  We show the regularity of, and derive a-priori estimates for (weakly)
harmonic maps from a Riemannian manifold into a Euclidean sphere under the
assumption that the image avoids some neighborhood of a half-equator. The
proofs combine constructions of strictly convex functions and the regularity
theory of quasi-linear elliptic systems.
  We apply these results to the spherical and Euclidean Bernstein problems for
minimal hypersurfaces, obtaining new conditions under which compact minimal
hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces
are trivial.
