If $u(x,y)=\frac{x^2}{x^2+y^2}$, then $u$ is twice continuously differentiable on $D$ and $\triangle u=0$ for all $(x,y) \in D =\{\ (x,y): |x|=|y| \}\ \backslash \{(0,0) \}$.
Can I say that $u$ is harmonic on $D$? even $D$ is not open and I think it is disconnected (I'm not sure about connectivity).