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ON THE MEAN-VALUE PROPERTY OF HARMONIC FUNCTIONS t) . KURAN In this paper we give a simple proof of the following: THEOREM. Let D be a domain ( = connected open set) of finite (Lebesgue) measure in the Euclidean space R" where n ^ 2. Suppose that there exists a point P in D such that, for every function h harmonic in D and integrable over D, the volume mean of h over D equals h(P ). Then D is an open ball (disc when n = 2) centred at P . 0 o This result was introduced and proved by Epstein [1] in the case where n = 2 and D is simply connected. Then Epstein and Schiffer [2] proved it in the case where the complement ~ D of D in R" has a non-empty interior. Recently, Goldstein and Ow [3] worked in the plane with the boundary 3D of D in R having at least one component which is a continuum. Clearly, there is a point P in ~D such that 0<P ^ i <P P (Pe~D). o O Hence, if B is the open ball of centre P and radius r = P P then
Bulletin of the London Mathematical Society – Wiley
Published: Nov 1, 1972
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