发表机构
The Institute of Mathematics, Polish Academy of Sciences; Warsaw University of Technology, Faculty of Mathematics and Information Science（波兰科学院数学研究所; 华沙理工大学数学与信息科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Grushin平面间调和映射的正则性、估计与Liouville定理，并联系全纯映射。
AI 中文摘要
我们研究了从$\alpha$-Grushin平面中的源域到$\beta$-Grushin平面中的目标域的调和映射，其中$\alpha$不一定等于$\beta$，且$\alpha\in[0,1)$。我们的调和映射源于与Dirichlet能量相关的Euler-Lagrange方程组，并在弱形式和强形式下进行研究。关键结果包括调和映射的二阶正则性、Bochner恒等式、调和映射坐标函数的弱Harnack估计，以及Caccioppoli估计和Liouville型定理。此外，我们讨论了几个（Grushin）调和映射的例子，并特别将它们与平面中的全纯映射联系起来。
英文摘要
We study harmonic mappings between the source domain in the $α$-Grushin plane and the target domain in the $β$-Grushin plane for $α$ not necessarily equal to $β$ and $α\in [0,1)$. Our harmonic mappings arise as a Euler--Lagrange system of equations related to the Dirichlet energy and are studied in weak and strong forms. The key results enclose the second order regularity of harmonic mappings, the Bochner identity, the weak Harnack estimates for coordinate functions of a harmonic mapping, as well as the Caccioppoli estimates and the Liouville type theorem. Furthermore, we discuss several examples of (Grushin) harmonic mappings and, in particular, relate them to holomorphic mappings in the plane.
Comments58 pages. The authors declare that no AI system was employed in any part of the reasoning in the manuscript
