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Inequalities for surface integrals of non-negative subharmonic functions

M. P. Aldred, David H. Armitage (1998)

Commentationes Mathematicae Universitatis Carolinae

Let ℋ denote the class of positive harmonic functions on a bounded domain Ω in ℝ N . Let S be a sphere contained in Ω ¯ , and let σ denote the ( N - 1 ) -dimensional measure. We give a condition on Ω which guarantees that there exists a constant K , depending only on Ω and S , such that ∫ S u d σ ≤ K ∫ ∂ Ω u d σ for every u ∈ ℋ ∩ C ( Ω ¯ ) . If this inequality holds for every such u , then it also holds for a large class of non-negative subharmonic functions. For certain types of domains explicit values for K are given. In particular the classical value...

Integral representation for a class of multiply superharmonic functions

Kohur Gowrisankaran (1973)

Annales de l'institut Fourier

Let Ω 1 , ... , Ω n be harmonic spaces of Brelot with countable base of completely determining domains. The elements of a subcone C of the cone of positive n -superharmonic functions in Ω 1 × ... × Ω n is shown to have an integral representation with the aid of Radon measures on the extreme elements belonging to a compact base of C . The extreme elements are shown to be the product of extreme superharmonic functions on the component spaces and the measure representing each element is shown to be unique. Necessary and sufficient...

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