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Capillary $L_p$-curvature problem

Yingxiang Hu, Mohammad N. Ivaki

Abstract

We prove a gradient estimate for a class of capillary curvature equations in the half-space. As an application, we prove the existence of an even, smooth, strictly convex solution to the even capillary $L_p$-curvature problem for all $1<p<k+1$ and all contact angles $θ\in(0,π/2)$.

Capillary $L_p$-curvature problem

Abstract

We prove a gradient estimate for a class of capillary curvature equations in the half-space. As an application, we prove the existence of an even, smooth, strictly convex solution to the even capillary -curvature problem for all and all contact angles .
Paper Structure (3 sections, 4 theorems, 53 equations)

This paper contains 3 sections, 4 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The gradient estimate

Key Result

Theorem 1.1

Let $\theta \in (0,\frac{\pi}{2})$, $1 < p < k+1$ and $1 \leq k < n$. Suppose $0 < \phi \in C^{\infty}(\mathcal{C}_\theta)$ is an even function. Then there exists an even, smooth, strictly convex capillary hypersurface $\Sigma \subset \overline{\mathbb{R}^{n+1}_{+}}$ whose $k$-th elementary symmetri

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 3.1: Main Lemma
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof