Partial Hölder continuity for Q-valued energy minimizing maps

Abstract

We consider multivalued maps between $\Omega \subset {\mathbb{R}}^N$ open ($N\ge 2$) and a smooth, compact Riemannian manifold $\mathcal{N}$ locally minimizing the Dirichlet energy. An interior partial Hölder regularity result 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 $N-3$. F. Almgren's original theory includes a global interior Hölder continuity result if the minimizers are valued into some ${\mathbb{R}}^m$. It cannot hold in general if the target is changed into a Riemannian manifold, since it already fails for "classical" single valued harmonic maps.

30 pages

References in corpus (8)

• Q-valued functions revisited
• Lipschitz extension of multiple Banach-valued functions in the sense of Almgren
• Lower semicontinuous functionals for Almgren's multiple valued functions
• A regularity theory for multiple-valued Dirichlet minimizing maps
• An Energy Reducing Flow for Multiple-Valued Functions
• Analysis on Metric Space Q
• A Theorem on Frequency Function for Multiple-Valued Dirichlet Minimizing Functions
• Regularity of area minimizing currents I: gradient L^p estimates