凸集外部的加权等温不等式与 Sobolev 不等式
偏微分方程分析
2025-11-10 v2
摘要
本文通过 -ABP 方法，在凸集外部建立加权毛细等温不等式。权函数 假设为正的、偶函数且齐次为 次，这样 在 上为凹函数。基于加权等温不等式，我们发展了凸集外部的毛细 Schwarz 镜像对称化技术，建立了加权 P\'{o}lya-Segm\'{o} 原理和锋利的加权毛细 Sobolev 不等式。我们的结果可视为对 Ciraolo-Figalli-Roncoroni 在半空间上建立的加权 Sobolev 不等式的延伸。
引用
@article{arxiv.2510.01647,
title = {The weighted isoperimetric inequality and Sobolev inequality outside convex sets},
author = {Lu Chen and Jiali Lan},
journal= {arXiv preprint arXiv:2510.01647},
year = {2025}
}
备注
First, we add the equality case. Second, we have revised the assumption on the weight function w. The original version required radial symmetry of w, which contradicts the concavity of w^{1/\alpha}. To resolve this, the new analysis focuses on a weighted isoperimetric inequality in the half-space, only requiring w to be even with respect to x_n