A representation formula for the p-Energy of metric space valued Sobolev maps

TL;DR

This paper provides an explicit formula for the p-energy of Sobolev maps into metric spaces, connecting different definitions and enhancing understanding of their structure.

Contribution

It introduces a new explicit representation formula for the p-energy of metric space valued Sobolev maps, linking Korevaar-Schoen and Ambrosio's approaches.

Findings

Explicit p-energy formula for Sobolev maps into metric spaces

Connects Korevaar-Schoen and Ambrosio definitions

Enhances understanding of Sobolev map energies in metric spaces

Abstract

We give an explicit representation formula for the p-energy of Sobolev maps with values in a metric space as defined by Korevaar and Schoen (Comm. Anal. Geom. 1 (1993), no.~3-4, 561--659). The formula is written in terms of the Lipschitz compositions introduced by Ambrosio (Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4) 3 (1990), n.17, 439--478), thus further relating the two different definitions considered in the literature.