Is the scalar product of the gradient of two harmonic functions an harmonic function?

Question

I am carrying out some research on the topological derivative, and I came across that $DT=\nabla u \cdot \nabla v$, with $u=u(x,y)$ and $v=v(x,y)$ harmonic functions, which are in general different, that is $u\not = v$. I was wondering whether I can state that $DT$ is also harmonic and if not, what would it be a counterexample of that. I would be grateful if you could please point me to some reference or give me hint to how to prove or build a counterexample. Thanks.

Answer 1

In general, the scalar product of the gradients of two harmonic function is not a harmonic function. The following shows that in the special case $u=v$, $DT$ is not a harmonic function, except for linear $u$.

Choose $u=v$. If $DT = |\nabla u|^2 \ge 0$ is harmonic on $\mathbb{R}^2$, then it is necessarily constant. Then also the partial derivatives of $u$ are bounded and therefore constant. In consequence, $u$ is a linear function.

On the other hand, there are some nonlinear harmonic functions, e.g. $u(x,y) = x^2-y^2$.