The capillary $L_p$-Minkowski problem

TL;DR

This work extends the L_p Minkowski theory to capillary convex bodies in the Euclidean half-space by formulating a capillary L_p surface area measure and reducing the capillary L_p-Minkowski problem to a Monge–Ampère equation with Robin boundary on the spherical cap $\mathcal{C}_{\theta}$. For $p>1$ and $\theta\in(0,\tfrac{\pi}{2})$, the authors establish existence (and in most regimes uniqueness) of smooth capillary convex solutions, with precise a priori estimates that depend on the position of $p$ relative to $n+1$; in particular a novel logarithmic gradient bound at the critical exponent $p=n+1$ is developed. The results are obtained via a continuity method anchored by C^0, C^1, and C^2 estimates, including a boundary-sensitive strategy to handle the Robin condition and the capillary geometry. This work provides a capillary counterpart to the classical $L_p$-Minkowski theory and underpins a capillary Brunn–Minkowski framework with potential links to capillary geometric flows and surface-energy models.

Abstract

This paper is a continuation of our recent work [54] concerning the capillary Minkowski problem. We propose, in this paper, a capillary $L_p$-Minkowski problem for $p\geq 1$, which seeks to find a capillary convex body with a prescribed capillary $L_p$-surface area measure in the Euclidean half-space. This formulation provides a natural Robin boundary analogue of the classical $L_p$-Minkowski problem introduced by Lutwak [43]. For $p>1$, we resolve the capillary $L_p$-Minkowski problem in the smooth category by reducing it to a Monge-Ampère equation with a Robin boundary condition on the unit spherical cap.

Theorems & Definitions (25)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Proposition 2.1
• proof
• Definition 2.2: Capillary $L_p$-surface area measure
• Proposition 2.2
• Theorem 3.1
• Lemma 3.2