On the mean value property of superharmonic and subharmonic functions (Q2468977)

The following theorem was established by \textit{B. Epstein} and \textit{M. Shiffer} [J. Anal. Math. 14, 109--111 (1965; Zbl 0131.10003)] in the theory of harmonic functions: Let \(\Omega\subset\mathbb R^n\) \((n\geq 1)\) be a bounded open set. Suppose that there exists \(x_0\in\Omega\), such that \(u(x_0)=(1/|\Omega|)\int_\Omega u(x)\,dx\) for every \(u\in H(\Omega)\cap L^1(\Omega)\). Then \(\Omega\) is a ball with center \(x_0\). The author generalizes this theorem proving the following one: Let \(\Omega\subset \mathbb{R}^n\) \((n\geq 1)\) be a bounded open set. Suppose that there exists \(x_0\in\Omega\) such that \(u(x_0)\geq(\leq) (1/|\Omega|)\int_\Omega u(x)\,dx\) for every \(u\in SH(\Omega)\cap L^1(\Omega)\setminus H(\Omega)(sH(\Omega)\cap L^1(\Omega)\setminus H(\Omega))\). Then \((\Omega)\) is a ball with center \(x_0\). In the above expressions the following notations were used: \(H(\Omega)\) -- the space of harmonic functions in \(\Omega\), \(L^1(\Omega)\) -- the space of Lebesgue integrable functions on \(\Omega\), \(|\Omega|\) -- the Lebesgue measure of \(\Omega\), \(SH(\Omega)\) --the subset of \(C^1(\Omega)\) consisting of the superharmonic functions in \(\Omega\), \(sh(\Omega)\) -- the subset of \(C^2(\Omega)\) consisting of the subharmonic functions in \(\Omega\).