Stationary $p$-harmonic maps approaching planar singular harmonic maps to the circle

TL;DR

This paper demonstrates that certain singular harmonic maps into the circle, which are critical points of a renormalised energy, can be approximated by stationary p-harmonic maps as p approaches 2, revealing a connection between these maps.

Contribution

It establishes that singular harmonic maps into the circle are limits of stationary p-harmonic maps as p approaches 2, linking harmonic and p-harmonic map theories.

Findings

Singular harmonic maps are limits of p-harmonic maps as p approaches 2.

Topologically nondegenerate critical points are considered.

The result applies to maps in bounded planar domains.

Abstract

Given a bounded planar domain $\Omega \subset \mathbb{R}^2$, we show that any singular harmonic map into the circle $\mathbb{S}^1$ corresponding to a topologically nondegenerate critical point of the renormalised energy in the sense of Bethuel, Brezis and H\'elein is a limit of stationary $p$-harmonic maps for $p < 2$ as $p \to 2$