Grushin平面上的调和映射

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations Pub Date : 2025-10-08 DOI:10.1016/j.jde.2025.113806

Tomasz Adamowicz , Marcin Walicki , Ben Warhurst

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引用次数: 0

摘要

我们研究了Hölder正则向量场给出的Grushin空间，该空间具有carnot - carathacimodory距离，具有自然加权n-Lebesgue测度。结果表明，该测度为n-Ahlfors正则，p-Muckenhoupt。我们研究了相关的2-Dirichlet能量和调和映射，并将注意力集中在调和函数和从Grushin平面上的域到欧几里德空间Rm的映射上，当m≥1时。我们的结果包括二阶Sobolev正则，Bochner恒等式及其对调和映射正则性的推论，liouville型定理和Lewy定理的对应。进一步研究了Grushin平面上映射的korevar - schoen能量，并证明了其等价于2-Dirichlet能量。用几个例子来说明讨论。我们推广了Ferrari-Valdinoci[23]和Franchi-Serapioni[30]的一些几何和正则性结果。

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Harmonic mappings on Grushin planes

We study the Grushin spaces given by Hölder regular vector fields with the Carnot–Carathéodory distance, equipped with the natural weighted n-Lebesgue measure. It turns out that such a measure is n-Ahlfors regular and p-Muckenhoupt for

p > 1

. We investigate the related 2-Dirichlet energy and the harmonic mappings, and focus our attention on harmonic functions and mappings from domains in a Grushin plane to Euclidean spaces

R^{m}

, for

m \geq 1

. Our results enclose the second order Sobolev regularity, the Bochner identity and its consequences for the regularity of harmonic mappings, the Liouville-type theorems and a counterpart of the Lewy theorem. Furthermore, the Korevaar–Schoen energy of mappings on Grushin planes is studied and proven to be equivalent to the 2-Dirichlet energy. The discussion is illustrated by several examples.

We generalize some geometric and regularity results by Ferrari–Valdinoci [23] and Franchi–Serapioni [30].

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来源期刊

Journal of Differential Equations 数学-数学

CiteScore

4.40

自引率

8.30%

发文量

543

审稿时长

9 months

期刊介绍： The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics