Ph.D Thesis | |

Ph.D Student | Poliakovsky Arkady |
---|---|

Subject | Lifting in BV Spaces and Related Problems |

Department | Department 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.