The Dirichlet Problem for the Minimal Surface Equation, with Infinite Data

takes the boundary values plus infinity on the vertical sides of the square \x\ < T T / 2 , \y\ <7r/2, and the boundary values minus infinity on the horizontal sides. This suggests the possibility of posing a boundary value problem for the minimal surface equation in which infinite data is assigned on certain boundary arcs of D. I t is a consequence of previous results of the authors (cf. [3, Lemma 6]) that if u is a solution in a convex domain D which assumes the value plus infinity or minus infinity on a boundary arc of D> 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