For the hyperbolic PDE u_xy = f with mixed boundary data on curves M and N intersecting at the origin, when M is below N there are multiple local solutions and existence holds under Lipschitz f.
Melvin Lieberstein, Theory of partial differential equations , Academic Press, New York-London
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A mixed boundary value problem for $u_{xy}=f(x,y,u,u_x,u_y)$
For the hyperbolic PDE u_xy = f with mixed boundary data on curves M and N intersecting at the origin, when M is below N there are multiple local solutions and existence holds under Lipschitz f.