An Inverse Problem in Potential Theory for Picone Elliptic-Parabolic PDEs.

Let $\Omega$ be a domain in ${\mathbb{R}^N$. A density with the mean value property for non-negative harmonic functions in $\Omega$ is a positive l.s.c. function $w$ such that, for a suitable $ x_0 \in \Omega $, $$ u(x0) = \frac{1}{w(Ω)} \nt_{\Omega} u(y)w(y)dy $$ for every non-negative harmonic function $u$ in $\Omega$. In this case we say that $(\Omega,w,x_0)$ is a $\Delta$-triple. Existence of $\Delta$-triples on every sufficently smooth domain has been proved in 1994-1995, by Hansen and Netuka, and by Aikawa. Very recently, we have given positive answers to the following inverse problem: “Let $ (\Omega,w,x_0)$ and $(D,w',x_0)$ be $\Delta$-triples such that $\frac{w }{w(\Omega)= \frac {w'}{w'(D)} in $D ∩Ω$. Then is it true that $ \Omega = D$?” Our result contains, as particular cases, several classical potential theoretical characterizations of the Euclidean balls. Densities with the mean value property for solutions to wide classes of Picone’s elliptic-parabolic PDEs have appeared in literature since the 1954 pioneering work by B.Pini on the mean value property for caloric functions. In this talk we present an abstract inverse problem Theorem allowing to extend the previously recalled result on the $ \Delta$-triples to elliptic, parabolic and sub-elliptic PDEs. The results have been obtained in collaboration with Giovanni Cupini (Universita' di Bologna).