Constraint maps: singularities vs free boundaries

TL;DR

This paper studies energy-minimizing constraint maps with obstacles, revealing their complex singularities and free-boundary behavior, and establishes new regularity results and the existence of novel singularities due to branch points.

Contribution

It introduces a new quantitative unique continuation principle for constraint maps and characterizes the location of singularities, also discovering new types of singularities from branch points.

Findings

Topological singularities lie only in the interior of the contact set.

Established continuity and higher-order regularity near the free boundary.

Discovered new singularities arising from branch points.

Abstract

Energy-minimizing constraint maps are a natural extension of the obstacle problem within a vectorial framework. Due to inherent topological constraints, these maps manifest a diverse structure that includes singularities similar to harmonic maps, branch points reminiscent of minimal surfaces, and the intricate free-boundary behavior of the obstacle problem. The complexity of these maps poses significant challenges to their analysis. In this paper, we first focus on constraint maps with uniformly convex obstacles and establish continuity (and therefore higher-order regularity) within a uniform neighborhood of the free boundary. More precisely, thanks to a new quantitative unique continuation principle near singularities (which is new even in the setting of classical harmonic maps), we prove that, in the uniformly convex setting, topological singularities can only lie in the interior of the contact set. We then establish the optimality of this result. Second, while exploring the structure of the free boundary, we investigate the presence of branch points and show how they lead to completely new types of singularities not present in the scalar case.