On the surface average for harmonic functions: a stability inequality

In this article we present some of the main aspects and the most recent results related to the following question: If the surface mean integral of every harmonic function on the boundary of an open set D is "almost" equal to the value of these functions at x0 in D, then is D "almost" a ball with center x0? This is the stability counterpart of the rigidity question (the statement above, without the two "almost") for which several positive answers are known in literature. A positive answer to the stability problem has been given in a paper by Preiss and Toro, by assuming a condition that turns out to be sufficient for ∂D to be geometrically close to a sphere. This condition, however, is not necessary, even for small Lipschitz perturbations of smooth domains, as shown in our recent paper, in which a stability inequality is obtained by assuming only a local regularity property of the boundary of D in at least one of its points closest to x0.