Interior regularity for two-dimensional stationary $Q$-valued maps

TL;DR

This paper proves interior regularity and bounds the singular set dimension for 2D stationary Q-valued maps, introducing a strong concentration-compactness theorem applicable in all dimensions.

Contribution

It establishes regularity results for 2D stationary Q-valued maps and introduces a new concentration-compactness theorem for outer variation stationary maps.

Findings

Q-valued maps are Hölder continuous in 2D

Singular set dimension is at most one

Strong concentration-compactness theorem holds in all dimensions

Abstract

We prove that $2$-dimensional $Q$-valued maps that are stationary with respect to outer and inner variations of the Dirichlet energy are H\"older continuous and that the dimension of their singular set is at most one. In the course of the proof we establish a strong concentration-compactness theorem for equicontinuous maps that are stationary with respect to outer variations only, and which holds in every dimensions.