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

Giulio Colombo, Alberto Farina, Marco Magliaro, Luciano Mari, Marco Rigoli

TL;DR

The paper develops an energy-based framework to classify capillary graphs with prescribed mean curvature on domains in Euclidean and non-Euclidean spaces. By establishing moderate energy growth, a stability-derived monotonicity along Killing fields, and a geometric Poincaré formula, it proves a splitting theorem that forces solutions to be 1D along a product direction, yielding rigidity results that rank half-planes or cylindrical graphs as the only possibilities under broad conditions. The approach extends to general quasilinear operators and manifolds with nonnegative Ricci curvature, broadening classical Serrin-type results for capillary and CMC graphs. These results provide a unified, flexible toolkit for rigidity and classification in capillarity problems across dimensions and geometries, including unbounded boundaries and multiple boundary components.

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.

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

TL;DR

The paper develops an energy-based framework to classify capillary graphs with prescribed mean curvature on domains in Euclidean and non-Euclidean spaces. By establishing moderate energy growth, a stability-derived monotonicity along Killing fields, and a geometric Poincaré formula, it proves a splitting theorem that forces solutions to be 1D along a product direction, yielding rigidity results that rank half-planes or cylindrical graphs as the only possibilities under broad conditions. The approach extends to general quasilinear operators and manifolds with nonnegative Ricci curvature, broadening classical Serrin-type results for capillary and CMC graphs. These results provide a unified, flexible toolkit for rigidity and classification in capillarity problems across dimensions and geometries, including unbounded boundaries and multiple boundary components.

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 and . We classify both the underlying domain and the resulting solution, providing general splitting theorems in this setting.

Paper Structure

This paper contains 7 sections, 29 theorems, 308 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The general setting
  3. Moderate energy growth
  4. Stability and monotonicity
  5. The geometric Poincaré formula and splitting
  6. Euclidean space and the critical set of $u$
  7. Proof of the main theorems for capillary graphs

Key Result

Theorem 3

Let $\Omega \subseteq \mathbb R^2$ be a $C^2$ domain supporting a non-constant solution $u \in C^2(\overline\Omega)$ to for some $H,b_j,c_j \in \mathbb R$.

Figures (2)

  • Figure 1: Cases $(i)$ and $(ii)$
  • Figure 2: Cases $(iii)$ and $(iv)$

Theorems & Definitions (77)

  • Remark 1
  • Definition 2: Mildly $\mathbf{v}$-transverse boundary
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • Remark 9
  • Remark 10
  • ...and 67 more