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Capillary curvature images

Yingxiang Hu, Mohammad N. Ivaki

TL;DR

The paper addresses the existence of solutions to the even capillary $L_p$-Minkowski problem in a halfspace for $-n<p<1$ and capillary angle $\theta\in(0,\tfrac{\pi}{2})$. It introduces capillary curvature image operators $\Lambda_p^{\phi}$ built on the $p=1$ capillary Minkowski solution and proves a monotone framework for associated functionals, enabling a discrete-flow like iterative scheme with uniform $C^m$ bounds. The main result is the existence of an even, smooth, strictly convex capillary hypersurface $\Sigma$ whose capillary support function and Gauss curvature satisfy $\frac{s^{1-p}}{\mathcal{K}\circ\tilde{\nu}^{-1}}=\phi$, achieved as a fixed point of the iteration via Minkowski equality. This approach bypasses intractable continuity/variational methods in this range of $p$ and extends capillary Minkowski theory to $-n<p<1$ with potential for further geometric applications.

Abstract

In this paper, we solve the even capillary $L_p$-Minkowski problem for the range $-n < p < 1$ and $θ\in (0,\fracπ{2})$. Our approach is based on an iterative scheme that builds on the solution to the capillary Minkowski problem (i.e., the case $p = 1$) and leverages the monotonicity of a class of functionals under a family of capillary curvature image operators. These operators are constructed so that their fixed points, whenever they exist, correspond precisely to solutions of the capillary $L_p$-Minkowski problem.

Capillary curvature images

TL;DR

The paper addresses the existence of solutions to the even capillary -Minkowski problem in a halfspace for and capillary angle . It introduces capillary curvature image operators built on the capillary Minkowski solution and proves a monotone framework for associated functionals, enabling a discrete-flow like iterative scheme with uniform bounds. The main result is the existence of an even, smooth, strictly convex capillary hypersurface whose capillary support function and Gauss curvature satisfy , achieved as a fixed point of the iteration via Minkowski equality. This approach bypasses intractable continuity/variational methods in this range of and extends capillary Minkowski theory to with potential for further geometric applications.

Abstract

In this paper, we solve the even capillary -Minkowski problem for the range and . Our approach is based on an iterative scheme that builds on the solution to the capillary Minkowski problem (i.e., the case ) and leverages the monotonicity of a class of functionals under a family of capillary curvature image operators. These operators are constructed so that their fixed points, whenever they exist, correspond precisely to solutions of the capillary -Minkowski problem.
Paper Structure (3 sections, 7 theorems, 64 equations)

This paper contains 3 sections, 7 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Capillary curvature image operators
  3. An iterative scheme

Key Result

Theorem 1.1

Let $\theta \in (0, \frac{\pi}{2})$. Suppose $0 < \phi \in C^2(\mathcal{C}_\theta)$ satisfies where $\{E_{i}\}_{i=1,\ldots,n-1}$ is the horizontal basis of $\partial \mathbb{R}^{n}_+$. Then there exists a $C^{3,\alpha}$ strictly convex capillary hypersurface $\Sigma \subset \overline{\mathbb{R}^{n}_{+}}$ such that its Gauss-Kronecker curvature $\mathcal{K}$ satisfies Moreover, $\Sigma$ is unique

Theorems & Definitions (14)

  • Theorem 1.1: Capillary Minkowski problem in halfspace
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 4 more