Minimizing 1/2-harmonic maps into spheres

Vincent Millot (1) , Marc Pegon (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

In this article, we improve the partial regularity theory for minimizing 1/2-harmonic maps of [30, 33] in the case where the target manifold is the (m − 1)-dimensional sphere. For m>2, we show that minimizing 1/2-harmonic maps are smooth in dimension 2, and have a singular set of codimension at least 3 in higher dimensions. For m=2, we prove that, up to an orthogonal transformation , x/|x| is the unique non trivial 0-homogeneous minimizing 1/2-harmonic map from the plane into the circle S1. As a corollary, each point singularity of a minimizing 1/2-harmonic maps from a 2d domain into S1 has a topological charge equal to ±1.

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Millot, Marc Pegon. Minimizing 1/2-harmonic maps into spheres. Calculus of Variations and Partial Differential Equations, 2020, 59 (2), pp.55. ⟨10.1007/s00526-020-1704-z⟩. ⟨hal-01985037⟩