光滑度量测度空间上Escobar-Riemann映射型问题的推广
微分几何
2018-07-10 v2 偏微分方程分析
摘要
本文引入一个类似于Case, J.所考虑的Yamabe型问题的问题，它推广了带边界的光滑度量测度空间上的Escobar-Riemann映射问题。后者将被称为Escobar-Riemann映射型问题。为此，我们考虑由Bolley等人推导的Sobolev迹不等式的推广。该迹不等式使我们能引入Escobar商及其下确界。此下确界称为Escobar加权常数。带边界流形上光滑度量测度空间的Escobar-Riemann映射型问题，在于寻找一个达到Escobar加权常数的函数。此外，当Escobar加权常数为负时，我们解决了该问题。最后，我们得到一个Aubin型不等式，将紧致光滑度量测度空间的加权Escobar常数与Bolley等人所得迹不等式的最优常数联系起来。
引用
@article{arxiv.1805.03694,
title = {A generalization of Escobar-Riemann mapping type problem on smooth metric measure spaces},
author = {Jhovanny Muñoz Posso},
journal= {arXiv preprint arXiv:1805.03694},
year = {2018}
}
备注
We have been informed by Nguyen Van Hoang that the Trace inequality, even in a general version, has been proven already by Bolley, F. et al. in 2018. However, the proof we did do in this particular case is different. In this version, we have changed the title and we have made some arrangements to give the credits. The other results work for $m\geq0$ instead of $m\in\mathbf{N}\cup\{0\}$