Interior partial regularity for minimal Lp-vectorfields with integer fluxes

TL;DR

This paper proves that minimal Lp-vectorfields with integer fluxes are H"older continuous outside a locally finite singular set, using a new combinatorial approach to establish optimal interior partial regularity.

Contribution

It introduces a novel combinatorial technique to establish optimal interior partial regularity for minimal Lp-vectorfields with integer fluxes, advancing the understanding of their regularity properties.

Findings

Minimal Lp-vectorfields are H"older continuous outside a locally finite set.

The regularity result is optimal and improves previous partial regularity theorems.

The paper provides a self-contained proof extending prior collaborative work.

Abstract

We use a new combinatorial technique to prove the optimal interior partial regularity result for Lp-vectorfields with integer fluxes minimizing the Lp-energy. More precisely, we prove that the minimal vectorfields are H\"older outside a set which is locally finite inside the domain. The results continue the program started in collaboration with Tristan Rivi\`ere, but this paper is self-contained.