Laplacian and harmonic functions

Let \(\Omega\subseteq\RR^2\) be open and let \(u\colon\Omega\to\RR\) be a \(C^2\) function of \((x,y)\). The Laplacian of \(u\) is \[ \Delta u\coloneqq\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}, \] and \(u\) is harmonic on \(\Omega\) if \(\Delta u=0\) on \(\Omega\).