A note on n-axially symmetric harmonic maps from B3 to S2 minimizing the relaxed energy

AbstractFor any n⩾2 we provide an explicit example of an n-axially symmetric map u∈H1(B2,S2)∩C0(B¯2∖B¯1), where Br={p∈R3:|p|<r}, with degu|∂B2=0, “strictly minimizing in B1” the relaxed Dirichlet energy of Bethuel, Brezis and CoronF(u,B2):=12∫B2|∇u|2dxdydz+4πΣ(u,B2), and having Σ(u,B2)>0, u|B1≢const. Here Σ(u,B2) is (in a generalized sense) the lenght of a minimal connection joining the topological singularities of u. By “strictly minimizing in B1” we intend that F(u,B2)