The weighted isoperimetric inequality and Sobolev inequality outside convex sets

In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the λw-ABP method. The weight function w is assumed to be positive, even, and homogeneous of degree $α$, such that w1/α is concave on ⁿ. Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted Pólya-Szegö 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 {CFR}.