Asymptotic behavior of minimizing $p$-harmonic maps when $p \nearrow 2$ in dimension 2

TL;DR

This paper investigates the limiting behavior of p-harmonic maps from planar domains into Riemannian manifolds as p approaches 2, showing convergence to singular harmonic maps influenced by topological constraints.

Contribution

It provides a detailed analysis of the asymptotic behavior of p-harmonic maps near p=2, including convergence results and the role of topological obstructions in singularity formation.

Findings

Convergence of p-harmonic maps to singular harmonic maps as p approaches 2.

Identification of singularities governed by topological obstructions.

Establishment of uniform weak-L^p bounds for the maps.

Abstract

We study $p$--harmonic maps with Dirichlet boundary conditions from a planar domain into a general compact Riemannian manifold. We show that as $p$ approaches $2$ from below, they converge up to a subsequence to a minimizing singular renormalizable harmonic map. The singularities are imposed by topological obstructions to the existence of harmonic mappings; the location of the singularities being governed by a renormalized energy. Our analysis is based on lower bounds on growing balls and also yields some uniform weak-$L^p$ bounds (also known as Marcinkiewicz or Lorentz $L^{p,\infty}$).