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

Julia Dobrosotskaya, Andrea Bertozzi
2010 Interfaces and free boundaries (Print)  
This paper considers a wavelet analogue of the classical Ginzburg-Landau energy, where the H 1 seminorm is replaced by the Besov seminorm defined via an arbitrary regular wavelet. We prove that functionals of this type Γ -converge 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
more » ... e tension energy in the 2D Wulff problem and its minimizers are represented by the corresponding Wulff shapes. This fact as well as the Γ -convergence results are illustrated with a series of computational examples. characteristic functions of finite-perimeter sets. LEMMA 2.6 Let u n → u in L 1 , where u = χ E and E ⊂ R N is some measurable set. Denote Since the estimate is independent of the scale J , R(u) can be defined via any sequence u n → u in L 1 : Proof. Fix an arbitrary scale J . Then where a is the constant from the seminorm equivalence a| · | H 1 | · | B b| · | H 1 . Now, since |u n J − u J | B O(2 J u − u n L 1 ) (from the definition of the B seminorm) and |u J | B = O(2 J ) (from Lemma 2.5), we conclude |u n J
doi:10.4171/ifb/243 fatcat:qqiuc5pvyrbanjg6ojq46k7jzu