The Dirichlet problem for the minimal surface equation, with infinite data

Abstract

takes the boundary values plus infinity on the vertical sides of the square \x then the arc must necessarily be straight. This being the case, the most general boundary value problem with infinite data takes the following form. Let D be a bounded convex domain whose boundary contains two families of open straight segments Ai, • • • , Ak and B\% • • • , Bu such that no two segments Ai and no two segments Bi have common endpoints. The remainder of the boundary then consists of open convex arcs Ci, • • • , Cm and endpoints of the segments Ai and Bi. It is now required to find a solution of the minimal surface equation in D which takes the value plus infinity on each segment Ai, the value minus infinity on each segment B^ and assigned continuous {though not necessarily bounded) values on the remaining arcs d. The solution of Scherk, for example, corresponds to the case where D is a square with plus infinity assigned on the horizontal sides and minus infinity assigned on the vertical sides, the family {d} being empty. Notwithstanding this example, one might a t first suppose tha t the problem as stated is not well posed. This turns out, however, not to