Suppose $u_{xx}+u_{yy} = 0$ for some positive function $u$ on $\Bbb R^2$. Show that $u$ is constant.
I am not sure how to show this but I think this has something to do with the Cauchy Riemann Equations. Since $u_x = v_y$ we have $$u_{xx} = v_{yx}$$
Likewise, $u_y = −v_x$ implies $$u_{yy} = −v_{xy}$$ Since $v_{xy} = v_{yx}$, we have $$u_{xx} + u_{yy} = v_{yx} −v_{xy} = 0$$. Thus $u$ is harmonic. But I do not know how to show $u$ is constant. This just shows $u$ is harmonic. What are the steps do I have to follow to show what is required? Do I have to assume a holomorphic function $f$ such that $f(z) = u(x,y) + iv(x,y) $ and then work my way?