Changes in a domain’s
geometry can force striking changes in the capillary surface lying above it. Concus
and Finn [1] first studied capillary surfaces above domains with corners, in the
presence of gravity. Above a corner with interior angle 𝜃 satisfying 𝜃 < π − 2γ, they
showed that a capillary surface making contact angle γ with the bounding wall must
approach infinity as the vertex is approached. In contrast, they showed that for
𝜃 ≧ π − 2γ the solution u(x,y) is bounded, uniformly in 𝜃 as the corner is closed.
Since their paper appeared, the continuity of u at the vertex has been an open
problem in the bounded case. In this note we show by example that for
any 𝜃 > π and any γ≠π∕2 there are domains for which u does not extend
continuously to the vertex. This is in contrast to the case π > 𝜃 > π − 2γ; here
independent results of Simon [5] show that u actually must extend to be C1 at the
vertex.
