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The isoperimetric problem in the Heisenberg group $$\pmb {\mathbb {H}^n}$$ H n with density
Analysis and Mathematical Physics ( IF 1.4 ) Pub Date : 2020-05-17 , DOI: 10.1007/s13324-020-00367-2
Guoqing He , Peibiao Zhao

We study the isoperimetric problem for the axially symmetric sets in the Heisenberg group \(\mathbb {H}^n\) with density \(|z|^p\). At first, we prove the existence of weighted isoperimetric sets. Then, we characterize weighted isoperimetric sets uniquely as bubble sets. Finally, we deduce an interesting result that, up to a constant multiplicator, \(|z|^p\) is the only horizontal radial density for which bubble sets can be weighted isoperimetric sets.

中文翻译:

海森堡群$$ \ pmb {\ mathbb {H} ^ n} $$ H n的等密度问题

我们研究密度为\(| z | ^ p \)的Heisenberg群\(\ mathbb {H} ^ n \)中的轴对称集的等距问题。首先,我们证明了加权等长集的存在。然后,我们将加权等测集独特地描述为气泡集。最后,我们得出一个有趣的结果,直到一个恒定的乘数,\(| z | ^ p \)是唯一可以将气泡集加权为等渗集的水平径向密度。
更新日期:2020-05-17
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