Question

Find all harmonic polynomials u(x,y) of degree three, that is, u(x,y)=ax3+bx2y+cxy2+dy3.

Answer 1

Compute the Laplacian:
\[\begin{align*}\nabla^2u(x,y)&=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\\[1ex]
&=\bigg(6ax+2by\bigg)+\bigg(2cx+6dy\bigg)\end{align*}\]
\(u(x,y)\) is harmonic if \(\nabla^2u=0\).