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The Anisotropic Capillary $L_p$-Minkowski Problem

Shanwei Ding, Jinyu Gao, Guanghan Li, Mengliang Liu

Abstract

This paper introduces the \textit{anisotropic $ω_0$-capillary $p$-sum} of two hypersurfaces in $\mathbb{R}_+^{n+1}$, and establishes a theory for anisotropic capillary convex bodies. For a smooth convex hypersurface $Σ$ with anisotropic $ω_0$-capillary boundary, we compute the variation of its anisotropic capillary $k$-th quermassintegral via this $p$-sum, thereby defining the associated anisotropic $ω_0$-capillary $k$-th $p$-surface area measure on the capillary Wulff shape $\mathcal{C}_{ω_{0}}$. This motivates us to propose and solve the anisotropic capillary $L_{p}$-Minkowski problem for $p\geq1$.

The Anisotropic Capillary $L_p$-Minkowski Problem

Abstract

This paper introduces the \textit{anisotropic -capillary -sum} of two hypersurfaces in , and establishes a theory for anisotropic capillary convex bodies. For a smooth convex hypersurface with anisotropic -capillary boundary, we compute the variation of its anisotropic capillary -th quermassintegral via this -sum, thereby defining the associated anisotropic -capillary -th -surface area measure on the capillary Wulff shape . This motivates us to propose and solve the anisotropic capillary -Minkowski problem for .
Paper Structure (10 sections, 12 theorems, 81 equations, 1 figure)

This paper contains 10 sections, 12 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Wulff shape and dual Minkowski norm
  4. Anisotropic curvature
  5. New metric and anisotropic formulas
  6. Anisotropic capillary support function
  7. General quermassintegrals for anisotropic capillary hypersurfaces
  8. Anisotropic $\omega_0$-capillary convex bodies in $\overline{\mathbb{R}^{n+1}_+}$
  9. Anisotropic capillary convex bodies and convex functions
  10. Anisotropic capillary $p$-sum for capillary convex bodies

Key Result

Theorem 1.2

Let $\omega_0 \in (-F(E_{n+1}), F(-E_{n+1}))$ such that the boundary of $\omega_0$-capillary Wulff shape $\partial \mathcal{C}_{\omega_0} \subset \mathcal{C}_{\omega_0}$ is anisotropically convex, and $f(\xi) \in C^{2}(\mathcal{C}_{\omega_0})$ be a positive function satisfying Then there is a $C^{3,\gamma}\ (0<\gamma<1)$ strictly convex $\omega_0$-capillary hypersurface $\Sigma\subset\overline{\

Figures (1)

  • Figure 1: Translation of $\mathcal{T}$

Theorems & Definitions (19)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.1
  • Lemma 2.1: Xia13*Lemma 2.5
  • Lemma 2.2
  • Proposition 2.3: Arxiv1*Theorem 6.2
  • Lemma 2.4: arxiv2*Lemma 3.11
  • Proposition 2.5: book-convex-body*Theorem 7.2.3
  • ...and 9 more