Abstract
We prove a converse of the mean value property for superharmonic and subharmonic functions. The case of harmonic functions was treated by Epstein and Schiffer.
Highlights
We prove a converse of the mean value property for superharmonic and subharmonic functions
The case of harmonic functions was treated by Epstein and Schiffer
In this note we give a proof of the following converse
Summary
We prove a converse of the mean value property for superharmonic and subharmonic functions. The case of harmonic functions was treated by Epstein and Schiffer. Recall that a function u is harmonic (superharmonic, subharmonic) in an open set U ⊂ Rn (n ≥ 1) if u ∈ C2(U) and Δu = 0 (Δu ≤ 0, Δu ≥ 0) on U. Denote by H(U) the space of harmonic functions in U and SH(U) (sH(U)) the subset of C2(U) consisting of superharmonic (subharmonic) functions in U.
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