A mixed boundary value problem for u_xy = f(x,y,u,u_x,u_y)

Consider a single hyperbolic PDE u_xy=f(x,y,u,u_x,u_y), with locally prescribed data: u along a non-characteristic curve M and u_x along a non-characteristic curve N. We assume that M and N are graphs of one-to-one functions, intersecting only at the origin, and located in the first quadrant of the (x,y)-plane. It is known that if M is located above N, then there is a unique local solution, obtainable by successive approximation. We show that in the opposite case, when M lies below N, the uniqueness can fail in the following strong sense: for the same boundary data, there are two solutions that differ at points arbitrarily close to the origin. In the latter case, we also establish existence of a local solution (under a Lipschitz condition on the function f). The construction, via Picard iteration, makes use of a careful choice of additional u-data which are updated in each iteration step.