Lipschitz continuity of harmonic maps between Alexandrov spaces

Abstract

In 1997, Jost (Calc Var PDE 5:1–19, 1997) and Lin (Collection of papers on geometry, analysis and mathematical physics, World Sci. Publ., River Edge, 1997), independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Holder continuous. Lin (1997) proposed an open problem: can the Holder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces [see page 38 in Jost (in: Jost, Kendall, Mosco, Rockner, Sturm (eds) New directions in Dirichlet forms, International Press, Boston, 1998)]. The main theorem of this paper gives a complete resolution to it.