Compactness of special functions of bounded higher variation

Ambrosio Luigi - Ghiraldin Francesco

year: 2012
journal: Analysis and Geometry in Metric Spaces
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.

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