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Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere

Antoine Detaille, Katarzyna Mazowiecka

TL;DR

The paper proves that boundary data yielding non-uniqueness of minimizing harmonic maps from the unit ball $B^3$ to the sphere ${\mathbb S}^2$ are dense in $W^{1,p}({\mathbb S}^2,{\mathbb S}^2})$ for any $p<2$. The authors introduce a novel controlled homotopy construction that keeps the deformation of boundary maps (supported in a small region) within a prescribed $W^{1,p}$-radius. Starting from a datum exhibiting a Lavrentiev gap, they produce a nearby datum with more singularities and propagate non-uniqueness along a carefully engineered homotopy. Together with stability and compactness results for minimizing harmonic maps, this yields density of boundary data that admit at least two energy minimizers with different singularity structures, strengthening prior results on generic non-uniqueness.

Abstract

In this note, we study non-uniqueness for minimizing harmonic maps from $B^3$ to $\mathbb{S}^2$. We show that every boundary map can be modified to a boundary map that admits multiple minimizers of the Dirichlet energy by a small $W^{1,p}$-change for $p<2$. This strengthens a remark by the second-named author and Strzelecki. The main novel ingredient is a homotopy construction, which is the answer to an easier variant of a challenging question regarding the existence of a norm control for homotopies between $ W^{1,p} $ maps.

Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere

TL;DR

The paper proves that boundary data yielding non-uniqueness of minimizing harmonic maps from the unit ball to the sphere are dense in for any . The authors introduce a novel controlled homotopy construction that keeps the deformation of boundary maps (supported in a small region) within a prescribed -radius. Starting from a datum exhibiting a Lavrentiev gap, they produce a nearby datum with more singularities and propagate non-uniqueness along a carefully engineered homotopy. Together with stability and compactness results for minimizing harmonic maps, this yields density of boundary data that admit at least two energy minimizers with different singularity structures, strengthening prior results on generic non-uniqueness.

Abstract

In this note, we study non-uniqueness for minimizing harmonic maps from to . We show that every boundary map can be modified to a boundary map that admits multiple minimizers of the Dirichlet energy by a small -change for . This strengthens a remark by the second-named author and Strzelecki. The main novel ingredient is a homotopy construction, which is the answer to an easier variant of a challenging question regarding the existence of a norm control for homotopies between maps.
Paper Structure (3 sections, 4 theorems, 23 equations)

This paper contains 3 sections, 4 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Homotopy construction
  3. Proof of the generic non-uniqueness

Key Result

Theorem 1.1

Let $\varphi \in W^{1,2}({\mathbb S}^2,{\mathbb S}^2)$. For every $\varepsilon>0$, there exists $\psi\in W^{1,2}({\mathbb S}^2,{\mathbb S}^2)$ such that $\left\|{\varphi - \psi}\right\|_{W^{1,2}({\mathbb S}^2)} < \varepsilon$ and for which there exists exactly one energy minimizer $u\colon B^3 \to {

Theorems & Definitions (6)

  • Theorem 1.1: AL88
  • Theorem 1.2
  • Proposition 2.1
  • Corollary 2.2
  • proof : Proof of Proposition \ref{['proposition:controlled_homotopies']}
  • proof : Proof of \ref{['th:main']}