Calculus of Variations and Geometric Measure Theory

D. Mucci

Graphs of vector valued maps: decomposition of the boundary

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


Unpublished Paper

Inserted: 21 dec 2011
Last Updated: 11 nov 2013

Year: 2011

Some of the results of this unpublished paper are contained in the paper "A structure property of ``vertical" integral currents, with an application to the distributional determinant" by the same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


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.