调和拟共形映射相关性质的研究

It is well known that harmonic mappings have a close relationship with quasiconformal mappings and PDEs etc. Hence, the related research has attracted more and more concern. This project mainly studies the following three aspects: (1) The Hölder continuity of mappings of finite distortion: As a generalization of quasiconformal mappings, the mappings of finite distortion are a class of mappings with finite dilatation. We plan to study its Hölder continuity by establishing the relationship between the length and area. (2) The norm estimate of solutions to hyperbolic Poisson’s equations: The hyperbolic Poisson's equations can be known as a generalization of the classical Poisson’s equations in metric. We will use hypergeometric function theory to estimate the norm of the corresponding hyperbolic Green integral operator. (3) The bi-Lipschitz continuity of harmonic quasiconformal mappings: We shall use the distance function to discuss the open problem regarding the bi-Lipschitz continuity of harmonic quasiconformal mappings in R^n raised by Kalaj in 2013. Hence, this study has the very important theoretical significance.

调和映射与拟共形映射、偏微分方程等研究领域有着紧密联系，从而它的相关研究引起了数学研究者的广泛关注。本项目主要研究以下三个方面的内容：(1)有限偏差映射的Hölder连续性：作为拟共形映射的推广，有限偏差映射是伸缩商有限的一类映射。我们将通过建立弧长与面积的关系来研究其Hölder连续性。(2)双曲Poisson方程解的范数估计：双曲Poisson方程是经典Poisson方程在度量上的推广。我们将利用超几何函数理论来估计其相应的双曲Green积分算子的范数。(3)调和拟共形映射的bi-Lipschitz连续性：我们将利用距离函数来研究Kalaj于2013年提出的关于R^n中调和拟共形映射bi-Lipschitz连续的公开问题。因此，本项目的研究具有重要的理论意义。

双曲Poisson方程的解是Poisson方程的解在双曲度量下的推广，而Dirichlet型能量又是共形能量在度量空间中的推广。本报告主要研究了以下两个方面的内容。一方面，我们借助Gauss超几何函数来研究单位球上双曲Poisson方程解的相关性质。通过建立双曲Laplace算子的径向特征函数的等价刻画，给出了Hardy空间中双曲调和映射的最佳点态估计；建立了Hardy空间中双曲调和映射的Schwarz引理；讨论了双曲Poisson方程解的存在性问题，并在此基础上给出了双曲Poisson方程解的若干等价模。另一方面，我们将偏微分方程与变分法相结合，解决了Kalaj于《Adv. Calc. Var., 2021》上提出的关于n维环域上某类Dirichlet型能量的极值问题的猜测。这些成果和研究方法对函数空间、调和映射与偏微分方程的研究起到了一定的促进作用。