A curious behavior of capillary surfaces

Abstract

It is known that for a given base domainΩ and «contact angle»γ, there will be a critical γ0, 0 ⩽γ0, ⩽π/2., such that a solution overΩ of the scalar capillarity equation in the absence of external (e.g. gravity) force field will exist if γ0, 0 ⩽γ, ⩽π/2, and will fail to exist if 0 ⩽γ ⩽γ0. In the particular case for whichΩ is a circular disk, a solution exists for everyγ, and is unique up to an additive constant. For a piecewise smoothΩ that is not circular, there will be a boundary pointP of maximal inward directed curvatureKM (at a protruding corner we defineKM=∞). We suppose that a solution exists for such a domain, and ask whether a solution continues to exist if the domain is made closer to circular by smoothing with an inscribed interior arc of constant curvatureK<KM. That is so in many cases, for example it is true ifΩ is a rectangle, and in fact in that case smoothing decreases γ0. In the present note, we show that the answer can also be negative. To that effect, we give an example of a convex domain Ω at which the valueKM is achieved only at a single isolated point, and for which smoothing at that point changes existence to non-existence.