I am studying converse of mean value theorem from the book Elliptic Partial Differential Equations of Second Order by Gilbarg and Trudinger. Here I find some difficulty. Here's this $:$
I know that if $u \in \Omega \subset_{\mathrm {domain}} \Bbb R^n$ then for any open ball $B \subset \subset \Omega$ $\exists$ a harmonic function $h$ such that $u=h$ on $\partial B$. From here how did the authors claim that the function $w:=u-h$ satisfies mean value property on any ball in $B$? (See the second sentence of Theorem 2.7 ).
I cannot convince myself by this reasoning. Why does this happen? Would anybody please point it out?
Thank you very much.
If $u$ is harmonic, it has the mean value property (and presumably the book proved this already). So the proof of this theorem is only of the MVP $\implies$ harmonic direction. So it assumes that $u$ has MVP. (And then it follows that $u-h$ also does since $h$ does.)
I agree that it's not very clear that this is what it's doing.