Ph.D Thesis

Ph.D StudentPoliakovsky Arkady
SubjectLifting in BV Spaces and Related Problems
DepartmentDepartment of Mathematics
Supervisor PROF. Itai Shafrir


     The main objective of this thesis is the study of singular perturbation problems involving -valued functions in BV-space, i.e., functions of bounded variation  with values in the circle. We study mainly two problems of this type in which we are concerned with the asymptotic behavior of the minimizers for an energy functional depending on a small parameter , as this parameter goes to . Our main contribution to the study of such problems is a construction method of an upper-bound for the energy which uses a special class of mollifiers.

  One of these problems involves the Aviles-Giga energy functional

with certain boundary conditions, where  is a smooth bounded domain in  and  is a function in . Energies similar to  appear in different physical situations: smectic liquid crystals, blisters in thin films, micromagnetics. Our main result for this problem asserts that for  satisfying  and  a.e., there exists a family  satisfying:  as  in  and .

   The second singular perturbation problem studied is related to the question of optimal lifting for BV-maps with values in , for which we prove a -convergence result. Let  be a bounded domain in with Lipschitz boundary. An optimal lifting of a function  is a function  realizing the minimum for  A natural approach to approximating optimal liftings is to consider, for a fixed parameter , the family of energy functionals defined on  by

For this problem we prove a -convergence result, and in particular we find the limiting functional. But somewhat surprisingly, our result shows that optimal liftings are obtained in the limit, in general, only for . However, if we restrict ourselves to , any limit is an optimal lifting.