Partial Hölder continuity for Q-valued energy minimizing maps

Abstract

ABSTRACTWe consider multivalued maps between Ω ⊂ ℝN open (N ≥ 2) and a smooth, compact Riemannian manifold 𝒩 locally minimizing the Dirichlet energy. An interior partial Hölder regularity results in the spirit of R. Schoen and K. Uhlenbeck is presented. Consequently a minimizer is Hölder continuous outside a set of Hausdorff dimension at most N − 3. Almgren's original theory includes a global interior Hölder continuity result if the minimizers are valued into some ℝm. It cannot hold in general if the target is changed into a Riemannian manifold, since it already fails for “classical” single valued harmonic maps.