Weak monotonicity inequality and partial regularity for harmonic maps (Q1305568)

The paper is concerned with weakly harmonic maps from a domain in \(\mathbb{R}^3\) to \(S^2\). It is shown that for such maps a local weak form of the monotonicity inequality is equivalent to certain bounds for the energy on balls centered in singular points. This is used to prove the existence of non-minimizing solutions to the Dirichlet problem having at most isolated singularities. By analyzing axially symmetric maps, it is also shown that the local form of the weak monotonicity inequality is not equivalent to the global one. Furthermore, examples with non-isolated singularities are constructed.