Harmonic mappings between singular metric spaces

In this paper, we survey the existence, uniqueness and interior regularity of solutions to Dirichlet problem associated with various energy functionals in setting mappings between singular metric spaces. Based on known ideas techniques, separate necessary analytical assumptions axiomatizing theory setting. More precisely, (1) extend existence result Guo Wenger (Comm Anal Geom 28(1):89–112, 2020) for Korevaar–Schoen functional more general purely (2) When Y has non-positive curvature sense Alexandrov (NPC), show that Jost (Calc Var Partial Differ Equ 5(1):1–19, 1997) Lin (Analysis spaces, collection papers geometry, analysis mathematical physics, World Science Publishers, River Edge, pp 114–126, can be adapted yield local Hölder continuity Kuwae–Shioya. (3) We Liouville theorem Sturm (J Reine Angew Math 456:173–196, 1994) harmonic functions (4) Mayer 6:199–253, 1998) mapping flow solve corresponding initial boundary value problem. Combing these ideas, or less standard techniques from spaces based upper gradients, leads new results when consider \({{\,\mathrm{RCD}\,}}(K,N)\) into NPC Similar Kuwae–Shioya gradient are also derived.