On the classification of capillary graphs in Euclidean and non-Euclidean spaces

TL;DR

This paper classifies capillary graphs in Euclidean and non-Euclidean spaces, establishing rigidity results and splitting theorems for graphs with prescribed mean curvature related to capillarity phenomena.

Contribution

It introduces new classification and splitting theorems for capillary graphs in Riemannian manifolds, extending understanding beyond Euclidean spaces.

Findings

Classification of capillary graphs in Euclidean and non-Euclidean spaces

Rigidity results for graphs with prescribed mean curvature

General splitting theorems for capillarity problems

Abstract

We prove some rigidity and classification results for graphs with prescribed mean curvature and locally constant Dirichlet and Neumann data, for instance as they appear in capillarity problems. We consider domains in Riemannian manifolds, with emphasis on $\mathbb{R}^2$ and $\mathbb{R}^3$. We classify both the underlying domain and the resulting solution, providing general splitting theorems in this setting.