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

For 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 deg u|∂B2 = 0, “strictly minimizing in B1” the relaxed Dirichlet energy of Bethuel, Brezis and Coron F(u,B2) := 1 2 B2 |∇u|2 dx dy dz +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) < F(v,B2) for every v ∈ H1(B2,S2) with v|B2\B1 = u|B2\B1 and v ≡ u. This result, which we shall also rephrase in terms of Cartesian currents (following Giaquinta, Modica and Souˇcek) stands in sharp contrast with a results of Hardt, Lin and Poon for the case n = 1, and partially answers a long standing question of Giaquinta, Modica and Souˇcek. In particular it is a first example of a minimizer of the relaxed energy having non-trivial minimal connection. We explain how this relates to the regularity of minimizers of F. © 2011 Elsevier Inc. All rights reserved.