So, the fact that must imply that there is no min/max inside the region or that is constant on . However, I can't seem to get to the vanishes part.
I am not getting anywhere on this problem
Let be a solution of
of class in the closed unit disk in the -plane. Let on the boundary of Prove that vanishes
identically. (Hint: Show that using conditions for a maximum at a boundary point.)
Since we have been using method of characteristics, I assume we need to use that.
So let Then the characteristic ODE is
Therefore, . So if then is a strictly decreasing function with respect to . If then is a strictly increasing function with respect to . However, I am not sure how this even helps me.
EDIT: I will even take some brain storming from people. I could use any help offered.