This is an interesting question I found online about Laplace equation.
We define a function $u$ :$R^N\to R$'s graph to be the set $\{x,u(x):\,x\in R^N\}\subset R^{N+1}$. Then I want to prove that any tangent hyperplane to the graph of an entire harmonic function intersects the graph more than once.
I worked on this for a while but I don't really have a clue. First of all, given $x_0\in R^N$, what is the tangent hyperplane, call it $S(x_0)$, to the graph at $(x_0, \,u(x_0))$? How we write this tangent hyperplane explicitly? And after that, what the key in the proving in this problem? What the properties of harmonic function should we use?
You can write the tangent plane explicitly as the graph of the function $$ T(x)=u(x_0)+Du(x_0)(x-x_0). $$ Now simply note that $T$ is a harmonic function (being a degree 1 polynomial). If the tangent plane only intersects the graph of $u$ at $x_0$ then the tangent plane is everywhere above (or below) the graph of $u$, i.e. $T(x)\geq u(x)$ for all $x\in \mathbb{R}^n$. Now apply the strong maximum principle.