Some results of ferro-magnetic spin systems and related issues

In this thesis we investigate variational lattice systems. First we provide a general framework for the design of surface energies on lattices. We prove sharp bounds for the homogenization of discrete systems describing mixtures of ferromagnetic interactions by constructing optimal micro-geometries, and we prove a localization principle which allows to reduce to the periodic setting in the general nonperiodic case. Secondly we prove the crystallinity of the homogenized energy density assuming that the interactions are periodic and of finite range. In two dimensions we use a dual description on curves in order to show that the energy density is already determined by its values of a finite number on energetically convenient directions. In higher dimensions we utilize convex analysis tools in order to obtain the result. The next problem consists in determining the Γ -limits of inhomogeneous weak- membrane energies as the lattice spacing ε tends to 0 and show that they coincide with an anisotropic, possible inhomogeneous, version of the Mumford Shah functional. By using blow-up and localization techniques we prove that the bulk and surface contribution decouple and we prove that the energy density describing the singular part with respect to the Lebesgue measure agrees with the energy density appearing in the variational limit of the lattice spin systems introduced in the previous chapter. Finally we give conditions to ensure, that the Γ -limit of a discrete system takes a specific integral form. These quite general conditions allow for multi-body, even infinite body interactions. The limit takes an integral form that is commonly used in continuum mechanics.