Capillary Orlicz-Minkowski flow in the upper half-space

TL;DR

The paper addresses the capillary Orlicz-Minkowski problem for convex capillary hypersurfaces in the upper half-space by employing an anisotropic Gauss curvature flow with capillary boundary. It derives uniform a priori estimates and a monotone functional to prove long-time existence and smooth convergence to a stationary solution that satisfies $\phi(\xi, \frac{h}{\ell})\det(h_{ij}+h\delta_{ij})=f$ alongside $\nabla_{\mu}h=\cot\theta\,h$, thereby solving the problem without the evenness assumption. The results extend capillary Minkowski theory and provide a constructive, flow-based approach to smooth solutions of the capillary Orlicz-Minkowski equation. The use of a carefully designed functional $J(\Sigma_t)$ links geometric evolution to nonlinear elliptic equations with Robin boundary conditions, offering tools for broader capillary problems in convex geometry.

Abstract

In this paper, we study the long-time existence and asymptotic behavior of an anisotropic capillary Gauss curvature flow. By studying this flow and proving its convergence to a stationary solution, we establish a new existence result for the capillary Orlicz-Minkowski problem without the evenness assumption, and provide a flow approach to the existence of smooth solutions.

Theorems & Definitions (8)

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• proof : Proof of Theorem \ref{['maintherorem2']}
• proof : Proof of Theorem \ref{['maintherorem1']}