The weighted isoperimetric inequality and Sobolev inequality outside convex sets

TL;DR

The paper addresses weighted capillary isoperimetry outside convex sets with a homogeneous density $w$, formulating the problem via the weighted capillary energy $J_{w,\lambda}$ and proving a sharp exterior inequality against the half-space benchmark using a $\lambda_w$-ABP framework. It reformulates the capillary energy as a weighted anisotropic perimeter with capillary gauge $\widetilde{F}_{\lambda,w}$ and develops a robust ABP-driven proof that yields equality only for spherical-cap extremals, with automatic ABP in the half-space. The authors then extend the result from smooth to general finite-perimeter domains through careful approximation, and derive powerful consequences: a weighted capillary Schwarz rearrangement, a weighted Pólya-Szegö principle, and a sharp weighted capillary Sobolev inequality outside convex sets, including explicit extremals and best constants. Overall, the work generalizes half-space results to arbitrary convex exterior domains, providing a versatile variational framework for weighted capillarity and anisotropic perimeters with potential geometric analysis applications.

Abstract

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

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Theorem 1.4
• Remark 2.1
• Lemma 2.1
• proof
• Definition 3.1
• Proposition 3.1