An investigation of capillary surfaces at non-convex corners

Abstract

In this thesis we take a close look at the paper CMC Capillary Surfaces at Reentrant Corners [33] a central feature of which is the question of when does the "central fan" of radial limits exist for a capillary graph in a vertical cylinder Ω × R ⊂ R3. The geometry of our cylinder will be examined under the condition that a non-convex (reentrant) corner P ∈ ∂Ω is present in the domain, the existence of said reentrant corner at O makes the determination of the continuity (i.e. the behavior of the radial limits at O) of the solution problematic. Given that continuity is equivalent to the existence of a "central fan" of radial limits under particular conditions, the determination of necessary and sufficient conditions for the existence of a central fan is a very important open question in the mathematical theory of capillarity. A secondary objective is to ascertain the feasibility of certain computations to develop insight into (and perhaps conjectures about) solutions of a boundary value problem for a class of non linear elliptic partial differential equations.