M.P. Aldred, D.H. Armitage
Inequalities for surface integrals of non-negative subharmonic functions

Comment.Math.Univ.Carolinae 39,1 (1998) 101-113.

Abstract:Let ${\Cal H}$ denote the class of positive harmonic functions on a bounded domain $\Omega $ in $\Bbb R^N$. Let $S$ be a sphere contained in $\overline {\Omega }$, and let $\sigma $ denote the $(N-1)$-dimensional measure. We give a condition on $\Omega $ which guarantees that there exists a constant $K$, depending only on $\Omega $ and $S$, such that $\int _Su d\sigma \le K\int _{\partial \Omega }u d\sigma $ for every $u\in {\Cal H}\cap C(\overline {\Omega })$. 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 $K=2$ for convex domains is slightly improved.

Keywords: subharmonic, surface integral
AMS Subject Classification: 31B05