Compactness of special functions of bounded higher variation
Abstract
Given an open set \Omega\subset\R^m and n>1, we introduce the new spaces GB_nV(\Omega) of Generalized functions of bounded higher variation and GSB_nV(\Omega) of Generalized special functions of bounded higher variation that generalize, respectively, the space B_nV introduced by Jerrard and Soner and the corresponding SB_nV space studied by De Lellis. In this class of spaces, which allow the description of singularities of codimension n, the distributional jacobian Ju need not have finite mass: roughly speaking, finiteness of mass is not required for the (m-n)-dimensional part of Ju, but only finiteness of size. In the space GSB_nV we are able to provide compactness of sublevel sets and lower semicontinuity of Mumford-Shah type functionals, in the same spirit of the codimension 1 theory.
Keywords
Cite
@article{arxiv.1210.6473,
title = {Compactness of special functions of bounded higher variation},
author = {Luigi Ambrosio and Francesco Ghiraldin},
journal= {arXiv preprint arXiv:1210.6473},
year = {2013}
}
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28 pages