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Suppose $f,g,h:\mathbb{R}^n \to \mathbb{R}$ are functions so that $f$ and $g$ are harmonic and not identically zero, $f=g\cdot h$ and $h\geq 0$. Is $h$ a constant function?

EDIT: Someone voted to close presumably because of a lack of context. The context is that I bounced on this problem while trying to flesh out a derivation of the Ginibre formula of the GUE ensemble. As for arguments why the statement is plausible, let me prove that $h$ cannot reach a strict local maximum or strict local minimum at a (isolated) point $x_0\in O:=\{x\in \mathbb{R}^n:\,f(x)> 0\}$. Suppose, anticipating a contradiction, that $h$ did reach such an (isolated) local minimum at $x_0$, then by the mean value property (fixing the radius $R>0$ sufficiently small so that $B(x_0,R)\subseteq O$) $$g(x_0)h(x_0)=f(x_0)=\frac{\int_{B(x_0,R)}dx\,f(x)}{\int_{B(x_0,R)}dx\,} = \frac{\int_{B(x_0,R)}dx\,g(x)h(x)}{\int_{B(x_0,R)}dx\,}>\frac{\int_{B(x_0,R)}dx\,g(x)h(x_0)}{\int_{B(x_0,R)}dx\,}=\frac{h(x_0)\int_{B(x_0,R)}dx\,g(x)}{\int_{B(x_0,R)}dx\,}=g(x_0)h(x_0),$$ a contradiction.

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1 Answers1

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For $n>2$, the family of entire harmonic functions $$h_{a,b}:\mathbb{R}^n \to \mathbb{R}:(x_1,...,x_n)\mapsto \sin(x_1)\exp(ax_2)\exp(bx_3),\qquad a^2+b^2=1$$ provides a counterexample. This example came from this paper, which alleges on p.255-256 that such a counterexample does not exist for $n=2$.

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