Actually what I wrote does not fit your problem. The boundary here is not smooth, it is made out of two pieces. But I am sure that is okay, I very certain what I wrote above easily generalizes to what you are doing. However, there is still another problem. It

*can* be constant on the outer boundary and on the inner boundary as long as the constant differ. If they happen to be the same then the constant solution is the only solution.

Maybe you can argue that a solution to such an equation can be extened to include the whole disk, in such a case you should be constant on the boundary of the full disk. Which will mean it is constant. But I am not sure how to do that, or if that is even possible.

I do not know much about harmonic functions,

, I would like to.