Geometric criteria for the existence of capillary surfaces in tubes

Abstract

We review some geometric criteria and prove a refined version, that yield existence of capillary surfaces in tubes $\Omega \times \mathbb{R}$ in a gravity free environment, in the case of physical interest, that is, for bounded, open, and simply connected $\Omega \subset {\mathbb{R}}^2$. These criteria rely on suitable weak one-sided bounds on the curvature of the boundary of the cross-section $\Omega$.

arXiv comment16 pages, 4 figures

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• The prescribed mean curvature equation in weakly regular domains
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