Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Capillary John ellipsoid theorem with applications to capillary curvature problems

Jinrong Hu, Bo Yang

Abstract

In this paper, we apply a capillary John ellipsoid theorem for capillary convex bodies in the Euclidean half-space $\overline{\mathbb{R}^{n+1}_{+}}$. This theorem yields a non-collapsing estimate for capillary hypersurfaces, which provides a new approach to obtaining $C^{0}$ estimates for solutions to some capillary curvature problems (including the capillary $L_{p}$ Christoffel-Minkowski problem and the capillary $L_{p}$ curvature problem), based on the corresponding gradient estimates. As an application, we study the capillary $L_{p}$ dual Minkowski problem. By deriving a gradient estimate, refining a $C^{2}$ estimate, and combining these with the non-collapsing estimate, we establish existence in the case $1<p\leq q\leq 3$ and improve upon the existing existence result for the case $p > q$ in $\overline{\mathbb{R}^3_{+}}$.

Capillary John ellipsoid theorem with applications to capillary curvature problems

Abstract

In this paper, we apply a capillary John ellipsoid theorem for capillary convex bodies in the Euclidean half-space . This theorem yields a non-collapsing estimate for capillary hypersurfaces, which provides a new approach to obtaining estimates for solutions to some capillary curvature problems (including the capillary Christoffel-Minkowski problem and the capillary curvature problem), based on the corresponding gradient estimates. As an application, we study the capillary dual Minkowski problem. By deriving a gradient estimate, refining a estimate, and combining these with the non-collapsing estimate, we establish existence in the case and improve upon the existing existence result for the case in .

Paper Structure

This paper contains 8 sections, 16 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Basics of capillary geometry
  3. A capillary John ellipsoid theorem
  4. A non-collapsing estimate of capillary hypersurfaces
  5. Proof of Theorem \ref{['maintheo2']}
  6. Proof of Theorem \ref{['maintheo2']} (i)
  7. Proof of Theorem \ref{['maintheo2']} (ii)
  8. Proof of Theorem \ref{['maintheo2']} (iii)

Key Result

Theorem 1.1

Let $n=2$ and $\theta \in (0,\frac{\pi}{2})$. Let $f$ be a positive and smooth function on $\mathcal{C}^2_{\theta}$. (i) If $1<p<q\leq 3$, further assume that $f$ is even, then there exists an even, smooth and strictly convex solution $h$ to Eq. Mong-Eq. (ii) If $p>q$, then there exists a unique, po

Theorems & Definitions (30)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • ...and 20 more