Lower semicontinuous functionals for Almgren's multiple valued functions

TL;DR

This paper characterizes lower semicontinuous integral functionals on Sobolev spaces of Almgren's multiple valued functions, extending previous results and answering a question posed by Mattila.

Contribution

It provides a complete characterization of semicontinuous functionals and generalizes earlier findings in the context of multiple valued functions.

Findings

Characterization of lower semicontinuous functionals

Extension of Mattila's results

Positive answer to Mattila's open question

Abstract

We consider general integral functionals on the Sobolev spaces of multiple valued functions, introduced by Almgren. We characterize the semicontinuous ones and recover earlier results of Mattila as a particular case. Moreover, we answer positively to one of the questions raised by Mattila in the same paper.