# Graphs of vector valued maps: decomposition of the boundary

created by mucci on 21 Dec 2011
modified on 11 Nov 2013

[BibTeX]

Unpublished Paper

Inserted: 21 dec 2011
Last Updated: 11 nov 2013

Year: 2011
Notes:

Some of the results of this unpublished paper are contained in the paper "A structure property of vertical" integral currents, with an application to the distributional determinant" by the same author

Abstract:

We consider non-smooth vector valued maps such that the current carried by the graph has finite mass. We give a suitable decomposition of the boundary of the graph-current, provided that it has finite mass, too. Every such a component is a nice current whose support projects on a subset of the domain space that has integer dimension. This structure property is a consequence of a more general one that is proved for vertical" integer multiplicity rectifiable currents satisfying a null-boundary condition. As a consequence, for wide classes of Sobolev maps such that the graph is a normal current, we shall prove that the singular part of the distributional minors of order $k$ is concentrated on a countably rectifiable set of codimension $k$, hence no Cantor-type" part appears. The corresponding class of functions of bounded higher variation is studied, too. Finally, we discuss a possible notion of singular set in our framework, and illustrate several examples.