ON THE HARMONIC CHARACTERIZATION OF DOMAINS VIA MEAN VALUE FORMULAS

The Euclidean ball have the following harmonic characterization, via Gauss-mean value property: Let D be an open set with finite Lebesgue measure and let x(0) be a point of D. If u(x(0)) = 1/|D| int_{D} u(y)dy for every nonnegative harmonic function u in D, then D is a Euclidean ball centered at x(0). On the other hand, on every sufficiently smooth domain D and for every point x(0) in D there exist Radon measures mu such that u(x(0)) = int_{D} u(y)d mu(y)for every nonnegative harmonic function u in D. In this paper we give sufficient conditions so that this last mean value property characterizes the domain D.