Limiting Behavior of Minimizing p-Harmonic Maps in 3d as p Goes to 2 with Finite Fundamental Group

Abstract

We study the limiting behavior of minimizing p-harmonic maps from a bounded Lipschitz domain \(\Omega \subset \mathbb {R}^{3}\) to a compact connected Riemannian manifold without boundary and with finite fundamental group as \(p \nearrow 2\). We prove that there exists a closed set \(S_{*}\) of finite length such that minimizing p-harmonic maps converge to a locally minimizing harmonic map in \(\Omega \setminus S_{*}\). We prove that locally inside \(\Omega \) the singular set \(S_{*}\) is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains. Furthermore, we establish local and global estimates for the limiting singular harmonic map. Under additional assumptions, we prove that globally in \(\overline{\Omega }\) the set \(S_{*}\) is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains, which is defined by a given boundary datum and \(\Omega \).