In this paper, we survey 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. Based on known ideas and techniques, we separate the necessary analytical assumptions to axiomatizing the theory in the singular setting. More precisely, - We extend the existence result of Guo and Wenger [25] for solutions for the Dirichlet problem of Korevaar and Schoen to the purely singular setting. - When Y has non-positive curvature in the sense of Alexandrov (NPC), we show that the ideas of Jost [40] and Lin [52] can be adapted to the purely singular setting to yield local Holder continuity of solutions. - We extend the Liouville theorem of Sturm [67] for harmonic functions to harmonic mappings between singular metric spaces. - We extend the theorem of Mayer [57] 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 RCD(K,N) spaces into NPC spaces. One advantage of this type of axiomatization is that it works for minimizers of other Dirichlet energy functional. In particular, as applications of the established theory, we deduce similar results for the Dirichlet problem based on the Kuwae-Shioya energy functional and for the Dirichlet problem based on upper gradients.
Quantitative gradient estimates for harmonic maps into singular spaces
本文给出了到度量空间$(X,d_X)$中调和映射的Yau梯度估计，该度量空间的曲率上界为常数$κ$，$κ\geq0$，在Alexandrov意义下.作为直接应用，给出了这类调和映射的Liouville定理。这扩展了S. Y. Cheng [4]和H. I. Choi [5]对奇异空间中调和映射的研究。
$(n,ρ)-$harmonic mappings and energy minimal deformations between annuli
我们推广了Iwaniec和Onninen在Memoirs of the AMS（2012）中的主要结果。本文解决了欧几里得空间$\mathbf{R}^n$中两同心环之间Sobolev同胚的$(ρ,n)$能量极小化问题。这里，$ρ$是在图像环中定义的径向度量。证明的关键在于径向调和映射的欧拉-拉格朗日方程的求解。这是对著名的Nitsche猜想的一个新的贡献。
The Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）；Warsaw University of Technology, Faculty of Mathematics and Information Science（华沙理工大学数学与信息科学学院）
