Consider $\Omega$ a open, bounded, convex domain in $R^n.$
I am trying to justify this:
Let $u, v$ non negative harmonic functions in $\Omega$ (in the Sobolev sense). Suppose that $ u = v =0$ in some neighbourhood of $x \in \partial \Omega$ and $u \geq v$ in $\Omega.$ Then
$$ lim \ sup_{y \rightarrow x} |\nabla u(y)| \geq lim \ sup_{y \rightarrow x} |\nabla v(y)| .$$
In the remark 1 of this paper:
http://hal.archives-ouvertes.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf
The authors say that the clam is true. I really dont understand the justificative of the paper.
Someone could explain to me please (or give a reference different of the article with a proof)?
Any help will be appreciated.
thanks in advance.