Serrin’s overdetermined problem and constant mean curvature surfaces

Serrin’s Overdetermined Problem and Constant Mean Curvature Surfaces

For all N ≥ 9, we find smooth entire epigraphs in R , namely smooth domains of the form Ω := {x ∈ R / xN > F (x1, . . . , xN−1)}, which are not half-spaces and in which a problem of the form ∆u+ f(u) = 0 in Ω has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on ∂Ω. This answers negatively for large dimensions a question by Berestycki, Caffarelli ...

New Constant Mean Curvature Surfaces

We use the DPW construction [5] to present three new classes of immersed CMC cylinders, each of which includes surfaces with umbilics. The first class consists of cylinders with one end asymptotic to a Delaunay surface. The second class presents surfaces with a closed planar geodesic. In the third class each surface has a closed curve of points with a common tangent plane. An appendix, by the t...

Coplanar Constant Mean Curvature Surfaces

We consider constant mean curvature surfaces with finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors [GKS2, GKS1]. Here we extend the arguments to the case of an arbitrary number of ends, under the assumption that the asymptotic axes of the ends lie in a common plane...

Surfaces of Constant Mean Curvature

Constant mean curvature surfaces and integrable equations