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arXiv · 2505.07746

The capillary $L_p$-Minkowski problem

Xinqun Mei·Guofang Wang·Liangjun Weng
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Abstract

This paper is a continuation of our recent work [Adv. Math. 469 (2025), Paper No. 110230] concerning the capillary Minkowski problem. We propose, in this paper, a capillary $L_p$-Minkowski problem for $p\in \mathbb{R}$, 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 [J. Differential Geom. 38 (1993), no. 1, 131--150]. For $p>1$, we resolve the capillary $L_p$-Minkowski problem in the smooth category by reducing it to a Monge--Amp\`ere equation with a Robin boundary condition on the unit spherical cap.

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Xinqun Mei, Guofang Wang, Liangjun Weng. 2025-05-12. The capillary $L_p$-Minkowski problem. https://doi.org/10.1090/tran%2F9866

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