New examples of capillary surfaces in polyhedral regions

By Antonio Alarcón (U. Granada)

Consider a closed region B in the Euclidean space R^3. A capillary surface in B is a compact H-surface (i.e., with  constant mean curvature H) with non-empty boundary, which is C^1 up to the boundary and meeting the frontier ∂B of B at a constant angle θ ∈ [0,π] along its boundary. Capillary surfaces are stationary surfaces for an energy functional under a volume constraint. In the physical interpretation, capillary surfaces model incompressible liquids inside a container in the absence of gravity.

In this talk we will provide a large new family of embedded capillary surfaces inside convex polyhedral regions in the Euclidean space. The angle of contact of the examples we will give is prescribed to be any value in (π,π] and it is allowed to vary from one boundary component to the other (in the physical interpretation, one allows the bounding faces of the polyhedral container to be composed of different homogeneous materials). We will also discuss a classification result for these examples. This talk is based on the papers [1, 2] by the authors.


References

[1] A. Alarcon, R. Souam: The Minkowski problem, new constant curvature surfaces in R^3, and some applications. J. Reine Angew. Math., in press.

[2] A. Alarcon, R. Souam: Capillary surfaces inside polyhedral regions. Preprint, January 2014.