Published Paper
Inserted: 16 oct 2012
Last Updated: 26 jan 2013
Journal: Analysis and Geometry in Metric Spaces
Volume: 1
Pages: 1–30
Year: 2012
Doi: 10.2478/agms-2012-0001
Abstract:
Given an open set $\Omega\subset\mathbb{R}^m$ and $n>1$, we introduce the new spaces $GB_nV(\Omega)$ of {\it Generalized functions of bounded higher variation} and $GSB_nV(\Omega)$ of {\it 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. 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.
Tags:
GeMeThNES
Keywords:
distributional jacobian, Mumford-Shah
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