Some regularity results for [formula omitted]-harmonic mappings between Riemannian manifolds

Abstract

Let M be a C2-smooth Riemannian manifold with boundary and N a complete C2-smooth Riemannian manifold. We show that each stationary p-harmonic mapping u:M→N, whose image lies in a compact subset of N, is locally C1,α for some α∈(0,1), provided that N is simply connected and has non-positive sectional curvature. We also prove similar results for minimizing p-harmonic mappings with image being contained in a regular geodesic ball. Moreover, when M has non-negative Ricci curvature and N is simply connected with non-positive sectional curvature, we deduce a gradient estimate for C1-smooth weakly p-harmonic mappings from which follows a Liouville-type theorem in the same setting.