Minimisers of supremal functionals and mass-minimising 1-currents

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.

References in corpus (2)

• Eigenvalue Problems in ${\mathrm{L}}^{\infty }$: Optimality Conditions, Duality, and Relations with Optimal Transport
• Variational problems in $L^{\infty }$ involving semilinear second order differential operators