Sharp capillary Sobolev inequality and Moser-Trudinger inequality outside convex domain

TL;DR

This paper establishes sharp capillary Sobolev and Moser-Trudinger inequalities outside convex domains, extending geometric inequalities in Finsler manifolds using rearrangement techniques.

Contribution

It introduces new sharp capillary inequalities outside convex domains, including Sobolev and Moser-Trudinger inequalities, using capillary Pólya-Szegő rearrangement methods.

Findings

Established sharp capillary Sobolev inequality outside convex domain.

Proved sharp capillary Moser-Trudinger inequality outside convex domain.

Extended geometric inequalities to Finsler manifold context.

Abstract

The theory of sharp geometric inequality in $\mathbb{R}^n$ and inside convex cone has been well-developed, much less known for sharp capillary geometric inequality outside convex domain. Recently, Fusco-Julin-Morini-Pratelli \cite{FJMP} obtained sharp capillary isoperimetric inequality and make it possible to obtain the sharp capillary geometric inequality outside convex domain. In this paper, we establish the sharp capillary Sobolev inequality and Moser-Trudinger inequality outside convex domain, which can be seen as geometric inequality on the Finsler manifold to some extent. Our method is based on constructing capillary P\'{a}lya-Szeg\"{o} rearrangement inequality outside convex domain. Finally, we also consider the capillary Talenti-Comparison principle and Bossel-Daners inequality.