Accepted Paper

Inserted: 30 dec 2024
Last Updated: 30 dec 2024

Journal: Arch. Ration. Mech. Anal.
Year: 2024

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 $\Omega$.

Keywords: Plateau problem, Singular harmonic map, $p$-minimizer, $p$-stationary map, varifold, systole, topological obstruction, finite homotopy group, stress-energy tensor, extension of Sobolev mappings