Find a harmonic function on two concentric balls?

Question

My attempt:

I thought about using Poisson Integral formula since the area is two concentric balls. Then I get something like the following: $u(x)=\frac{1}{nw_nR}\int_{\partial B_R(p)}(\frac{R^2-x^2}{|y-x|^2}*b)dy$ on the big ball and $u(x)=\frac{1}{nw_nR}\int_{\partial B_R(p)}(\frac{R^2-x^2}{|y-x|^2}*a)dy$ on the small ball.

Then I got stuck. How to deal with the boundary condition at $u=a$, the Poisson Integral formula doesn't work well.

I totally have no clue about this problem. Can anyone give me some answers or clues?

Thanks so much about your help!! :)

Answer 1

Following up on comments by Daniel Fischer:

The boundary values are rotationally symmetric, hence so will be $u$

You don't actually need a proof of this statement (since your task is to find one solution, not all of them) so I would rather say:

The boundary values are rotationally symmetric, hence we should look for a solution that is rotationally symmetric.

In the pool of rotationally symmetric harmonic functions we find constants and power $2-n$ of distance function (replaced with the logarithm of distance function when $n=2$). You can also make linear combinations of these, since the PDE is linear. Try to fit a linear combination to the boundary conditions.