Proposition

Suppose $u(x, y)$ and $v(x, y)$ are harmonic in $G$, and $c \in \mathbb{R}$. Prove that $u(x, y) + cv(x, y)$ is also harmonic in $G$.

Solution

\((u + cv)_{xx} + (u + cv)_{yy} = u_{xx} + cv_{xx} + u_{yy} + cv_{yy} = (u_{xx} + u_{yy}) + c(v_{xx} + v_{yy}) = 0 + 0 = 0\).