
Here is the uniqueness theorem in Evans PDE :

I can show u is harmonic, however I don't know how to show it = 0 on the surface and the contradiction.

Here is the uniqueness theorem in Evans PDE :

I can show u is harmonic, however I don't know how to show it = 0 on the surface and the contradiction.
Hint:
What is the value of $y$ on $\partial U$?