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Logarithmic Convexity for Supremum Norms of Harmonic Functions

Abstract: "We prove the following convexity property for supremum norms of harmonic functions. Let [omega] be a domain in R[superscript n], [omega]0 and E a subdomain and a compact subset of [omega], respectively. There exists a constant [alpha] = [alpha](E, [omega]0, [omega]) [element of] (0,1] such that for all harmonic functions u on [omega] the inequality [formula] is valid. The case of concentric balls [omega]0 [contained within] E [contained within] [omega] plays a key role in the proof. For positive harmonic functions on such balls we determine the sharp constant [alpha] in the inequality."

Abstract: "We prove the following convexity property for supremum norms of harmonic functions.