay
between non-convexity, non-locality and discreteness, represents the main goal
of this study. While in general one cannot expect that ground states in such
problems possess global properties, such as periodicity, in some cases the
appropriately defined global solutions exist, and are sufficient to describe
the corresponding continuum (homogenized) limits. We interpret those cases as
complying with a Generalized Cauchy-Born (GCB) rule, and present a new class of
problems with geometrical frustration which comply with GCB rule in one range
of (loading) parameters while being strictly outside this class in a
complementary range. A general approach to problems with such mixed behavior is
developed.
