Consider a nonparametric capillary or prescribed mean curvature surface z = f(x) defined in a cylinder Omega x R over a two-dimensional region Omega whose boundary has a corner at O with an opening angle of 2 alpha. Suppose the contact angle approaches limiting values gamma(1) and gamma(2) in (0, pi) as O is approached along each side of the opening angle. We will prove the nonconvex Concus-Finn conjecture, determine the exact sizes of the radial limit fans of f at O when (gamma(1), gamma(1)) is an element of D-1(+/-) boolean OR D-2(+/-) and discuss the continuity of the Gauss map.