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

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.

💡 Research Summary

The paper investigates overdetermined boundary value problems for graphs whose mean curvature is prescribed by a nonlinear function, encompassing both capillary surfaces (where the source term is linear in the height) and constant‑mean‑curvature (CMC) surfaces. The authors consider a domain Ω in a Riemannian manifold (with particular emphasis on ℝ² and ℝ³) and a function u solving

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