Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lu Chen, Jiali Lan

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 Pálya-Szegö rearrangement inequality outside convex domain. Finally, we also consider the capillary Talenti-Comparison principle and Bossel-Daners inequality.

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

Abstract

The theory of sharp geometric inequality in 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 Pálya-Szegö rearrangement inequality outside convex domain. Finally, we also consider the capillary Talenti-Comparison principle and Bossel-Daners inequality.
Paper Structure (7 sections, 14 theorems, 205 equations)

This paper contains 7 sections, 14 theorems, 205 equations.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries
  3. Anisotropic perimeter
  4. Capillary Schwartz symmetrization outside convex domain
  5. Proof of the Theorem \ref{['theorem4.1']}
  6. Proof of Theorem \ref{['theorem4.4']} and Theorem \ref{['theorem1.6']}
  7. Proof of the Theorem \ref{['theorem4.2']}, Theorem \ref{['corollary4.1']}

Key Result

Theorem 1.1

(Capillary Pólya-Szegö principle outside convex domain) For $1\leq p<\infty$, let $u\in W_0^{1,p}(\Omega;E^c)$ be a non-negative function satisfying the following anisotropic Neumann boundary condition: Then, the following inequality holds: where $F_{\lambda}(\xi)=|\xi|-\lambda\left<\xi,e_n\right>$ with $e_n$ being the n-th coordinate unit vector, and $u^*$ is the capillary Schwartz symmetrizati

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Proposition 2.2
  • ...and 18 more