The weighted isoperimetric inequality and Sobolev inequality outside convex sets
Abstract
In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the -ABP method. The weight function is assumed to be positive, even, and homogeneous of degree , such that is concave on . Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted P\'{o}lya-Szeg\"{o} principle and a sharp weighted capillary Sobolev inequality outside convex domain. Our result can be seen as an extension of the weighted Sobolev inequality in the half-space established by Ciraolo-Figalli-Roncoroni in \cite{CFR}.
Keywords
Cite
@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}
}
Comments
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