Harmonic function product, Knowing that one is Harmonic implies something about the other?

Question

Hello I have been fighting with this problems for a couples of days and cant get anything better, I will thanks a lot for your help, for little hint, some clue or idea :

Let $A$ be an Harmonic function and we know that $A\phi$ is harmonic.

And also we know that $A >0$

(Both function $A,\phi:R^2\rightarrow R$)

$\nabla ^2(\phi A) = A(\nabla ^2 \phi)+2\nabla A.\nabla \phi+\phi(\nabla ^2 A) $

and so: $0 = A(\nabla ^2 \phi)+2\nabla A.\nabla \phi$

With the Green identities we can say that:

$\int{A (\nabla\phi.n) dS }=-\int{ (\nabla A.\nabla \phi) dV } $

or

$\int{A (\nabla\phi.n) dS }=-\int{\phi (\nabla A.n) dS } $

Does that implies that $A$ is subharmonic or harmonic ?

Thanks!

Answer 1

Another way to phrase the question: what can we say about the ratio of two harmonic functions? I'll use different notation: $f =u/v$ where $u$ and $v$ are harmonic. Then:

1. $f$ is real-analytic in its domain of definition (where $ v\ne 0$)
2. Every level set of $f$ is the zero set of a harmonic function ($f=c$ means $u-cv=0$). In particular, a level set of $f$ cannot contain a closed curve (unless the domain of definition of $u,v$ is multiply connected). Zero sets of harmonic function are quite special: for example, they cannot contain a piece of cubic curve such as $y=x^3$. See this paper for more.
3. $f$ has no local extrema unless it is constant. Again, this follows by applying the maximum principle to $u-cv$.

On the other hand, $f$ need not be harmonic or sub- or super- harmonic. A simple example is $1/x$ which is subharmonic in the right halfplane and superharmonic in the other. It is less trivial to find an example where $\Delta f$ changes sign within a connected component of the domain of $f$. Here is one: $f(x,y)=\dfrac{xy}{x^2-y^2}$ has $\Delta f = \dfrac{8xy(x^2+y^2)}{(x^2-y^2)^3}$.