If $u=ax^3+by^3$ and $u$ is harmonic, find values of $a$ and $b$. Also find the harmonic conjugate of $u$.

Question

If $u=ax^3+by^3$ and $u$ is harmonic, find values of $a$ and $b$. Also find the harmonic conjugate of $u$.

I could not find any confirmation regarding this solution of $a$ and $b$.

Answer 1

I suppose that the domain $D \ne \emptyset$ of $u$ is an open subset of $ \mathbb R^2.$

We have $u_{xx}(x,y)=6ax$ and $u_{yy}(x,y)=6by.$

Then $u$ is harmonic on $D \iff ax+by=0$ for all $(x,y) \in D \iff a=b=0.$