Existence and behavior of the radial limits of a bounded capillary surface at a corner

Abstract

Consider a bounded capillary surface defined on a twodimensional region Ω that has a corner point at 0, with opening angle 2a. If the contact angle is bounded away from 0 and π, then the radial limits exist as 0 is approached from any direction in Ω. If the contact angle approaches limiting values as 0 is approached along each portion of the boundary, then there exist fans of directions adjacent to the two tangent directions at 0 in which the radial limits are constant. Other properties of the radial limit function are given and these results are used to show continuity of the solution up to 0 under certain conditions. For a convex corner, the solution is continuous up to 0 when the limiting angles 7Q, 7^ satisfy fa - 7o - 7όΊ < 2α and 2a + \*y£ - 7^ < π.