# Harmonic mappings between singular metric spaces

We provide a unified approach to the existence, uniqueness and interior regularity of solutions to the Dirichlet problem of Korevaar and Schoen in the setting of mappings between singular metric spaces. More precisely, under mild conditions on the metric spaces $X$ and $Y$, we obtain the existence of solutions for the Dirichlet problem of Korevaar and Schoen. When $Y$ has non-positive curvature in the sense of Alexandrov (NPC), solutions are shown to be unique and local H\"older continuous. We further apply the method of Mayer to show the existence of the harmonic mapping flow and solve the corresponding initial boundary value problem. Finally, we deduce similar results for the Dirichlet problem based on the Kuwae-Shioya energy functional and for the Dirichlet problem based on upper gradients.

**Comments:**36 pages

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