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

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 $p \to 2$