Let $D$ be either the unit ball $B_1\left(0\right)$ or the half ball $B_1^{+}\left(0\right),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem:$$\Delta u\left(x\right)={\chi }_{{}_{\Omega }}\left(x\right)f\left(x\right)\text{in}D,0\in \partial \Omega ,u=\left|\nabla u\right|=0\text{in}{\Omega }^c,$$where ${\chi }_{{}_{\Omega }}$ denotes the characteristic function of $\Omega ,$${\Omega }^c$ denotes the set $D\setminus \Omega ,$ and the equation is satisfied in the...