Calculus of Variations and Geometric Measure Theory

L. Ambrosio - F. Ghiraldin

Compactness of special functions of bounded higher variation

created by ghiraldin on 16 Oct 2012
modified on 26 Jan 2013


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


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