Uniqueness of capillary disks in three-dimensional domains

TL;DR

This paper proves the uniqueness of capillary disks in 3D domains modeled by elliptic PDEs, extending classical results for constant mean curvature surfaces in Euclidean space.

Contribution

It generalizes Nitsche's and Hopf's theorems to more complex 3D domains with specific surface families, broadening the scope of uniqueness results.

Findings

Uniqueness of capillary disks in certain 3D domains established

Extension of classical Euclidean results to more general elliptic PDE settings

Provides conditions under which capillary disks are uniquely determined

Abstract

We prove uniqueness results for capillary disks in three-dimensional domains that are modeled by an elliptic PDE, under the assumption that the domain admits a family of surfaces with suitable properties. Our main theorem generalizes Nitsche's result for capillary constant mean curvature disks in the Euclidean ball and is inspired by the extension of Hopf's uniqueness theorem for constant mean curvature spheres in Euclidean space due to G\'alvez and Mira.