The consistency of a nonlocal anisotropic Ginzburg-Landau type functional for data classification and clustering is studied. The Ginzburg-Landau objective functional combines a double well potential, that favours indicator valued function, and the @math -Laplacian, that enforces regularity. Under appropriate scaling between the two terms minimisers exhibit a phase transition on the order of @math where @math is the number of data points. We study the large data asymptotics, i.e. as @math , in the regime where @math . The mathematical tool used to address this question is @math -convergence. In particular, it is proved that the discrete model converges to a weighted anisotropic perimeter.
 We obtain the r(L*(Q)) limit of the sequence where E£ is the family of anisotropic perturbations Ee(u) := j W(u(x)) dx + e J h*(x.Vu(x)) dx of the nonconvex functional of vector-valued functions The proof relies on the blow-up argument introduced by Fonseca and Muller [FMl]. We study the effect of a singular perturbation on certain nonconvex variational problems. The goal is to characterize the limit of minimizers as some perturbation parameter ɛ → 0. The technique utilizes the notion of “Γ-convergence” of variational problems developed by De Giorgi. The essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem. In so doing, one characterizes the limit of minimizers as the solution of a new variational problem. For the cases considered here, these new problems have a simple geometric nature involving minimal surfaces and geodesics. We obtain the Γ(L 1 (Ω) - )-limit for a class of variational problems which arise as an «anisotropic» singular perturbation of a nonconvex functional. Any limit of minimizers of the perturbed problems must minimize the new variational problem which arises in the Γ-limit, and the primary goal is to understand the dependence of this new variational problem on the type of perturbation chosen In this paper I prove some conjectures of GURTIN [15] concerning the Van der Waals-Cahn-Hilliard theory of phase transitions. Consider a fluid, under isothermal conditions and confined to a bounded container 12 Q R', whose Gibbs free energy, per unit volume, is a prescribed function Wo of the density distribution u. The classical problem (cf. GURTIN [16]) of determining the stable configurations of the fluid is to minimize the total energy of the fluid, E(u) = f Wo(u(x)) dx, t2  On generalise la theorie du gradient des transitions de phase au cas a valeur vectoriel. On considere la famille des perturbations E e (u)=∫ Ω W(u)dx+e 2 ∫ Ω |⊇u| 2 dx de la fonctionnelle non convexe E 0 (u):=∫ Ω W(u)dx, ou W:R N →R supporte deux phases et N≥1 Given thatα, β are two Lipschitz continuous functions of Ω to ℝ+ and thatf(x, u, p) is a continuous function of ( ) × ℝ × ℝ N to [0, + ∞[ such that, for everyx, f(x,·, 0) reaches its minimum value 0 at exactly two pointsα(x) andβ(x), we prove the convergence ofF e (u) = (1 e)∫ Ω f (x, u, eDu) dx when the perturbation parametere goes to zero. A formula is given for the limit functional and a general minimal interface criterium is deduced for a wide class of two-phase transition models. Earlier results of [19], [21], and [22] are extended with new proofs.  