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

2024/03/19 by Detaille, Antoine, Mazowiecka, Katarzyna
#46E35 #58E20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2403.12662

Abstract

In this note, we study non-uniqueness for minimizing harmonic maps from B³ to \mathbbS². 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.

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