Question:
$$\frac{\partial u}{\partial x} \frac{\partial u}{\partial y}=1 \qquad \qquad u=0 \; \text{ when } \; x+y=1$$
Find all possible solutions and state where each one exists.
Attempt:
Using the method of characteristics (Charpit's equations), we end up with
$$x=s \pm t \;,\; y=1-s \pm t \;,\; \frac{\partial u}{\partial x} = \pm 1 \;,\; \frac{\partial u}{\partial y}= \pm 1 \;,\; u=2t$$
The set of all solutions is then
$$u(x,y) = \pm(x+y-1)$$
However, how do you know where these solution does/doesn't "exist"?
There seems to be no problems as far as the characteristic projections are concerned (they are a set of straight lines perpendicular to the initial data), and the Jacobian is $J= \pm 2$ which is non-zero.
So is $u$ just supposed to be a multi-valued function in this case?