Consider a nonparametric
capillary or prescribed mean curvature surface z = f(x,y) defined in a cylinder
Ω × ℝ over a two-dimensional region Ω that has a boundary corner point
at O with an opening angle of 2α. Suppose 2α ≤ π and the contact angle
approaches limiting values γ1 and γ2 in (0,π) as O is approached along
each side of the opening angle. Our results yield a proof of the Concus–Finn
conjecture, which provides the last piece of the puzzle of determining the
qualitative behavior of a capillary surface at a convex corner. We find that
if (γ1,γ2) satisfies 2α+|γ1−γ2| > π, then f is bounded but discontinuous
at O and has radial limits at O from all directions in Ω and, these radial
limits behave in a prescribed way;
if (γ1,γ2) satisfies |γ1+ γ2− π| > 2α, then f is unbounded in every
neighborhood of O; and
