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Abstract

In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the $\lambda_w$-ABP method. The weight function $w$ is assumed to be positive, even, and homogeneous of degree $\alpha$, such that $w^{1/\alpha}$ is concave on $\R^n$. 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}.