On the surface average for harmonic functions: a stability inequality

Abstract

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