Wavelet analogue of the Ginzburg-Landau energy and its Γ-convergence

This paper considers a wavelet analogue of the classical Ginzburg-Landau energy, where the Hseminorm is replaced by the Besov seminorm defined via an arbitrary regular wavelet. We prove that functionals of this type converge, in the Γ-sense, to a weighted analogue of the TV functional on characteristic functions of finite-perimeter sets. The Γ-limiting functional is defined explicitly, in terms of the wavelet that is used to define the energy. We show that the limiting energy is none other than the surface tension energy in the 2D Wulff problem and its minimizers are represented by corresponding Wulff shapes. This fact as well as the Γ-convergence results are illustrated with a series of computational examples.