modified:

My solution is **$U=x^2+y^2-z^2-\frac{1}{3} (x^2+y^2+z^2-1)$**, the laplacian of this equation is zero and it equal g(x,y,z) at the boundary. How can I write this as a sum of harmonic homogenous polynomials, since there is a factor of 1/3 : $U=2/3 x^2+ 2/3 y^2- 4/3 z^2+ \frac{1}{3}$

By the way, g is $x^2+y^2-z^2$