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 is an element of H-1(B-2, S-2) boolean AND C-0 ((B) over bar (2) (B) over bar (1)), where B-r = {p is an element of R-3: vertical bar p vertical bar < r}, with deg u vertical bar(partial derivative B2) = 0, "strictly minimizing in B-1" the relaxed Dirichlet energy of Bethuel, Brezis and Coron F(u, B-2): = 1/2 integral(B2) vertical bar del u vertical bar(2)dxdydz + 4 pi Sigma(u, B-2), and having Sigma(u, B-2) > 0, u vertical bar B-1 not equivalent to const. Here Sigma(u, B-2) is (in a generalized sense) the lenght of a minimal connection joining the topological singularities of u. By "strictly minimizing in B-1" we intend that F(u, B-2) < F(v, B-2) for every v is an element of H-1 (B-2, S-2) with v vertical bar B-2B1 = u vertical bar(B2B1) and v not equivalent to u. This result, which we shall also rephrase in terms of Cartesian currents (following Giaquinta, Modica and Soueek) 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 SouCek. 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. (C) 2011 Elsevier Inc. All rights reserved.