Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere (Q6639464)

The authors explore the issue of non-uniqueness in the minimization of harmonic maps from the unit ball in 3-dimensional space to the 2-dimensional sphere. This study builds upon a previous observation made by Mazowiecka and Strzelecki in [\textit{K. Mazowiecka} and \textit{P. Strzelecki}, Adv. Calc. Var. 10, No. 3, 303--314 (2017; Zbl 1369.58012)], which suggested that generic non-uniqueness arises when considering small perturbations of the boundary data within the topology of the space \(W^{1,p}\) for \(p<2\). The research demonstrates that any boundary map can be adjusted to allow for multiple minimizers of the Dirichlet energy through a slight modification in the \(W^{1,p}\) space for \(p<2\). Additionally, it is proven that the boundary data leading to non-uniqueness is densely populated within \(W^{1,p}(\mathbb{S}^2, \mathbb{S}^2)\), ensuring the existence of at least one boundary map where non-uniqueness occurs. A key component of this study is the introduction of a homotopy construction, which addresses a simplified version of a complex question regarding the control of norms in homotopies between \(W^{1,p}\) maps.