Symmetric boundary values for the Dirichlet problem for harmonic maps from the disc into the 2-sphere

Abstract

Let Aut(D) denote the group of biholomorphic diffeormorphisms from the unit disc D onto itself and O(3) the group of orthogonal transformations of the unit sphere S2. The existence of multiple solutions to the Dirichlet problem for harmonic maps from D into S2 is related to the symmetries (if any) of the boundary value γ : ∂D → S2, by invariance of the Dirichlet energy under the action of Aut(D) × O(3). In this paper, we classify the stabilizers in Aut(D) × O(3) of boundary values in H1/2(S1, S2) and \({\mathcal{C}}^0(S^1, S^2)\) . We give two applications to the Dirichlet problem for harmonic maps.