On the stability of the Gauss mean value formula for harmonic functions

Among the various rigidity properties of the Euclidean balls one of the best known examples is the Gauss mean value formula for harmonic functions. This property raises the question of its stability. i.e. : if $D$ is an open set with finite measure and $x_0$ is a point of $D$ such that $u(x_0)$ is close to the average of $u$ on $D$ for every integrable harmonic functions $u$ in $D$, is it true that $D$ is close to a ball centered at $x_0$? In this talk we present some positive answers to this question, obtained in collaboration with Giovanni Cupini, Nicola Fusco and Xiao Zhong