Find a C-harmonic function in the unit disk with boundary values $x^3-xy$. I know the answer is $u(x,y)=\frac{(x^3-3xy^2)}{4} + \frac{3x}{4} - xy$ but don't know how to solve it Any hint or help is appreciated Thanks (C-harmonic function is a function which is harmonic in the interior of a domain and continuous the closure)
I found a solution but I couldn't understand much. "Soluiton: Since $z^3 = (x^3$-$3xy^2$)+i($3x^2$-$y^3$) is analytic on the unit disk, its real part is harmonic there with boundary values $x^3-3xy^2$=$4x^3-3x$. Clearly xy is harmonic everywhere. Therefore the required function is $u(x,y)=\frac{(x^3-3xy^2)}{4} + \frac{3x}{4} - xy$."
I think in the solution "Poisson Kernel" is used but I can't figure out where it has been used.
You have to verify that your candidate function verifies the boundary values requirement (it is straightforward).
The real (and imaginary) part of a holomorphic function is harmonic (it is a theorem) and the sum of harmonic functions is harmonic. $xy$ is harmonic because it is the imaginary part of $z \mapsto z^2$.