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Limiting behavior of minimizing $p$-harmonic maps in 3d as $p$ goes to $2$ with finite fundamental group

Bohdan Bulanyi, Jean Van Schaftingen, Benoît Van Vaerenbergh

TL;DR

The paper analyzes the asymptotics of minimizing $p$-harmonic maps from a bounded 3D domain into a compact manifold $\\mathcal N$ with finite $\\pi_1(\\mathcal N)$ as $p\\nearrow 2$. It proves that a subsequence converges to a locally harmonic map $u_*$ on $\\Omega\\setminus S_*$, where $S_*$ is a finite-length singular set that, inside the domain, is a finite union of straight segments minimizing a homotopical mass. The limiting stress-energy measures converge to a stationary 1-dimensional rectifiable varifold $V_*$ supported on $S_*$, with densities drawn from a finite set of singular energies associated to boundary loops. Under extra regularity and boundary convexity, a boundary repulsion phenomenon ties $S_*$ to the boundary data, ensuring $S_*\\cap\\partial\\Omega$ matches prescribed singularities $S(g)$. Together, these results provide a detailed structure for energy concentration, singular set geometry, and boundary interactions in the 3D, non-simply connected target setting as $p\to 2$.

Abstract

We study the limiting behavior of minimizing $p$-harmonic maps from a bounded Lipschitz domain $Ω\subset \mathbb{R}^{3}$ 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 $Ω\setminus S_{*}$. We prove that locally inside $Ω$ 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 $\overlineΩ$ 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 $Ω$.

Limiting behavior of minimizing $p$-harmonic maps in 3d as $p$ goes to $2$ with finite fundamental group

TL;DR

The paper analyzes the asymptotics of minimizing -harmonic maps from a bounded 3D domain into a compact manifold with finite as . It proves that a subsequence converges to a locally harmonic map on , where is a finite-length singular set that, inside the domain, is a finite union of straight segments minimizing a homotopical mass. The limiting stress-energy measures converge to a stationary 1-dimensional rectifiable varifold supported on , with densities drawn from a finite set of singular energies associated to boundary loops. Under extra regularity and boundary convexity, a boundary repulsion phenomenon ties to the boundary data, ensuring matches prescribed singularities . Together, these results provide a detailed structure for energy concentration, singular set geometry, and boundary interactions in the 3D, non-simply connected target setting as .

Abstract

We study the limiting behavior of minimizing -harmonic maps from a bounded Lipschitz domain to a compact connected Riemannian manifold without boundary and with finite fundamental group as . We prove that there exists a closed set of finite length such that minimizing -harmonic maps converge to a locally minimizing harmonic map in . We prove that locally inside the singular set 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 the set 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 .
Paper Structure (29 sections, 69 theorems, 378 equations, 1 figure)

This paper contains 29 sections, 69 theorems, 378 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions and Notation
  4. Embedding and nearest point retraction
  5. Sobolev mappings into manifolds
  6. Topological resolution of the boundary datum
  7. Singular p-energy
  8. Renormalized energy of renormalizable maps
  9. Several extension results
  10. Minimizing $p$-harmonic maps
  11. Extension from triangulation
  12. Lower bounds for the energy
  13. Lipschitz triangulations of 𝕊²
  14. Convergence to a harmonic map
  15. The singular set
  16. ...and 14 more sections

Key Result

Theorem 1.1

If $\Omega \subset \mathbb{R}^{3}$ is a bounded Lipschitz domain, $\pi_1 (\mathcal{N})$ is finite, $g \in W^{1/2, 2} (\partial \Omega, \mathcal{N})$, $(p_{n})_{n\in \mathbb{N}}\subset [1,2)$, $p_{n}\nearrow 2$ as $n\to +\infty$ and if for each $n\in \mathbb{N}$, $u_{n}\in W^{1,p_{n}}(\Omega, \mathca where $C=C(\partial \Omega, \mathcal{N})>0$. Furthermore, there exists a closed set $S_{*} \subset

Figures (1)

  • Figure 5.1: A uniform Lipschitz triangulation of the lateral surface of a cylinder.

Theorems & Definitions (162)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 152 more