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

在本文中，我们调查了在奇异度量空间之间的映射设置中与各种能量泛函相关的狄利克雷问题解的存在性、唯一性和内部规律性。基于已知的思想和技术，我们将必要的分析假设分开，以在奇异设置中公理化理论。更准确地说，(1) 我们将 Guo 和 Wenger (Comm Anal Geom 28(1):89–112, 2020) 的存在性结果扩展到 Korevaar-Schoen 能量泛函的 Dirichlet 问题的解决方案到更一般的纯奇异能量泛函环境。(2) 当Y具有 Alexandrov (NPC) 意义上的非正曲率，我们证明了 Jost（Calc Var Partial Differ Equ 5(1):1-19, 1997）和 Lin（奇异空间分析，论文集几何、分析和数学物理学，世界科学出版社，River Edge，第 114-126 页，1997 年）可以适用于纯奇异设置，以产生 Korevaar-Schoen 和 Kuwae-Shioya 的 Dirichlet 问题解的局部 Hölder 连续性。(3) 我们将调和函数的 Sturm 的刘维尔定理 (J Reine Angew Math 456:173–196, 1994) 扩展到奇异度量空间之间的调和映射。(4) 我们扩展了 Mayer (Comm Anal Geom 6:199–253, 1998) 关于调和映射流的存在性的定理，并求解相应的初始边值问题。结合这些已知的想法，\({{\,\mathrm{RCD}\,}}(K,N)\)空格转换为 NPC 空格。与 Kuwae-Shioya 能量泛函和上梯度泛函相关的狄利克雷问题也得到了类似的结果。