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The capillary Christoffel-Minkowski problem

Xinqun Mei, Guofang Wang, Liangjun Weng

TL;DR

The paper advances the capillary Christoffel-Minkowski problem by introducing a $k$-th capillary area measure for capillary convex bodies in the half-space and reformulating the problem as a Hessian equation with Robin boundary. Using the method of continuity, it proves existence and uniqueness of a strictly convex capillary hypersurface whose capillary area measure matches a prescribed function under a natural connectivity condition in a weighted function class. Central to the approach are new a priori estimates (C^0, gradient, and C^2) for the Hessian equation, convexity preservation via Guan-Ma’s constant rank theorem, and a compatibility orthogonality condition. The results extend capillary Minkowski-type theory to intermediate $k$-th measures and rely on capillary Steiner formulas and oblique boundary estimates to handle the Robin boundary.

Abstract

In this article, we introduce a $k$-th capillary area measure for capillary convex bodies in the Euclidean half-space, which serves as a boundary counterpart to the classical concept of area measure (see, e.g., \cite[Chapter 8]{Sch}). We then propose a Christoffel-Minkowski problem for capillary convex bodies, to find a capillary convex body in the Euclidean half-space with a prescribed $k$-th capillary area measure. This problem is equivalent to solving a Hessian-type equation with a Robin boundary value condition. We then establish the existence and uniqueness of a smooth solution under a natural sufficient condition.

The capillary Christoffel-Minkowski problem

TL;DR

The paper advances the capillary Christoffel-Minkowski problem by introducing a -th capillary area measure for capillary convex bodies in the half-space and reformulating the problem as a Hessian equation with Robin boundary. Using the method of continuity, it proves existence and uniqueness of a strictly convex capillary hypersurface whose capillary area measure matches a prescribed function under a natural connectivity condition in a weighted function class. Central to the approach are new a priori estimates (C^0, gradient, and C^2) for the Hessian equation, convexity preservation via Guan-Ma’s constant rank theorem, and a compatibility orthogonality condition. The results extend capillary Minkowski-type theory to intermediate -th measures and rely on capillary Steiner formulas and oblique boundary estimates to handle the Robin boundary.

Abstract

In this article, we introduce a -th capillary area measure for capillary convex bodies in the Euclidean half-space, which serves as a boundary counterpart to the classical concept of area measure (see, e.g., \cite[Chapter 8]{Sch}). We then propose a Christoffel-Minkowski problem for capillary convex bodies, to find a capillary convex body in the Euclidean half-space with a prescribed -th capillary area measure. This problem is equivalent to solving a Hessian-type equation with a Robin boundary value condition. We then establish the existence and uniqueness of a smooth solution under a natural sufficient condition.

Paper Structure

This paper contains 12 sections, 18 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Motivation and set up
  3. Main result
  4. Preliminaries
  5. Basic properties of elementary symmetric functions.
  6. Some properties of strictly convex capillary hypersurfaces in half-space
  7. A priori estimates
  8. estimate
  9. estimate
  10. estimate
  11. Convexity
  12. Proof of Theorem \ref{['main theorem']}

Key Result

Theorem 1.1

Let $\theta\in (0,\frac{\pi}{2}]$ and $1\leq k\leq n-1$. Suppose $f\in C^{2}(\mathcal{C}_{\theta})$ is connected to $1$ in $\mathcal{L}_{-\frac{1}{k}}$. Then there exists a $C^{3, \gamma}$$(\gamma\in(0,1))$ strictly convex capillary hypersurface $\Sigma$ in $\overline{{\mathbb{R}}^{n+1}_{+}}$, such

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • ...and 25 more