Densities with the mean value property for harmonic functions in a Lipschitz domain

Let $D$ be a bounded domain in $\mathbb {R}^{n}$, $n\ge 2$, and let $x_{0}\in D$. We consider positive functions $w$ on such that $h( x_{0}) = (\int _{D}w dx)^{-1} \int _{D} h w dx$ for all bounded harmonic functions $h$ on $D$. We determine Lipschitz domains $D$ having such $w$ with $\inf _{D} w>0$.