CHARACTERIZATION OF BALLS THROUGH CONCAVITY PROPERTIES OF SOLUTIONS TO ELLIPTIC EQUATIONS

I will present two (unconventional) overdetermined problems. Let $n\geq 3$ and $\Omega$ be a bounded domain in $R^n$. First: if the Newtonian potential $u$ of $\Omega$ has two homothetic convex level sets, then $\Omega$ is a ball. Second: if the Newtonian potential $u$ of $\Omega$ is $\frac{1}{2-n}$-concave (i.e. $u^{(1/(2−n)}$ is convex), then Ω is a ball. The result can be extend to the $p$-capacity potential for $p\in(1,n)$.