Symmetry breaking for the Dirichlet problem for harmonic maps from the disc into the 2-sphere (Q2504049)

The author studies the relations between symmetry and degree, for the Dirichlet problem for harmonic maps from the disc into the 2-sphere. The remark that many homotopy classes which are invariant by some symmetries of a given boundary value do not contain any symmetric map, and thus contain several harmonic maps -- if any, allows to further exhibit multiple solutions in many homotopy classes for a wide range of boundary values with symmetries.