On the Harmonic characterization of domains via mean value formulas
Abstract
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 x0 be a point of D. If
for every nonnegative harmonic function u in D, then D is a Euclidean ball centered at x0. On the other hand, on every sufficiently smooth domain D and for every point x0 in D there exist Radon measures μ such that
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.
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