I'm fascinated (probably by my lack of understanding of the topic) by the following result discussed in this paper
Let $u,v$ be two harmonic functions on a compact domain $K$ such that their set of zeroes are exactly the same. Let $f=\frac{u}{v}$. Then $\sup|f|\leq C_1 \inf|f|$, where $C_1$ depends only on the compact domain $K$. Moreover, $|\nabla f|\leq C_2\inf |f|$, where $C_2$ again only depends on $K$.
I'm not much of an analyst. But my basic hunch is that this works because a harmonic function is very restricted in how it "looks like" locally- it can only look like a saddle, at least in two dimensions. So my questions are the following:
(1) Can I get example of harmonic functions $u,v$ such that they have the same set of zeros on $\Bbb{R}^2$, but are not scalar multiples of each other? I'm trying to generate more examples.
(2) Shouldn't this also be true for functions that satisfy $$\sum\limits_{i=1}^n \frac{\partial^3 u}{\partial x_i^3}=0$$ and other such nice constraints on the magnitude of the derivatives, shape of the functions, etc?