Variational Properties of Unbounded Order Parameters

Order parameters in physical and biological systems can sometimes become unbounded as the size of an underlying system increases. It is proposed that such a quantity be modeled as a minimizer of the energy functional Iε(u) = − ∫ [ ε2 2 |∇u| − 1 2 log(1 + |u|) ] dx, where u is constrained by a side condition, and ε > 0 is a parameter that is inversely proportional to the linear size of the system. It is shown that a minimizer of Iε exists; the minimum value of Iε scales as log ε; and both the L 2 and H1 norms of any minimizer of Iε are of the order O(1/ε), indicating the unboundedness of the order parameter. It is also shown that the renormalized energy functionals Jε(v) = Iε (v ε ) − log ε Γ-converge to the functional J(v) = − ∫ ( 1 2 |∇v| − log |v| ) dx. Minimizers of this Γ-limit for scalar order parameters with the Dirichlet boundary condition are well characterized. 2000 Mathematics Subject Class: 49J45, 49S05, 74G65.