Minimisers of supremal functionals and mass-minimising 1-currents

TL;DR

This paper investigates vector-valued functions that minimize the supremal norm of their derivatives, constructing a mass-minimizing 1-current that captures the solutions' common support, extending existing scalar theories.

Contribution

It introduces a novel construction of a mass-minimizing 1-current for vector-valued functions, extending scalar theories to vector cases using p-harmonic approximation.

Findings

Constructed a mass-minimizing 1-current for vector-valued functions.

Showed all solutions coincide on the current's support.

Extended scalar theories of Evans and Yu to vector-valued functions.

Abstract

We study vector-valued functions that minimise the $L^\infty$-norm of their derivatives for prescribed boundary data. We construct a vector-valued, mass minimising $1$-current (i.e., a generalised geodesic) in the domain such that all solutions of the problem coincide on its support. Furthermore, this current can be interpreted as a streamline of the solutions. The construction relies on a $p$-harmonic approximation. In the case of scalar-valued functions, it is closely related to a construction of Evans and Yu. We therefore obtain an extension of their theory.