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On the characterization of harmonic and subharmonic functions via mean-value properties. (English) Zbl 1187.31003
The paper studies the relation between mean-value properties on the one side and harmonicity and subharmonicity on the other side. If \(G\) is a bounded domain with sufficiently smooth boundary and \(u\) is a continuous function on its closure \(\overline G\), denote by \(B_u(G)\) the mean value of \(u\) over \(G\), and by \(S_u(G)\) the mean value of \(u\) over \(\partial G\). The first result is the following: Let \(u\) be a continuous function on a domain \(\Omega \). Then \(u\) is harmonic (or subharmonic) in \(\Omega \) if and only if \(B_u(B)=S_u(B)\) (or \(B_u(B)\leq S_u(B)\)) for every ball \(B\) whose closure \(\overline B\) is contained in \(\Omega \). The second result is the following: Let \(\Omega \) be a bounded domain with sufficiently smooth boundary. Then there exist constants \(0<c_1 \leq 1\leq c_2 <\infty \) such that \(c_1S_u (\Omega ) \leq B_u (\Omega ) \leq c_2 S_u(\Omega )\) for all non-negative harmonic functions \(u\in C^1(\overline \Omega )\). If \(c_1 =1\) or \(c_2=1\), then \(\Omega \) is a ball.

MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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